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tegral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. <span>A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the

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tion (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. <span>The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. Contents [hide] 1 Notation 2 Definition 3 Properties 3.1 Existence and uniqueness 3.2 Basic properties 3.2.1 Identities 3.3 Reduction to Hermitian case 3.4 Products 3.5

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; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

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| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr

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, {\displaystyle v_{i}\,\,(i=1,\dots ,N),} can also be used as the columns of Q. That can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q −1 . <span>The decomposition can be derived from the fundamental property of eigenvectors: A v = λ v {\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} } and thus A Q = Q Λ {\displaystyle \mathbf {A} \mathbf {Q} =\mathbf {Q} \mathbf {\Lambda } } which yields A = Q Λ Q − 1 {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}} . Example[edit source] Taking a 2 × 2 real matrix A = [

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f them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. <span>Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution (which is the marginal distribution at that poi

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μ ℓ {\displaystyle \mu _{\ell }} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] <span>A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of t

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} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha

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initeness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] <span>Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. If the process depends only on |x − x'|, the Euclidean distance (not the dire

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stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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r the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] <span>If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and

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en we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable. <span>Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)). Usual covariance functions[edit source] [imagelink] The effect of choosing different kernels on the prior function distribution of the Gaussian process. Left is a squared expon

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{\displaystyle \nu } and Γ ( ν ) {\displaystyle \Gamma (\nu )} is the gamma function evaluated at ν {\displaystyle \nu } . <span>Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand. Clearly, the inferential results are dependent on the values of the hyperparameters θ (e.g. ℓ and σ) defining the model's behaviour. A popular choice for θ is to provide maximum a pos

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Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand.

{\displaystyle \nu } and Γ ( ν ) {\displaystyle \Gamma (\nu )} is the gamma function evaluated at ν {\displaystyle \nu } . <span>Importantly, a complicated covariance function can be defined as a linear combination of other simpler covariance functions in order to incorporate different insights about the data-set at hand. Clearly, the inferential results are dependent on the values of the hyperparameters θ (e.g. ℓ and σ) defining the model's behaviour. A popular choice for θ is to provide maximum a pos

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Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)).

en we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable. <span>Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u(x) = (cos(x), sin(x)). Usual covariance functions[edit source] [imagelink] The effect of choosing different kernels on the prior function distribution of the Gaussian process. Left is a squared expon

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If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present.

r the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] <span>If we expect that for "near-by" input points x and x' their corresponding output points y and y' to be "near-by" also, then the assumption of continuity is present. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the be

stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.

stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic.

stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer.

stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of <span>the observer. <span><body><html>

stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. <span>If the process depends only on |x − x'|, the Euclidean distance (not the direction) between x and x', then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; [7] in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [5] If we expect that for "ne

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y> Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. <body><html>

initeness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] <span>Stationarity refers to the process' behaviour regarding the separation of any two points x and x' . If the process is stationary, it depends on their separation, x − x', while if non-stationary it depends on the actual position of the points x and x'. For example, the special case of an Ornstein–Uhlenbeck process, a Brownian motion process, is stationary. If the process depends only on |x − x'|, the Euclidean distance (not the dire

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if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion.

} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha

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if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour.

} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha

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head> if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. <html>

} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, <span>if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loeve expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. [5] [6] Stationarity refers to the process' beha

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A key fact of Gaussian processes is that they can be completely defined by their second-order statistics.

μ ℓ {\displaystyle \mu _{\ell }} can be shown to be the covariances and means of the variables in the process. [3] Covariance functions[edit source] <span>A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. [4] Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of t

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Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.

f them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. <span>Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution (which is the marginal distribution at that poi

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Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data.

f them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. <span>Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution (which is the marginal distribution at that poi

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The eigendecomposition can be derived from the fundamental property of eigenvectors: and thus which yields .

, {\displaystyle v_{i}\,\,(i=1,\dots ,N),} can also be used as the columns of Q. That can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q −1 . <span>The decomposition can be derived from the fundamental property of eigenvectors: A v = λ v {\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} } and thus A Q = Q Λ {\displaystyle \mathbf {A} \mathbf {Q} =\mathbf {Q} \mathbf {\Lambda } } which yields A = Q Λ Q − 1 {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}} . Example[edit source] Taking a 2 × 2 real matrix A = [

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The eigendecomposition can be derived from the fundamental property of eigenvectors: and thus which yields .

