# Problem

The assignment we will explore today is to create a simulation that will allow us to apply a pattern recognition system based around maximum likelihood classification. The entire process is a simulation thus we will be required to first generate our own Gaussian distrusted data, then transform it using principle component analysis as we explored in CA12, and lastly classify it. In our assignment we will focus on analyzing five different cases. In each case we will only change our sigma2 parameter. The parameters we analyze our laid out below:

`Class 1: μ1=[1; 1] ; C1=[1 0;0 1]`

`Class 2: μ2=[−1; −1] ;C2=[sigma2 0.5; 0.5 sigma2] sigma2→0.25,0.5,1.0,2.0,4.0`

As can be seen our first class parameter does not change. It is a Gaussian distributed multivariate array with equal energy in all directions. It has a mean that centers around [1,1]. Our second class energy changes but the correlation between our multivariate array stays constant at 0.5 Our second class will be center around [-1, -1].

From classification we use maximum likelihood classification. Before performing the classification we transform each set of data using principal component analysis. The process will convert our data sets so our data is not independently correlated. Thus our covariance matrix will have an off diagonal of zeros. By performing the transformation we can perform a Euclidean distance calculation (instead of a Mahalanobis distance) to the means of our data sets. What we wish to see is how the energy of our data correlates to the classification success of our data.

# Approach and Results

We begin our analysis by create multiple cases to compare. The results are laid out below. We plotted both the original un-transformed data (left) with the transformed scatter plot of our classes (right).

The above case is our first test. We see our two sets of data and how the principal component analysis transforms our data so that they become uncorrelated. Our transformation process is performed to the expected means of data sets where we see there is hardly any difference for our first set of data. For our second class our mean differs slightly. The error can be produced in our data creation process. We still see that even though we our mean is off by 23% our classification error rate is low for this case.