Multivariate Analysis

Section- A


 1-Discuss about the Wishart distribution. Also find its additive Property.

 The Wishart distribution is a multivariate continuous distribution which generalizes the Gamma distribution. It is a family of probability distributions defined over symmetric, nonnegative-definite random matrices (i.e. matrix-valued random variables).

 The Wishart distribution is a multivariate generalisation of the univariate χ2 distribution, and it plays an analogous role in multivariate statistics. In inferential statistics, the Wishart distribution is also defined as the distribution of the sample covariance matrix. 


The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a multivariate normal distribution.

The Wishart distribution has an additive property, which states that

 if M1∼Wp(Σ,n1) and M2∼Wp(Σ,n2) are independent, 

then M1+M2∼Wp(Σ,n1+n2).

 In other words, the sum of two independent Wishart distributed matrices is also Wishart distributed, with the degrees of freedom equal to the sum of the degrees of freedom of the individual matrices.

 

2-Discuss about the Maholanobis D2 with its various applications.

Mahalanobis D^2 is a 2x2 matrix which can be used for various applications such as calculating genetic distances, estimating population parameters, and clustering data.

Identify the entries of the second row of the Mahalanobis D^2 matrix. The entries are the first two eigenvalues of the matrix that are negative. These eigenvalues are associated with correlations in the population which are not due to common ancestry or environmental factors, but are mainly due to historical effects such as migration and bottlenecks.

The second row contains two negative eigenvalues with corresponding eigenvectors (EV) {E11} and {E22}. These EVs are orthogonal to each other and therefore correlate negatively with each other. Thus, these two EVs represent the two eigenvectors that have the highest correlation with the other EVs representing the four eigenvalues in the first row. 

If we denote the eigenvalue associated with the fourth eigenvector by λ4, then λ4 corresponds to the first entry of the second row. Similarly, the first entry of the third row is λ3 = √{(E11)2 + (E22)2}, 

which corresponds to the second eigenvalue of the first row. The third row also has two negative eigenvalues, corresponding to the second and fourth eigenvectors of the first row. 

Therefore, λ3 = √{(E12)2 + (E21)2} corresponds to the entry of the third row.

For the n-dimensional vector X having N components xi we can compute its Mahalanobis distance with respect to the original observations as:

d_{M}\left(X;Y\right)=\sqrt{\sum_{k=0}^{N}{\left( Xi-Yk \right)}^2} 

where Y represents the vector of original observations. This distance measure is used to find the distance between the data points and the nearest cluster center in a data set. In k-means clustering, the Mahalanobis distance between the data points and the center of the cluster is used to select the cluster with the minimum distance for each data point. It can also be used to select the optimal number of clusters in a data set by assigning each data point to the nearest cluster center based on its Mahalanobis distance from the data points to all the cluster centers. The process of finding the best cluster for a given data point is called search optimization.

This post will explore how to use D^2 matrices to aid in selecting the best data clustering algorithm to use based on a given data set, thus resulting in a more accurate and accurate model of the given dataset.

Clustering is the process of grouping a set of objects into subsets such that objects in the same subset are more similar to each other than to those in other subsets. A clustering technique may be deterministic or stochastic. Deterministic techniques are based on predefined criteria and algorithmic methods while stochastic techniques use random sampling techniques to group similar objects together.

 

3-Discuss about the Hoteing’s T2 distribution and its applications.

 Hotelling's T-squared distribution is a statistical model that describes the relationship between two variables. It is used to test for significant differences between two groups, or to compare the relative strength of two variables. Hotelling's T-squared distribution has applications in many fields, including:

 Psychology

 Economics

 Biology

 Sociology

Hotelling's T-squared distribution is a statistical distribution that describes the relationship between two variables. It was developed by Harold Hotelling in 1933, who used it to describe the relationship between price and quantity for a given product.

 Hotelling's T-squared distribution can be used to describe the relationship between two continuous variables, such as temperature and humidity or height and weight. It can also be used to describe the relationship between two discrete variables, like gender and age.

In this article, we will explore the Hotelling's T-squared distribution and it's applications. We will start with a simple example of how to use Hotelling's T-squared distribution to determine the number of customers who would leave your business if they were given a free ice cream cone. Then we'll discuss how this can be applied in more complex situations.

 Hotelling's T-squared distribution is a probability density function that describes the relationship between the prices of two different products. The T-squared distribution is used to determine whether a pair of prices is an equilibrium.

 The T-squared distribution has applications in many areas, including marketing, finance, and economics. Here are some examples:

 Marketing: Hotelling's T-squared distribution can be used to determine whether a pair of prices is an equilibrium by determining whether both variables have a Pareto distribution. This allows marketers to determine whether their products will be successful or not based on the prices they set for those products. For example, if Coca-Cola sets their price at $1 per bottle but Pepsi sets theirs at $1.50 per bottle, then Coca-Cola's product will be less popular than Pepsi's because there is not enough demand for both products at their respective prices.

