highspeed-analysis/code/highspeed-analysis-oddball.Rmd

## Decoding on slow trials

### Initialization

We load the data and relevant functions:

```{r, warning=FALSE, message=FALSE, echo=TRUE}
# find the path to the root of this project:
if (!requireNamespace("here")) install.packages("here")
if ( basename(here::here()) == "highspeed" ) {
  path_root = here::here("highspeed-analysis")
} else {
  path_root = here::here()
}
# create a list of participants to exclude based on behavioral performance:
sub_exclude <- c("sub-24", "sub-31", "sub-37", "sub-40")
```

The next step is only evaluated during manual execution:

```{r, eval=FALSE}
# source all relevant functions from the setup R script:
source(file.path(path_root, "code", "highspeed-analysis-setup.R"))
```

### Decoding accuracy

#### Mean decoding accuracy

We calculate the mean decoding accuracy:

```{r, results="hold", echo=TRUE}
calc_class_stim_acc <- function(data, mask_label) {
  # calculate the mean decoding accuracy for each participant:
  dt_odd_peak = data %>%
    # select data for the oddball peak decoding
    # leave out the redundant "other" class predictions:
    filter(test_set == "test-odd_peak" & class != "other" & mask == mask_label) %>%
    # filter out excluded participants:
    filter(!(id %in% sub_exclude)) %>%
    # only check classifier of stimulus presented on a given trial:
    filter(class == stim) %>%
    # calculate the decoding accuracy by comparing true and predicted label:
    mutate(acc = 0) %>%
    transform(acc = ifelse(pred_label == stim, 1, acc)) %>%
    setDT(.) %>%
    # calculate average decoding accuracy for every participant and classification:
    .[, by = .(classification, id), .(
      mean_accuracy = mean(acc) * 100, 
      num_trials = .N
    )] %>%
    verify(all(num_trials <= 480)) %>%
    # verify if the number of participants matches:
    verify(.[classification == "ovr", .(num_subs = .N)]$num_subs == 36)
  return(dt_odd_peak)
}
```

```{r, results="hold", echo=TRUE}
calc_max_prob_acc <- function(data, mask_label) {
  dt_odd_peak <- data %>%
    # select data for the oddball peak decoding
    # leave out the redundant "other" class predictions:
    filter(test_set == "test-odd_peak" & class != "other" & mask == mask_label) %>%
    # filter out excluded participants:
    filter(!(id %in% sub_exclude)) %>%
    # calculate the decoding accuracy by comparing true and predicted label:
    setDT(.) %>%
    # calculate average decoding accuracy for every participant and classification:
    .[, by = .(classification, id, trial, stim), .(
      max_prob_label = class[which.max(probability)],
      num_classes = .N
    )] %>%
    verify(num_classes == 5) %>%
    .[, "acc" := ifelse(stim == max_prob_label, 1, 0)] %>%
    .[, by = .(classification, id), .(
      mean_accuracy = mean(acc) * 100,
      num_trials = .N
    )] %>%
    verify(num_trials <= 480)
  return(dt_odd_peak)
}
```

We print the results for the decoding accuracy:

```{r, results="hold", warning=FALSE, message=FALSE, echo=TRUE}
decoding_accuracy <- function(accuracy, alt_label){
  # print average decoding accuracy:
  print(sprintf("decoding accuracy: m = %0.2f %%", mean(accuracy)))
  # print sd decoding accuracy:
  print(sprintf("decoding accuracy: sd = %0.2f %%", sd(accuracy)))
  # print range of decoding accuracy:
  print(sprintf("decoding accuracy: range = %s %%",
          paste(round(range(accuracy),0), collapse = "-")))
  # define decoding accuracy baseline (should equal 20%):
  acc_baseline = 100/5
  print(sprintf("chance baseline = %d %%", acc_baseline))
  # perform effect size (cohen`s d)
  cohens_d = (mean(accuracy) - acc_baseline) / sd(accuracy)
  print(sprintf("effect size (cohen's d) = %0.2f", cohens_d))
  # perform shapiro-wilk test to test for normality of the data
  print(shapiro.test(accuracy))
  # perform one-sided one-sample t-test against baseline:
  results = t.test(accuracy, mu = acc_baseline, alternative = alt_label)
  print(results)
  # perform one-sample wilcoxon test in case of non-normality of the data:
  print(wilcox.test(accuracy, mu = acc_baseline, alternative = alt_label))
}
```

```{r, results="hold", warning=FALSE, message=FALSE, echo=TRUE}
#dt_odd_peak <- calc_class_stim_acc(data = dt_pred, mask_label = "cv")
#dt_odd_peak_hpc <- calc_class_stim_acc(data = dt_pred, mask_label = "cv_hpc")
dt_odd_peak <- calc_max_prob_acc(data = dt_pred, mask_label = "cv")
dt_odd_peak_hpc <- calc_max_prob_acc(data = dt_pred, mask_label = "cv_hpc")
decoding_accuracy(
  accuracy = subset(dt_odd_peak, classification == "ovr")$mean_acc,
  alt_label = "greater")
decoding_accuracy(
  accuracy = subset(dt_odd_peak_hpc, classification == "ovr")$mean_accuracy,
  alt_label = "two.sided")
```

#### Figure 2a / S2a

We plot the mean decoding accuracy in occipito-temporal and hippocampal data:

```{r, echo=TRUE, class.source=NULL, fig.width=1, fig.height=3}
plot_odd_peak <- function(data) {
  plot.odd = ggplot(data = data, aes(
    x = "mean_acc", y = as.numeric(mean_accuracy))) +
    geom_bar(stat = "summary", fun = "mean", fill = "lightgray") +
    geom_dotplot(binaxis = "y", stackdir = "center", stackratio = 0.5,
                 color = "black", fill = "lightgray", alpha = 0.5,
                 inherit.aes = TRUE, binwidth = 2) +
    geom_errorbar(stat = "summary", fun.data = "mean_se", width = 0.0, color = "black") +
    ylab("Accuracy (%)") + xlab("Condition") +
    geom_hline(aes(yintercept = 20), linetype = "dashed", color = "black") +
    scale_y_continuous(labels = label_fill(seq(0, 100, 20), mod = 1),
                       breaks = seq(0, 100, 20)) +
    coord_capped_cart(left = "both", expand = TRUE, ylim = c(0, 100)) +
    theme(axis.ticks.x = element_blank(), axis.line.x = element_blank()) +
    theme(axis.title.x = element_blank(), axis.text.x = element_blank()) +
    annotate("text", x = 1, y = 15, label = "Chance", color = "black",
             size = rel(2.5), family = "Helvetica", fontface = "plain") +
    theme(panel.border = element_blank(), axis.line = element_line()) +
    theme(axis.line = element_line(colour = "black"),
          panel.grid.major = element_blank(),
          panel.grid.minor = element_blank(),
          panel.border = element_blank(),
          panel.background = element_blank())
  return(plot.odd)
}
fig_odd_peak <- plot_odd_peak(subset(dt_odd_peak, classification == "ovr"))
fig_odd_peak_hpc <- plot_odd_peak(subset(dt_odd_peak_hpc, classification == "ovr"))
fig_odd_peak; fig_odd_peak_hpc
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_average_decoding.pdf",
       plot = fig_odd_peak, device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", height = 5, width = 2)
```

