diff --git a/_quarto.yml b/_quarto.yml
index 3f504c47..2165fec2 100755
--- a/_quarto.yml
+++ b/_quarto.yml
@@ -64,6 +64,7 @@ book:
         - analyze-text-data.qmd
         - analyze-time-series-data.qmd
         - analyze-spatial-data.qmd
+        - analyze-areal-data.qmd
     - part: "优化建模"
       chapters:
         - statistical-computation.qmd
@@ -73,6 +74,7 @@ book:
       chapters:
         - probabilistic-reasoning-framework.qmd
         - generalized-linear-models.qmd
+        - hierarchical-normal-models.qmd
         - mixed-effects-models.qmd
         - generalized-additive-models.qmd
         - gaussian-processes-regression.qmd
diff --git a/analyze-areal-data.qmd b/analyze-areal-data.qmd
new file mode 100644
index 00000000..7780a5ea
--- /dev/null
+++ b/analyze-areal-data.qmd
@@ -0,0 +1,137 @@
+# 区域数据分析 {#sec-analyze-areal-data}
+
+## 苏格兰唇癌数据分析 {#sec-scotland-lip-cancer}
+
+> Everything is related to everything else, but near things are more related than distant things.
+>
+> --- Waldo Tobler [@Tobler1970]
+
+::: {#spatial-areal-data .callout-note title="空间区域数据分析"}
+空间区域数据的贝叶斯建模
+
+-   Bayesian spatial and spatio-temporal GLMMs with possible extremes [glmmfields](https://github.com/seananderson/glmmfields)
+-   Bayesian spatial analysis [geostan](https://github.com/ConnorDonegan/geostan/)
+-   [Spatial Models in Stan: Intrinsic Auto-Regressive Models for Areal Data](https://mc-stan.org/users/documentation/case-studies/icar_stan.html)
+-   [Exact sparse CAR models in Stan](https://github.com/mbjoseph/CARstan) [网页文档](https://mc-stan.org/users/documentation/case-studies/mbjoseph-CARStan.html)
+-   [Spatial Models in Stan: Intrinsic Auto-Regressive Models for Areal Data](https://github.com/stan-dev/example-models/tree/master/knitr/car-iar-poisson) [网页文档](https://mc-stan.org/users/documentation/case-studies/icar_stan.html) 原始数据和代码，接上面苏格兰唇癌数据分析，用 CmdStanR 更新后的[代码](https://github.com/stan-dev/example-models/tree/master/knitr/car-iar-poisson)
+-   [Spatial modeling of areal data. Lip cancer in Scotland](https://www.paulamoraga.com/book-geospatial/sec-arealdataexamplespatial.html) INLA 建模
+-   [CAR models Scotland Lip cancer dataset](https://rafaelcabral96.github.io/nigstan/sar-and-car-models.html#car-models) Stan 建模
+-   空间计量 [区域数据分析](https://rsbivand.github.io/emos_talk_2304/bivand_emos_230419.pdf) [on-the-use-of-r-for-spatial-econometrics](https://github.com/rsbivand/emos_talk_2304)
+:::
+
+响应变量服从泊松分布
+
+-   BYM-INLA [@blangiardo2013;@moraga2020]
+-   BYM-Stan [@morris2019; @donegan2022; @cabral2022]
+
+记录 1975-1986 年苏格兰 56 个地区的唇癌病例数，这是一个按地区汇总的数据。
+
+```{r}
+library(sf)
+scotlips <- st_read('data/scotland/scotland.shp', crs = st_crs("EPSG:27700"))
+str(scotlips)
+```
+
+```{r}
+#| label: fig-lip-cancer-map
+#| fig-cap: 苏格兰各地区唇癌病例数分布
+#| fig-width: 5
+#| fig-height: 5
+#| fig-showtext: true
+
+library(ggplot2)
+ggplot() +
+  geom_sf(data = scotlips, aes(fill = Observed)) +
+  scale_fill_viridis_c() +
+  theme_minimal()
+```
+
+## 美国各州犯罪率分析
+
+响应变量服从高斯分布的调查数据 [@bivand2001]
+
+数据集 USArrests 记录 1973 年美国各州每 10 万居民中因谋杀 Murder、袭击 Assault 和强奸 Rape 被警察逮捕的人数以及城市人口所占百分比（可以看作城市化率）。
+
+```{r}
+#| echo: false
+#| label: tbl-us-arrests
+#| tbl-cap: "数据集 USArrests（部分）"
+
+us_arrests <- data.frame(
+  state_name = rownames(USArrests),
+  state_region = state.region,
+  USArrests, check.names = FALSE
+)
+
+knitr::kable(head(us_arrests), col.names = c(
+  "州名", "区域划分", "谋杀犯", "袭击犯", "城市化率", "强奸犯"
+), row.names = FALSE)
+```
+
+```{r}
+#| label: fig-us-arrests-sf
+#| fig-cap: 因袭击被逮捕的人数分布
+#| fig-showtext: true
+#| fig-width: 7
+#| fig-height: 4
+
+library(sf)
+# 州数据
