NMST539 | Lab Session 4Wishart Distribution(application for confidence bands and statistical tests)LS 2017 | Tuesday 21/03/2017Rmd file (UTF8 encoding)The R-software is available for download from the website: https://www.r-project.org A user-friendly interface (one of many): RStudio. Manuals and introduction into R (in Czech or English):
Wishart distributionThe Wishart distribution is, as an analogy to the \(\chi^2\) distribution and the variance inference in the univariate case, a very useful tool for the analysis of the variance-covariance matrix of same random sample \(X_{1}, \dots, X_{n}\) for \(n \in \mathbb{N}\) where each \(X_{i}\) in the sample is a \(p\)-dimensional random vector (\(p\) different covariates recorded on each subject). Thus, it is probability distributions defined over symmetric and nonnegative-definite random matrices. In general, the Wishart distribution is a multi-variate generalization of the \(\chi^2\) distribution: for instance, for a normaly distributed random sample \(\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n}\) drawn from some multivariate distribution \(N_{p}(\boldsymbol{0}, \Sigma)\), with the zero mean vector and \(\Sigma\) being some variance-covariance matrix we have that \[ \mathbb{X}^{\top}\mathbb{X} \sim W_{p}(\Sigma, n), \] for \(\mathbb{X} = (\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n})^{\top}\). Similarly, for a random sample \(X_{1}, \dots, X_{n}\) drawn from \(N(0,1)\) we have that \[ \boldsymbol{X}^{\top}\boldsymbol{X} \sim \chi_{n}^2, \] where \(\boldsymbol{X} = (X_{1}, \dots, X_{n})^\top\). Thus, for a sample of size \(n \in \mathbb{N}\) drawn from \(N(0,1)\) the corresponding distribution of \(\mathbb{X}^\top\mathbb{X}\) is \(W_{1}(1, n) \equiv \chi_{n}^2\). The corresponding density function of the Wishart distribution takes the form \[ f(\mathcal{X}) = \frac{1}{2^{np/2} |\Sigma|^{n/2} \Gamma_{p}(\frac{n}{2})} \cdot |\mathcal{X}|^{\frac{n - p - 1}{2}} e^{-(1/2) tr(\Sigma^{-1}\mathcal{X})}, \] for \(\mathcal{X}\) being a \(p\times p\) random matrix and \(\Gamma_{p}(\cdot)\) is a multivariate generalization of the Gamma function \(\Gamma(\cdot)\) In the R software there are various options (packages) on how to use and apply the Wishart distribution. To Do by Yourself (Theoretical and Practical)
A simple random sample from Wishart distribution \(W_{1}(\Sigma = 1, n = 10)\) (equivalently a \(\chi^2\) distribution with \(n = 10\) degrees of freedom) can be obtained, for instance, as Using a standard approach for generating a univariate random sample we would use the R function
To Do by Yourself
Hotelling’s \(\boldsymbol{T^2}\) DistributionIn a similar manner as we define a classical \(t\)-distribution in an univariate case (i.e. standard normal \(N(0,1)\) variable devided by a square root of a \(\chi^2\) variable normalized by its degrees of freedom) we define a multivariate generalization (Hotelling’s \(T^{2}\) distribution) as \[ n \boldsymbol{Y}^{\top} \mathbb{M}^{-1} \boldsymbol{Y} \sim T^{2}(n, p), \] to be a random variable with the Hotelling’s \(T^2\) distribution with \(p \in \mathbb{N}\) to be the dimension of \(Y \sim N_{p}(0, \mathbb{I})\) and \(n \in \mathbb{N}\) being the parameter of the Wishart distribution of \(\mathbb{M} \sim W_{p}(\mathbb{I}, n)\). A special case for \(p = 1\) gives the standard Fisher distribution with one and \(n\) degrees of freedom (equivalently a square of the \(t\)-distribution with \(n\) degrees of freedom). The Hotelling’s \(T^2\) distribution with parameters \(p, n \in \mathbb{N}\) can be identivied with the Fisher distribution using the following expression \[ T^{2}(p, n) \equiv \frac{n p}{n - p + 1}F_{p, n - p + 1}. \] In general, the relationship betwenn the Hotelling’s \(T^2\) distribution and the Fisher distribution is given by the same expression \[ T^{2}(p, n) \equiv \frac{n p}{n - p + 1}F_{p, n - p + 1}. \] The effect of different parameter settings can be (a little) visualized, for instance, by the following figure:
