---
title: "Statistical Properties of the N-Back Sequences"
output:
  html_notebook: default
  pdf_document: default
---

# 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


## Formulating Generating the N-Back Sequences as a CSP instance

$P=\langle V,D,C,W\rangle$

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

$D=\{\}$


Constraints:

$$
\\

x_n = N, W_n = 1 - |10 \times dnorm(x_n-N,sd=4)|

\\\\

x_t = T \times trials, W_t = 1 - |10\times dnorm(T\times trials-x_t,sd=4)|

\\\\

x_{tl} = {T \times w \over trials}, W_{tl} = 1 - |10\times dnorm(x_{tl} - {T \over trials} \times w,sd=4)|

\\\\

x_{l} = L \times trials
\\\\

x_{ll} = L \times w
\\\\

x_{v} = |V|
\\

x_{ul} = w
\\\\

x_{s} = {trials \over |V|}
\\\\

x_{sl} = max(1, {w \over |V|})
\\\\

x_{g} = {trials \over w}

\\\\

x_{vl} = min(|V|, w)
$$

```{r libraries, message=FALSE, include=FALSE, paged.print=FALSE}
library(ggplot2)
library(tidyverse)
library(stringi)
library(plsRglm)
library(plsdof)
library(caret)
```

```{r params}
load('./data/CL2015.RData')

window_size <- 8
```



```{r history}

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))
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(rt ~ ., data=NB2[,c("rt","x_sl","x_ul","x_t","x_l")],3)
plsResult <- plsR(correct ~ ., data=NB2[,c("correct","x_sl","x_ul","x_t","x_l")],3)


plsResult
```