---
title: "tf Vectors and Operations"
author: "Jeff Goldsmith, Fabian Scheipl"
date: "`r Sys.Date()`"
output: 
    rmarkdown::html_vignette:
      fig_width: 7
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{tf Vectors and Operations}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "##",
  fig.width = 6,
  fig.height = 4,
  dpi = 72,
  fig.retina = 1,
  out.width = "90%"
)

library("tidyverse")
library("viridisLite")

theme_set(theme_minimal() + theme(legend.position = "bottom"))

options(
  ggplot2.continuous.colour = "viridis",
  ggplot2.continuous.fill = "viridis"
)
scale_colour_discrete <- scale_colour_viridis_d
scale_fill_discrete <- scale_fill_viridis_d

library("tidyfun")

pal_5 <- viridis(7)[-(1:2)]
set.seed(1221)
```

This vignette introduces the `tf` class, as well as the `tfd` and `tfb` subclasses, and focuses on vectors of this class. It also illustrates operations for `tf` vectors.

The functions in a `tf` vector are *scalar-valued*, i.e., $f: \mathbb{R} \to \mathbb{R}$. For **vector-valued** functional data $f: \mathbb{R} \to \mathbb{R}^d$ -- coupled signals sharing a common argument, like the (hip, knee) joint angles over a gait cycle or the (longitude, latitude) track of a hurricane -- `tf` provides the analogous `tfd_mv` and `tfb_mv` classes, which bundle $d$ such `tf` vectors. These are introduced in the [vector-valued functions vignette](https://tidyfun.github.io/tidyfun/articles/x07_Vector-valued_Functions.html).

# `tf`-Class: Definition

##  `tf`-class

**`tf`** is a new data type for (vectors of) functional data: 

- an abstract superclass for functional data in 2 forms:
    - as (argument, value)-tuples: subclass **`tfd`**, also irregular or sparse
    - or in basis representation: subclass **`tfb`** represents each observed function as a weighted sum
    of a fixed dictionary of known "basis functions".
    
- basically, a `list` of numeric vectors  
  (... since `list`s work well as columns of data frames ...)

- with additional attributes that define *function-like* behavior:
    - how to **evaluate** the given "functions" for new arguments
    - their **domain**: the range of valid argument values 

- `S3` based

## Example Data

First we extract a `tf` vector from the `tidyfun::dti_df` dataset containing fractional anisotropy tract profiles for the corpus callosum (`cca`). When printed, `tf` vectors show the first few `arg` and `value` pairs for each subject.  

```{r load-dti}
data("dti_df")

cca <- dti_df$cca
cca
```

We also extract a simple 5-element vector of functions on a regular grid:

```{r ex-def}
cca_five <- cca[1:5, seq(0, 1, length.out = 93), interpolate = TRUE]
rownames(cca_five) <- LETTERS[1:5]
cca_five <- tfd(cca_five, signif = 2)
cca_five
```

For illustration, we plot the vector `cca_five` below. 

```{r, ex-fig}
plot(cca_five, xlim = c(-0.15, 1), col = pal_5)
text(x = -0.1, y = cca_five[, 0.07], labels = names(cca_five), col = pal_5)
```

## **`tf`** subclass: **`tfd`**

**`tfd`** objects contain "raw" functional data: 

 - represented as a list of **`evaluations`** $f_i(t)|_{t=t'}$ and corresponding **`arg`**ument vector(s) $t'$
 - has a **`domain`**:  the range of valid **`arg`**s.

```{r}
cca_five |>
  tf_evaluations() |>
  str()
cca_five |>
  tf_arg() |>
  str()
cca_five |> tf_domain()
```

- each **`tfd`**-vector contains an **`evaluator`** function that defines how to inter-/extrapolate `evaluations` between `arg`s

```{r}
tf_evaluator(cca_five) |> str()
tf_evaluator(cca_five) <- tf_approx_spline
```

- **`tfd`** has two subclasses: one for regular data with a common grid and one for irregular or sparse data. The `cca` data are irregular (values are missing for some subjects at some arguments) but the example below more clearly illustrates support for sparse and irregular data using CD4 cell counts from a longitudinal study included in **`refund`**.

