---
title: "Bootstrapping many curves using rTPC"
author: "Daniel Padfield"
output: rmarkdown::html_vignette
date: "`r Sys.Date()`"
vignette: >
  %\VignetteIndexEntry{Bootstrapping many curves using rTPC}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
#### A brief example of how many curves can be bootstrapped when fitting models to TPCs using rTPC, nls.multstart, car and the tidyverse.

***
## Things to consider 

- For a comprehensive run-through of the types of bootstrapping that can be done within the *rTPC* workflow, please see `vignette("bootstrapping_models")`
- This vignette is written as an example of how to run your chosen bootstrapping method on multiple models

***

```{r setup, message=FALSE}
# load packages
library(boot)
library(car)
library(rTPC)
library(nls.multstart)
library(broom)
library(tidyverse)
library(patchwork)
library(minpack.lm)

```

## The problem

This vignette is inspired by an email I got from someone who was struggling to implement the bootstrapping approach using the package **car** on multiple curves. First I will demonstrate how the approach would be done using the approach of using the **tidyverse** and **car**, and how it breaks. I will fit the `gaussian_1987()` model to the first three curves of the `chlorella_tpc` dataset.

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  #tidy.opts=list(width.cutoff=60),
  #tidy=TRUE,
  fig.align = 'center',
  warning=FALSE
)
```

```{r load_data, fig.height=3.5, fig.width=9}
# load in data
data("chlorella_tpc")

# keep just a single curve
d <- filter(chlorella_tpc, curve_id <= 3)

# fit
d_fits <- nest(d, data = c(rate, temp)) %>%
  mutate(gaussian = map(data, ~nls_multstart(rate ~ gaussian_1987(temp, rmax, topt, a),
                                             data = .x,
                                             iter = c(3,3,3),
                                             start_lower = get_start_vals(.x$temp, .x$rate, model_name = 'gaussian_1987') - 1,
                                             start_upper = get_start_vals(.x$temp, .x$rate, model_name = 'gaussian_1987') + 1,
                                             lower = get_lower_lims(.x$temp, .x$rate, model_name = 'gaussian_1987'),
                                             upper = get_upper_lims(.x$temp, .x$rate, model_name = 'gaussian_1987'),
                                             supp_errors = 'Y',
                                             convergence_count = FALSE)))

# create high resolution predictions
d_preds <- mutate(d_fits, new_data = map(data, ~tibble(temp = seq(min(.x$temp), max(.x$temp), length.out = 100)))) %>%
  select(., -data) %>%
  mutate(preds = map2(gaussian, new_data, ~augment(.x, newdata = .y))) %>%
  select(curve_id, growth_temp, process, flux, preds) %>%
  unnest(preds)

# show the data
ggplot(d, aes(temp, rate)) +
  geom_point(size = 2) +
  geom_line(aes(temp, .fitted), d_preds) +
  theme_bw(base_size = 12) +
  labs(x = 'Temperature (ºC)',
       y = 'Metabolic rate',
       title = 'Metabolic rate across temperatures') +
  facet_wrap(~curve_id)

```

Using the pipeline used previously, we would extract the coefficients of each model, fun **minpack.lm::nlsLM()** on each one and then run **car::Boot()**. However, this time the output of **minpack.lm::nlsLM()** and **car::Boot()** need to be stored in the list column within `d_fits`.

Extracting the coefficients and refitting the models using **minpack.lm::nlsLM()** works fine.

```{r refit_nlsLM}
# get coefs
d_fits <- mutate(d_fits, coefs = map(gaussian, coef))

# fit with nlsLM instead
d_fits <- mutate(d_fits, nls_fit = map2(data, coefs, ~nlsLM(rate ~ gaussian_1987(temp, rmax, topt, a),
                                        data = .x,
                                        start = .y,
                                        lower = get_lower_lims(.x$temp, .x$rate, model_name = 'gaussian_1987'),
                                        upper = get_upper_lims(.x$temp, .x$rate, model_name = 'gaussian_1987'))))

head(d_fits)

d_fits$nls_fit[[1]]

```

However, using **car::Boot()** currently gives an error.