, {\displaystyle v_{i}\,\,(i=1,\dots ,N),} can also be used as the columns of Q. That can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q −1 . <span>The decomposition can be derived from the fundamental property of eigenvectors: A v = λ v {\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} } and thus A Q = Q Λ {\displaystyle \mathbf {A} \mathbf {Q} =\mathbf {Q} \mathbf {\Lambda } } which yields A = Q Λ Q − 1 {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}} . Example[edit source] Taking a 2 × 2 real matrix A = [

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In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonal

| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr

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In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in

| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr

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In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

| ocultar ahora Eigendecomposition of a matrix From Wikipedia, the free encyclopedia (Redirected from Eigendecomposition) Jump to: navigation, search <span>In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. Contents [hide] 1 Fundamental theory of matrix eigenvectors and eigenvalues 2 Eigendecomposition of a matrix 2.1 Example 2.2 Matrix inverse via eigendecomposition 2.2.1 Pr

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The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition.

tion (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. <span>The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. Contents [hide] 1 Notation 2 Definition 3 Properties 3.1 Existence and uniqueness 3.2 Basic properties 3.2.1 Identities 3.3 Reduction to Hermitian case 3.4 Products 3.5

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A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution

tegral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. <span>A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the

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nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d

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ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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In linear algebra, the singular-value decomposition (SVD) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar deco

nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form

ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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om. In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. <span>Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be s

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Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to f

om. In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. <span>Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be s

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f references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (September 2010) (Learn how and when to remove this template message) <span>Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various

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s sorts of relationships among the three types of random variables, as might be described by a graphical model. As is typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". <span>Variational Bayesian methods are primarily used for two purposes: To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables. To derive a lower bound for the marginal likelihood (sometimes called the "evidence") of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserved variables). This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data. (See also the Bayes factor article.) In the former purpose (that of approximating a posterior probability), variational Bayes is an alternative to Monte Carlo sampling methods — particularly, Markov chain Monte Carlo met

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he parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be solved analytically. <span>For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed. However, deriving the set of equations used to iteratively update the parameters often requires a large amount of work compared with deriving the comparable Gibbs sampling equations. Th

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m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem

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3; ( AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); ( A + is a weak inverse for the multiplicative semigroup); ( AA + is Hermitian); and ( A + A is also Hermitian). <span>Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, . When has linearly independent rows (matrix is invertible), can be computed as: This is a right inverse, as . <span><body><html>

; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

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ml> Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, . <html>

; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

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e exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as<span>: This particular pseudoinverse constitutes a left inverse, since, in this case, . <span><body><html>

; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

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SVD as change of coordinates The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases,

m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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ollows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to <span>zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. <span><body><html>

m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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h that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with <span>non-negative real diagonal entries. <span><body><html>

m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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Verbs: The Conditional Simple Usage: To ask politely: ¿Podrías pasarme ese plato, por favor? (Could you pass me that plate, please?).

Verbs: The Conditional Simple Usage: To ask politely: ¿Podrías pasarme ese plato, por favor? (Could you pass me that plate, please?). To express wishes: ¡Me encantaría ir de viaje a Australia! (I would love to go on a trip to Australia!). To suggest: Creo que deberías ir al médico a verte ese dolor de espalda (I think

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If is the singular value decomposition of A , then . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in p

A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem

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A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If is the singular value decomposition of A , then .

A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem

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pseudo datapoint based approximation methods for DGPs trade model complexity for a lower computational complexity of \(O(NLM^ 2 ) \) where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints.

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible f

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design. </sp

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design.

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Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

f references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (September 2010) (Learn how and when to remove this template message) <span>Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various

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This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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head><head> In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the p

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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ter science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, <span>solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving c

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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nomics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and <span>storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expens

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of <span>recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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blem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply <span>looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on t

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby <span>saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is <span>indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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tion, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on <span>the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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y Real estate Reinsurance Over-the-counter (off-exchange) Forwards Options Spot market Swaps Trading Participants Regulation Clearing Related areas Banks and banking Finance corporate personal public v t e <span>In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or

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or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th

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In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option.

y Real estate Reinsurance Over-the-counter (off-exchange) Forwards Options Spot market Swaps Trading Participants Regulation Clearing Related areas Banks and banking Finance corporate personal public v t e <span>In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or

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In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option.

y Real estate Reinsurance Over-the-counter (off-exchange) Forwards Options Spot market Swaps Trading Participants Regulation Clearing Related areas Banks and banking Finance corporate personal public v t e <span>In finance, an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or

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An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. <

or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th

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An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. <html>

or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th

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An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but <span>the call option is more frequently discussed. <span><body><html>

or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. <span>An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Both are commonly traded, but the call option is more frequently discussed. The seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to th

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Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of E

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives <span>a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate)

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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nt instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that <span>the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). <span><body><html>