 Finance: Hotelling's T-squared distribution can also be used in finance applications when determining whether or not risky investments are profitable or not. For example, if you invest in an investment property with a mortgage debt of $200K and annual rent of $2K per month, then your expected.

 One of the most famous statistics is Hotelling's T-squared distribution.

 It is used to calculate the spread between the expected value and the observed value of a variable.

 The T-squared distribution has two parameters, which are defined by:

 1) The number of observations (n)

 2) The mean of each variable (μ)

Hotelling's T-squared distribution is used to test a hypothesis about the relationship between two variables. It can be used in business, finance, and economics to determine if there is a link between two variables. The T-squared test can also be used to determine if two variables are independent of one another.

 Hotelling's T-squared distribution is named after Ragnar Hotelling, who introduced it in 1924. He was concerned with testing whether two random variables were related or independent.

Hotelling's T-squared distribution is a statistical model that describes how two variables are related. It was first introduced by Frank H. Hotelling in 1932, and it's been used in many different scientific fields since then.

 One of the most common applications of Hotelling's T-squared distribution is in business research. It can be used to predict how two salespeople will perform as a team or how two products will sell together.

 Another common application of Hotelling's T-squared distribution is to describe correlations between variables when they have been measured in different ways.

Hotelling's T-squared distribution is a statistical distribution that gives us information about the relationship between two variables, such as the amount of investment in a product and how much the product costs.

 The Hotelling's T-squared distribution is useful for making predictions about how much a variable will change, given changes to other variables. For example, if you know that your company makes products that cost $10 each, you might want to predict whether or not people will buy them at an all-time low price of $5 each. If the price drops, then it seems likely that people will buy more of your product.

 

Section - B

 1-Find the characteristic function of MMD.

 The characteristic function of the maximum mean discrepancy (MMD) is the Fourier transform of the probability density function (PDF) of the MMD. It is defined as the expected value of the function of the MMD's random variable, taken over all possible values of the MMD's random variable. The characteristic function is often used to find the MMD's PDF, and it is also used in the computation of the probability of an event occurring under the MMD's distribution. The characteristic function of the MMD is given by: C(t) = E[exp(itMMD)] where t is the random variable. The characteristic function can be used to calculate the probability of an event occurring under the MMD's distribution. For example, to calculate the probability that the MMD is greater than a given value x, the characteristic function can be used to calculate the PDF of the MMD, which can then be integrated to find the probability that the MMD is greater than x.

 

2-Describe about the multiple and partial short.

In statistics, multiple comparison refers to the problem of comparing more than two groups or samples at the same time. This can lead to an increased risk of making a type I error, or falsely rejecting the null hypothesis. There are several methods for controlling this error rate, such as the Bonferroni correction, the Tukey test, and the Holm-Bonferroni method.

Partial correlation is a statistical technique that measures the correlation between two variables while controlling for the effects of one or more other variables. This allows for the determination of the unique relationship between the two variables of interest, as opposed to the relationship that may be confounded by the effects of other variables. It can be calculated using statistical software. 

3-Write short notes on Discriminate Analysis.

 Discriminant analysis is a statistical technique used to classify observations into non-overlapping groups, based on scores on one or more quantitative predictor variables. It is used to analyze data when the dependent variable is categorical and the independent variable is interval in nature. Discriminant analysis builds a predictive model for group membership. The model is composed of a discriminant function (or, for more than two groups, a set of discriminant functions) that is used to assign observations to groups. There are four main types of discriminant analysis: linear discriminant analysis, multiple discriminant analysis, quadratic discriminant analysis, and non-parametric discriminant analysis. Discriminant analysis is used in a variety of fields, such as psychology, marketing, and medical research.

 4-Maximum likelihood estimates of mean vector.

 Maximum likelihood estimates (MLEs) are a method for estimating the unknown parameters of a statistical model, given a set of observed data. In the case of the mean vector, the MLEs of the mean vector for a multivariate normal distribution are the sample mean of the observations.

The likelihood function for a multivariate normal distribution with a mean vector of μ and a covariance matrix of Σ is given by:

L(μ,Σ|X) = (2π)^(-n/2) |Σ|^(-1/2) exp[-1/2(X-μ)^T Σ^(-1) (X-μ)]

Where X is the nxm matrix of the observations, n is the number of observations and m is the number of variables. 

The MLEs of the mean vector are found by taking the partial derivative of the log-likelihood function with respect to μ and setting it to zero.

Then, solving this equation for the mean vector, we get the MLEs of the mean vector as the sample mean of the observations.

μ_MLE = 1/n Σ X_i

This is the point in the parameter space that maximizes the likelihood function given the observations.

In summary, the MLE of the mean vector for a multivariate normal distribution is the sample mean of the observations. It is the point in the parameter space that maximizes the likelihood function given the observations.

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