#### Source Data File Fig. 2a

```{r, echo=TRUE}
subset(dt_odd_peak, classification == "ovr") %>%
  select(-num_trials, -classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_2a.csv"),
            row.names = FALSE)
```

#### Source Data File Fig. S2a

```{r, echo=TRUE}
subset(dt_odd_peak_hpc, classification == "ovr") %>%
  select(-num_trials, -classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_s2a.csv"),
            row.names = FALSE)
```

### Fold-wise decoding accuracy

We calculate the mean decoding accuracy for each of the eight folds of the cross-validation procedure:

```{r, results="hold", echo=TRUE}
dt_odd_peak_run = dt_pred %>%
  # select data for the oddball peak decoding
  # leave out the redundant "other" class predictions:
  filter(test_set == "test-odd_peak" & class != "other" & mask == "cv") %>%
  # filter out excluded participants:
  filter(!(id %in% sub_exclude)) %>%
  setDT(.) %>%
  # calculate average decoding accuracy for every participant and classification:
  .[, by = .(classification, id, run_study, trial, stim), .(
    max_prob_label = class[which.max(probability)],
    num_classes = .N
  )] %>%
  # check if there are five classifier predictions per trial:
  verify(num_classes == 5) %>%
  # determine accuracy if label with highest probability matches stimulus:
  .[, "accuracy" := ifelse(stim == max_prob_label, 1, 0)]
# print results table:
rmarkdown::paged_table(dt_odd_peak_run)
```

```{r, results="hold", echo=TRUE}
# calculate average decoding accuracy for every participant and classification:
dt_odd_peak_run_mean <- dt_odd_peak_run %>%
  .[, by = .(classification, id, run_study), .(
    mean_accuracy = mean(accuracy) * 100,
    num_trials = length(unique(trial))
  )] %>%
  # verify if the number of participants matches for every classification and run:
  verify(.[, by = .(classification, run_study),
           .(num_subs = .N)]$num_subs == 36) %>%
  setorder(., classification, id, run_study)
# print the table:
rmarkdown::paged_table(dt_odd_peak_run_mean)
```


```{r, echo=FALSE, class.source=NULL}
fig_odd_run = ggplot(data = subset(dt_odd_peak_run_mean, classification == "ovr"), aes(
  x = run_study, y = as.numeric(mean_accuracy))) +
  geom_bar(stat = "summary", fun = "mean", fill = "lightgray", color = "black") +
  geom_dotplot(aes(color = "black", fill = id),
               binaxis = "y", stackdir = "center", stackratio = 0.5, alpha = 0.5,
               inherit.aes = TRUE, binwidth = 2) +
  geom_line(aes(group = id, color = id)) +
  geom_line(aes(group = 1), stat = "summary", fun = "mean", color = "black") +
  geom_errorbar(stat = "summary", fun.data = "mean_se", width = 0.0, color = "black") +
  ylab("Accuracy (%)") + xlab("Task run") +
  geom_hline(aes(yintercept = 20), linetype = "dashed", color = "black") +
  scale_y_continuous(labels = label_fill(seq(0, 100, 20), mod = 1), breaks = seq(0, 100, 20)) +
  coord_capped_cart(left = "both", expand = TRUE, ylim = c(0, 100)) +
  annotate("text", x = 8, y = 16, label = "Chance", color = "black",
           size = rel(2.5), family = "Helvetica", fontface = "plain") +
  theme(legend.position = "none") +
  theme(panel.border = element_blank()) +
  theme(axis.text = element_text(color = "black")) +
  theme(axis.ticks = element_line(color = "black")) +
  theme(axis.line.y = element_line(colour = "black"),
        axis.ticks.x = element_blank(),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_odd_run
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_average_decoding_run.pdf",
       plot = fig_odd_run, device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", height = 3, width = 5)
```

```{r, results="hold"}
lme_odd_peak_run <- lmerTest::lmer(
  mean_accuracy ~ run_study + (1 | id), control = lcctrl,
  data = subset(dt_odd_peak_run_mean, classification == "ovr")
)
summary(lme_odd_peak_run)
anova(lme_odd_peak_run)
emmeans_results = emmeans(lme_odd_peak_run, list(pairwise ~ run_study))
emmeans_results
```

## Single-trial decoding time courses

We calculate the multivariate decoding time courses on single slow trials:

```{r, echo=TRUE}
dt_odd_long_sub = dt_pred %>%
  # select data for the oddball long decoding
  # leave out the redundant "other" class predictions:
  filter(test_set == "test-odd_long" & class != "other" & mask == "cv") %>% 
  # filter out excluded participants:
  filter(!(id %in% sub_exclude)) %>% setDT(.) %>%
  # average across stimuli and runs for each participant
  # create an index for the true stimulus presented on each trial:
  .[, by = .(classification, id, seq_tr, class, stim), .(
    num = .N,
    mean_prob = mean(probability * 100),
    current_stim = as.numeric(class == unique(stim))
  )]

dt_odd_long_mean = dt_odd_long_sub %>% 
  # average across participants:
  .[, by = .(classification, seq_tr, class, stim, current_stim), .(
    mean_prob = mean(mean_prob),
    num_subs = .N,
    sem_upper = (mean(mean_prob) + (sd(mean_prob)/sqrt(.N))),
    sem_lower = (mean(mean_prob) - (sd(mean_prob)/sqrt(.N)))
  )] %>%
  # create an additional variable that expresses time in seconds:
  mutate(time = (seq_tr - 1) * 1.25) %>%
  # check if the number of participants is correct:
  verify(all(num_subs == 36))
```

