Multi Variant Statistical Analysis

created : 2021-09-24T03:38:16+00:00
modified : 2021-10-15T14:07:06+00:00

Chapter 0. Introduction

0.1 Visualization of Multivariate Data

library(psych) # Scatter Plot Matrix

# Chernoff's Faces
faces(USArrests, face.type=1, cex=0.5))

# Star plot

# 3-D scatter plot
scatterplot3d(USArrests[, -1], type="h", highlight.3d=TRUE, angle=55, scale.y=0.7, pch=16, main="USArrests")

# 3-D rotated plot

# Profile plot
parcoord(USArrests, col=c(1+(1:50)), var.label=T)

# Growth curves for longitudinal data
p <- ggplot(data = Orthodont, aes(x = age, y = distance, group = Subject, colour=Subject))
p + geom_line()

p + geom_line() + facet_grid(. ~ Sex)

Summary of Introudction

Chapter 1. Linear algebra

1.1 Scalars, vectors, matrices

1.2 Operations of matrices

1.3 Trace and determinant for square matrcies

1.4 Rank of a matrix

1.5 Inverse matrix

1.6 Partitioned matrices

Example 1.6.2

1.7 Positive definite matrix

1.8 Orthogonal vectors and matrices

1.9 Eigenvalues and eigenvectors

1.10 Spectral decomposition

1.11 Cauchy-Schwarz inequality

1.12 Differentiation in Vectors and Matrices

1.13 Some useful quantities

1.14 Random vectors and matrices

1.14.1 Parameter vectors and matrices

2. Chapter 2 Multivariate Normal Distribtuion

2.1 Definitions

2.2 Properties of multivariate normal distribution

2.3 Estimation for sampling from a multivariate normal distributions

2.3.1 Likelihood function of a sample from a multivariate normal distribution

2.3.2 Maximum likelihood estimations (MLEs) from a multivariate normal distribution

2.4 Sampling distributions of $\bar X$ and $S$

2.5 Definition and Properties of the Wishart Distirubiton

2.6 Large sample distributions for $\bar X$ and $S$

2.7 Assessing the assumption of multivariate normality

2.8 Transformations to near normality

  1. Power transformations: When all observations are nonnegative, we may consider a family of power transformations. If some measurements are negative, then we first add a constant to all measurements and then apply a power transformation.:
    • $x_i + c \rightarrow (x_i + c)^{\lambda}$
  2. Box-Cox transformations: The Box-Cox transformation family is similar to the power transformation. This family continuously connects the logarithmic transform as the power $\lambda$ approaches zero.:
    • $x^{(\lambda)} = \begin{cases} \frac{x ^{\lambda} - 1}{\lambda} & for \lambda \not = 0 \\ log x & for \lambda = 0 \end{cases}$
    • for $x > 0$. We choose $\lambda$ by maximizing the log-likelihood function:
    • $l(\lambda) = - \frac{n}{2} log [\frac{1}{n} \sum_{i = 1}^n (x_i^{(\lambda)} - \bar {x ^ {(\lambda)}})^2] + (\lambda - 1) \sum_{i=1}^{n} log x_I$
  3. Note that we should not expect some transformation can always make the data close to normality.

Chapter 3 Hypothesis tests

3.1 Review of hypothesis tests for a univariate normal mean

3.1.1 When $\sigma^2$ is known

3.1.2 When $\sigma^2$ is unknown

3.2 Hypothesis test on one sample multivariate normal mean vector

3.2.1 When the covariance matrix $\Sigma$ is known

3.2.2 Hotelling’s $T^2$ Statistic: when $\Sigma$ is unknown