# Statistics and PDF output

This pages discusses how to extract statistics from a Walker simulation. See also the Walker examples.

## 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.

- Examples of ordinary moments:
- 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