on 04-Mar-2017 (Sat)

The transp ose of a matrix pro duct has a simple form: \((AB)^{T} = B^{T}A^{T}\) . (2.9) This allo ws us to demonstrate equation , b y exploiting the fact that the v alue 2.8 of suc h a pro duct is a scalar and therefore equal to its o wn transp ose:\( x^Ty = (x^Ty)^T = y^T x\)

W e can now solve equation Ax = b by the following steps:
1. A^-1Ax = A⁻¹b
2. I_nx = A⁻¹b

The “evidence” for the model, p(D), is the overall probability of the data according to the model, determined by averaging across all possible parameter values weighted by the strength of belief in those parameter value

The residual r=A*x-b is now r= -0.3333 -0.3333 0.3333 What happens in this case is that MATLAB produces the least squares best fit. This is the value of x which makes the magnitude of r, i.e., r(1) ² + r(2) ² + r(3)² , as small as possible.

The space of possibilities is the joint parameter space involving all combinations of parameter values. If we represent each parameter with a comb of, say, 1,000 values, then for P parameters there are 1,000 ^P combinations of parameter values.

The L 1 norm is commonly used in machine learning when the difference b etw een zero and nonzero elements is v ery imp ortan t. Every time an element of x mo v es a w a y from 0 by \(\epsilon\), the L 1 norm increases b y \(\epsilon\).

T o compute diag(v)x , we only need to scale each element x_i b y v_i . In other w ords, diag(v)x = \(x\cdot y\)

A unit vector is a vector with unit norm ||x||₂ = 1

A vector x and a vector y are orthogonal to each other if x^Ty = 0 .

An eigenvector of a square matrix A is a non-zero vector v suc h that m ulti- plication b y A alters only the scale of v: Av = λv

Av= λv
The scalar λ is kno wn as the eigen v alue corresp onding to this eigenv ector. (One can also find a left eigen v ector suc h that v A = λ v , but we are usually concerned with righ t eigenv ectors

SVD of A = UVD^T

The elemen ts along the diagonal of D are kno wn as the singular v alues of the matrix A . The columns of U are known as the left-singular v ectors . The columns of V are kno wn as as the right-singular v ectors

W e can actually interpret the singular v alue decomp osition of A in terms of the eigendecomposition of functions of A . The left-singular v ectors of A are the eigen v ectors of AA^T . The righ t-singular v ectors of A are the eigen vectors of A^TA . The non-zero singular v alues of A are the square ro ots of the eigen v alues of A^TA . The same is true for AA^T

In other words, the normalizer for the beta distribution is the beta function \(B(a,b) = \int d\theta \space \theta^{a-1}(1-\theta)^{b-1}\)

In R, beta(θ|a, b) is dbeta(θ,a,b),and B(a, b) is beta(a,b)

The standard deviation of the beta distribution is \(\sqrt{μ(1 − μ)/(a + b +1)}\). Notice that the standard deviation gets smaller when the concentration κ = a + b gets larger.

Waste management or Waste disposal is all the activities and actions required to manage waste from its inception to its final disposal.^[1] This includes amongst other things, collection, transport, treatment and disposal of waste together with monitoring and regulation. It also encompasses the legal and regulatory framework that relates to waste management encompassing guidance on recycling etc.

W e can identi fy all of the n um b ers with v ertical co ordinate i b y writing a “ ” for the horizontal

To be useful for life, sulfate must be reduced to sulfide (S 2– )