Multivariate Normal Distribution Tutorial

Overview

• This is the first tutorial focusing on multivariate normal distribution and clustering algorithms.
• The multivariate normal distribution is determined by parameters: mu (mean) and sigma (covariance matrix).

Key Concepts

Probability Density Function

• Probability of x is given by:
  $P(X | ext{params})$
  where
  • mu is in k-space (means)
  • sigma is in k x k space (covariance matrix)

Covariance

• Covariance between two variables x and y is calculated as: $$Cov(X,Y) = \frac{(X - E[X])(Y - E[Y])}{n - 1}$$
  where E[X] and E[Y] are the expected values.

Implementation Steps

1. Import Libraries

   • Import numpy and matplotlib for calculations and plotting.

2. Generate Toy Dataset

   • Create a multivariate normal distribution with:
     • Mean mu = (0, 0)
     • Covariance sigma = [[1, 0.5], [0.5, 1]]
   • Generate 100 data points.
   • Check shape of x, should be (100, 2).

3. Define Class

   • Create a class MultivariateNormal.
   • Initialize parameters mu and sigma to None.

4. Fit Method

   • Define fit(self, x) method to calculate mu and sigma.
   • Convert x into column vectors as needed.
   • Set mu as the mean of x along the appropriate axis.
   • Calculate covariance matrix sigma:
     • Subtract mu from x.
     • Use numpy functions to compute covariance matrix and normalize by n-1.

5. Probability Calculation

   • Define prob(self, x) method to predict probabilities:
     • Calculate factors for the probability density function based on mu and sigma.
     • Use numpy.einsum for efficient calculations.

6. Plotting the Distribution

   • Generate a mesh grid for plotting probabilities.
   • Reshape data to appropriate formats for plotting.
   • Create contour plots of the probability density function.
   • Overlay the original data points on the contour plot.

Conclusion

• Successfully implemented a Gaussian distribution fitting procedure using multivariate normal distribution concepts.
• Visualization confirms the model fits the generated data well.
• Questions and comments are welcome for further clarification.