The covariance matrix has many interesting properties, and it can be found in mixture models, component analysis, Kalman filters, and more. Developing an intuition for how the covariance matrix operates is useful in understanding its practical implications. This article will focus on a few important properties, associated proofs, and then some interesting practical applications, i.e., extracting transformed polygons from a Gaussian mixture's covariance matrix.

I have often found that…

Data scientists and predictive modelers often use 1-D and 2-D aggregate statistics for exploratory analysis, data cleaning, and feature creation. Higher dimensional aggregations, i.e., 3 dimensional and above, are more difficult to visualize and understand. High density regions are one example of these N-dimensional statistics. High density regions can be useful for summarizing common characteristics across multiple variables. Another use case is to validate a forecast prediction’s…

I recently created a ‘particle optimizer’ and published a pip python package called kernelml. The motivation for making this algorithm was to give analysts and data scientists a generalized machine learning algorithm for complex loss functions and non-linear coefficients. The optimizer uses a combination of simple machine learning and probabilistic simulations to search for optimal parameters using a loss function, input and output matrices, and (optionally) a random…