sampling.Rmd

# (PART) 建模篇 {-}

# 模拟与抽样 {#sampling}


```{r sampling-1, message = FALSE, warning = FALSE}
library(tidyverse)
```

本章目的是在tidyverse的架构下，介绍一些模拟和抽样的知识。先回顾下Hadley Wickham提出的数据科学tidy原则，tidy思想体现在:

- 任何数据都可以规整为数据框 
- 数据框的一列代表一个**变量**，数据框的一行代表一次**观察**
- 函数处理数据时，数据框进、数据框出

## 模拟

### 生成随机数

比如生成5个高斯分布的随机数，高斯分布就是正态分布，R语言里我们用`rnorm()`函数产生正态分布的随机数
```{r sampling-2}
rnorm(n = 5, mean = 0, sd = 1)
```

事实上，R内置了很多随机数产生的函数

| Distrution 	| Notation 		| R |
|:----------	|:-----------:|:-------------:|
|Uniform | $\text{U}(a, b)$ | `runif`|
|Normal  | $\text{N}(\mu, \sigma)$ | `rnorm`|
|Binormal  | $\text{Bin}(n, p)$ | `rbinorm`|
|Piosson  | $\text{pois}(\lambda)$ | `rpois`|
|Beta  | $\text{Beta}(\alpha, \beta)$ | `rbeta`|

如果大家查看帮助文档`?runif`，会发现每种**分布**都有对应的四个函数

- `d`:density 
- `p`:cumulative probability 
- `q`:quantile 
- `r`:random 


```{r sampling-3}
dnorm(seq(0.1, 0.5, length.out = 10), mean = 0, sd = 1)
```


在tidyverse的框架下，我们喜欢在数据框(data.frame)下运用这些函数，因为这样我们可以方便使用ggplot2来可视化，

- 例子1，我们生成100个正态分布的点，然后看看其分布
```{r sampling-4}
tibble(
  x = rnorm(n = 100, mean = 0, sd = 1)
) %>%
  ggplot(aes(x = x)) +
  geom_density()
```


我们将模拟的正态分布和理论上正态分布画在一起
```{r sampling-5}
tibble(
  x = rnorm(n = 100, mean = 0, sd = 1)
) %>%
  ggplot(aes(x = x)) +
  geom_density() +
  stat_function(
    fun = dnorm,
    args = list(mean = 0, sd = 1),
    color = "red"
  )
```
如果我们模拟点再增加点，会越来越逼近理论上的分布。



- 例子2，在数据框(data.frame)下，建立模拟$x$和$y$的线性关系

$$ y_i  = 4 + 3.2\, x_i$$
现实中，观察值往往会带入误差，假定误差服从正态分布，那么$x$和$y$的线性关系重新表述为
$$ y_i  = \beta_0 + \beta_1\, x_i + \epsilon_i, \quad \epsilon \in \text{Normal}(\mu =0, \sigma =1) $$

```{r sampling-6}
beta_0 <- 4
beta_1 <- 3.2
epsilon <- rnorm(n = 1000, mean = 0, sd = 1)

sim_normal <- tibble(
  # x_vals = runif(1000, 0, 10)
  x_vals = seq(from = 0, to = 5, length.out = 1000),
  y_vals = beta_0 + beta_1 * x_vals + epsilon,
)

sim_normal %>% head()
```




```{r sampling-7}
sim_normal %>%
  ggplot(aes(x = x_vals, y = y_vals)) +
  geom_point()
```


有时候为了方便，可以写简练点
```{r sampling-8}
tibble(
  a = runif(1000, 0, 5),
  b = 4 + rnorm(1000, mean = 3.2 * a, sd = 1)
) %>%
  ggplot(aes(x = a, y = b)) +
  geom_point()
```



## MASS::mvrnorm

`MASS::mvrnorm(n = 1, mu,  Sigma)`产生多元高斯分布的随机数，每组随机变量高度相关。
比如人的身高服从正态分布，人的体重也服从正态分布，同时身高和体重又存在强烈的关联。


