Statistical Properties of the N-Back Sequences

Problems

Statistical properties of n-back sequences bias behaviors. These bias, under specified structure, allows multiple cognitive strategies, producing heterogeneous behavior in a “gold standard” cognitive task.

Gaps

• Unclear how to parameterize interesting variations for sequence generation
• How do we model these multiple strategies (which requires identifying which sequence variations matter)
  • local vs. global properties, which one matters the most?
  • Local: lumpiness, short sequence patterns -> could be exploited by “reactive”/automaticity
  • Global: No lures, large vocabulary -> pattern repeats implies a target

load('./data/CL2015.RData')
window_size <- 8

\(P=\{V,D,C,W\}\)

\(V=\{x_N,x_{T},x_{T,local},x_L,x_{L,local},x_V,x_U,x_S,x_{S,local},x_G\}\)

\(D=\{\}\)

with_lures <- function(stimulus, stimulus_type, n) {
  sapply(1:length(stimulus), function(i) {
    lures <- c(as.character(stimulus[i-n-1]), as.character(stimulus[i-n+1]))
    are_valid_trials <- i>n && all(!is.na(c(lures,stimulus[i])))
    ifelse(are_valid_trials && stimulus[i] %in% lures,
           "lure", 
           as.character(stimulus_type[i]))
  })
}
with_history <- function(stimuli, length=16, fixed=F) {
  seq <- paste(stimuli, collapse = '')
  
  sapply(1:length(stimuli), function(i) {
    stri_reverse(str_sub(seq, max(1,i-length+1), i))
  })
  #ifelse(fixed, h[str_length(h)==size], h)
}
# $x_{s,local}$
with_skewness <- function(history) {
  sapply(history, function(h) {
    freqs <- table(unlist(str_split(h,"")))
    sum(sort(freqs, decreasing = T)[1:2]) - 1
  })
}
# $x_{u,local}$
with_lumpiness <- function(history) {
  sapply(history, function(h) {
    freqs <- table(unlist(str_split(h,"")))
    max(freqs) - 1
  })
}
with_targets_ratio <- function(stimulus_type, length=16) {
  sapply(1:length(stimulus_type), function(i) {
    trials <- stimulus_type[max(1,i-length):i]
    length(trials[trials=="target"]) / length(trials)
  })
}
with_lures_ratio <- function(stimulus_type, length=16) {
  sapply(1:length(stimulus_type), function(i) {
    trials <- stimulus_type[max(1,i-length):i]
    length(trials[trials=="lure"]) / length(trials)
  })
}
NB2 <- NB %>%
  filter(participant=="P13") %>%
  group_by(participant, condition, block) %>%
  mutate(n = ifelse(condition=='2-back',2,3)) %>%
  mutate(stimulus_type = with_lures(stimulus, stimulus_type, n)) %>%
  mutate(history = with_history(stimulus, window_size)) %>%
  mutate(x_sl = with_skewness(history)) %>%
  mutate(x_ul = with_lumpiness(history)) %>%
  mutate(x_t = with_targets_ratio(stimulus_type, window_size)) %>%
  mutate(x_l = with_lures_ratio(stimulus_type, window_size)) %>%
  ungroup()
  
pca <- prcomp(~x_sl+x_ul+x_t+x_l, NB2, center = TRUE,scale. = TRUE, na.action=na.exclude)
NB2 <- NB2 %>% mutate(pc1=pca$x[,'PC1'], pc2=pca$x[,'PC2'])
# caret
library(caret)
# Compile cross-validation settings
any(is.na(NB2))

[1] TRUE

NB2 <- na.omit(NB2)
# set.seed(100)
# trainingfold <- createMultiFolds(NB2@correct, k = 5, times = 10)
# 
# # PLS
# mod1 <- train(correct ~ ., data = NB2[,c("correct","x_sl","x_ul","x_t","x_l")],
#  method = "pls",
#  metric = "Accuracy",
#  tuneLength = 20,
#  trControl = trainControl("repeatedcv", index = trainingfold, selectionFunction = "oneSE"),
#  preProc = c("zv","center","scale"))
#  
# # Check CV
# plot(mod1)
plsResult <- plsR(correct ~ ., data=NB2[,c("correct","x_sl","x_ul","x_t","x_l")],3)

____************************************************____
____Component____ 1 ____
____Component____ 2 ____
____Component____ 3 ____
____Predicting X without NA neither in X nor in Y____
****________________________________________________****

plsResult

Number of required components:
[1] 3
Number of successfully computed components:
[1] 3
Coefficients:
                  [,1]
Intercept  1.008159108
x_sl      -0.012385995
x_ul       0.019020579
x_t       -0.009732895
x_l        0.015588407
Information criteria and Fit statistics:
                AIC     RSS_Y        R2_Y   R2_residY RSS_residY  AIC.std  DoF.dof
Nb_Comp_0 -400.8879 0.9942529          NA          NA   173.0000 496.7877 1.000000
Nb_Comp_1 -399.2749 0.9920439 0.002221709 0.002221709   172.6156 498.4007 4.020014
Nb_Comp_2 -397.5844 0.9902807 0.003995089 0.003995089   172.3088 500.0912 5.000000
Nb_Comp_3 -396.2011 0.9867774 0.007518680 0.007518680   171.6993 501.4745 5.000000
          sigmahat.dof     AIC.dof     BIC.dof  GMDL.dof DoF.naive sigmahat.naive
Nb_Comp_0   0.07580980 0.005780156 0.005884498 -441.1050         1     0.07580980
Nb_Comp_1   0.07617155 0.005969500 0.006392969 -427.5293         2     0.07594536
Nb_Comp_2   0.07632287 0.006026049 0.006554844 -423.3755         3     0.07609938
Nb_Comp_3   0.07618775 0.006004731 0.006531655 -423.6749         4     0.07618775
            AIC.naive   BIC.naive GMDL.naive
Nb_Comp_0 0.005780156 0.005884498  -441.1050
Nb_Comp_1 0.005833993 0.006043423  -436.3376
Nb_Comp_2 0.005890962 0.006206384  -431.7944
Nb_Comp_3 0.005938011 0.006359551  -427.5722