+us_state_sf <- readRDS("data/us-state-map-2010.rds")
+# 观测数据
+us_state_df <- merge(x = us_state_sf, y = us_arrests,
+  by.x = "NAME", by.y = "state_name", all.x = TRUE)
+
+ggplot() +
+  geom_sf(
+    data = us_state_df, aes(fill = Assault), color = "gray80", lwd = 0.25) +
+  scale_fill_viridis_c(option = "plasma", na.value = "white") +
+  theme_void()
+```
+
+1973 年美国各州因袭击被逮捕的人数与城市化率的关系：相关分析
+
+```{r}
+#| label: fig-us-arrests-point
+#| fig-cap: 逮捕人数比例与城市化率的关系
+#| fig-width: 7
+#| fig-height: 5.5
+#| code-fold: true
+#| echo: !expr knitr::is_html_output()
+#| fig-showtext: true
+
+library(ggrepel)
+ggplot(data = us_arrests, aes(x = UrbanPop, y = Assault)) +
+  geom_point(aes(color = state_region)) +
+  geom_text_repel(aes(label = state_name), size = 3, seed = 2022) +
+  theme_classic() +
+  labs(x = "城市化率（%）", y = "因袭击被逮捕人数", color = "区域划分")
+```
+
+阿拉斯加州和夏威夷州与其它州都不相连，属于孤立的情况，下面在空间相关性的分析中排除这两个州。
+
+```{r}
+# 州的中心
+centers48 <- subset(
+  x = data.frame(x = state.center$x, y = state.center$y),
+  subset = !state.name %in% c("Alaska", "Hawaii")
+)
+# 观测数据
+arrests48 <- subset(
+  x = USArrests,
+  subset = !rownames(USArrests) %in% c("Alaska", "Hawaii")
+)
+```
+
+```{r}
+#| message: false
+
+library(spData)
+library(spdep)
+# KNN
+k4.48 <- knn2nb(knearneigh(as.matrix(centers48), k = 4))
+# Moran I test
+moran.test(x = arrests48$Assault, listw = nb2listw(k4.48))
+# Permutation test for Moran's I statistic
+moran.mc(x = arrests48$Assault, listw = nb2listw(k4.48), nsim = 499)
+```
diff --git a/common-statistical-tests.qmd b/common-statistical-tests.qmd
index 344d7a32..d608a189 100644
--- a/common-statistical-tests.qmd
+++ b/common-statistical-tests.qmd
@@ -111,6 +111,20 @@ qnorm(p = 1 - 0.05, mean = 0, sd = 1)
 1 - pnorm(q = u)
 ```
 
+::: callout-important
+随机变量 $X$ 服从标准正态分布，它的概率分布函数如下：
+
+$$
+P(X \leq u)= \phi(u) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{u}\mathrm{e}^{-t^2/2}\mathrm{dt}
+$$
+
+若已知概率 $p = 0.95$ ，则对应的下分位点可用函数 `qnorm()` 计算。
+
+```{r}
+qnorm(p = 0.95, mean = 0, sd = 1)
+```
+:::
+
 #### 方差未知
 
 $$
@@ -150,6 +164,25 @@ qt(p = 1 - 0.05, df = n - 1)
 
 ::: callout-note
 英国统计学家 William Sealy Gosset (1876-1937) 于 1908 年在杂志 《Biometrics》 上以笔名 Student 发表论文《The Probable Error of a Mean》[@Gosset1908]，论文中展示了独立同正态分布的样本 $x_1, \ldots, x_n \stackrel{i.i.d}{\sim} \mathcal{N}(\mu,\sigma^2)$ 的样本方差 $s^2$ 和样本标准差 $s$ 的抽样分布，根据均值和标准差不相关的性质导出 t 分布，宣告 t 分布的诞生，因其在小样本领域的突出贡献，W. S. Gosset 进入世纪名人录 [@Heyde2001]。
+
+```{r}
+#| label: tbl-t-quantile
+#| tbl-cap: $t$ 分布的分位数表
+#| echo: false
+
+vec_prob <- c(
+  0.75, 0.80, 0.90, 0.95,
+  0.975, 0.99, 0.995, 0.999
+)
+vec_df <- 1:10
+
+tmp <- mapply(FUN = qt,
+  p = vec_prob,
+  MoreArgs = list(df = vec_df), SIMPLIFY = TRUE
+)
+row.names(tmp) <- vec_df
+knitr::kable(tmp, row.names = TRUE, col.names = vec_prob, digits = 4)
+```
 :::
 
 ### 正态总体方差检验
@@ -181,6 +214,41 @@ qchisq(p = 1 - 0.05, df = n -1)
 1 - pchisq(q = chi, df = n -1)
 ```
 
+::: callout-important
+R 软件提供很多统计分布的计算，因此，不再需要查分位数表，现算即可。计算自由度为 $n$ 概率为 $p$ 的 $\chi^2$ 分布的分位数 $\chi^2_p(n)$ ，即
+
+$$
+P(\chi^2(n) \leq \chi^2_p(n)) = p
+$$
+
+若已知自由度为 1 ，概率为 0.05，则可借助分位数函数 `qchisq()` 计算对应的（下）分位点。
+
+```{r}
+qchisq(p = 0.05, df = 1)
+```
+
+同理，也可以获得 $\chi^2$ 分布的分位数 @tbl-chisq-quantile ，计算出来的分位数保留 4 位小数。
+
+```{r}
+#| label: tbl-chisq-quantile
+#| tbl-cap: $\chi^2$ 分布的分位数表
+#| echo: false
+
+vec_prob <- c(
+  0.005, 0.01, 0.025, 0.05, 0.1,
+  0.9, 0.95, 0.975, 0.99, 0.995
+)
+vec_df <- 1:10
+
+tmp <- mapply(FUN = qchisq,
+  p = vec_prob,
+  MoreArgs = list(df = vec_df), SIMPLIFY = TRUE
+)
+row.names(tmp) <- vec_df
+knitr::kable(tmp, row.names = TRUE, col.names = vec_prob, digits = 4)