- `n`: 随机样本的大小
- `mu`: 多元高斯分布的均值向量
- `Sigma`: 协方差矩阵，主要这里是大写的S (Sigma)，提醒我们它是一个矩阵，不是一个数值

```{r sampling-9}
a <- 3.5
b <- -1
sigma_a <- 1
sigma_b <- 0.5
rho <- -0.7

mu <- c(a, b)
cov_ab <- sigma_a * sigma_b * rho # 协方差
```



```{r sampling-10}
# 构建协方差矩阵
sigma <- matrix(c(
  sigma_a^2, cov_ab,
  cov_ab, sigma_b^2
), ncol = 2)
```



```{r sampling-11}
d <- MASS::mvrnorm(1000, mu = mu, Sigma = sigma) %>%
  data.frame() %>%
  set_names("group_a", "group_b")

head(d)
```




```{r sampling-12}
d %>%
  ggplot(aes(x = group_a)) +
  geom_density(
    color = "transparent",
    fill = "dodgerblue3",
    alpha = 1 / 2
  ) +
  stat_function(
    fun = dnorm,
    args = list(mean = 3.5, sd = 1),
    linetype = 2
  )
```


```{r sampling-13}
d %>%
  ggplot(aes(x = group_b)) +
  geom_density(
    color = "transparent",
    fill = "dodgerblue3",
    alpha = 1 / 2
  ) +
  stat_function(
    fun = dnorm,
    args = list(mean = -1, sd = 0.5),
    linetype = 2
  )
```


```{r sampling-14}
d %>%
  ggplot(aes(x = group_a, y = group_b)) +
  geom_point() +
  stat_ellipse(type = "norm", level = 0.95)
```


我们回头验算一下


```{r sampling-15}
d %>% summarise(
  a_mean = mean(group_a),
  b_mean = mean(group_b),
  a_sd = sd(group_a),
  b_sd = sd(group_b),
  cor = cor(group_a, group_b),
  cov = cov(group_a, group_b)
)
```



## 蒙特卡洛

这是我研究生时候老师布置的一个的题目，当时我用的是C语言代码，现在我们有强大的tidyverse


```{r sampling-16, out.width = '50%', fig.align='left', echo = FALSE}
knitr::include_graphics(path = "images/pi.jpg")
```



```{r sampling-17}
set.seed(2019)

n <- 50000
points <- tibble("x" = runif(n), "y" = runif(n))
```





```{r sampling-18}
points <- points %>%
  mutate(inside = map2_dbl(.x = x, .y = y, ~ if_else(.x**2 + .y**2 < 1, 1, 0))) %>%
  rowid_to_column("N")
```

正方形的面积是1，圆的面积是$\pi r^2 = \frac{1}{4} \pi$，如果知道两者的比例，就可以估算$\pi$

```{r sampling-19}
points <- points %>%
  mutate(estimate = 4 * cumsum(inside) / N)
```

```{r sampling-20}
points %>% tail()
```


```{r sampling-21}
points %>%
  ggplot() +
  geom_line(aes(y = estimate, x = N), colour = "#82518c") +
  geom_hline(yintercept = pi)
```


## 抽样与样本


### 总体分布
假定一个事实，川师男生总体的平均身高和身高的标准差分别为
```{r sampling-22}
true.mean <- 175.7
true.sd <- 15.19
```

那么我们可以模拟分布情况如下

```{r sampling-23}
pop.distn <-
  tibble(
    height = seq(100, 250, 0.5),
    density = dnorm(height, mean = true.mean, sd = true.sd)
  )

ggplot(pop.distn) +
  geom_line(aes(height, density)) +
  geom_vline(
    xintercept = true.mean,
    color = "red",
    linetype = "dashed"
  ) +
  geom_vline(
    xintercept = true.mean + true.sd,
    color = "blue",
    linetype = "dashed"
  ) +
  geom_vline(
    xintercept = true.mean - true.sd,
    color = "blue",
    linetype = "dashed"
  ) +
  labs(
    x = "Height (cm)", y = "Density",
    title = "川师男生身高分布"
  )
```

### 样本
假定我们从中抽取30个男生身高样本

```{r sampling-24}
sample.a <-
  tibble(height = rnorm(n = 30, mean = true.mean, sd = true.sd))
```

然后看看样本的直方图

```{r sampling-25}
sample.a %>%
  ggplot(aes(x = height)) +
  geom_histogram(aes(y = stat(density)),
    fill = "steelblue",
    alpha = 0.75,
    bins = 10
  ) +
  geom_line(
    data = pop.distn,
    aes(x = height, y = density),
    alpha = 0.25, size = 1.5
  ) +
  geom_vline(xintercept = true.mean, linetype = "dashed", color = "red") +
  geom_vline(xintercept = mean(sample.a$height), linetype = "solid")
```

红色的虚线代表分布的总体的均值，黑色实线代表30个样本的均值，

```{r sampling-26}
sample.a %>%
  summarize(
    sample.mean = mean(height),
    sample.sd = sd(height)
  )
```
也就是说，基于这30个观察值的样本，我们认为川师男生的身高均值为`r mean(sample.a$height)`cm，方差为`r sd(sample.a$height)`


可能有同学说，这个样本太少了，计算的均值还不够科学，会以偏概全。于是又重新找了30个男生，和上次类似，用`rnorm`函数模拟，我们记为样本b

```{r sampling-27}
sample.b <-
  tibble(height = rnorm(30, mean = true.mean, sd = true.sd))
```

