联系我们： 手动添加方式: 微信>添加朋友>企业微信联系人>13262280223 或者 QQ: 1483266981
这是一篇英国的多元数据分析数学作业

Question 1. (20 points)
Consider a 2 Gaussian random vectorx = (X,X,)T and a matrix A∈3×2; assume that

with c∈ . Introduce z= (Z1 , Z2 , Z3)T = Ax.
(a) Give the condition on cfor the matrix ∑xto be a covariance matrix. [2]
(b) For which value of cdo we have X1 X2 [2]
Assume that c is such that the matrix ∑x is a covariance matrix.
(c) Compute E(z) and var(z), and give the distribution of z. [7]
Hint: Compute each entry of the vector E(z) and of the matrix var(z); note that var(z) should depend on c.
Consider the vector v = (1,1, 1) ; note that ATv= 0.
(d) Use the vector v to defifine a principal component of z, and give the variance of this principal component. [3]
(e) What is the fraction of variance explained by the fifirst two principal components of the random vector z For which values of c does the fifirst principal component of the random vector z explain 100% of its variance [4]
Hint: You may consider det(∑x).
(f) Give an expression for Z3 in terms of Z1 and Z2 . [2]
Question 2. (11 points)
Consider a centered Gaussian process (Zx)x∈ with covariance kernel K(x, x′ ) = e|xx′|, for x and x′ ∈ .
Assume that the realisation Z0 = z0 was observed, with z0 ∈ (i.e. observation at x = 0). Consider x and x′ ∈ .
(a) Give the distribution of the random variable Zx|Z0=z0 . Draw a scheme illustrating the appearance of the map x E (Zx|Z0= z0) , withx ∈. [7]
(b) Compute c-cov (Zx, Zx′ | |Z0 = z0) ; what can be concluded forx < 0 and x′ > 0 [4]
Question 3. (21 points)
Consider two independent random samples x1, , xnxi.i.d. x ～ Nd(Ux, ∑), and y1, , yny i.i.d.y ～ Nd(uy, ∑). Note that these two samples are defifined from Gaussian distributions withsamecovariance matrix, but with potentially difffferent means; the sample related to x is of size nx, and theone related to y is of size ny(and the random vectors xiand yiare independent for alli ∈ {1, , nx}and j∈ {1, , ny}). Let {x, Sx} and {y, Sy} be the corresponding sample mean and corrected (i.e.unbiased) sample covariance estimators. Consider the pooled variance estimator