Gaussian random variables inr n

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1 Gaussian vectors Lecture 5 Gaussian random variables inr n One-dimensional case One-dimensional Gaussian density with mean and standard deviation (called N, ): fx x exp. Proposition If X N,, then ax b is again Gaussian. Precisely Proof since The function EaX b ax b t ax b dx dt/a ax b Na b,a. x t x exp a ax b a b a. dx t a b exp a t a b exp a a is the density of a Gaussian Na b,a. The proof is complete. Corollary If X N,, then Z : X N0,. In other words, any Gaussian r.v. X N, can be represented in the form where Z is canonical (standard). X Z Remark For any random variable X (not necessarily Gaussian), the transformation X Z : is called standardization. The r.v. Z has always mean zero and standard deviation. However, if X belongs to some class (ex. Weibull), Z does not necessarily belong to the same class. Unless X is Gaussian. This is one of the reasons why the Gaussian part of probability theory is called the linear theory (invariance by linear, or even affine, transformations). Exercise (Theoretical) Show that Weibull class is not invariant. The general property of the previous remark is based on the linearity of the expectation: dt

2 and the quadratic property of the variance: EaX byc aex bey c VaraX b a VarX which hold true for all random variables and constants. See Appunti teorici terza parte, section. Multidimensional case We give the definition of multidimensional Gaussian variable reversing the previous procedure. Definition Canonical (standard) gaussian density inr n : or in vector notations: fx,...,x n x e... e fx x exp n/ where x is the Euclidean norm of x x,...,x n. xn x...x n e n/ A random vector Z Z,...,Z n with density fx will be called a canonical Gaussian vector. A picture of the canonical Gaussian density in dimension n was given in the first lecture. A sample of 00 points from a -D Gaussian is: z z. Definition General Gaussian random vector X X,...,X n: any random vector of the form X AZ where Z Z,...,Z k is a canonical Gaussian vector inr k, for some k, A is a matrix with k input (columns) and n output (rows), and is a n-vector. In plain words: Gaussian vectors: linear (affine) transformations of canonical Gaussian vectors. translation. A: several possibilities: rotation, stretching in some direction... It plays the role of ( large A means large dispersion), but it is multidimensional. Let us see a few -D examples: translation by, multiplication by 0 0

3 multiplication by 0 0 followed by 45 rotation, namely multiplication by A 0 0 / / x[, ] x[, ] Proposition Let Q AA T (n n square, symmetric, matrix). If detq 0, then the density of X is fx n/ Level curves of the density: fx C They are ellipsoids. Covariance matrix detq exp x T Q x x T Q x R. More on independence Recall that two events A and B are called independent if PA B PAPB (more or less equivalently, if PA B PA and PB A PB). Two random variables X, Y are called independent if PX I,Y J PX IPY J for every interval I,J. If they have a densities f Xx, f Yy (called marginals), and joint density fx,y, then the identity is equivalent to independence of X, Y. fx,y f Xx f Yy Remark Z Z,...,Z n canonical Gaussian vectorz,...,z n independent -d Gaussian standard. Proposition If X, Y are independent, then EXY EXEY..

4 This is not a characterization of independence: it may happen that EXY EXEY but X, Y are not independent (the average is only a summary of the density, so a propriety of product of averages does not imply product of densities). However, such examples must be cooked with intention, they do not happen at random. Moreover: Proposition If X, Y are jointly gaussian and they are independent. EXY EXEY (a posteriori of thi lecture we could prove this claim). Definition Given two random variables X,Y, we call covariance the number CovX,Y EX EXYEY EXY EXEY. It is a generalization of the Variance: CovX,X VarX. We see that: CovX,Y 0 EXY EXEY. Definition We say that X and Y are uncorrelated if CovX,Y 0, or equivalently if EXY EXEY. Corollary Independent implies uncorrelated. Uncorrelated and jointly gaussian implies independent. The number CovX,Y gives a measure of the relation between two random variables. More closely we could see that it describes the degree of linear relation (regression theory). Large CovX,Y correspondes to high degree of linear correlation. A drawback of CovX,Y is that it depends on the unit of measure of X and Y: large CovX,Y is relative to the order of magnitude of the other quantities of the problem. The correlation coefficient X,Y CovX,Y X Y is independent of the unit of measure (it is absolute ), and X,Y. Again, X,Y 0 means uncorrelated. High degree of correlation becomesx,y close to or (positive or negative linear correlation). Proposition In general, VarX Y VarX VarY CovX,Y. Hence, if X,Y are uncorrelated, then VarX Y VarX VarY. This is not linearity of the variance. The first identity comes simply from the property a b a b ab. Proposition Cov is linear in both arguments: CovaX bx c,y acovx,y bcovx,y

5 and similarly in the second argument (it is symmetric). Notice that additive constants c disappear (as in the variance). (proof: elementary) What is Q AA T Let us understand better Q AA T. Write X AZin components: and compute X A Z A Z... X A Z A Z CovX,X In general, Cov i A i A j CovZ i,z j i,j A i Z i, A j Z j j A i A i AA T, Q, i CovX h,x k AA T h,k Q h,k. Proposition Q AA T is the covariance matrix (matrix of covariances). Covariance is generalization of variance. Q is generalization of from one-dimensional to multi-dimensional. Example For the example we have Q / / A The covariance between X and X is 3. / / / / Spectral theorem Any symmetric matrix, hence Q in particular, can be diagonalized: there exists a new orthonormal basis ofr n where Q is diagonal. The elements of such basis are eigenvectors of Q, the elements of Q on the diagonal are the corresponding eigenvalues: Qv i i v i

