---
title: "Statistics of Fuzzy Random Variables -- A Short Tour"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Statistics of Fuzzy Random Variables}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 6,
                      fig.height = 4, dpi = 80)
library(FuzzyRandom)
set.seed(2024)
```

## Fuzzy numbers and samples

A fuzzy number is the alpha-cut envelope `[X^-(alpha), X^+(alpha)]` with a
non-decreasing lower and non-increasing upper endpoint.

```{r fuzzy-numbers}
tri <- triangular_fuzzy(0, 1, 2)
trap <- trapezoidal_fuzzy(-1, 0, 1, 3)
plot(tri)
```

A `FuzzySample` is an `(n, m, 2)` array of cuts for `n` i.i.d. fuzzy random
variables observed on `m` alpha levels.

```{r sample}
s <- FuzzySample(replicate(60, triangular_fuzzy(0, 1, 2), simplify = FALSE))
plot(s)
```

## Level-wise moments

```{r moments}
expectation(s)
variance(s)
correlation(s, FuzzySample(replicate(60, triangular_fuzzy(1, 2, 3),
                                  simplify = FALSE)))
```

## Empirical distribution and goodness of fit

The empirical CDFs and a level-wise Kolmogorov--Smirnov test against a
reference distribution:

```{r ecdf-ks}
fe <- fuzzy_ecdf(s, alpha = c(0, 0.5, 1))
plot(fe)
ks_test(s, p0 = "pnorm")
```

## The fuzzy normal and the central limit theorem

`fuzzy_normal()` builds a level-wise bivariate-normal fuzzy number. `clt()`
applies the level-wise CLT to a sample, and `simulate()` draws from it:

```{r clt}
cl <- clt(s)
summary(cl)
plot(simulate(cl, nsim = 80))
```

`lln()` studies the shrinkage of the sample-mean width against sample size:

```{r lln}
w <- lln(s)
plot(w)
```