Moreover, using the relationship between the Hotelling’s \(T^2\) distribution and the Fisher’s F distriubtion we can effectively use the \(F\)-distribution to draw the corresponding critical values when needed for some statistical tests. For some multivariate random sample \(\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n} \sim N_{p}(\boldsymbol{\mu}, \Sigma)\) for some unknown mean vector \(\boldsymbol{\mu} \in \mathbb{R}^{p}\) and some variance-covariance matrix \(\Sigma\), we have that \[ (n - 1)\Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}\Big)^\top \mathcal{S}^{-1} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}\Big) \sim T^2(p, n - 1), \] which can be also equivalently expressed as \[ \frac{n - p}{p} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}\Big)^\top \mathcal{S}^{-1} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}\Big) \sim F_{p, n - p}. \] This can be now used to construct confidence regions for the unknown mean vector \(\boldsymbol{\mu}\) of for testing hypothesis about the true value of the vector of parameters \(\boldsymbol{\mu} = (\mu_{1}, \dots, \mu_{p})^{\top}\). However, rather than construction a confidence region for \(\boldsymbol{\mu}\) (which can be impractical in higher dimensions for even slightly larger values of \(p\)) one focusses on construction confidence intervals for the elements of \(\boldsymbol{\mu}\) such that the mutual coverage is under control (usually we require a simultaneous coverage of \((1 - \alpha)\times 100~\%\) for some small \(\alpha \in (0,1)\)). For a hypothesis test \[ H_{0}: \boldsymbol{\mu} = \boldsymbol{\mu}_{0} \in \mathbb{R}^{p} \] \[ H_{1}: \boldsymbol{\mu} \neq \boldsymbol{\mu}_{0} \in \mathbb{R}^{p} \] we can use the following test statistic: \[ (n - 1)\Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}_{0}\Big)^\top \mathcal{S}^{-1} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}_{0}\Big), \] which, under the null hypothesis, follows the \(T^2(p, n - 1)\) distribution. Equivalently we also have that \[ \frac{n - p}{p} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}_{0}\Big)^\top \mathcal{S}^{-1} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}_{0}\Big) \] follows the Fisher \(F\)-distribution with \(p\) and \(n - p\) degrees of freedom. In the R software we can use the library and we can use the function Using the same approach we can also construct the confidence elipsoid for \(\boldsymbol{\mu} \in \mathbb{R}^p\) - it holds that \[ \frac{n - p}{p} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}_{0}\Big)^\top \mathcal{S}^{-1} \Big(\overline{\boldsymbol{X}} - \boldsymbol{\mu}_{0}\Big) \sim F_{p, n - p}, \] and therefore, the following set \[ \left\{\boldsymbol{\mu} \in \mathbb{R}^p;~ \Big( \overline{\boldsymbol{X}} - \boldsymbol{\mu}\Big)^\top \mathcal{S}^{-1} \Big( \overline{\boldsymbol{X}} - \boldsymbol{\mu}\Big) \leq \frac{p}{n - p} F_{p, n- p}(1 - \alpha) \right\} \] is a confidence region at the confidence level of \(\alpha = 1 - \alpha\) for the vector of parameters \(\boldsymbol{\mu} \in \mathbb{R}^p\) - an interior of iso-distance ellipsoid in \(\mathbb{R}^p\). A brief example on how it works (example from the lecture notes):
(which is the 95% confidence region the true mean vector) To Do by Yourself
In an analogous way one can also construct a two sample Hotelling test to compare two population means. Difference of two multivariate means with the same variance-covariance matrixWe assume a multivariate sample \(\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n} \sim N_{p}(\boldsymbol{\mu}_{1}, \Sigma)\) and some another sample \(\boldsymbol{Y}_{1}, \dots, \boldsymbol{Y}_{M} \sim N_{p}(\boldsymbol{\mu}_{2}, \Sigma)\), with some generally different mean parameter \(\boldsymbol{\mu}_{1} \neq \boldsymbol{\mu}_{2}\). We are interested in testing the null hypothesis \[
H_{0}: \boldsymbol{\mu} = \boldsymbol{\mu}_{0}
\] against the alternative hypothesis, that the null does not hold. We have that the following holds: \[
(\overline{X}_{1} - \overline{X}_{2}) \sim N_{p}\Bigg(\boldsymbol{\Delta}, \frac{n + m}{n m} \Sigma\Bigg),
\] and also \[
n\mathcal{S}_{1} + m\mathcal{S}_{2} \sim W_{p}(\Sigma, n + m - 2),
\] where \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) respectively are the empirical estimates of the variance-covariance matrix \(\Sigma\), given the first sample and the second sample respectively. Then the rejection region is defined as \[
\frac{nm(n + m - p - 1)}{p(n + m)^2}(\overline{\boldsymbol{X}}_{1} - \overline{\boldsymbol{X}}_{2})^{\top} \mathcal{S}^{-1} (\overline{\boldsymbol{X}}_{1} - \overline{\boldsymbol{X}}_{2}) \geq F_{p, n + m - p - 1}(1 - \alpha).