#### Figure 2b

We plot the single-trial multi-variate decoding time courses on slow trials:

```{r}
plot.odd.long = ggplot(data = subset(dt_odd_long_mean, classification == "ovr"), aes(
  x = as.factor(seq_tr), y = as.numeric(mean_prob))) +
  facet_wrap(~ as.factor(stim)) +
  geom_line(data = subset(dt_odd_long_sub, classification == "ovr"), aes(
    group = as.factor(interaction(id, class)), color = fct_rev(as.factor(current_stim))), alpha = 0.0) +
  geom_line(data = subset(dt_odd_long_sub, classification == "ovr" & current_stim == 0), aes(
    group = as.factor(interaction(id, class)), color = fct_rev(as.factor(current_stim))), alpha = 0.3) +
  geom_line(data = subset(dt_odd_long_sub, classification == "ovr" & current_stim == 1), aes(
    group = as.factor(interaction(id, class)), color = fct_rev(as.factor(current_stim))), alpha = 0.3) +
  geom_ribbon(
    data = subset(dt_odd_long_mean, classification == "ovr"),
    aes(ymin = sem_lower, ymax = sem_upper, fill = fct_rev(as.factor(current_stim)),
        group = as.factor(class)), alpha = 0.0, color = NA) +
  geom_line(
    data = subset(dt_odd_long_mean, classification == "ovr", alpha = 0),
    aes(color = fct_rev(as.factor(current_stim)), group = as.factor(class))) +
  geom_ribbon(
    data = subset(dt_odd_long_mean, classification == "ovr" & current_stim == 1),
    aes(ymin = sem_lower, ymax = sem_upper, fill = fct_rev(as.factor(current_stim)),
        group = as.factor(class)), alpha = 0.3, color = NA) +
  geom_line(
    data = subset(dt_odd_long_mean, classification == "ovr" & current_stim == 1),
    aes(color = fct_rev(as.factor(current_stim)), group = as.factor(class))) +
  ylab("Probability (%)") + xlab("Time from stimulus onset (TRs)") +
  scale_fill_manual(name = "Classified class", values = c("black", "lightgray"), labels = c("true", "other")) +
  scale_color_manual(name = "Classified class", values = c("black", "lightgray"), labels = c("true", "other")) +
  coord_capped_cart(left = "both", bottom = "both", expand = TRUE, ylim = c(0, 80)) +
  theme(legend.position = c(1, 0), legend.justification = c(1, 0)) +
  #theme(legend.position = c(0.95, 0.8), legend.justification = c(1, 0)) +
  #theme(legend.title = element_text(size = 8), legend.text = element_text(size = 8)) +
  #theme(legend.key.size = unit(0.7, "line")) +
  scale_x_discrete(labels = label_fill(seq(0, 7, 1), mod = 1), breaks = seq(0, 7, 1)) +
  guides(color = guide_legend(ncol = 2), fill = guide_legend(ncol = 2)) +
  theme(strip.text.x = element_text(margin = margin(b = 2, t = 2))) +
  theme(panel.border = element_blank(), axis.line = element_line()) +
  theme(axis.line = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
#theme(strip.text.x = element_blank())
plot.odd.long
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_single_trial_activation.pdf",
       plot = plot.odd.long, device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 3.5, height = 3)
```

#### Source Data File Fig. 2b

```{r, echo=TRUE}
subset(dt_odd_long_sub, classification == "ovr") %>%
  select(-num, -classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_2b.csv"),
            row.names = FALSE)
```

We compare the mean classification probability of the current stimulus
versus the mean of all other stimuli for each TR:

```{r}
dt_odd_long_mean_stat = dt_pred %>%
  # select data for the oddball long decoding
  # leave out the redundant "other" class predictions:
  filter(test_set == "test-odd_long" & class != "other" & mask == "cv") %>%
  setDT(.) %>%
  # filter out excluded participants:
  filter(!(id %in% sub_exclude)) %>% setDT(.) %>%
  # average across stimuli and runs for each participant
  # create a new variable that indicated whether stimulus is current or not:
  .[, by = .(classification, id, seq_tr, class, stim), .(
    num_stim = .N,
    mean_prob = mean(probability * 100),
    stim_type = ifelse(class == unique(stim), "current", "other")
  )] %>%
  verify(num_stim <= 96) %>%
  # average across stimulus type:
  .[, by = .(classification, id, seq_tr, stim, stim_type), .(
    num_class = .N,
    mean_prob = mean(mean_prob)
  )] %>%
  verify(num_class %in% c(1, 4)) %>%
  select(., -num_class) %>%
  # turn into wide format:
  spread(key = stim_type, value = mean_prob) %>% setDT(.) %>%
  # calculate a paired t-test at every TR for each unique stimulus to
  # compare probability of the current vs. the mean other class probabilities:
  .[, by = .(classification, seq_tr, stim), {
    results = t.test(current, other, paired = TRUE, alternative = "two.sided")
    list(
      tvalue = results$statistic,
      df = results$parameter,
      pvalue = results$p.value,
      cohens_d = (mean(current) - mean(other)) / sd(current - other),
      num_subs = .N,
      mean_current = mean(current),
      mean_other = mean(other),
      pvalue_round = round_pvalues(results$p.value)
    )
  }] %>%
  # ajdust the p-value for multiple comparisons using Bonferroni correction:
  .[, by = .(classification), ":=" (
    num_comp = .N,
    pvalue_adjust = p.adjust(pvalue, method = "bonferroni", n = .N)
  )] %>%
  # check if correction was done for 5 (stimuli) by 7 (TRs) = 35 comparisons:
  verify(all(num_comp == 5 * 7)) %>%
  # verify if number of participants matches:
  verify(all(num_subs == 36)) %>%
  # check if degrees-of-freedom = n - 1
  verify(all(df == num_subs - 1)) %>%
  # round the bonferroni-adjusted p-values for clarity:
  mutate(pvalue_adjust_round = round_pvalues(pvalue_adjust)) %>%
  # sort the datatable according to classification, stimulus and TR:
  setorder(., classification, stim, seq_tr) %>%
  # only look at the fourth TR where probabilities are expected to peak
  filter(seq_tr == 4 & classification == "ovr")
# print the data table:
rmarkdown::paged_table(dt_odd_long_mean_stat)
```

```{r, echo=FALSE, class.source = NULL}
plot_grid(fig_odd_peak, plot.odd.long, labels = c('a', 'b'), ncol = 2,
          rel_widths = c(1.5, 5), hjust = c(0, 0), label_fontface = "bold")
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_oddball.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 6.5, height = 3.5)
ggsave(filename = "wittkuhn_schuck_figure_s2.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 6.5, height = 3.5)
```

