on 10-Jan-2018 (Wed)

Specifically, Probabilistic PCA assumes that each latent variable is normally distributed, \(\mathbf{z}_n \sim N(\mathbf{0}, \mathbf{I})\).
The corresponding data point is generated via a projection \( \mathbf{x}_n \mid \mathbf{z}_n \sim N(\mathbf{W}\mathbf{z}_n, \sigma^2\mathbf{I}) \),

error covariance structure in PPCA is σ 2 I σ2I \sigma^2 \mathbf I and in FA it is an arbitrary diagonal matrix Ψ Ψ \boldsymbol \Psi .

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In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem^[1][2] is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval.

When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

The functions f and g are orthogonal when this integral is zero: As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

Marcus Aurelius ( / ɔː ˈ r iː l i ə s / ; Latin: Marcus Aurelius Antoninus Augustus ;^{[6][notes 1][9]} 26 April 121 – 17 March 180 AD) was Roman emperor from 161 to 180

The polar decomposition of a square complex matrix A is a matrix decomposition of the form

A = U P {\displaystyle A=UP\,}

where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix.^[1] Intuitively, the polar decomposition separates A into a component that stretches the space along a set of orthogonal axes, represented by P, and a rotation (with possible reflection) represented by U. The decomposition of the complex conjugate of A {\displaystyle A} is given by A ¯ = U ¯ {\displaystyle {\overline {A}}={\overline {U}}} P ¯ {\displaystyle {\overline {P}}} .

The fourth page of the booklets provided six five-point rating scales on which participants were asked to rate the best answer to the following questions: (1) How good are you at learning languages? (1 ¼ very good–5 ¼ very bad). (2) How interesting do you find learning languages? (1 ¼ very interesting–5 ¼ very uninteresting). (3) How easy did you find this method of learning languages? (1 ¼ very easy–5 ¼ very hard). (4) How quick did you find this method of learning languages? (1 ¼ very quick–5 ¼ very slow). (5) How enjoyable did you find this method of learning languages? (1 ¼ very enjoyable–5 ¼ very boring). (6) How useful did you find this method of learning languages? (1 ¼ very useful–5 ¼ useless).