A review of entropy, conditional entropy, and mutual information, with the basic proofs. I then set up how I plan to use mutual information to relate spatial mappings between cortical regions to their temporal coupling.
Principal angles between two subspaces come from the SVD of the product of their orthonormal bases. Most subspace distances, including the Grassmann distance, are functions of those angles.
A friend asked how to tell whether a biomarker can predict cardiotoxicity. I build ROC curves by hand on synthetic case and control data, then use them to assess the diagnostic power of simulated dose-response curves.
Deriving the delta method from a Taylor expansion and simulating its use for variance stabilizing transformations of Poisson and exponential data and for standard errors in polynomial regression.
Using the squared Mahalanobis distance, which is chi-squared distributed under multivariate normality, to compare connectivity fingerprints between one cortical region and every other region, with Ledoit-Wolf covariance estimates and a check that the empirical distances follow the expected distribution.
Working through the definition of convergence in probability from Casella and Berger, using Chebyshev's inequality to show the weak law of large numbers for the sample mean and variance, and simulating normal samples to watch both estimates converge.
A sum of independent Poisson variables is Poisson with the summed rate. Conditioned on that sum, the individual counts follow a multinomial distribution.
An introduction to dynamic mode decomposition, which splits spatiotemporal data into spatial modes and their time dynamics. I derive the algorithm using a truncated SVD, with resting-state MRI as the motivating use case.