Last updated: 2017-07-26

1. Simulation and Approximation Methods
2. Properties of the Complexity Coefficients
3. Seizure Prediction
4. Computing the Complexity Coefficients

The \(\varepsilon\)-complexity of a continuous function is, roughly speaking, the amount of information needed to reconstruct that function within an absolute error \(\varepsilon\). The theory and procedure for estimating \(\varepsilon\)-complexity was introduced by Darkhovsky and Piryatinska¹. In previous work, the complexity coefficients have been shown to be useful features for the segmentation and classification of time series².

The purpose of these analyses was three-fold:

1. To improve the estimation of the complexity coefficients by enlarging the set of approximation methods used in the estimation procedure.

2. To explore the relation between the complexity coefficients, the Hölder condition and fractal dimension.

3. An application of the complexity coefficients to the segmentation and classification of EEG.

4. Along with an overview of the results of each of these analyses we outline the procedure for estimating the complexity coefficients.

library(devtools)
load_all()
knitr::read_chunk("../code/eeg-segment-plot.R")

  par(mar = c(0, 0, 2, 0))
  library(ggplot2)
  library(dplyr)
  library(viridis)
  library(ecomplex)
  prefix    = "eeg"
  seed      = 2017
  
  # Derived features. 30 Trials : 6 channels : 15 features
  feature_df <-readRDS(cache_file("mod_all_features", prefix))
  # Load meta data 
  trial_df <- readRDS(cache_file("mod_trial_segments", "eeg"))
  
  df <- do.call(rbind, feature_df) %>% 
        cbind(., trial_df) %>% 
        dplyr::filter(., window == 240)
  
  df$trial <- 1:26
  
  ch = 1; colnums = c(5:9,  13:15); tnum = 8
  
  dfch <- df[[ch]][[tnum]]
  res <- palarm(dfch$ecomp.cspline_B)
  
  seg_plot <- function(startn, stopn, title ="", ...){
    segcol = adjustcolor("tomato3", 0.9)
    pal  = c("gray20", segcol, viridis::viridis(256)[c(1, 30, 60, 90, 120)])
  
    plot(dfch$ecomp.cspline_B + 3, 
                    type = 'l',  
                    col = adjustcolor("gray20", 0.6), 
                    lwd = 2,
                    lty = 1, 
                    cex = 0.5,
                    ylim = c(0,2.2), 
                    xlim = c(0,120), 
                    xlab = "Time",
                    yaxt = "n",
                    ylab = "", 
                    main = title,
                    ...)
    lines(res$means + 3, type = 'l', xlim=c(0,120), col=segcol, lwd=2, lty= 1 )
    for(k in startn:stopn) lines(dfch[ , 4 + k] + 0.5*(k-startn), 
                       lwd = 2, 
                       col = pal[k + 2])
    for(k in res$kout) abline(v = k, col = segcol, lwd = 1.5, lty = 2)
  }
  
  # pdf(file.path(getwd(), paste0("figures/", prefix, "-segment-plot.pdf")), 
  #     width = 9, height = 4)
  par(mfrow = c(1,1), mar = c(1,1,1,1))
  seg_plot(3,5, xaxt = "n")