# d2nat.d1nat: Probability Mass Function Calculator for Matrices In stheoreme: Klimontovich's S-Theorem Algorithm Implementation and Data Preparation Tools

## Description

Function `d2nat.d1nat` is applied to a pair of matrices and generates then the pair of corresponding probability mass functions by calling `d1nat`

## Usage

 `1` ```d2nat.d1nat(d2arr0, d2arr1, band = c(0, 0), brks = 64, method = "default") ```

## Arguments

 `d2arr0` sample matrix `d2arr1` sample matrix `band` two border values to set a range of considered values in matrices. The default c(0,0) sets full entire range i.e. `range(d2arr0, d2arr1)` `brks` value giving a number of bins (in a same manner as the number of cells for the histogram). The default value sets the number of bins automatically equal to 64. `method` specifies selection of matrix elements `method='default'` simply to call `d1nat` and apply it to ensemble of all matrix elements as to 1d vector of outcomes `method='cols'` to create 1d array with elements being the mean values of original matrix columns and then simply to call `d1nat` function `method='rows'` to create 1d array with elements being the mean values of original matrix rows and then simply to call `d1nat` function

## Details

It works similarly to `d1nat` function but for pair of matrices. It is recommended for use as a data preparation step before following Klimontovich's S-theorem based analysis. For instance, it can be used for image analysis.

## Value

 `f0 ` probability vector representing state0 of a system `f1 ` probability vector representing state1 of a system `midpoints ` vector of the centres of bins where probability values are calculated

## Author(s)

Vitaly Efremov <[email protected]>

## References

A.N.Herega. On One Criterion of the Relative Degree of Ordering in Images. Technical Physics, 2010, Vol.55, No.5, pp.741-742.

G.B.Bagci, U.Tirnakli. Self-organization in dissipative optical lattices. CHAOS. 19, 033113. 2009.

`crit.stheorem`, `cxds.stheorem`, `d1nat`, `utild2group`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```#two modelling arrays: random with randomness distorted by power s0<-array(runif(256,0,1)^2, c(16,16)) s1<-array(runif(512,0,1)^3, c(16,8)) b<-d2nat.d1nat(d2arr0=s0,d2arr1=s1); b b<-d2nat.d1nat(s0,s1,brks=256); b b<-d2nat.d1nat(s0,s1,brks=18,band=c(0.1,0.5),method='rows'); b #example of 3-step data analysis with Klimontovich's S-theorem # step a. Split matrices to regions with radius 1, create new matrices # of region means a<-utild2group(s0, s1, radius=1) # step b. Create probability vectors b<-d1nat(a\$group0,a\$group1,brks=8,band=c(0.1,0.8)) # step c. Compare samples with Klimontovich's S-theorem crit.stheorem(b\$f0,b\$f1) cxds.stheorem(b\$f0,b\$f1) ```