Statistics and PDF output

Supported statistics and PDFs

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. Examples:
  • <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 PDFs can be done using either ordinary or central sample space variables. Examples:
  • 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

Example control file section for statistics output

statistics
  interval 2  # Output statistics every 2nd time step
  <X1> <X2> <x1x1> <x2x2> <x1x2>
  <R> <rr> <R2> <r2r2> <R3> <r3r3> <r1r2> <r1r3> <r2r3>
  <K1> <k1k1> <k2k2> <K1K1> <k3>
  #<H1> <H2> <h1h1> <h2h2> <h1h2>
  #<x1z2Uy2>
  <Y2>
  <y1y1>
  <y2y2>
  <y1y2>
  #<x1x2Z1z2>
end

Example control file section for PDF output

pdfs
  interval   10             # Output PDFs every 10th time step
  filetype   txt            # Use txt file output
  policy     overwrite      # Overwrite previous time step with new one
  centering  elem           # Use element-centering for sample space
  format     scientific     # Use 'scientific' floats in txt file output
  precision  4              # Use 4 digits percision for floats in txt output

  # Univariate PDF "O2" of the full variable O2 with bin size 0.05 and
  # explicitly specified sample space extents 0.0 and 1.0 (min and max)
  O2( O2 : 5.0e-2 ; 0 1 )

  # Bivariate PDF "f2" of the fluctuating variables o1 and o2 with bin sizes
  # 0.05 in both sample space dimensions
  f2( o1 o2 ; 5.0e-2 5.0e-2 )

  # Bivariate PDF "mypdf" of the fluctuating variables o1 and o2 with bin sizes
  # 0.05 in both sample space dimensions and explicitly specified sample space
  # extents, { xmin, xmax, ymin, ymax } = { -2, 2, -2, 2 }
  mypdf( o1 o2 : 5.0e-2 5.0e-2 ; -2 2 -2 2 )

  # Trivariate PDF "f3" of full variables O1, O2, and O3 with bin sizes 0.1 in
  # all dimensions of the sample space
  f3( O1 O2 O1 : 1.0e-1 1.0e-1 1.0e-1 )

  # Trivariate PDF "newpdf" of full variables O1, O2, and O3 with bin sizes
  # 0.1 in all dimensions of the sample space and explicitly specified sample
  # space extents, { xmin, xmax, ymin, ymax, zmin, zmax } = { 0, 1, 0, 1,
  # -0.5, -0.5 }
  newpdf( O1 O2 O1 : 1.0e-1 1.0e-1 1.0e-1 ; 0 1 0 1 -0.5 0.5  )
end