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A geometric interpretation of the covariance matrix

In this article, we provide an intuitive, geometric interpretation of the covariance matrix, by exploring the relation between linear transformations and the resulting data covariance. Most textbooks explain the shape of data based on the concept of covariance matrices. Instead, we take a backwards approach and explain the...

April 24, 2014 Vincent SpruytLinear algebra47 Comments

How to draw a covariance error ellipse?

In this post, I will show how to draw an error ellipse, a.k.a. confidence ellipse, for 2D normally distributed data. The error ellipse represents an iso-contour of the Gaussian distribution, and allows you to visualize a 2D confidence interval. The following figure shows a 95% confidence ellipse for...

April 3, 2014 Vincent SpruytStatistics62 Comments

Why divide the sample variance by N-1?

In this article, we will derive the well known formulas for calculating the mean and the variance of normally distributed data, in order to answer the question in the article’s title. However, for readers who are not interested in the ‘why’ of this question but only in the...

March 7, 2014 Vincent SpruytStatistics18 Comments

What are eigenvectors and eigenvalues?

Eigenvectors and eigenvalues have many important applications in computer vision and machine learning in general. Well known examples are PCA (Principal Component Analysis) for dimensionality reduction or EigenFaces for face recognition. An interesting use of eigenvectors and eigenvalues is also illustrated in my post about error ellipses. Furthermore,...