## Response functions

### Define response functions

We define the sine wave response function:

```{r}
sine_truncated <- function(params, time) {
  if (!is.list(params)) {
    params = as.list(params)
    names(params) = c('frequency', 'amplitude', 'shift', 'baseline')
  }
  y = rep(0, 13)
  y = params$amp/2 * sin(2*pi*params$freq*time - 2*pi*params$freq*params$shift - 0.5*pi) + params$baseline + params$amp/2
  # flatten response function after one cycle:
  y[time < (params$shift)] = params$baseline
  y[time > (params$shift + 1/params$freq)] = params$baseline  
  return(y) 
}
```

We define a function to evaluate during optimization:

```{r}
sine_truncated_eval = function(params, time, data) {
  y = sine_truncated(params, time)
  SSE = sum((data - y)^2)
  return(SSE)
}
```

### Mean parameters

We calculate the multivariate decoding time courses for each stimulus class: 

```{r}
# calculate mean probability for every stimulus for every TR:
dt_odd_long_mean_class = dt_pred %>%
  # filter for oddball long predictions and only select ovr classifier:
  filter(test_set == "test-odd_long" & class != "other" &
           classification == "ovr" & mask == "cv") %>%
  # filter out excluded participants:
  filter(!(id %in% sub_exclude)) %>% setDT(.) %>%
  # select only predictions for the class currently shown on any given trial:
  filter(class == stim) %>% setDT(.) %>%
  # average the probability across trials
  .[, by = .(id, classification, stim, seq_tr), .(
    mean_probability = mean(probability, na.rm = TRUE)
  )]
```

We fit the truncated sine wave response function to data from every participant:

```{r}
# set optimization parameters:
opts = list("algorithm" = "NLOPT_LN_COBYLA", "xtol_rel" = 1.0e-8, "maxeval" = 1.0e+5)
default_params = c(0.2, 0.6, 0, 0.1)
lower_bounds = c(0.01, 0.1, 0, 0)
upper_bounds = c(0.5, 1, 5, 0.3)
time = 0:6
```


```{r, results="hold"}
dt_odd_long_fit = dt_odd_long_mean_class %>%
  # fit the truncated sine wave to probabilities of every decoding timecourse:
  .[, by = .(id, classification, stim), {
    res <- nloptr(
      x0 = default_params, eval_f = sine_truncated_eval, time = time,
      data = mean_probability, lb = lower_bounds, ub = upper_bounds, opts = opts)
    list(
      num_trs = .N,
      frequency = res$solution[1],
      wavelength = 1/res$solution[1],
      amplitude = res$solution[2],
      shift = res$solution[3],
      baseline = res$solution[4],
      params = list(res$solution)
    )}] %>%
  verify(all(num_trs == length(time)))
# print the mean model parameters:
summary(dt_odd_long_fit[, c("frequency", "wavelength", "amplitude", "shift", "baseline")])
# calculate the mean parameters across all participants to be used later:
mean_parameters = dt_odd_long_fit %>%
  # average the fitted parameters across participants:
  .[, by = .(classification), .(
    mean_freq = mean(frequency),
    mean_wavelength = mean(wavelength),
    mean_amplitude = mean(amplitude),
    mean_shift = mean(shift),
    mean_baseline = mean(baseline)
  )]
```

We fit the sine response function for every participant using individual parameters:

```{r}
dt_odd_long_fit_single = dt_odd_long_fit %>%
  # calculate a sine-wave respone function timecourse for every participant:
  .[, by = .(id, classification, stim), .(
    csine = sine_truncated(params = unlist(params), seq(0, 6, 0.1))
  )] %>%
  # add a counter that is used for plotting below:
  .[, by = .(classification, id, stim), ":=" (
    t = seq(0, 6, 0.1))]
```

#### Figure S4a

We plot the single sine wave fits for three example participants (supplement):

```{r, echo=FALSE}
# set seed to reproduce random id sampling:
set.seed(18)
num_sub_select = 3
# select random example participants:
select_id = sample(x = unique(dt_odd_long_fit_single$id), size = num_sub_select)
fig_s1 = ggplot(data = subset(dt_odd_long_fit_single, id %in% select_id),
                aes(x = t, y = csine * 100)) +
  geom_point(aes(color = "Model"), alpha = 0.5) +
  geom_line(data = subset(dt_odd_long_mean_class, id %in% select_id),
            aes(x = (seq_tr - 1), y = mean_probability * 100, color = "Data")) +
  geom_point(data = subset(dt_odd_long_mean_class, id %in% select_id),
             aes(x = (seq_tr - 1), y = mean_probability * 100, color = "Data")) +
  scale_colour_manual(name = "", values = c("gray", "black")) +
  facet_grid(vars(as.factor(id)), vars(as.factor(stim))) +
  coord_capped_cart(left = "both", bottom = "both") +
  xlab("Time from stimulus onset (in TRs)") + ylab("Probability (%)") +
  scale_x_continuous(labels = label_fill(seq(1, 7, 1), mod = 1), breaks = seq(0, 6, 1)) +
  theme(legend.position = "top", legend.direction = "horizontal",
        legend.justification = "center", legend.margin = margin(0, 0 ,0, 0),
        legend.box.margin = margin(t = 0, r = 0, b = -10, l = 0)) +
  theme(strip.text = element_text(margin = margin(b = 2, t = 2, r = 2, l = 2))) +
  theme(axis.title.x = element_blank()) +
  theme(axis.line = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_s1
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highsspeed_plot_decoding_oddball_single_sine_fits.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 6, height = 4)
```

#### Source Data File Fig. S4a

```{r, echo=TRUE}
dt_odd_long_mean_class %>%
  select(-classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_s4a.csv"),
            row.names = FALSE)
```

We calculate the mean shape of the sine wave response function across participants:

```{r, echo=TRUE}
dt_odd_long_fit_mean = dt_odd_long_fit %>%
  # average the fitting parameters across participants:
  .[, by = .(classification, stim), .(
    num_subs = .N,
    mean_freq = mean(frequency),
    mean_amplitude = mean(amplitude),
    mean_shift = mean(shift),
    mean_baseline = mean(baseline)
  )] %>% verify(all(num_subs == 36)) %>%
  .[, by = .(classification, stim), .(
    csine = sine_truncated(params = c(mean_freq, mean_amplitude, mean_shift, mean_baseline),
      seq(0, 6, 0.1)))] %>%
  .[, by = .(classification, stim), ":=" (
    t = seq(0, 6, 0.1))]
```