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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rential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of <span>the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). <span><body><html>

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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mula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with <span>the risk-neutral rate). <span><body><html>

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to th

h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior obse

h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.

h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. <body><html>

h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is

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Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. </sp

translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is

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kov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as <span>a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. <span><body><html>

translations!] Gauss–Markov process From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Gauss–Markov theorem of mathematical statistics. <span>Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] The stationary Gauss–Markov process (also known as a Ornstein–Uhlenbeck process) is a very special case because it is unique, except for some trivial exceptions. Every Gauss–Markov process X(t) possesses the three following properties: If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process If f(t) is

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Delante significa ‘en la parte anterior’, ‘en frente’ o ‘ante alguien’, se usa por lo general para indicar la situación de alguien o algo. Adelante , por su parte, equivale a ‘más allá’, ‘hacia allá’, o ‘hacia enfrente’, y se emplea para indicar la existencia de un movimiento, sea real o figurado. La forma alante , por otro lado, es incorrecta.

amp;cj=1"> [imagelink] Palabras Homófonas Palabras Parónimas Fonética y fonología Uso Grafía Léxicas Ver más Latinismos Extranjerismos Barbarismos Ultracorrecciones Dudas de uso Delante o adelante Delante significa ‘en la parte anterior’, ‘en frente’ o ‘ante alguien’, se usa por lo general para indicar la situación de alguien o algo. Adelante , por su parte, equivale a ‘más allá’, ‘hacia allá’, o ‘hacia enfrente’, y se emplea para indicar la existencia de un movimiento, sea real o figurado. La forma alante , por otro lado, es incorrecta. Cuándo usar delante Delante es un adverbio de lugar; se emplea con el significado de ‘en la parte anterior’, ‘en frente’ o ‘en presencia de alguien’. Por lo general, es un adverbio que

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2 ) {\displaystyle K_{\operatorname {SE} }(x,x')=\exp {\Big (}-{\frac {\|d\|^{2}}{2\ell ^{2}}}{\Big )}} <span>Ornstein–Uhlenbeck: K OU ( x , x ′ ) = exp ( − | d | ℓ ) {\displaystyle K_{\operatorname {OU} }(x,x')=\exp \left(-{\frac {|d|}{\ell }}\right)} Matérn: K Matern ( x , x ′ )

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{\displaystyle K_{\operatorname {Matern} }(x,x')={\frac {2^{1-\nu }}{\Gamma (\nu )}}{\Big (}{\frac {{\sqrt {2\nu }}|d|}{\ell }}{\Big )}^{\nu }K_{\nu }{\Big (}{\frac {{\sqrt {2\nu }}|d|}{\ell }}{\Big )}} <span>Periodic: K P ( x , x ′ ) = exp ( − 2 sin 2 ( d 2 ) ℓ 2 ) {\displaystyle K_{\operatorname {P} }(x,x')=\exp \left(-{\frac {2\sin ^{2}\left({\frac {d}{2}}\right)}{\ell ^{2}}}\right)} Rational quadratic: K RQ ( x , x ′

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2 ) {\displaystyle K_{\operatorname {P} }(x,x')=\exp \left(-{\frac {2\sin ^{2}\left({\frac {d}{2}}\right)}{\ell ^{2}}}\right)} <span>Rational quadratic: K RQ ( x , x ′ ) = ( 1 + | d | 2 ) − α , α ≥ 0 {\displaystyle K_{\operatorname {RQ} }(x,x')=(1+|d|^{2})^{-\alpha },\quad \alpha \geq 0} Here d = x − x ′ {\displaystyle d=x-x'} . The parameter ℓ is the character

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centurie

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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grand periods in the history of logic, two of them, the ancient period and the medieval scholastic period, were closely connected to the idea that the primary application of logic is for practices of debating such as dialectical disputations. <span>The third of them, in contrast, exemplifies an entirely different rationale for logic, namely as a foundational branch of mathematics, not in any way connected to the ordinary languages in which debates are typically conducted. The hiatus between the second and third periods can be explained by the fall from grace of

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rving of the title ‘logic’ than the work done under this heading in the early modern period, given that it comes closer to the level of rigour and formal sophistication that came to be associated with logic from the late 19th century onwards. <span>In the modern period, a number of philosophers came to see the nature of logic in terms of the faculties of mind. To be sure, this is again a theme present in medieval scholastic thought (in the work of the 14th-century author Pierre d’Ailly, for example), but in the early modern period it became t

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ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which

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s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