+```
+:::
+
 ### 总体未知均值检验
 
 有了均值和方差，为什么还要位置参数和尺度参数？为了更一般地描述问题，扩展范围。特别是在总体分布未知或知之甚少的情况下做检验，不再仅限于均值和方差这样的特征量。
@@ -531,7 +599,7 @@ flowchart LR
   B1 --> C1(均值检验)
   C1 --> D2(方差相等) --> E2(F 检验)
   C1 --> D3(方差不等) --> E3(F 检验)
-  B1 --> C2(方差检验) --> E4(Bartlett 检验)
+  B1 --> C2(方差检验) --> E4(Hartley 检验\n Bartlett 检验\n 修正的 Bartlett 检验\n Levene 检验)
   B2 --> C3(均值检验) --> E5(Kruskal-Wallis 秩和检验\n Friedman 秩和检验\n Quade 检验)
   B2 --> C4(方差检验) --> E7(Fligner-Killeen 检验)
 ```
@@ -644,7 +712,16 @@ logLik(fit_gls)
 
 ### 正态总体方差检验
 
-后面总体分布未知的情况下的方差检验也都可以用在这里。
+总体服从正态分布，有四种常见的参数检验方法：
+
+1.  Hartley 检验：各组样本量必须相等。
+2.  Bartlett 检验：各组样本量可以相等或不等，但每个组的样本量必须不低于 5。
+3.  修正的 Bartlett 检验：在样本量较大或较小、相等或不等场合都可使用。
+4.  Levene 检验：相当于单因素组间方差分析，相比于 Bartlett 检验，Levene 检验更加稳健。
+
+::: callout-tip
+在总体分布未知的情况下，检验方差齐性的非参数方法也都可以用在这里。
+:::
 
 设 $x_1,\cdots,x_{n_1}$ 是来自总体 $\mathcal{N}(\mu_1,\sigma_1^2)$ 的样本，设 $y_1,\cdots,y_{n_2}$ 是来自总体 $\mathcal{N}(\mu_2,\sigma_2^2)$ 的样本，设 $z_1,\cdots,z_{n_3}$ 是来自总体 $\mathcal{N}(\mu_3,\sigma_3^2)$ 的样本。
 
@@ -652,7 +729,7 @@ $$
 \sigma_1^2 = \sigma_2^2 = \sigma_3^2 \quad vs. \quad \sigma_1^2,\sigma_2^2,\sigma_3^2 \quad  \text{不全相等}
 $$
 
-Bartlett （巴特利特）检验 `bartlett.test()` 要求总体的分布为正态分布，检验各个组的方差是否有显著性差异，即方差齐性检验，属于参数检验，适用于多个样本的情况。相比于 Bartlett 检验，Levene 检验更加稳健。
+Bartlett （巴特利特）检验 `bartlett.test()` 要求总体的分布为正态分布，检验各个组的方差是否有显著性差异，即方差齐性检验，属于参数检验，适用于多个样本的情况。
 
 ```{r}
 # 三样本
@@ -1018,11 +1095,11 @@ apply(Wish, MARGIN = 1:2, var)
 | [W. Kruskal](https://en.wikipedia.org/wiki/William_Kruskal)              | 美国       | 1919-10-10 | 2005-04-21 | 85   | Kruskal-Wallis 检验            |
 | [George E. P. Box](https://en.wikipedia.org/wiki/George_E._P._Box)       | 英国、美国 | 1919-10-18 | 2013-03-28 | 93   | Box-Pierce 检验                |
 | [C. R. Rao](https://en.wikipedia.org/wiki/C._R._Rao)                     | 印度、美国 | 1920-09-10 | 2023-08-22 | 102  | Score 检验                     |
-| [M. Wilk](https://en.wikipedia.org/wiki/Martin_Wilk)                     | 加拿大     | 1922-12-18 | 2013-02-19 | 90   | Shapiro-Wilk 正态性检验        |
+| [M. Wilk](https://en.wikipedia.org/wiki/Martin_Wilk)                     | 加拿大     | 1922-12-18 | 2013-02-19 | 90   | Shapiro-Wilk 检验              |
 | [J. Durbin](https://en.wikipedia.org/wiki/James_Durbin)                  | 英国       | 1923-06-30 | 2012-06-23 | 88   | Durbin 检验                    |
 | [L. Le Cam](https://en.wikipedia.org/wiki/Lucien_Le_Cam)                 | 法国       | 1924-11-18 | 2000-04-25 | 75   | 渐近理论                       |
 | [H. Lilliefors](https://en.wikipedia.org/wiki/Hubert_Lilliefors)         | 美国       | 1928-06-14 | 2008-02-23 | 80   | Lilliefors 检验                |
-| [S. S. Shapiro](https://en.wikipedia.org/wiki/Samuel_Sanford_Shapiro)    | 美国       | 1930-07-13 | \-         | 93   | Shapiro-Wilk 正态性检验        |
+| [S. S. Shapiro](https://en.wikipedia.org/wiki/Samuel_Sanford_Shapiro)    | 美国       | 1930-07-13 | \-         | 93   | Shapiro-Wilk 检验              |
 
 : 对假设检验理论有重要贡献的学者 {#tbl-statistican tbl-colwidths="\[20,15,15,15,7,28\]"}
 
diff --git a/data/scotland/scotland.dbf b/data/scotland/scotland.dbf
new file mode 100755
index 00000000..44ee8812
Binary files /dev/null and b/data/scotland/scotland.dbf differ
diff --git a/data/scotland/scotland.shp b/data/scotland/scotland.shp
new file mode 100755
index 00000000..d826e4fb
Binary files /dev/null and b/data/scotland/scotland.shp differ
diff --git a/data/scotland/scotland.shx b/data/scotland/scotland.shx
new file mode 100755
index 00000000..7c787ada
Binary files /dev/null and b/data/scotland/scotland.shx differ
diff --git a/hierarchical-normal-models.qmd b/hierarchical-normal-models.qmd