```{r}
cd4_vec <- tfd(refund::cd4)

cd4_vec[1:2]
cd4_vec[1:2] |>
  tf_arg() |>
  str()
cd4_vec[1:20] |> plot(pch = "x", col = viridis(20))
```

## **`tf`** subclass: **`tfb`**

Functional data in basis representation: 

 - represented as a list of **`coefficients`** and a common **`basis_matrix`** of basis function evaluations on a vector of `arg`-values.
 - contains a **`basis`** function that defines how to evaluate the basis functions for new **`arg`**s and how to differentiate or integrate it.
- (internal) flavors: 
    - `tfb_spline`: uses **`mgcv`**-spline bases 
    - `tfb_fpc`: uses functional principal components 
- significant memory and time savings:

```{r}
refund::DTI$cca |>
  object.size() |>
  print(units = "Kb")
cca |>
  object.size() |>
  print(units = "Kb")
cca |>
  tfb(verbose = FALSE) |>
  object.size() |>
  print(units = "Kb")
```

### **`tfb_spline`**: spline basis

- default for `tfb()`
- accepts all arguments of **`mgcv`**'s `s()`-syntax: basis type `bs`, basis dimension `k`, penalty order `m`, etc...
- also does non-Gaussian fits: `family` argument 
    - all exponential families
    - but also: $t$-distribution, ZI-Poisson, Beta, ... 

```{r, message = TRUE}
cca_five_b <- cca_five |> tfb()
cca_five_b[1:2]
cca_five[1:2] |> tfb(bs = "tp", k = 55)

# functions represent ratios in (0,1), so a Beta-distribution is more appropriate:
cca_five[1:2] |>
  tfb(bs = "ps", m = c(2, 1), family = mgcv::betar(link = "cloglog"))
```

### Penalization: 

**Function-specific (default), none**, prespecified (`sp`), or global: 

```{r}
layout(t(1:2))
cca_five |> plot()
cca_five_b |> plot(col = "red")
cca_five |>
  tfb(k = 35, penalized = FALSE) |>
  lines(col = "blue")
cca_five |>
  tfb(sp = 0.001) |>
  lines(col = "orange")
```

Right plot shows smoothing with function-specific penalization in red, without penalization in blue, 
and with manually set strong smoothing (`sp` $\to 0$) in orange.


**"Global" smoothing**:  

1. estimate smoothing parameters for subsample (~10\%) of curves
2. apply geometric mean of estimated smoothing parameters to smooth *all* curves

**Advantages:**

- (much) faster than optimizing penalization for each curve
- should scale well for largish datasets

**Disadvantages**

- no real borrowing of information across curves (very sparse or functional fragment data, e.g.)
- still requires more observations than basis functions *per curve*
- subsample could miss small subgroups with different roughness, over-/undersmooth parts of the data, see below.

```{r, echo = FALSE}
set.seed(1212)
raw <- c(
  tf_rgp(5, scale = 0.2, nugget = 0.05, arg = 101L) - 5,
  tf_rgp(5, scale = 0.02, nugget = 0.05, arg = 101L),
  tf_rgp(5, scale = 0.002, nugget = 0.05, arg = 101L) + 5
)
```

Dataset with heterogeneous roughness:
```{r}
layout(t(1:3))
clrs <- scales::alpha(sample(viridis(15)), 0.5)
plot(raw, main = "raw", col = clrs)
plot(tfb(raw, k = 55), main = "separate", col = clrs)
plot(tfb(raw, k = 55, global = TRUE), main = "global", col = clrs)
```

###  **`tfb`** FPC-based

- uses first few eigenfunctions computed from a simple unregularized (weighted) SVD of the data matrix by default
- corresponding FPC basis and mean function saved as `tfd`-object
- observed functions are linear combinations of those.
- amount of "smoothing" can be controlled (roughly!) by setting the 
minimal *percentage of variance explained* `pve` 

```{r}
cca_five_fpc <- cca_five |> tfb_fpc(pve = 0.999)
cca_five_fpc

cca_five_fpc_lowrank <- cca_five |> tfb_fpc(pve = 0.6)
cca_five_fpc_lowrank
```

```{r}
layout(t(1:2))
cca_five |> plot()
cca_five_fpc |> plot(col = "red", ylab = "tfb_fpc(cca_five)")
cca_five_fpc_lowrank |> lines(col = "blue", lty = 2)
```

`tfb_fpc` is currently only implemented for data on identical 
(but possibly non-equidistant) grids. The **`{refunder}`** `rfr_fpca`-functions 
provide FPCA methods appropriate for highly irregular and sparse data and regularized/smoothed FPCA.


# `tf`-Class: Methods

**`tidyfun`** implements almost all types of operations that are available for conventional
numerical or logical vectors for `tf`-vectors as well, so you can:

### subset & subassign:

```{r}
cca_five[1:2]
cca_five[1:2] <- cca_five[2:1]
cca_five
```

### compare & compute:

```{r, echo = FALSE}
n_cca_five <- names(cca_five)
cca_five <- unname(cca_five)
```

```{r}
cca_five[1] + cca_five[1] == 2 * cca_five[1]
log(exp(cca_five[2])) == cca_five[2]
(cca_five - (2:-2)) != cca_five
```

```{r, echo = FALSE}
names(cca_five) <- n_cca_five
```

### summarize across a vector of functions: 

Compute functional summaries like mean functions, functional standard deviations or variances or functional data depths over a vector of functional data:
```{r}
c(mean = mean(cca_five), sd = sd(cca_five))

tf_depth(cca_five) ## Modified Band-2 Depth (a la Sun/Genton/Nychka, 2012), others to come.
median(cca_five) == cca_five[which.max(tf_depth(cca_five))]
summary(cca_five)
```

### summarize each function over its domain: 

Compute summaries for each function like its mean or extreme values, quantiles, etc.
```{r}
tf_fmean(cca_five) # mean of each function's evaluations
tf_fmax(cca_five) # max of each function's evaluations
# 25%-tile of each f(t) for t > .5:
tf_fwise(cca_five, \(x) quantile(x$value[x$arg > 0.5], prob = 0.25)) |> unlist()
```
`tf_fwise` can be used to define custom statistics for each function that can depend on both its `value` and its `arg`.


In addition, **`tidyfun`** provides methods for operations that are specific for functional data:

## Methods for "functional" operations

### evaluate:

`tf`-objects have a special `[`-operator: Its second argument specifies 
`arg`ument values at which to evaluate the functions and has some additional options, 
so it's easy to get point values for `tf` objects, in `matrix` or `data.frame` formats: 

```{r, warning  = FALSE}
cca_five[1:2, seq(0, 1, length.out = 3)]
cca_five["B", seq(0, 0.15, length.out = 3), interpolate = FALSE]
cca_five[1:2, seq(0, 1, length.out = 7), matrix = FALSE] |> str()
```


### (simple, local) smoothing

```{r}
layout(t(1:3))
cca_five |> plot(alpha = 0.2, ylab = "lowess")
cca_five |>
  tf_smooth("lowess") |>
  lines(col = pal_5)

cca_five |> plot(alpha = 0.2, ylab = "rolling median (k=5)")
cca_five |>
  tf_smooth("rollmedian", k = 5) |>
  lines(col = pal_5)

cca_five |> plot(alpha = 0.2, ylab = "Savitzky-Golay (quartic, 11 steps)")
cca_five |>
  tf_smooth("savgol", fl = 11) |>
  lines(col = pal_5)
```

### differentiate & integrate:

```{r}
layout(t(1:3))
cca_five |> plot(col = pal_5)
cca_five |>
  tf_smooth() |>
  tf_derive() |>
  plot(col = pal_5, ylab = "tf_derive(tf_smooth(cca_five))")
cca_five |>
  tf_integrate(definite = FALSE) |>
  plot(col = pal_5)
```

```{r}
cca_five |> tf_integrate()
```

### query

**`tidyfun`** makes it easy to find (ranges of) `arg`uments $t$ satisfying a condition on `value` $f(t)$ (and `arg`ument $t$):

```{r}
cca_five |> tf_anywhere(value > 0.65)
cca_five[1:2] |> tf_where(value > 0.6, "all")
cca_five[2] |> tf_where(value > 0.6, "range")
cca_five |> tf_where(value > 0.6 & arg > 0.5, "first")
```

### zoom & query

```{r, ex-fig2}
cca_five |> plot(xlim = c(-0.15, 1), col = pal_5, lwd = 2)
text(x = -0.1, y = cca_five[, 0.07], labels = names(cca_five), col = pal_5, cex = 1.5)
median(cca_five) |> lines(col = pal_5[3], lwd = 4)
```

```{r}
# where are the first maxima of these functions?
cca_five |> tf_where(value == max(value), "first")

# where are the first maxima of the later part (t > .5) of these functions?
cca_five[c("A", "D")] |>
  tf_zoom(0.5, 1) |>
  tf_where(value == max(value), "first")

# which f_i(t) are below the functional median anywhere for 0.2 < t < 0.6?
# (t() needed here so we're comparing column vectors to column vectors...)
cca_five |>
  tf_zoom(0.2, 0.6) |>
  tf_anywhere(value <= t(median(cca_five)[, arg]))
```