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cond intentions’, roughly what we call second-order concepts, or concepts of concepts. But as late as in the 16th century, the Spanish theologian Domingo de Soto could write with confidence that ‘dialectic is the art or science of disputing’. <span>The tight connection between traditional logic and debating practices dates back to the classical Hellenistic period. Intellectual activity then was quintessentially a dialogical affair, as registered in Plato’s dialogues. In these dialogues, Socrates regularly engages in the practice of refutation (el

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as tr

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitab

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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span> Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the <span>linear functions defined on these spaces and respecting these structures in a suitable sense. <span><body><html>

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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amp;0\\1&3\\\end{bmatrix}}{\begin{bmatrix}-2c&0\\c&d\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&3\\\end{bmatrix}},[c,d]\in \mathbb {R} } Matrix inverse via eigendecomposition[edit source] Main article: Inverse matrix <span>If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by A − 1 = Q Λ − 1 Q − 1 {\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: [ Λ

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{\displaystyle A=A^{*}} ), which implies that it is also complex normal, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle. Real symmetric matrices[edit source] <span>As a special case, for every N×N real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other. Thus a real symmetric matrix A can be decomposed as A = Q Λ Q T {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}} where Q is an orthogonal matrix, and Λ is a diagonal matrix whose entries are the eigenvalues of A. Useful facts[edit source] Useful facts regarding eigenvalues[edit source] The product of the eigenvalues is equal to the determinant of A det

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language

s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. <body><html>

s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which

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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which

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tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima

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Instead, early modern authors emphasise the role of novelty and individual discovery

tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima

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without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. <span>Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous. Up to Descartes’s time, the ch

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Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition , which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know.

without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. <span>Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous. Up to Descartes’s time, the ch

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

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Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

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tml> In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. <html>

Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

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algebra v t e In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] <span>Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras. The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. The language of c

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oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T

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and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, [3] biology, [4] ecology, [5] geology, [6] physics, [7] economics, [8] image processing, [9] and telecommunications. [10] [11] <span>The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in tim

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e, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in time. <span>In the plane, the point process, also known as a spatial Poisson process, [13] can represent the locations of scattered objects such as transmitters in a wireless network, [10] [14] [15] [16] particles colliding into a detector, or trees in a forest. [17] In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, [18] stochastic geometry, [1] spatial statistics [18] [1

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Its name (Poisson Process) derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.

oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T

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Its name (Poisson Process) derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution.

oint processes, some of which are constructed with the Poisson point process, that seek to capture such interaction. [22] The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. <span>Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. [23] [24] T

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The Poisson point process is often defined on the real line, where it can be considered as a stochastic process.

and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, [3] biology, [4] ecology, [5] geology, [6] physics, [7] economics, [8] image processing, [9] and telecommunications. [10] [11] <span>The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in tim

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In the plane, the point process, also known as a spatial Poisson process, [13] can represent the locations of scattered objects such as transmitters in a wireless network, [10] [14] [15] [16] particles colliding into a detector, or trees in a forest.

e, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory [12] to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in time. <span>In the plane, the point process, also known as a spatial Poisson process, [13] can represent the locations of scattered objects such as transmitters in a wireless network, [10] [14] [15] [16] particles colliding into a detector, or trees in a forest. [17] In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, [18] stochastic geometry, [1] spatial statistics [18] [1

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of the elements; a variable used this way is sometimes called an accumulator. Adding up the elements of a list is such a common operation that Python provides it as a built-in function, sum: >>> t = [1, 2, 3] >>> sum(t) 6 <span>An operation like this that combines a sequence of elements into a single value is sometimes called reduce. Sometimes you want to traverse one list while building another. For example, the following function takes a list of strings and returns a new list that contains capitalized strings:

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ngs: def capitalize_all(t): res = [] for s in t: res.append(s.capitalize()) return res res is initialized with an empty list; each time through the loop, we append the next element. So res is another kind of accumulator. <span>An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize) onto each of the elements in a sequence. Another common operation is to select some of the elements from a list and return a sublist. For example, the following function takes a list of strings and returns a list that cont

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ontains only the uppercase strings: def only_upper(t): res = [] for s in t: if s.isupper(): res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. <span>An operation like only_upper is called a filter because it selects some of the elements and filters out the others. Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you kno

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res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. An operation like only_upper is called a filter because it selects some of the elements and filters out the others. <span>Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you know the index of the element you want, you can use pop: >>> t = ['a', 'b', 'c']

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An operation like this that combines a sequence of elements into a single value is sometimes called reduce .