#### Figure S4b

```{r, echo=TRUE}
fig_s2 = ggplot(data = dt_odd_long_mean_class) +
  geom_line(data = dt_odd_long_mean_class,
            aes(x = (seq_tr - 1), y = mean_probability * 100, group = as.factor(id),
                color = "Data"), alpha = 0.3) +
  geom_line(data = dt_odd_long_fit_mean, aes(x = t, y = csine * 100, color = "Model"),
            size = 1) +
  facet_wrap(~ as.factor(stim), nrow = 1) +
  coord_capped_cart(left = "both", bottom = "both", ylim = c(0, 80)) +
  xlab("Time from stimulus onset (in TRs)") + ylab("Probability (%)") +
  scale_colour_manual(name = "", values = c("gray", "black"), guide = FALSE) +
  scale_x_continuous(labels = label_fill(seq(1, 7, 1), mod = 1), breaks = seq(0, 6, 1)) +
  #theme(plot.margin = unit(c(t = 1, r = 1, b = 1, l = 1), "pt")) +
  theme(legend.position = "bottom", legend.direction = "horizontal",
        legend.justification = "center", legend.margin = margin(0, 0, 0, 0),
        legend.box.margin = margin(t = -10, r = 0, b = 0, l = 0)) +
  theme(strip.text = element_text(margin = margin(b = 2, t = 2, r = 2, l = 2))) +
  theme(axis.line = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_s2
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highsspeed_plot_decoding_oddball_mean_sine_fits.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 5, height = 3)
```

#### Source Data File Fig. S4b

```{r, echo=TRUE}
dt_odd_long_mean_class %>%
  select(-classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_s4b.csv"),
            row.names = FALSE)
```

#### Figure S4

```{r}
plot_grid(fig_s1, fig_s2, nrow = 2, ncol = 1, hjust = c(0, 0),
          rel_heights = c(4, 2), labels = c("a", "b"), label_fontface = "bold")
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highsspeed_plot_decoding_oddball_supplement.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 7, height = 7)
ggsave(filename = "wittkuhn_schuck_figure_s4.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 7, height = 7)
```

We evaluate the response function for two time-shifted events:

```{r}
# horizontal phase shift (in TRs) that will be added to the second sine wave:
add_shift = 0.5
time = seq(0, 8, 0.1)
dt_odd_long_fit_shift = dt_odd_long_fit %>%
  # average the fitted parameters across participants:
  .[, by = .(classification), .(
    mean_freq = mean(frequency),
    mean_amplitude = mean(amplitude),
    mean_shift = mean(shift),
    mean_baseline = mean(baseline)
  )] %>%
  # add variable that adds shift (in TRs) for the second sine wave:
  mutate(mean_shift_shifted = mean_shift + add_shift) %>%
  mutate(amp = mean_amplitude * sin(add_shift * mean_freq * pi)) %>%
  mutate(freq = mean_freq / (add_shift * mean_freq + 1)) %>% setDT(.) %>%
  # calculate first and second response filter time-courses
  .[, by = .(classification), .(
    first = sine_truncated(params = c(mean_freq, mean_amplitude, mean_shift, mean_baseline), time),
    second = sine_truncated(params = c(mean_freq, mean_amplitude, mean_shift_shifted, mean_baseline), time),
    cosine = amp * sin(2 * pi * time * freq - 2 * pi * freq * mean_shift),
    t = time
  )] %>%
  # shift the cosine wave by 0.6 (only for illustrational purposes):
  mutate(cosine = cosine - 0.6) %>% setDT(.) %>%
  # signify the tails of the sine wave (that will otherwise be flattened):
  .[time < mean_parameters$mean_shift, ":=" (
    cosine_tails = cosine,
    cosine = NA)] %>%
  .[time > (mean_parameters$mean_shift + add_shift + 1/mean_parameters$mean_freq), ":=" (
    cosine_tails = cosine,
    cosine = -NA)] %>%
  # calculate the difference between the first and second wave
  # adjust by 0.2 for illustrational purposes only:
  mutate(difference = first - second - 0.2) %>%
  gather(key = "event", value = "probability", -classification, -t)
```

We calculate the mean probability timecourses for the true stimulus class across participants:

```{r, ech0=TRUE}
# calculate the mean decoding time course across all classes and participants:
dt_odd_long_mean = dt_pred %>%
  # filter for oddball multivariate data and one-versus-rest classification only:
  filter(test_set == "test-odd_long" & class != "other" &
           classification == "ovr" & mask == "cv") %>%
  # filter out excluded participants:
  filter(!(id %in% sub_exclude)) %>%
  # filter for all trials where the predicted class matches the true stimulus:
  filter(class == stim) %>% setDT(.) %>%
  # calculate the mean decoding time courses for each participant:
  .[, by = .(id, classification, stim, seq_tr), .(
    mean_probability = mean(probability, na.rm = TRUE) * 100)] %>%
  # average mean decoding time courses across participants:
  .[, by = .(classification, stim, seq_tr), .(
    mean_probability = mean(mean_probability),
    num_subs = .N,
    sem_upper = mean(mean_probability) + (sd(mean_probability)/sqrt(.N)),
    sem_lower = mean(mean_probability) - (sd(mean_probability)/sqrt(.N))
  )] %>%
  verify(all(num_subs == 36))
```

