Aaron Schlegel's Notebook of Interesting Things

Multiplicative congruential generators, also known as Lehmer random number generators, is a type of linear congruential generator for generating uniform pseudorandom numbers. The multiplicative congruential generator, often abbreviated as MLCG or MCG, is defined as a recurrence relation similar to the LCG.

Linear congruential generators (LCGs) are a class of pseudorandom number generator (PRNG) algorithms used for generating sequences of random-like numbers. The generation of random numbers plays a large role in many applications ranging from cryptography to Monte Carlo methods. Linear congruential generators are one of the oldest and most well-known methods for generating random numbers primarily due to their comparative ease of implementation and speed and their need for little memory.

The set operations 'union' and 'intersection' should ring a bell for those who've worked with relational databases and Venn Diagrams. The 'union' of two of sets A and B represents a set that comprises all members of A and B (or both).

Sets define a 'collection' of objects, or things typically referred to as 'elements' or 'members.' The concept of sets arises naturally when dealing with any collection of objects, whether it be a group of numbers or anything else.

Hierarchical clustering is a widely used and popular tool in statistics

The iterated principal factor method is an extension of the principal

As discussed in a previous post on the principal component method of factor analysis, the \(\hat{\Psi}\) term in the estimated covariance matrix \(S\), \(S = \hat{\Lambda} \hat{\Lambda}' + \hat{\Psi}\), was excluded and we proceeded directly to factoring \(S\) and \(R\). The principal factor method of factor analysis (also called the principal axis method) finds an initial estimate of \(\hat{\Psi}\) and factors \(S - \hat{\Psi}\), or \(R - \hat{\Psi}\) for the correlation matrix.

In the first post on factor analysis, we examined computing the estimated covariance matrix \(S\) of the rootstock data and proceeded to find two factors that fit most of the variance of the data. However, the variables in the data are not on the same scale of measurement, which can cause variables with comparatively large variances to dominate the diagonal of the covariance matrix and the resulting factors. The correlation matrix, therefore, makes more intuitive sense to employ in factor analysis.

The goal of factor analysis, similar to principal component analysis, is to reduce the original variables into a smaller number of factors that allows for easier interpretation. PCA and factor analysis still defer in several respects. One difference is principal components are defined as linear combinations of the variables while factors are defined as linear combinations of the underlying latent variables.

Image compression with principal component

Often, it is not helpful or informative to only look at all the variables in a dataset for correlations or covariances. A preferable approach is to derive new variables from the original variables that preserve most of the information given by their variances. Principal component analysis is a widely used and popular statistical method for reducing data with many dimensions (variables) by projecting the data with fewer dimensions using linear combinations of the variables, known as principal components.

The method of image compression with singular value decomposition is

SVD underpins many statistical and real-world

Cholesky decomposition, also known as Cholesky factorization, is a

The inverse of a number is its reciprocal. For example, the inverse of 8