Statistics class declaration.
This file implements a statistics class that can be used to estimate statistics from an ensemble. Supported at this time are ordinary and central statistical moments of arbitrary-length products and arbitrary number of 1D, 2D, and 3D probability density functions (PDF) with sample spaces of ordinary and/or central sample space variables.
Definitions and nomenclature:
- Upper-case letters denote a full random variable, e.g., X
- Lower-case letters denote a fluctuation about the mean, i.e., x = X - <X>
- Letters can be augmented by a field ID, i.e., X2 is the full variable of the second component of the vector X, while x1 = X1 - <X1> is the fluctuation about the mean of the first component of vector X.
- If the field ID is unspecified, it defaults to the first field, i.e., X = X1, x = x1, etc.
Statistical moments of arbitrary-length products can be computed.
- <X> - mean,
- <xx> - variance,
- <xxx> - third central moment,
- <xy> - covariance of X and Y,
- <x1y2> - covariance of the first component of vector X and the second component of vector Y
- In general, arbitrary-length products can be estimated that make up a statistical moment, using any number and combinations of upper and lower-case letters and their field IDs < [A-Za-z][1-9] ... >.
- A statistical moment is ordinary if and only if all of its terms are ordinary. A central moment has at least one term that is central, i.e., a fluctuation about its mean.
- Examples of ordinary moments: <X>, <XX>, <XYZ>, etc.
- Examples of central moments: <x1x2>, <Xy>, <XYz>, etc.
Estimation of the PDFs can be done using either ordinary or central sample space variables.
- p(X) denotes the univariate PDF of the full variable X,
- f(x1,x2) denotes the bivariate joint PDF of the fluctuations of the variables x1 and x2 about their respective means,
- g(X,y,Z2) denotes the trivariate joint PDF of variables X, y = Y - <Y>, and Z2.
- namespace tk
- Toolkit declarations and definitions for general purpose utilities.