#### Figure 2c

```{r, echo=FALSE}
fig_a = ggplot(data = dt_odd_long_mean, aes(
  x = (seq_tr - 1), y = mean_probability, group = stim)) +
  geom_ribbon(aes(ymin = sem_lower, ymax = sem_upper), alpha = 0.1, color = NA) +
  geom_line(mapping = aes(color = "Data"), alpha = 1) +
  geom_point(mapping = aes(color = "Data"), alpha = 1, pch = 16) +
  geom_line(data = subset(dt_odd_long_fit_shift, event == "first"),
            aes(x = t, y = probability * 100, color = "Model"),
            size = 2, inherit.aes = FALSE) +
  coord_capped_cart(left = "both", bottom = "both", ylim = c(0, 100),
                    xlim = c(0, 6), expand = TRUE) +
  xlab("Time (TRs)") + ylab("Probability (%)") +
  scale_colour_manual(name = "", values = c("gray", "black")) +
  scale_fill_manual(name = "", values = c("gray", "black")) +
  scale_x_continuous(labels = label_fill(seq(1,7,1), mod = 1), breaks = seq(0,6,1)) +
  scale_y_continuous(labels = label_fill(seq(0, 80, 20), mod = 1), breaks = seq(0, 80, 20)) +
  theme(legend.position = "top", legend.direction = "horizontal",
        legend.justification = "center", legend.margin = margin(0, 0 ,0, 0),
        legend.box.margin = margin(t = 0, r = 0, b = -60, l = 0)) +
  guides(color = guide_legend(ncol = 2)) +
  # add the shift parameter:
  geom_segment(aes(x = 0, xend = mean_parameters$mean_shift, yend = 65, y = 65),
               color = "darkgray", linetype = "dashed") +
  annotate("text", x = mean_parameters$mean_shift/2, y = 70,
           label = paste("italic(d)"), color = "darkgray", hjust = 0.5, parse = TRUE) +
  # add the wavelength parameter:
  geom_segment(aes(x = mean_parameters$mean_shift, xend = mean_parameters$mean_shift + 1/mean_parameters$mean_freq,
                   yend = 70, y = 70), color = "darkgray", linetype = "dashed") +
  annotate("text", x = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq), y = 75,
           label = paste("italic(lambda)"), color = "darkgray", hjust = 0.5, parse = TRUE) +
  # add the amplitude parameter:
  geom_segment(aes(x = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq),
                   xend = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq),
                   y = mean_parameters$mean_baseline * 100,
                   yend = (mean_parameters$mean_baseline + mean_parameters$mean_amplitude) * 100),
               color = "darkgray", linetype = "dashed") +
  annotate("text", x = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq) - 0.25,
           y = (mean_parameters$mean_baseline + mean_parameters$mean_amplitude / 2) * 100,
           label = paste("italic(A)"), color = "darkgray", parse = TRUE) + 
  # add the baseline parameter:
  geom_segment(aes(x = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq) - 0.2,
                   xend = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq) - 0.2,
                   y = 0, yend = mean_parameters$mean_baseline * 100),
               color = "darkgray", linetype = "dashed") +
  annotate("text", x = mean_parameters$mean_shift + 1/(2 * mean_parameters$mean_freq) - 0.25 - 0.2,
           y = mean_parameters$mean_baseline / 2 * 100,
           label = paste("italic(b)"), color = "darkgray", parse = TRUE) +
  annotate("text", x = 6, y = 0, label = "1 TR = 1.25 s",
             hjust = 1, size = rel(2)) +
  theme(axis.line = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_a
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_oddball_sine_illustration_fig1.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 3, height = 3.1)
```

#### Source Data File Fig. 2c

```{r, echo=TRUE}
dt_odd_long_mean %>%
  select(-classification, -num_subs) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_2c.csv"),
            row.names = FALSE)
```

### Response model and difference dynamics 

```{r, echo=TRUE}
# get the average horizontal shift of the first sine wave:
t_first = mean(dt_odd_long_fit$shift)
# calculate the shift of the second sine wave (shift + half a wavelength)
t_second = mean(dt_odd_long_fit$shift) + add_shift + 1/mean(dt_odd_long_fit$freq)
# calculate the time point of cross over between the two sine functions:
t_crossover = 0.5/mean(dt_odd_long_fit$freq) + mean(dt_odd_long_fit$shift) + add_shift/2
# create a vector of time steps:
time = seq(0, 8, 0.1)

max_prob = max(dt_odd_long_fit_shift$probability[dt_odd_long_fit_shift$event == "difference"], na.rm = TRUE)
t_max_diff = dt_odd_long_fit_shift$t[dt_odd_long_fit_shift$probability == max_prob & !is.na(dt_odd_long_fit_shift$probability)]
a = dt_odd_long_fit_shift$probability[dt_odd_long_fit_shift$t == t_max_diff & dt_odd_long_fit_shift$event == "first"]
```