\] Difference of two multivariate means with unequal variance-covariance matrixNow, we assume a multivariate sample \(\boldsymbol{X}_{1}, \dots, \boldsymbol{X}_{n} \sim N_{p}(\boldsymbol{\mu}_{1}, \Sigma_{1})\) and some another sample \(\boldsymbol{Y}_{1}, \dots, \boldsymbol{Y}_{M} \sim N_{p}(\boldsymbol{\mu}_{2}, \Sigma_{2})\), with some generally different mean parameter \(\boldsymbol{\mu}_{1} \neq \boldsymbol{\mu}_{2}\). We are interested in testing the null hypothesis \[ H_{0}: \boldsymbol{\mu} = \boldsymbol{\mu}_{0} \] against the alternative hypothesis, that the null does not hold. Again, we have that the following holds: \[ (\overline{X}_{1} - \overline{X}_{2}) \sim N_{p}\Bigg(\boldsymbol{\Delta}, \frac{\Sigma_{1}}{n} + \frac{\Sigma_{2}}{m}\Bigg), \] and therefore, it also holds that \[ (\overline{X}_{1} - \overline{X}_{2})^\top \Big(\frac{\Sigma_{1}}{n} + \frac{\Sigma_{2}}{m}\Big)^{-1} (\overline{X}_{1} - \overline{X}_{2}) \sim \chi_{p}^{2}. \] To Do by Yourself
There is a dataset called The estimates for the mean vector and variance-covariance matrix can be obtained as and we can test whether the mean vector of concentrations in the first container ( For a two sample problem we can use the first covariate in the data to define two populations and we provide a test whether the mean vectors are equal or not. The corresponding Hotelling test statistic equals and the corresponding test is performed by Questions
The correponding simultaneous confidence intervals for all possible linear combinations \(\boldsymbol{a}^\top \boldsymbol{\mu}\) of the mean vector \(\boldsymbol{\mu} \in \mathbb{R}^{p}\) is given as \[ P\Big(\forall \boldsymbol{a} \in \mathbb{R}^{p};~ \boldsymbol{a}^\top \boldsymbol{\mu} \in \big( \boldsymbol{a}^\top\overline{\boldsymbol{X}} - \sqrt{K_{\alpha} \boldsymbol{a}^\top \mathcal{S} \boldsymbol{a}}, \boldsymbol{a}^\top\overline{\boldsymbol{X}} + \sqrt{K_{\alpha} \boldsymbol{a}^\top \mathcal{S} \boldsymbol{a}} \big) \big)\Big) = 1 - \alpha, \] where \(K_{\alpha}\) is the corresponding quantile obtained from the Fisher \(F\) distribution (\(K_{\alpha} = \frac{p}{n - p} F_{p, n - p}(1 - \alpha)\)) and \(\mathcal{S}\) is the sample variance-covariance matrix. To Do by Yourself
Homework Assignment(Deadline: fifth lab session / 04.04.2007)Use the command below to instal the Chose one dataset of your choise from the list of all available datasets in the package: There are 21 different datasets and you can load each of them by typing and running
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