# Multivariate Probability Distributions in R

Learn to analyze, plot, and model multivariate data.

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## Course Description

When working with data that contains many variables, we are often interested in studying the relationship between these variables using multivariate statistics. In this course, you'll learn ways to analyze these datasets. You will also learn about common multivariate probability distributions, including the multivariate normal, the multivariate-t, and some multivariate skew distributions. You will then be introduced to techniques for representing high dimensional data in fewer dimensions, including principal component analysis (PCA) and multidimensional scaling (MDS).

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### Reading and plotting multivariate data

**Free**In this introduction to multivariate data, you will learn how to read and summarize it. You will learn how to summarize multivariate data using descriptive statistics, such as the mean vector, variance-covariance, and correlation matrices. You'll then explore plotting techniques to provide insights into multivariate data.

Reading multivariate data50 xpReading multivariate data using read.table100 xpSpecifying datatypes for columns100 xpMean vector and variance-covariance matrix50 xpCalculating the mean vector100 xpCalculating the variance-covariance matrix100 xpCalculating the correlation matrix100 xpPlotting multivariate data50 xpPairs plot using base graphics and lattice100 xpPlotting multivariate data using ggplot100 xp3D plotting techniques100 xp - 2
### Multivariate Normal Distribution

This chapter will introduce you to the most important and widely used multivariate probability distribution, the multivariate normal. You will learn how to generate random samples from a multivariate normal distribution and how to calculate and plot the densities and probabilities under this distribution. You will also learn how to test if a dataset follows multivariate normality.

Multivariate normal distribution50 xpSamples from multivariate normal distributions100 xpIdentify the distribution of a normal sample50 xpDensity of a multivariate normal distribution50 xpCalculating the density of multivariate normal100 xpCalculating dmvnorm over a grid100 xpCumulative Distribution and Inverse CDF50 xpUsing the pmvnorm function100 xpCalculating probability contours using qmvnorm100 xpChecking normality of multivariate data50 xpGraphical tests for multivariate normality100 xpNumerical tests for multivariate normality100 xpTest multivariate normality by wine type50 xp - 3
### Other Multivariate Distributions

This chapter introduces a host of probability distributions to model non-normal data. In particular, you will be introduced to multivariate t-distributions, which can model heavier tails and are a generalization of the univariate Student's t-distribution. You will be introduced to various skew distributions, which are specifically designed to model data that are right or left skewed.

Other common Multivariate distributions50 xpGenerate samples from multivariate t-distribution100 xpIdentify the distribution50 xpDensity and cumulative density for multivariate-t50 xpDensity of multivariate t-distribution100 xpCumulative distributions and quantiles of t100 xpComparing normal and t probabilities50 xpMultivariate skewed distributions50 xpDrawing samples from skew distributions100 xpPlotting and testing of skewed-densities100 xpExamine skewness from contour plot50 xpParameter estimation for multivariate skew-normals100 xp - 4
### Principal Component Analysis and Multidimensional Scaling

In the final chapter, you will be introduced to techniques for analyzing high dimensional data, including principal component analysis (PCA) and multidimensional scaling (MDS). You will also learn to implement these techniques by analyzing data.

Principal Component Analysis50 xpNumber of PCs for the state.x77 dataset50 xpUsing the princomp function100 xpChoosing the number of components50 xpCalculating the proportion of variation explained100 xpChoosing the number of PCs100 xpNumber of components explaining 95% variation50 xpNumber of PCs using scree plot100 xpInterpreting PCA attributes50 xpLoadings and scores for the PCs100 xpVisualizing PCA using the factoextra library100 xpMulti-dimensional scaling50 xpMultidimensional scaling in two dimensions100 xpMultidimensional scaling in three dimensions100 xpCongratulations50 xp

Prerequisites

Foundations of Probability in R#### Surajit Ray

Senior Lecturer in Statistics, University of Glasgow

Surajit is a Professor of Statistics at the University of
Glasgow's School of Mathematics & Statistics. His research
interests are in the area of model selection, the theory and
geometry of mixture models, and functional data analysis. He is
especially interested in challenges presented by "large
magnitude", both in the dimension of data vectors and in the
number of vectors. He is the author of the R-package Modalclust.
He is also a founder board member and instructor for the Online
MSc in Data Analytics at the University of Glasgow.

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