new file mode 100644
index 00000000..37697033
--- /dev/null
+++ b/hierarchical-normal-models.qmd
@@ -0,0 +1,572 @@
+# 分层正态模型 {#sec-hierarchical-normal-models}
+
+```{r}
+#| echo: false
+
+source("_common.R")
+```
+
+以分层正态模型介绍 **rstan** 包的用法
+
+```{r}
+#| message: false
+
+library(StanHeaders)
+library(ggplot2)
+library(rstan)
+rstan_options(auto_write = TRUE)
+# 如果CPU和内存足够，设置成与马尔科夫链一样多
+options(mc.cores = max(c(floor(parallel::detectCores() / 2), 1L)))
+
+custom_colors <- c(
+  "#4285f4", # GoogleBlue
+  "#34A853", # GoogleGreen
+  "#FBBC05", # GoogleYellow
+  "#EA4335"  # GoogleRed
+)
+rstan_ggtheme_options(
+  panel.background = element_rect(fill = "white"),
+  legend.position = "top"
+)
+rstan_gg_options(
+  fill = "#4285f4", color = "white",
+  pt_color = "#EA4335", chain_colors = custom_colors
+)
+```
+
+## 8schools 数据 {#sec-eight-schools}
+
+分层正态模型
+
+$$
+\begin{aligned}
+y_j &\sim \mathcal{N}(\theta_j,\sigma_j) \quad
+\theta_j = \mu + \tau * \eta_j \\
+\theta_j &\sim \mathcal{N}(\mu, \tau) \quad
+\eta_j \sim \mathcal{N}(0,1)
+\end{aligned}
+$$
+
+### 拟合模型
+
+用 rstan 包来拟合模型
+
+```{r}
+# 编译模型
+eight_schools_fit <- stan(
+  model_name = "eight_schools",
+  # file = "code/stan/8schools.stan",
+  model_code = "
+  // saved as 8schools.stan
+  data {
+    int<lower=0> J;         // number of schools
+    real y[J];              // estimated treatment effects
+    real<lower=0> sigma[J]; // standard error of effect estimates
+  }
+  parameters {
+    real mu;                // population treatment effect
+    real<lower=0> tau;      // standard deviation in treatment effects
+    vector[J] eta;          // unscaled deviation from mu by school
+  }
+  transformed parameters {
+    vector[J] theta = mu + tau * eta;        // school treatment effects
+  }
+  model {
+    target += normal_lpdf(eta | 0, 1);       // prior log-density
+    target += normal_lpdf(y | theta, sigma); // log-likelihood
+  }
+  ",
+  data = list( # 观测数据
+    J = 8,
+    y = c(28, 8, -3, 7, -1, 1, 18, 12),
+    sigma = c(15, 10, 16, 11, 9, 11, 10, 18)
+  ),
+  warmup = 1000, # 每条链预处理迭代次数
+  iter = 2000,   # 每条链总迭代次数
+  chains = 2,    # 马尔科夫链的数目
+  cores = 2,     # 指定 CPU 核心数，可以给每条链分配一个
+  verbose = FALSE, # 不显示迭代的中间过程
+  refresh = 0,     # 不显示采样的进度
+  seed = 20190425  # 设置随机数种子，不要使用 set.seed() 函数
+)
+```
+
+### 模型输出
+
+```{r}
+print(eight_schools_fit, digits = 1)
+```
+
+提取任意一个参数的结果，如查看参数 $\tau$ 的 95% 置信区间。
+
+```{r}
+print(eight_schools_fit, pars = "tau", probs = c(0.025, 0.975))
+```
+
+从迭代抽样数据获得与 `print(fit)` 一样的结果。以便后续对原始采样数据做任意的进一步分析。rstan 包扩展泛型函数 `summary()` 以支持对 stanfit 数据对象汇总，输出各个参数分链条和合并链条的后验分布结果。
+
+### 操作数据
+
+合并四条马氏链的结果
+
+```{r}
+eight_schools_sim <- extract(eight_schools_fit, permuted = TRUE)
+```
+
+返回的结果是一个列表
+
+```{r}
+str(eight_schools_sim)
+class(eight_schools_sim)
+```
+
+计算参数 $\eta,\theta$ 的均值
+
+```{r}
+apply(eight_schools_sim$eta, 2, mean)
+apply(eight_schools_sim$theta, 2, mean)
+```
+
+计算参数 $\eta,\theta$ 的分位点
+
+```{r}
+t(apply(eight_schools_sim$eta, 2, quantile, probs = c(2.5, 25, 50, 75, 97.5) / 100))
+t(apply(eight_schools_sim$theta, 2, quantile, probs = c(2.5, 25, 50, 75, 97.5) / 100))
+```
+
+计算参数 $\mu,\tau$ 的均值
+
+```{r}
+lapply(eight_schools_sim["mu"], mean)
+lapply(eight_schools_sim["tau"], mean)
+lapply(eight_schools_sim["lp__"], mean)
+```
+
+计算参数 $\mu,\tau$ 的分位点
+
+```{r}
+lapply(eight_schools_sim["mu"], quantile, probs = c(2.5, 25, 50, 75, 97.5) / 100)
+lapply(eight_schools_sim["tau"], quantile, probs = c(2.5, 25, 50, 75, 97.5) / 100)
+lapply(eight_schools_sim["lp__"], quantile, probs = c(2.5, 25, 50, 75, 97.5) / 100)
+```
+
+### 采样诊断
