# Power Spectrum Probability Vector Calculator

### Description

Function `d1spec`

is applied to a pair of vectors (sample outcomes, observation data values, time series data, 1d signal values, etc.) and generates the pair of corresponding spectral power density functions

### Usage

1 |

### Arguments

`sample0` |
vector of values (sample outcomes) |

`sample1` |
vector of values (sample outcomes) |

`band` |
two border values to set a range of considered frequencies. The default c(0,0) sets full entire range i.e. from 0 to 0.5 Hz, where 1 Hz = [1/sampling_interval] |

`brks` |
value used in a same manner as the number of cells for the histogram. The default brks=0 sets the number of cells equal to |

`meansub` |
logical item defining if individual baseline (defined as individual mean value) is subtracted from original vectors before application of Fourier transform |

### Details

Spectral power density functions are often being used as probability vectors characterizing thermodynamic states of a system. Here fast Fourier transform algorithm is utilized and the frequency values are used as 1 Hz = [1/sampling_interval], i.e. in a range from 0 to 0.5 Hz. In general the function works similarly to `d1nat`

. As a bonus it prints basic statistics summary for power density functions alongside with technical plot. It is recommended for use as a data preparation step before following Klimontovich's S-theorem based analysis.

### Value

`f0 ` |
spectral power density function as a probability vector representing state0 of a system |

`f1 ` |
spectral power density function as a probability vector representing state1 of a system |

`freqs ` |
vector of coresponding frequency values |

### Author(s)

Vitaly Efremov <vitaly.efremov@dcu.ie>

### References

T.G.Anishchenko, P.I.Saparin, N.B.Igosheva, V.S.Anishchenko. Sex differences in human cardiovascular responses to external excitation. Il Nuovo Cimento D. Luglio-Agosto 1995, Volume17, Issue7-8, pp.699-707.

E.J.Groth. Probability distributions related to power spectra. Astrophysical Journal, 1975. Suppl. Ser., Vol.29, No.286, p. 285-302.

T.Anishchenko, N.Igosheva, T.Yakusheva, O.Glushkovskaya-Semyachkina, O.Khokhlova. Normalized entropy applied to the analysis of interindividual and gender-related differences in the cardiovascular effects of stress. Eur J Appl Physiol. 2001 Aug; 85(3-4):287-98.

### See Also

`crit.stheorem`

,
`cxds.stheorem`

,
`d2spec`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
s0 <- 2+sin(c(1:128))
s1<- array(c(rep(0,8),rep(1,8)), c(256))
b<-d1spec(sample0=s0,sample1=s1); b
b<-d1spec(s0, s1, band=c(0,0.25), brks=16, meansub=FALSE); b
#example of 3-step data analysis with Klimontovich's S-theorem
# step a. Clean samples from outliers (points out of 1.4 sigmas)
a<-utild1clean(s0, s1, method='both', nsigma=1.4)
# step b. Create probability vectors. It seems that s0 has lower level
# of orderliness (Shannon entropy is higher).
b<-d1spec(a$clean0,a$clean1); b
# step c. Compare samples with Klimontovich's S-theorem. Renormalized entropy indicates
# the opposite: s0 is more ordered and difference in Shannon entropy values was
# due to just "thermodynamic noise" (discretization noise in this case)
crit.stheorem(b$f0,b$f1)
cxds.stheorem(b$f0,b$f1)
``` |