```{r boot_error, error=TRUE}
# try and bootstrap # THIS BREAKS
d_fits <- mutate(d_fits, bootstrap = map(nls_fit, ~Boot(.x, method = 'residual')))

```
The error, that the object `.x` cannot be found, likely means that **car::Boot()** is incompatible with the **purrr::map()** method of using list columns to store model objects and get predictions and parameter estimates. Usually I would email the creators and maintainers of **car**, but having already emailed them multiple times with code problems/queries when trying to get **car::Boot()** to work with non-linear least squares regressions, I decided to try find a not-so-painful workaround.

## The solution

Instead of creating a list column with **mutate()** and **map()**, we can create an empty list column and then run a for loop to run **Boot()** on each model in the list column of the dataframe. We can then just place that result of **Boot()** into the right place of our empty list column. Because the error comes from the actual model fit, we need to run the **nlsLM()** model again each time. I really like this approach and have found it powerful numerous times now.

```{r work_around}
# create empty list column
d_fits <- mutate(d_fits, bootstrap = list(rep(NA, n())))

# run for loop to bootstrap each refitted model
for(i in 1:nrow(d_fits)){
  temp_data <- d_fits$data[[i]]
  temp_fit <- nlsLM(rate ~ gaussian_1987(temp, rmax, topt, a),
               data = temp_data,
               start = d_fits$coefs[[i]],
               lower = get_lower_lims(temp_data$temp, temp_data$rate, model_name = 'gaussian_1987'),
               upper = get_upper_lims(temp_data$temp, temp_data$rate, model_name = 'gaussian_1987'))
  boot <- Boot(temp_fit, method = 'residual')
  d_fits$bootstrap[[i]] <- boot
  rm(list = c('temp_fit', 'temp_data', 'boot'))
}

d_fits

```
Voila! There is now a list column of the bootstrapped parameter estimates for each model.

It is now possible to do all the other things in the pipeline. Firstly, we can get the 95% confidence intervals around our predictions. This heavily borrows from the code from `vignette("bootstrapping_models")`, but is a little more laborious as we are applying it to a list column. The function defined is not the prettiest but it does exactly the same job as in `vignette("bootstrapping_models")`.

```{r get_preds_and_plot, fig.height=3.5, fig.width=9}
# get the raw values of each bootstrap
d_fits <- mutate(d_fits, output_boot = map(bootstrap, function(x) x$t))

# calculate predictions with a gnarly written function
d_fits <- mutate(d_fits, preds = map2(output_boot, data, function(x, y){
  temp <- as.data.frame(x) %>%
    drop_na() %>%
    mutate(iter = 1:n()) %>%
    group_by_all() %>%
    do(data.frame(temp = seq(min(y$temp), max(y$temp), length.out = 100))) %>%
    ungroup() %>%
    mutate(pred = gaussian_1987(temp, rmax, topt, a))
  return(temp)
}))

# select, unnest and calculate 95% CIs of predictions
boot_conf_preds <- select(d_fits, curve_id, preds) %>%
  unnest(preds) %>%
  group_by(curve_id, temp) %>%
  summarise(conf_lower = quantile(pred, 0.025),
            conf_upper = quantile(pred, 0.975),
            .groups = 'drop')

ggplot() +
  geom_line(aes(temp, .fitted), d_preds, col = 'blue') +
  geom_ribbon(aes(temp, ymin = conf_lower, ymax = conf_upper), boot_conf_preds, fill = 'blue', alpha = 0.3) +
  geom_point(aes(temp, rate), d, size = 2) +
  theme_bw(base_size = 12) +
  labs(x = 'Temperature (ºC)',
       y = 'Rate') +
  facet_wrap(~curve_id)
```

Second, we can calculate the confidence intervals of the estimated parameters explicitly modelled in the regression.