#### Figure 2d

```{r, echo=TRUE}
# select colors used for plotting:
colors = c("darkgray", "darkgray", "black", "dodgerblue", "red", "black")
# plot sine wave difference illustration:
fig_b = ggplot(data = dt_odd_long_fit_shift, aes(
  x = time, y = probability * 100, group = event, color = event)) +
  # create rectangles to indicate the forward and backward phase:
  annotate("rect", xmin = t_first, xmax = t_crossover, ymin = -80, ymax = 95,
           alpha = 0.05, fill = "dodgerblue") +
  annotate("rect", xmin = t_crossover, xmax = t_second, ymin = -80, ymax = 95,
           alpha = 0.05, fill = "red") +
  # plot the onset of the first event:
  geom_vline(xintercept = t_first, color = "gray", linetype = "dashed") +
  # plot the offset of the second event (d + lambda + s in manuscript text):
  geom_vline(xintercept = t_second, color = "gray", linetype = "dashed") +
  # plot the crossover (d + 0.5 * (lambda + s) in manuscript text)
  geom_vline(xintercept = t_crossover, color = "gray", linetype = "dashed") +
  # plot the horizontal line for the difference time course:
  geom_hline(yintercept = -20, color = "gray") +
  # plot the horizontal line for the cosine wave:
  geom_hline(yintercept = -60, color = "gray") +
  # plot the first and second event time courses:
  geom_line(aes(x = t, linetype = fct_rev(event)), na.rm = TRUE) +
  coord_capped_cart(left = "both", bottom = "both", ylim = c(-80, 85)) +
  xlab("Time (in TRs)") + ylab("Probability (a.u.)") +
  scale_colour_manual(name = "Serial event", values = colors) +
  scale_linetype_manual(values = c(rep("solid", 3), "dashed", "solid")) +
  scale_x_continuous(labels = c(label_fill(seq(1,7,1), mod = 1), rep("", 2)),
                     breaks = seq(0,8,1)) +
  theme(axis.text.y = element_blank(), axis.ticks.y = element_line(colour = "white"),
        axis.line.y = element_line(colour = "white")) +
  theme(legend.position = "none") +
  annotate("label", x = 0.5, y = 45, label = "~1^'st'~event",
           color = "dodgerblue", parse = TRUE, size = 3) +
  annotate("label", x = 0.5, y = 30, label = "~2^'nd'~event",
           color = "red", parse = TRUE, size = 3) +
  annotate("label", x = 0.5, y = -7, label = "Difference",
           color = "black", size = 3) +
  annotate("label", x = 0.5, y = -35, label = "Difference\n(no flattening)",
           color = "darkgray", size = 3) +
  # add annotation for forward period:
  geom_segment(aes(x = t_first, xend = t_crossover, yend = 85, y = 85,
                   colour = "segment"), color = "dodgerblue") +
  annotate("label", x = t_first + (t_crossover - t_first)/2, y = 85,
           label = "Forward period",
           color = "dodgerblue", fill = "white", hjust = 0.5) +
  # add annotations for backward period:
  geom_segment(aes(x = t_crossover, xend = t_second, y = 85, yend = 85, 
                   colour = "segment"), color = "red") +
  annotate("label", x = t_crossover + (t_second - t_crossover)/2, y = 85,
           label = "Backward period",
           color = "red", fill = "white", hjust = 0.5) +
  # add annotation for early forward phase
  geom_segment(aes(x = t_first, xend = t_first + (t_crossover - t_first)/2, yend = 70, y = 70,
                   colour = "segment"), color = "gray") +
  annotate("label", x = (t_first + (t_crossover - t_first)/4),
           y = 70, label = "early",
           color = "gray", fill = "white", hjust = 0.5) +
  # add annotation for late forward phase
  geom_segment(aes(x = t_first + (t_crossover - t_first)/2, xend = t_crossover,
                   yend = 68, y = 68, colour = "segment"), color = "gray") +
  annotate("label", x = (t_first + (t_crossover - t_first)/4 * 3),
           y = 68, label = "late",
           color = "gray", fill = "white", hjust = 0.5) +
  # add annotation for early backward phase
  geom_segment(aes(x = t_crossover, xend = t_crossover + (t_second - t_crossover)/2,
                   yend = 70, y = 70, colour = "segment"), color = "gray") +
  annotate("label", x = (t_crossover + (t_second - t_crossover)/4),
           y = 70, label = "early", color = "gray", fill = "white", hjust = 0.5) +
  # add annotation for late backward phase
  geom_segment(aes(x = t_crossover + (t_second - t_crossover)/2, xend = t_second,
                   yend = 68, y = 68, colour = "segment"), color = "gray") +
  annotate("label", x = (t_crossover + (t_second - t_crossover)/4 * 3),
           y = 68, label = "late", color = "gray", fill = "white", hjust = 0.5) +
  # add annotation for delta:
  geom_segment(aes(x = t_first + (t_crossover - t_first)/2,
                   xend = t_first + (t_crossover - t_first)/2 + add_shift,
                   y = 35 , yend = 35, colour = "segment"), color = "darkgray", linetype = "dotted") +
  annotate("text", x = t_first + (t_crossover - t_first)/2 - 0.5 + add_shift / 2, y = 36,
           label = "delta", color = "darkgray", hjust = 0.5, parse = TRUE) +
  # add annotation for TRs:
  annotate("text", x = 8, y = -80, label = "1 TR = 1.25 s",
             hjust = 1, size = rel(2)) +
  theme(axis.line.x = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_b
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_oddball_sine_illustration_figb.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 5.5, height = 4.5)
```

#### Source Data File Fig. 2d

```{r, echo=TRUE}
dt_odd_long_fit_shift %>%
  select(-classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_2d.csv"),
            row.names = FALSE)
```

```{r, echo = TRUE}
ampfun = function(s, A) {A*cos(s*mean(dt_odd_long_fit$freq)*pi - 0.5*pi)}
cs = seq(0, 0.5/mean(dt_odd_long_fit$freq), 0.01)
ba = data.table(cs = cs, diff = ampfun(cs, 1))
fig_c = ggplot(data = ba, aes(x = cs, y = diff)) +
  geom_line() +
  xlab("Time-shift (in TRs)") + ylab("Max. diff. in amplitude") +
  coord_capped_cart(left = "both", bottom = "both", xlim = c(0,3)) +
  scale_x_continuous(labels = c("0", rep("", 2), "Resp. peak"), breaks = seq(0,3)) +
  scale_y_continuous(labels = c("0", rep("", 3), "Max")) +
  #theme(plot.margin = unit(c(t = 10, r = 30, b = 1, l = 1), "pt")) +
  theme(axis.text.y = element_text(angle = 90, hjust = 0.5)) +
  theme(axis.line = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_c
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_oddball_sine_illustration_fig3.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 3, height = 2.5)
```

We plot a combination of the previous plots:

```{r, echo=TRUE, fig.width = 7, fig.height = 3}
plot_grid(fig_a, fig_b, ncol = 2, hjust = c(0,0),
          labels = c("c", "d", "e"), rel_widths = c(2, 4),
          label_fontface = "bold", label_size = 14)
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_oddball_sine_illustration.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 7, height = 3)
```


### Forward and backward period for different speeds

Here, we calculate the expected forward and backward periods for two events
separated by different speed levels (different values for delta).

```{r}
calc_periods = function(speed, stim_dur = 0.1, num_stim = 5, tr = 1.25){
  # mean wavelength based on fitted models, in TRs, round to two digits:
  lambda = round(mean_parameters$mean_wavelength, 2)
  # mean phase-shift based on fitted models, in TRs, round to two digits:
  phase_shift = round(mean_parameters$mean_shift, 2)
  # delta, in sec (time shift depending on speed)
  delta_sec = stim_dur * (num_stim - 1) + (num_stim - 1) * speed
  # delta, in trs:
  delta_tr = delta_sec / tr
  # end of forward period, in TRs:
  fwd_end = 0.5 * (lambda + delta_tr) + phase_shift
  # end of the backward period, in TRs:
  bwd_end = lambda + phase_shift + delta_tr
  # forward period, in TRs:
  forward = list(seq(round(phase_shift) + 1, round(fwd_end - 0.5) + 1))
  # backward period, in TRs:
  backward = list(seq(round(fwd_end + 0.5) + 1, round(bwd_end) + 1))
  # entire relevant trial period, in rounded TRs:
  trial_period = list(seq(round(phase_shift) + 1, round(lambda + phase_shift + delta_tr) + 1))
  # save results as a data table and return results:
  dt = data.table(speed, lambda, delta_sec, delta_tr, fwd_end, bwd_end, forward, backward, trial_period)
  return(dt)
}
# get the relevant timeperiods for each sequence speed condition:
speeds = c(0.032, 0.064, 0.128, 0.512, 2.048)
dt_periods = do.call(rbind, lapply(speeds, calc_periods))
rmarkdown::paged_table(dt_periods)
```