+
+获取马尔科夫链迭代点列数据
+
+```{r}
+eight_schools_sim <- extract(eight_schools_fit, permuted = FALSE)
+```
+
+`eight_schools_sim` 是一个三维数组，1000（次迭代）\* 4 （条链）\* 19（个参数）。如果 `permuted = TRUE` 则会合并四条马氏链的迭代结果，变成一个列表。
+
+```{r}
+# 数据类型
+class(eight_schools_sim)
+# 1000（次迭代）* 4 （条链）* 19（个参数）
+str(eight_schools_sim)
+```
+
+提取参数 $\mu$ 的四条迭代点列
+
+```{r}
+eight_schools_mu_sim <- eight_schools_sim[, , "mu"]
+```
+
+```{r}
+#| label: fig-8schools-mu-base
+#| fig-cap: 参数 $\mu$ 的迭代轨迹
+#| fig-showtext: true
+#| par: true
+
+matplot(eight_schools_mu_sim,
+        xlab = "Iteration", ylab = expression(mu),
+        type = "l", lty = "solid", col = custom_colors
+)
+abline(h = apply(eight_schools_mu_sim, 2, mean), col = custom_colors)
+legend("bottomleft",
+       legend = paste0("chain:", 1:2), box.col = "white", inset = 0.01,
+       lty = "solid", horiz = TRUE, col = custom_colors
+)
+```
+
+或者使用 rstan 提供的 `traceplot` 函数或者 `stan_trace` 函数，rstan 大量依赖 ggplot2 绘图，所以如果你熟悉 GGplot2 可以很方便地定制属于自己的风格，除了 rstan 提供的 `rstan_ggtheme_options` 和 `rstan_gg_options` 两个函数外，还可以使用 ggplot2 自带的大量配置选项和主题，如 `theme_minimal` 主题，因为 `stan_trace`等作图函数返回的是一个 ggplot 对象。
+
+```{r}
+#| label: fig-8schools-mu-gg
+#| fig-cap: 马氏链的迭代序列
+#| fig-showtext: true
+
+stan_trace(eight_schools_fit, pars = "mu") +
+  theme_minimal() +
+  labs(x = "Iteration", y = expression(mu))
+```
+
+序列的自相关图，类似地，我们这里也使用 `stan_ac` 函数绘制自相关图
+
+```{r}
+#| label: fig-8schools-mu-acf
+#| fig-cap: 马氏链的自相关图
+#| fig-showtext: true
+#| fig-height: 4
+
+stan_ac(eight_schools_fit, pars = "mu", separate_chains = TRUE, color = "white") +
+  theme_minimal()
+```
+
+### 后验分布
+
+可以用 `stan_hist` 函数绘制参数 $\mu$ 的后验分布图，它没有 `separate_chains` 参数，所以不能分链条绘制
+
+```{r}
+#| label: fig-8schools-mu-hist
+#| fig-cap: 参数 $\mu$ 的后验分布
+#| fig-showtext: true
+
+stan_hist(eight_schools_fit, pars = "mu", bins = 30) + theme_minimal()
+```
+
+参数 $\mu$ 和 $\tau$ 的散点图
+
+```{r}
+#| label: fig-8schools-mu-tau
+#| fig-cap: 参数 $\mu$ 和 $\tau$ 的联合后验分布
+#| fig-showtext: true
+
+stan_scat(eight_schools_fit, pars = c("mu","tau")) + theme_minimal()
+```
+
+参数 $\mu$ 的后验概率密度分布图
+
+```{r}
+#| label: fig-8schools-mu-dens
+#| fig-cap: 参数 $\mu$ 的后验概率密度分布图
+#| fig-showtext: true
+
+stan_dens(eight_schools_fit, pars = "mu", separate_chains = TRUE) +
+  theme_minimal() +
+  labs(x = expression(mu), y = "Density")
+```
+
+**bayesplot** 包的函数 `mcmc_pairs()` 以矩阵图展示多个参数的分布。
+
+```{r}
+#| label: fig-parameters-posterior
+#| fig-cap: 参数 $\mu$，$\tau$ 和 $\mathrm{lp\_\_}$ 的后验分布图
+#| fig-showtext: true
+
+bayesplot::mcmc_pairs(eight_schools_fit, pars = c("mu", "tau", "lp__"))
+```
+
+### 模型导出
+
+rstan 包还支持从外部磁盘读取代码
+
+```{r}
+#| eval: false
+
+fit <- stan(file = 'code/stan/8schools.stan', ...)
+
+schools_dat <- read_rdump('data/8schools.rdump')
+source('data/8schools.rdump')
+```
+
+### 其它 R 包
+
+接下来，分别用 nlme 和 lme4 包拟合模型。
+
+```{r}
+# 成绩
+y <- c(28, 8, -3, 7, -1, 1, 18, 12)
+# 标准差
+sigma <- c(15, 10, 16, 11, 9, 11, 10, 18)
+# 学校编号
+g <- 1:8
+```
+
+首先，调用 **nlme** 包的函数 `lme()` 拟合模型。
+
+```{r}
+library(nlme)
+fit_lme <- lme(y ~ 1, random = ~ 1 | g, weights = varFunc(~ 1/sigma^2))
+summary(fit_lme)
+```
+
+接着，采用 lme4 包拟合模型，发现 lme4 包可以获得与 nlme 包一致的结果。
+
+```{r}
+control <- lme4::lmerControl(
+  check.conv.singular = "ignore",
+  check.nobs.vs.nRE = "ignore",
+  check.nobs.vs.nlev = "ignore"
+)
+fit_lme4 <- lme4::lmer(y ~ 1 + (1 | g), weights = 1 / sigma^2, control = control, REML = FALSE)
+summary(fit_lme4)
+```
+
+最后，使用 blme 包 [@Chung2013] ，blme 包基于 lme4 包，结果完全一致。
+
+```{r}
+### Example using a residual variance prior ###
+# This is the "eight schools" data set; 
+# the mode should be at the boundary of the space.