再来看看这次样本的分布
```{r sampling-28}
sample.b %>%
  ggplot(aes(x = height)) +
  geom_histogram(aes(y = stat(density)),
    fill = "steelblue", alpha = 0.75, bins = 10
  ) +
  geom_line(
    data = pop.distn, aes(x = height, y = density),
    alpha = 0.25, size = 1.5
  ) +
  geom_vline(xintercept = true.mean, linetype = "dashed", color = "red") +
  geom_vline(xintercept = mean(sample.a$height), linetype = "solid")
```

同样，我们计算样本b的均值和方差

```{r sampling-29}
sample.b %>%
  summarize(
    sample.mean = mean(height),
    sample.sd = sd(height)
  )
```


这次抽样的结果，均值为`r mean(sample.b$height)`cm，方差为`r sd(sample.b$height)`

和样本a比，有一点点变化。不经想问，我能否继续抽样呢？结果会有变化吗？为了避免重复写代码
，我把上面的过程整合到一起，写一个**子函数**，专门模拟抽样过程

```{r sampling-30}
rnorm.stats <- function(n, mu, sigma) {
  the.sample <- rnorm(n, mu, sigma)
  tibble(
    sample.size = n,
    sample.mean = mean(the.sample),
    sample.sd = sd(the.sample)
  )
}
```

于是，我们又可以继续模拟了。注意我们之前设定的总体分布的均值和方差
```{r sampling-31, eval= FALSE}
true.mean <- 175.7
true.sd <- 15.19
```

```{r sampling-32}
rnorm.stats(30, true.mean, true.sd)
```
yes，代码工作的很好，但不过只是代码减少了一点点，仍然只是一次抽样（这里30个样本为一次抽样），**我们的目的是反复抽样**， 抽很多次的那种喔。


那我们用`purrr`包的`rerun`函数偷个懒，

```{r sampling-33}
df.samples.of.30 <-
  purrr::rerun(2500, rnorm.stats(30, true.mean, true.sd)) %>%
  dplyr::bind_rows()
```
哇，一下子抽了2500个样本,全部装进了`df.sample.of.30`这个数据框， 偷偷看一眼呢
```{r sampling-34}
df.samples.of.30 %>% head()
```

回过头看看`df.samples.of.30`是什么：

 - 从川师的男生中随机抽取30个，计算这30个人身高的均值和方差，这叫一次抽样
 - 把上面的工作，重复2500次，得到2500个均值和方差
 - 2500个均值和方差，组成了一个数据框
 
 
 
我们发现每次抽样的均值都不一样，感觉又像一个分布(抽样的均值分布)，我们画出来看看吧
```{r sampling-35}
df.samples.of.30 %>%
  ggplot(aes(x = sample.mean, y = stat(density))) +
  geom_histogram(bins = 25, fill = "firebrick", alpha = 0.5) +
  geom_vline(xintercept = true.mean, linetype = "dashed", color = "red") +
  labs(
    title = "抽样2500次（每次30个男生）身高均值的分布",
    subtitle = "Distribution of mean heights for 2500 samples of size 30"
  )
```
注意到，这不是男生身高的分布，而是**每次抽样计算的均值**构成的分布.

为了更清楚的说明，我们把**整体的分布(灰色曲线)、样本a（蓝色直方图）、抽样的均值分布（红色直方图）**三者画在一起。


```{r sampling-36, message=FALSE, warning=FALSE}
df.samples.of.30 %>%
  ggplot(aes(x = sample.mean, y = stat(density))) +
  geom_histogram(bins = 50, fill = "firebrick", alpha = 0.5) +
  geom_histogram(
    data = sample.a,
    aes(x = height, y = stat(density)),
    bins = 11, fill = "steelblue", alpha = 0.25
  ) +
  geom_vline(xintercept = true.mean, linetype = "dashed", color = "red") +
  geom_line(data = pop.distn, aes(x = height, y = density), alpha = 0.25, size = 1.5) +
  xlim(125, 225)
```

样本的均值分布，是个很有意思的结果，比如，我们**再选30个男生**再抽样一次，我们可以断定，这次抽样的均值会落在了红色的区间之内。

然而，注意到，必须限定再次抽样的大小仍然是30个男生，以上这句话才成立。


```{r sampling-37}
df.samples.of.30 %>%
  summarize(
    mean.of.means = mean(sample.mean),
    sd.of.means = sd(sample.mean)
  )
```