We save the periods of interest for different speeds which will be used for the analysis of sequence and repetition trials:

```{r}
save(dt_periods, file = file.path(path_root, 'data', "tmp", "dt_periods.Rdata"))
```

### Probability differences at different speeds

We plot the probability differences at different speeds:

```{r}
time = seq(0, 12, 0.1)
speeds = c(0.032, 0.064, 0.128, 0.512, 2.048)
stim_dur = 0.1

dt_odd_seq_sim = dt_odd_long_fit %>% setDT(.) %>%
  # average the fitted parameters across stimuli for each participant:
  .[, by = .(classification, id), .(
    mean_freq = mean(frequency),
    mean_amplitude = mean(amplitude),
    mean_shift = mean(shift),
    mean_baseline = mean(baseline)
  )] %>%
  # repeat rows once for each speed condition (5 repetitions in total):
  .[rep(seq(1, nrow(.)), length(speeds))] %>%
  # add the different speed conditions for every participant:
  .[, by = .(classification, id), ":=" (speed = speeds, num_events = 1)] %>%
  # repeat rows once for each hypothetical event (here, 2 events):
  .[rep(seq(1, nrow(.)), num_events)] %>%
  # add a counter for the number of events:
  .[, by = .(classification, id, speed), ":=" (event = 5)] %>%
  # calculate a speed-specific shift for each event:
  .[, by = .(classification, id, speed, event), {
    shift = (speed * (event - 1) + (event - 1) * stim_dur)/1.25
    amp = mean_amplitude * sin(shift * 0.5 * mean_freq * pi)
    freq = (1/(1/mean_freq + shift))
    c_d = amp * sin(2 * pi * time * freq - 2 * pi * freq * mean_shift)
    # flatten the tails of the response function
    cidx = time < mean_shift | time > (mean_shift + (1 / freq))
    c_d[cidx] = 0
    # save the response function:
    list(time = time, probability = c_d)
  }] %>%
  filter(classification == "ovr") %>% setDT(.)

dt_odd_seq_sim_diff = dt_odd_seq_sim %>% 
  .[, by = .(classification, speed, time), .(
    num_subs = .N,
    mean_difference = mean(probability),
    sem_upper = mean(probability) + (sd(probability)/sqrt(.N)),
    sem_lower = mean(probability) - (sd(probability)/sqrt(.N))
  )] %>% verify(all(num_subs == 36))
```

```{r}
save(dt_odd_seq_sim_diff, file = file.path(
  path_root, "data", "tmp", "dt_odd_seq_sim_diff.Rdata"))
save(dt_odd_seq_sim, file = file.path(
  path_root, "data", "tmp", "dt_odd_seq_sim.Rdata"))
```

#### Figure 2e

```{r}
fig_seq_sim_diff = ggplot(data = dt_odd_seq_sim_diff, mapping = aes(
  x = time, y = as.numeric(mean_difference), color = as.factor(speed * 1000),
  fill = as.factor(speed * 1000))) +
  geom_hline(aes(yintercept = 0), linetype = "solid", color = "gray") +
  geom_ribbon(aes(ymin = sem_lower, ymax = sem_upper), alpha = 0.5, color = NA) +
  geom_line() +
  theme(legend.position = "top", legend.direction = "horizontal",
        legend.justification = "center", legend.margin = margin(0, 0, 0, 0),
        legend.box.margin = margin(t = 0, r = 0, b = -5, l = 0)) +
  xlab("Time from sequence onset (TRs)") + ylab("Probability difference") +
  scale_colour_viridis(name = "Speed (ms)", discrete = TRUE, option = "cividis") +
  scale_fill_viridis(name = "Speed (ms)", discrete = TRUE, option = "cividis") +
  coord_capped_cart(left = "both", bottom = "both", expand = TRUE,
                    ylim = c(-0.4, 0.4), xlim = c(0, 12)) +
  scale_x_continuous(labels = label_fill(seq(1, 13, 1), mod = 4), breaks = seq(0, 12, 1)) +
  guides(color = guide_legend(nrow = 1)) +
  geom_segment(aes(x = 0, xend = 0, y = 0.01, yend = 0.4),
               arrow = arrow(length = unit(5, "pt")), color = "darkgray") +
  geom_segment(aes(x = 0, xend = 0, y = -0.01, yend = -0.4),
               arrow = arrow(length = unit(5, "pt")), color = "darkgray") +
  annotate(geom = "text", x = 0.4, y = 0.2, label = "Forward order",
           color = "darkgray", angle = 90, size = 3) +
  annotate(geom = "text", x = 0.4, y = -0.2, label = "Backward order",
           color = "darkgray", angle = 90, size = 3) +
  annotate("text", x = 12, y = -0.4, label = "1 TR = 1.25 s",
             hjust = 1, size = rel(2)) +
  theme(axis.line = element_line(colour = "black"),
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank())
fig_seq_sim_diff
```

```{r, include=FALSE, eval=FALSE, echo=FALSE}
ggsave(filename = "highspeed_plot_decoding_oddball_sequence_predictions.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 5.5, height = 3)
```

#### Source Data File Fig. 2e

```{r, echo=TRUE}
dt_odd_seq_sim_diff %>%
  select(-num_subs, -classification) %>%
  write.csv(., file = file.path(path_sourcedata, "source_data_figure_2e.csv"),
            row.names = FALSE)
```

### Figure 2

```{r, echo=TRUE, fig.width=8, fig.height=3}
upper = plot_grid(fig_odd_peak, plot.odd.long, fig_a, labels = c("a", "b", "c"), ncol = 3,
                  rel_widths = c(0.8, 3.5, 2), label_fontface = "bold", hjust = c(0, 0))
lower = plot_grid(fig_b, fig_seq_sim_diff, labels = c("d", "e"), nrow = 1,
                  label_fontface = "bold", hjust = c(0, 0), rel_widths = c(0.45, 0.53))
plot_grid(upper, lower, nrow = 2, ncol = 1, rel_heights = c(2.5, 3))
```

```{r}
ggsave(filename = "highspeed_plot_decoding_oddball_data.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 10, height = 6.5)
ggsave(filename = "wittkuhn_schuck_figure_2.pdf",
       plot = last_plot(), device = cairo_pdf, path = path_figures, scale = 1,
       dpi = "retina", width = 10, height = 6.5)
```