6 Q n in the basis v,...,v n. The use is to order the eigenvalues in decreasing order. Example For the example A the eigenvectors are v / / and v, Q The covarance matrix Q AA T is also positive semi-definite: for all vectors x R n. This is equivalent to for i,...,n. Moreover, detq 0. We have x T Qx 0 i , with eigenvalues 4,. detq 0 i 0 for all i,...,n. In such a case, the level curves have the form y... y n n R where y,...,y n are the coordinates in the new basis v,...,v n. They are ellipses with axes v,...,v n and amplitudes along these axes equal to,..., n. The method of Principal Component Analysis (PCA) will be based on these remarks. Example For our usual example, since v, v, 4,.5 y x the ellipses have the form: the equation x T Q x R, x x,y T, namely -.5 which can be obtained also from

7 Q (R ). 0.65x 0.65y xy Generation of multivariate samples How to generate Gaussian samples with given covariance? In many applications Q is known, but A is not. We want to generate a sample under X AZ. Problem: Q A? We have to solve the equation (A is the unknown) AA T Q. The software R gives us the following solution: require(mgcv) A-mroot(Q) We may choose the dimension k of Z. Simplest choice: k n dimension of X. Thus A is a square matrix. We may choose A symmetric. Thus the equation is The solution is In practice? A Q. A Q. Exercise Assume to know the spectral decomposition of Q, namely the eigenvectors v i and the eigevalues i. Let U be the orthogonal matrix (U T U ) defined as follows: the i-th column of U is v i. Check that Q : U T QU is diagonal, with diagonal elements i. The matrix matrix with elements i. Then set Check that A is symmetric and A Q. A : U Q U T. Generation of non-gaussian samples Recall the theorem: Q is simply the diagonal Theorem i) If Y is a random variable with cdf F (continuous case), then the random variable is uniformly distributed on 0,. U : FY

8 ii) If U is a uniform random variable then is a random variable with cdf F. Application of both (i) and (ii) gives us: F U Corollary If Y is a random variable with cdf F and denotes the cdf of a standard normal, then Z : FY is a standard normal variable. And vice-versa, Y F Z. Algorithm to generate a sample from Y: generate a sample from a standard normal variable Z compute F Z. Nothing more than the old one based on uniform? No: multidimensional, correlated! Let Y, Y two r.v. with cdf F, F (continuous case). Compute X : F Y, X : F Y. They are standard normal, but not necessarily independent. Theoretical gap: no reason why X,X should be jointly gaussian (gaussian vector). Assume X,X jointly gaussian. Compute covariance matrix Q of X,X, and average,. Compute A Q as above (or any other solution of AA T Q). Simulate standard Gaussian vector Z,Z. Compute X,X fromz,z by means of A and. Anti-transform Y i F i X i, i,. This is a way to generate samples from non-gaussian correlated r.v. Y,Y. Multidimensional data fit We describe only simple rules. Assume a sample x,y,...,x n,y n is given. We cannot plot a joint histogram or cdf. Thus we cannot get a feeling about gaussianity or not. But we can plot -D marginals. A Gaussian vector has Gaussian marginals. If we want to model our data by a -D Gaussian (either because we see a good agreement with gaussinaity of the marginals, or because of simplicity), we estimate Q and simply by cov and mean, in R. Otherwise, if we want to describe marginals by non-gaussian distributions, we fit the marginals and find F, F, transform the data by x i F x i, y i F y i, i,...,n into a new sample x,y,...,x n,y n assume it is jointly gaussian (we only know that x,...,x n and y,...,y n are gaussian) compute cov and mean of x,y,...,x n,y n. This is the model, that we may use for simulation, computation of probabilities and

9 other purposes. Example Consider the following 0 points in the plane y x They have been produced artificially by two independent N, components. Let us ignore this fact. As an exercise, let us think they are the values of two physical quantities measured in 0 experiments. We want to solve the following problem: compute the probability that both components are positive. A simple answer is: we count the number of point with positive components, 3 in this example, and answer We clearly see that a number of points are close to the 0 boundary, thus the result suffers very much the peculiarity of the sample. We are sure that, if we repeat the experiments, this number may change considerably. Thus let us extract a model, a -D density, from data and compute the theoretical probability from it. We hope it is a more stabel result. For simplicity, let us choose a Gaussian fit from the beginning. Compute cov and mean of data, that in our case are: Q , We see that in this example the first component is fitted quite well with respect to the true N0, which generated the sample. The second is not: the second sample is poor. The correlation between the two samples is very small, good indication of independence. The (gaussian) model has been found. How to compute the required probability? By Monte Carlo. Using require(mgcv), A-mroot(Q), get A. Then produce N standard points z z,z, transform them by Az, yy xx compute the fraction with both positive components. This is a Monte Carlo approzimation of the required probability. At the end we find p 0.69.

10 It is not very different from But if we repeat a few times the whole procedure 0 we see that the second estimate is more stable than the first one (not so much, however, only roughly 0% better).