```{r get_params_and_plot, fig.height=3.5, fig.width=9}
# get tidied parameters using broom::tidy
# get confidence intervals of parameters
d_fits <- mutate(d_fits, params = map(nls_fit, broom::tidy),
                 cis = map(bootstrap, function(x){
                   temp <- confint(x, method = 'bca') %>%
                     as.data.frame() %>%
                     rename(conf_lower = 1, conf_upper = 2) %>%
                     rownames_to_column(., var = 'term')
                   return(temp)
                   }))

# join parameter and confidence intervals in the same dataset 
left_join(select(d_fits, curve_id, growth_temp, flux, params) %>% unnest(params),
          select(d_fits, curve_id, growth_temp, flux, cis) %>% unnest(cis)) %>%
  ggplot(., aes(curve_id, estimate)) +
  geom_point(size = 4) +
  geom_linerange(aes(ymin = conf_lower, ymax = conf_upper)) +
  theme_bw() +
  facet_wrap(~term, scales = 'free')

```

Finally, we can redo our **car::Boot()** procedure, but this time use **calc_params()** to bootstrap confidence intervals for the extra parameters such as $T_{opt}$ and $r_{max}$. For reasons that I currently do not understand, **Boot()** and **calc_params()** only calculates the activation energy, deactivation energy, and q10 when using `method = case` not `method = residual`, but actually it is not recommended to bootstrap these parameters from models where they are not explicitly included in the model formula anyway.

```{r calc_params_and_plot, fig_width = 8, fig_height = 5}
# create empty list column
d_fits <- mutate(d_fits, ci_extra_params = list(rep(NA, n())))

# run for loop to bootstrap extra params from each model
for(i in 1:nrow(d_fits)){
  temp_data <- d_fits$data[[i]]
  temp_fit <- nlsLM(rate ~ gaussian_1987(temp, rmax, topt, a),
               data = temp_data,
               start = d_fits$coefs[[i]],
               lower = get_lower_lims(temp_data$temp, temp_data$rate, model_name = 'gaussian_1987'),
               upper = get_upper_lims(temp_data$temp, temp_data$rate, model_name = 'gaussian_1987'))
  boot <- Boot(temp_fit, f = function(x){unlist(calc_params(x))}, labels = names(calc_params(temp_fit)), R = 20, method = 'case') %>%
  confint(., method = 'bca') %>%
  as.data.frame() %>%
  rename(conf_lower = 1, conf_upper = 2) %>%
  rownames_to_column(., var = 'param')
  d_fits$ci_extra_params[[i]] <- boot
  rm(list = c('temp_fit', 'temp_data', 'boot'))
}

# calculate extra params for each model and put in long format to begin with
d_fits <- mutate(d_fits, extra_params = map(nls_fit, function(x){calc_params(x) %>% pivot_longer(everything(), names_to =  'param', values_to = 'estimate')}))

left_join(select(d_fits, curve_id, growth_temp, flux, extra_params) %>% unnest(extra_params),
          select(d_fits, curve_id, growth_temp, flux, ci_extra_params) %>% unnest(ci_extra_params)) %>%
  ggplot(., aes(as.character(curve_id), estimate)) +
  geom_point(size = 4) +
  geom_linerange(aes(ymin = conf_lower, ymax = conf_upper)) +
  theme_bw() +
  labs(y = 'estimate', x = "curve id") +
  facet_wrap(~param, scales = 'free') +
  labs(title = 'Calculation of confidence intervals for extra parameters')

```
#### Further reading

- John Fox (author of car) on bootstrapping regression models in R
    - https://artowen.su.domains/courses/305a/FoxOnBootingRegInR.pdf
- A.C. Davison & D.V. Hinkley (2003) Bootstrap Methods and their Application.         - https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A
- Schenker, N., & Gentleman, J. F. (2001). On judging the significance of differences by examining the overlap between confidence intervals. The American Statistician, 55(3), 182-186.
- Puth, M. T., Neuhäuser, M., & Ruxton, G. D. (2015). On the variety of methods for calculating confidence intervals by bootstrapping. Journal of Animal Ecology, 84(4), 892-897.