+
+fit_blme <- blme::blmer(
+  y ~ 1 + (1 | g),
+  control = control, REML = FALSE,
+  resid.prior = point, cov.prior = NULL,
+  weights = 1 / sigma^2
+)
+summary(fit_blme)
+```
+
+## rats 数据 {#sec-thirty-rats}
+
+rats 数据最早来自 @gelfand1990 ，记录 30 只小鼠每隔一周的重量，一共进行了 5 周。第一次记录是小鼠第 8 天的时候，第二次测量记录是第 15 天的时候，一直持续到第 36 天。下面在 R 环境中准备数据。
+
+```{r}
+# 总共 30 只老鼠
+N <- 30
+# 总共进行 5 周
+T <- 5
+# 小鼠重量
+y <- structure(c(
+  151, 145, 147, 155, 135, 159, 141, 159, 177, 134,
+  160, 143, 154, 171, 163, 160, 142, 156, 157, 152, 154, 139, 146,
+  157, 132, 160, 169, 157, 137, 153, 199, 199, 214, 200, 188, 210,
+  189, 201, 236, 182, 208, 188, 200, 221, 216, 207, 187, 203, 212,
+  203, 205, 190, 191, 211, 185, 207, 216, 205, 180, 200, 246, 249,
+  263, 237, 230, 252, 231, 248, 285, 220, 261, 220, 244, 270, 242,
+  248, 234, 243, 259, 246, 253, 225, 229, 250, 237, 257, 261, 248,
+  219, 244, 283, 293, 312, 272, 280, 298, 275, 297, 350, 260, 313,
+  273, 289, 326, 281, 288, 280, 283, 307, 286, 298, 267, 272, 285,
+  286, 303, 295, 289, 258, 286, 320, 354, 328, 297, 323, 331, 305,
+  338, 376, 296, 352, 314, 325, 358, 312, 324, 316, 317, 336, 321,
+  334, 302, 302, 323, 331, 345, 333, 316, 291, 324
+), .Dim = c(30, 5))
+# 第几天
+x <- c(8.0, 15.0, 22.0, 29.0, 36.0)
+xbar <- 22.0
+```
+
+重复测量的小鼠重量数据 rats 如下 @tbl-rats 所示。
+
+```{r}
+#| label: tbl-rats
+#| tbl-cap: 小鼠重量数据（部分）
+#| echo: false
+
+rownames(y) <- 1:30
+knitr::kable(head(y), col.names = paste("第", c(8, 15, 22, 29, 36), "天"), row.names = TRUE)
+```
+
+小鼠重量数据的分布情况见下 @fig-rats ，由图可以假定 30 只小鼠的重量服从正态分布。
+
+```{r}
+#| label: fig-rats
+#| fig-cap: 30 只小鼠 5 次测量的数据
+#| fig-showtext: true
+#| par: true
+#| echo: false
+#| fig-width: 5
+#| fig-height: 4.5
+
+matplot(y, xlab = "小鼠编号", ylab = "小鼠重量")
+```
+
+由 @fig-rats-growth 可见， 30 只小鼠的重量增长趋势呈现一种线性趋势。
+
+```{r}
+#| label: fig-rats-growth
+#| fig-cap: 30 只小鼠 5 次测量的数据
+#| fig-showtext: true
+#| par: true
+#| echo: false
+#| fig-width: 5
+#| fig-height: 4.5
+
+matplot(t(y), xlab = "测量次数", ylab = "小鼠重量")
+```
+
+### rstan {#sec-rats-rstan}
+
+初始化模型参数，设置采样算法的参数。
+
+```{r}
+# 迭代链
+chains <- 4
+# 迭代次数
+iter <- 1000
+# 初始值
+init <- rep(list(list(
+  alpha = rep(250, 30), beta = rep(6, 30),
+  alpha_c = 150, beta_c = 10,
+  tausq_c = 1, tausq_alpha = 1,
+  tausq_beta = 1
+)), chains)
+```
+
+接下来，基于重复测量数据，建立线性生长曲线模型：
+
+$$
+\begin{aligned}
+\alpha_c &\sim \mathcal{N}(0,100) \quad \beta_c  \sim \mathcal{N}(0,100) \\
+\tau^2_c &\sim \mathrm{inv\_gamma}(0.001, 0.001) \\
+\tau^2_{\alpha} &\sim \mathrm{inv\_gamma}(0.001, 0.001) \\
+\tau^2_{\beta}  &\sim \mathrm{inv\_gamma}(0.001, 0.001) \\
+\alpha &\sim \mathcal{N}(\alpha_c, \tau_{\alpha})  \quad
+\beta  \sim \mathcal{N}(\beta_c, \tau_{\beta}) \\
+y_{nt} &\sim \mathcal{N}(\alpha_n + \beta_n * (x_t - \bar{x}), \tau_c) \\
+& n = 1,2,\ldots,N \quad t = 1,2,\ldots,T
+\end{aligned}
+$$
+
+其中， $\alpha_c,\beta_c,\tau_c,\tau_{\alpha},\tau_{\beta}$ 为无信息先验，$N = 30$ 和 $T = 5$ 分别表示实验中的小鼠数量和测量次数，下面采用 Stan 编码、编译、采样和拟合模型。
+
+```{r}
+rats_fit <- stan(
+  model_name = "rats",
+  model_code = "
+  data {
+    int<lower=0> N;
+    int<lower=0> T;
+    vector[T] x;
+    matrix[N,T] y;
+    real xbar;
+  }
+  parameters {
+    vector[N] alpha;
+    vector[N] beta;
+
+    real alpha_c;
+    real beta_c;          // beta.c in original bugs model
+
+    real<lower=0> tausq_c;
+    real<lower=0> tausq_alpha;
+    real<lower=0> tausq_beta;
+  }
+  transformed parameters {
+    real<lower=0> tau_c;       // sigma in original bugs model
+    real<lower=0> tau_alpha;
+    real<lower=0> tau_beta;
+
+    tau_c = sqrt(tausq_c);