of the elements; a variable used this way is sometimes called an accumulator. Adding up the elements of a list is such a common operation that Python provides it as a built-in function, sum: >>> t = [1, 2, 3] >>> sum(t) 6 <span>An operation like this that combines a sequence of elements into a single value is sometimes called reduce. Sometimes you want to traverse one list while building another. For example, the following function takes a list of strings and returns a new list that contains capitalized strings:

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An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize ) onto each of the elements in a sequence.

ngs: def capitalize_all(t): res = [] for s in t: res.append(s.capitalize()) return res res is initialized with an empty list; each time through the loop, we append the next element. So res is another kind of accumulator. <span>An operation like capitalize_all is sometimes called a map because it “maps” a function (in this case the method capitalize) onto each of the elements in a sequence. Another common operation is to select some of the elements from a list and return a sublist. For example, the following function takes a list of strings and returns a list that cont

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An operation like only_upper is called a filter because it selects some of the elements and filters out the others.

ontains only the uppercase strings: def only_upper(t): res = [] for s in t: if s.isupper(): res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. <span>An operation like only_upper is called a filter because it selects some of the elements and filters out the others. Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you kno

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Most common list operations can be expressed as a combination of map, filter and reduce.

res.append(s) return res isupper is a string method that returns True if the string contains only upper case letters. An operation like only_upper is called a filter because it selects some of the elements and filters out the others. <span>Most common list operations can be expressed as a combination of map, filter and reduce. 10.8 Deleting elements There are several ways to delete elements from a list. If you know the index of the element you want, you can use pop: >>> t = ['a', 'b', 'c']

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focus on the two most important examples of state space models, namely the hidden Markov model, in which the latent variables are discrete, and linear dynamical systems, in which the latent variables are Gaussian. Both models are described by <span>directed graphs having a tree structure (no loops) for which inference can be performed efficiently using the sum-product algorithm <span><body><html>

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In most applications of such (Markovian) models, the conditional distributions p(xn|xn−1) that define the model will be constrained to be equal, corresponding to the assumption of a stationary time series. The model is then known as a homogeneous Markov chain.

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= 1/4. 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4. Properties[edit] <span>The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolut

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The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as goes from 0 to 1, and take

= 1/4. 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4. Properties[edit] <span>The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolut

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The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as goes from 0 to 1, and takes on every value in between.

= 1/4. 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4, so c(200/243) = 3/4. Properties[edit] <span>The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, c ( x ) {\textstyle c(x)} goes from 0 to 1 as x {\textstyle x} goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolut

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Historically, the dividing line is 1933 when Grundbegriffe der Wahrscheinlichkeitsrech- nung (Foundations of the Theory of Probability) by Andrey Kolmogorov was published

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list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (May 2017) (Learn how and when to remove this template message) <span>A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning. [imagelink] An example of a graphical model. Each arrow indicates

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the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables

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ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a

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A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables.

list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (May 2017) (Learn how and when to remove this template message) <span>A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning. [imagelink] An example of a graphical model. Each arrow indicates

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In a Bayesian network, the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability satisfies where is

the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables

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In a Bayesian network, the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability <span>satisfies where is the set of parents of node . In other words, the joint distribution factors into a product of conditional distributions. <span><body><html>

the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables

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factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability satisfies where is the set of parents of node . In other words, the joint distribution factors into <span>a product of conditional distributions. <span><body><html>

the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. [1] Bayesian network[edit source] Main article: Bayesian network <span>If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} then the joint probability satisfies P [ X 1 , … , X n ] = ∏ i = 1 n P [ X i | p a i ] {\displaystyle P[X_{1},\ldots ,X_{n}]=\prod _{i=1}^{n}P[X_{i}|pa_{i}]} where p a i {\displaystyle pa_{i}} is the set of parents of node X i {\displaystyle X_{i}} . In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables

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A Markov random field, also known as a Markov network, is a model over an undirected graph.

ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a

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A Markov random field, also known as a Markov network, is a model over an undirected graph.

ne learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks. Markov random field[edit source] Main article: Markov random field <span>A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation. Other types[edit source] A factor graph is an undirected bipartite graph connecting variables a

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Formally, Bayesian networks are DAGs whose: nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that ar

ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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html> Formally, Bayesian networks are DAGs whose: nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are condition

ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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ntities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are <span>conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probabil

ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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re not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, <span>a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. <span><body><html>

ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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ayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) <span>the probability (or probability distribution, if applicable) of the variable represented by the node. <span><body><html>

ed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. <span>Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if m {\displaystyle m} parent nodes represent m {\displaystyle m} Boolean variables

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CAUSALITY - Discussion d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness"

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d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. <span>The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"). We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each el

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nce" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by <span>"connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account f

d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. <span>The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"). We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each el