这里计算的是抽样(样本大小为30)均值分布，而不是整体的均值分布。言外之意，样本大小可以是其它的呗， 那就把样本调整为50、100、250、500分别试试看
```{r sampling-38}
df.samples.of.50 <-
  rerun(2500, rnorm.stats(50, true.mean, true.sd)) %>%
  bind_rows()

df.samples.of.100 <-
  rerun(2500, rnorm.stats(100, true.mean, true.sd)) %>%
  bind_rows()

df.samples.of.250 <-
  rerun(2500, rnorm.stats(250, true.mean, true.sd)) %>%
  bind_rows()

df.samples.of.500 <-
  rerun(2500, rnorm.stats(500, true.mean, true.sd)) %>%
  bind_rows()
```
忍不住想画图看看，每次抽取的男生数量不同，均值的分布会有不同？
```{r sampling-39}
df.combined <-
  bind_rows(
    df.samples.of.30,
    df.samples.of.50,
    df.samples.of.100,
    df.samples.of.250,
    df.samples.of.500
  ) %>%
  mutate(sample.sz = as.factor(sample.size))
```


```{r sampling-40, fig.width = 10, fig.asp= 0.3}
df.combined %>%
  ggplot(aes(x = sample.mean, y = stat(density), fill = sample.sz)) +
  geom_histogram(bins = 25, alpha = 0.5) +
  geom_vline(xintercept = true.mean, linetype = "dashed") +
  facet_wrap(vars(sample.sz), nrow = 1) +
  scale_fill_brewer(palette = "Set1") +
  labs(
    x = "Sample means", y = "Density",
    title = "Distribution of mean heights for samples of varying size"
  )
```

随着样本大小由30增加到500，抽样的均值分布围绕着越来越聚合到实际的均值，或者说随着样本大小的增多，对均值估计的不确定性越小。 

```{r sampling-41}
sampling.distn.mean.table <-
  df.combined %>%
  group_by(sample.size) %>%
  summarize(
    mean.of.means = mean(sample.mean),
    sd.of.means = sd(sample.mean)
  )
sampling.distn.mean.table
```

有个统计学上的概念需要明确。

输出结果的第三列`sd.of.means` 是不同样本大小(30,50,100,250,500)下，反复抽样后平均数分布的标准差。


数学上，如果已知总体的标准差($\sigma$)，那么抽取无限多份大小为 $n$ 的样本，每个样本各有一个平均值，所有这个大小的样本之平均值的标准差可证明为

$$
\frac{\sigma}{\sqrt{n}}
$$
即，**平均值的标准误差**。



下面我们画图看看，模拟出来的$sd.of.means$和理论值$\frac{\sigma}{\sqrt{n}}$是否一致。

注意到这里的$\sigma$是总体的标准差，即最开始我们设定的川师男生身高的标准差`true.sd`. 也就说，理论上

```{r sampling-42}
df.se.mean.theory <- tibble(
  sample.size = seq(10, 500, 10)
) %>%
  mutate(std.error = true.sd / sqrt(sample.size))

df.se.mean.theory
```



```{r sampling-43}
sampling.distn.mean.table %>%
  ggplot(aes(x = sample.size, y = sd.of.means)) +
  geom_point() +
  geom_line(aes(x = sample.size, y = std.error),
    data = df.se.mean.theory,
    color = "red"
  ) +
  labs(
    x = "Sample size", y = "Std Error of Mean",
    title = "平均值标准误差随样本大小变化（理论值和模拟值对比）"
  )
```

两者吻合的很好。



刚刚我们看到的，**抽样均值分布**随着**样本大小**变化而变化。可以试想下，抽样的其他统计量分布（方差，中位数），是不是也随着样本大小变化而变化呢？



```{r sampling-44}
sampling.distn.sd.table <-
  df.combined %>%
  group_by(sample.size) %>%
  summarize(
    mean.of.sds = mean(sample.sd),
    sd.of.sds = sd(sample.sd)
  )

sampling.distn.sd.table
```

答案是肯定的，样本量的增多，抽样方差的不确定性减少。








## 扩展阅读

 - <https://learningstatisticswithr.com/book/>

<!-- - <https://bookdown.org/content/922/> -->




```{r sampling-45, echo = F}
# remove the objects
# rm(list=ls())
rm(
  a, b,
  beta_0, beta_1,
  cov_ab, d,
  df.combined, df.samples.of.100,
  df.samples.of.250, df.samples.of.30,
  df.samples.of.50, df.samples.of.500,
  df.se.mean.theory, epsilon,
  mu, n,
  points, pop.distn,
  rho, rnorm.stats,
  sample.a, sample.b,
  sampling.distn.mean.table, sampling.distn.sd.table,
  sigma, sigma_a,
  sigma_b, sim_normal,
  true.mean, true.sd
)
```

```{r sampling-46, echo = F, message = F, warning = F, results = "hide"}
pacman::p_unload(pacman::p_loaded(), character.only = TRUE)
```