+    tau_alpha = sqrt(tausq_alpha);
+    tau_beta = sqrt(tausq_beta);
+  }
+  model {
+    alpha_c ~ normal(0, 100);
+    beta_c ~ normal(0, 100);
+    tausq_c ~ inv_gamma(0.001, 0.001);
+    tausq_alpha ~ inv_gamma(0.001, 0.001);
+    tausq_beta ~ inv_gamma(0.001, 0.001);
+    alpha ~ normal(alpha_c, tau_alpha); // vectorized
+    beta ~ normal(beta_c, tau_beta);  // vectorized
+    for (n in 1:N)
+      for (t in 1:T)
+        y[n,t] ~ normal(alpha[n] + beta[n] * (x[t] - xbar), tau_c);
+  }
+  generated quantities {
+    real alpha0;
+    alpha0 = alpha_c - xbar * beta_c;
+  }
+  ",
+  data = list(N = N, T = T, y = y, x = x, xbar = xbar),
+  chains = chains, init = init, iter = iter,   
+  verbose = FALSE, refresh = 0, seed = 20190425
+)
+```
+
+模型输出结果如下：
+
+```{r}
+print(rats_fit, pars = c("alpha", "beta"), include = FALSE, digits = 1)
+```
+
+对于分量众多的参数向量，比较适合用岭线图展示后验分布，下面调用 bayesplot 包绘制参数向量 $\alpha,\beta$ 的后验分布。
+
+```{r}
+#| label: fig-rats-alpha
+#| fig-cap: 参数 $\alpha$ 的后验分布
+#| fig-showtext: true
+#| fig-width: 6
+#| fig-height: 8
+
+# plot(rats_fit, pars = "alpha", show_density = TRUE, ci_level = 0.8, outer_level = 0.95)
+bayesplot::mcmc_areas_ridges(rats_fit, pars = paste0("alpha", "[", 1:30, "]"))
+```
+
+```{r}
+#| label: fig-rats-beta
+#| fig-cap: 参数 $\beta$ 的后验分布
+#| fig-showtext: true
+#| fig-width: 6
+#| fig-height: 8
+
+# plot(rats_fit, pars = "beta", ci_level = 0.8, outer_level = 0.95)
+bayesplot::mcmc_areas_ridges(rats_fit, pars = paste0("beta", "[", 1:30, "]"))
+```
+
+### nlme {#sec-rats-nlme}
+
+**nlme** 包适合将长格式的数据，因此，先将小鼠数据整理成长格式。
+
+```{r}
+weight <- c(
+  151, 145, 147, 155, 135, 159, 141, 159, 177, 134,
+  160, 143, 154, 171, 163, 160, 142, 156, 157, 152, 154, 139, 146,
+  157, 132, 160, 169, 157, 137, 153, 199, 199, 214, 200, 188, 210,
+  189, 201, 236, 182, 208, 188, 200, 221, 216, 207, 187, 203, 212,
+  203, 205, 190, 191, 211, 185, 207, 216, 205, 180, 200, 246, 249,
+  263, 237, 230, 252, 231, 248, 285, 220, 261, 220, 244, 270, 242,
+  248, 234, 243, 259, 246, 253, 225, 229, 250, 237, 257, 261, 248,
+  219, 244, 283, 293, 312, 272, 280, 298, 275, 297, 350, 260, 313,
+  273, 289, 326, 281, 288, 280, 283, 307, 286, 298, 267, 272, 285,
+  286, 303, 295, 289, 258, 286, 320, 354, 328, 297, 323, 331, 305,
+  338, 376, 296, 352, 314, 325, 358, 312, 324, 316, 317, 336, 321,
+  334, 302, 302, 323, 331, 345, 333, 316, 291, 324
+)
+rats <- rep(1:30, each = 5)
+days <- rep(c(8, 15, 22, 29, 36), each = 30)
+rats_data <- data.frame(weight = weight, rats = rats, days = days)
+```
+
+小鼠的重量随时间增长，不同小鼠的情况又会有所不同。可用随机效应模型表示生长曲线模型，下面加载 **nlme** 包调用函数 `lme()` 拟合该模型。
+
+```{r}
+library(nlme)
+rats_lme <- lme(data = rats_data, fixed = weight ~ days, random = ~ 1 | rats)
+summary(rats_lme)
+```
+
+模型输出结果中，固定效应中的截距项 `(Intercept)` 对应 106.56762，斜率 `days` 对应 6.18571。Stan 模型中截距参数 `alpha0` 的后验估计是 106.332，斜率参数 `beta_c` 的后验估计是 6.188。对比 Stan 和 **nlme** 包的拟合结果，可以发现贝叶斯和频率方法的结果是非常接近的。
+
+### lme4
+
+当采用 **lme4** 包拟合数据的时候，发现参数取值落在参数空间的边界上，除此之外，输出结果与 **nlme** 包几乎相同。
+
+```{r}
+rats_lme4 <- lme4::lmer(weight ~ days + (1 | rats), data = rats_data)
+summary(rats_lme4)
+```
+
+**lme4** 包依赖 **nloptr** 包提供数值优化求解器，也做了更多的矩阵奇异性检查，对于复杂的重复测量模型，往往会发生奇异，需要适当降低复杂性，更多详情见 **lme4** 包的帮助文档 `help('isSingular')` 。
+
+### blme
+
+blme 包基于 lme4 包
+
+```{r}
+control <- lme4::lmerControl(
+  check.conv.singular = "ignore",
+  check.nobs.vs.nRE = "ignore",
+  check.nobs.vs.nlev = "ignore"
+)
+rats_blme <- blme::blmer(
+  weight ~ days + (1 | rats),
+  data = rats_data,
+  resid.prior = point, cov.prior = NULL,
+  control = control, REML = FALSE
+)
+summary(rats_blme)
+```
diff --git a/references.bib b/references.bib
index af6b9eb9..132a0633 100755
--- a/references.bib
+++ b/references.bib
@@ -1674,5 +1674,110 @@ @article{shapiro1965
 	volume = {52},
 	number = {3-4},
 	doi = {10.1093/biomet/52.3-4.591},
-	url = {https://doi.org/10.1093/biomet/52.3-4.591}
+}
+
+@article{blangiardo2013,
+  title = {Spatial and spatio-temporal models with {R-INLA}},
+  author = {Blangiardo, Marta and Cameletti, Michela and Baio, Gianluca and Rue, {Håvard}},
+  year = {2013},
+  month = {12},
+  date = {2013-12-01},
+  journal = {Spatial and Spatio-temporal Epidemiology},
+  pages = {39--55},
+  volume = {7},
+  doi = {10.1016/j.sste.2013.07.003},
+}
+
+@book{moraga2020,
+  title = {Geospatial health data: modeling and visualization with R-INLA and Shiny},
+  author = {Moraga, Paula},
+  year = {2020},
+  publisher = {Chapman and Hall/CRC},
+  address = {Boca Raton, Florida},
+}
+
+@article{donegan2022,
+  title = {{geostan}: An R package for Bayesian spatialanalysis},
+  author = {Donegan, Connor},
+  year = {2022},
+  month = {11},
+  date = {2022-11-16},
+  journal = {Journal of Open Source Software},
+  pages = {4716},
+  volume = {7},
+  number = {79},
+  doi = {10.21105/joss.04716},
+}
+
+@article{morris2019,
+  title = {Bayesian hierarchical spatial models: Implementing the Besag York Mollié model in stan},
+  author = {Morris, Mitzi and Wheeler-Martin, Katherine and Simpson, Dan and Mooney, Stephen J. and Gelman, Andrew and DiMaggio, Charles},
+  year = {2019},
+  month = {11},
+  date = {2019-11},
+  journal = {Spatial and Spatio-temporal Epidemiology},
+  pages = {100301},
+  volume = {31},
+  doi = {10.1016/j.sste.2019.100301},
+}
+
+@article{cabral2022,
+  title = {Controlling the Flexibility of Non-Gaussian Processes Through Shrinkage Priors},
+  author = {Cabral, Rafael and Bolin, David and Rue, {Håvard}},
+  year = {2022},
+  month = {01},
+  date = {2022-01},
+  journal = {Bayesian Analysis},
+  pages = {1--24},
+  volume = {-1},
+  number = {-1},
+  doi = {10.1214/22-BA1342},
+}
+
+@article{bivand2001,
+	title = {More on Spatial Data Analysis},
+	author = {Bivand, Roger},
+	year = {2001},
+	month = {09},
+	date = {2001-09-01},
+	journal = {R News},
+	pages = {13--17},
+	volume = {1},
+	number = {3},
+	url = {https://www.r-project.org/doc/Rnews/Rnews_2001-3.pdf}
+}
+
+@Article{Tobler1970,
+  title = {A computer movie simulating urban growth in the Detroit region},
+  author = {Waldo Tobler},
+  journal = {Economic Geography},
+  year = {1970},
+  volume = {46},
+  number = {Supplement},
+  pages = {234--240},
+  doi = {10.2307/143141},
+}
+
+@article{gelfand1990,
+  title = {Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling},
+  author = {Gelfand, Alan E. and Hills, Susan E. and Racine-Poon, Amy and Smith, Adrian F. M.},
+  year = {1990},
+  date = {1990},
+  journal = {Journal of the American Statistical Association},
+  pages = {972--985},
+  volume = {85},
+  number = {412},
+  doi = {10.2307/2289594},
+}
+
+@Article{Chung2013,
+  title = {A nondegenerate penalized likelihood estimator for variance parameters in multilevel models},
+  author = {Yeojin Chung and Sophia Rabe-Hesketh and Vincent Dorie and Andrew Gelman and Jingchen Liu},
+  year = {2013},
+  journal = {Psychometrika},
+  volume = {78},
+  number = {4},
+  pages = {685--709},
+  publisher = {Springer},
+  doi = {10.1007/s11336-013-9328-2},
 }
diff --git a/regression-and-correlation.qmd b/regression-and-correlation.qmd
index 1e139d94..e915188b 100644
--- a/regression-and-correlation.qmd
+++ b/regression-and-correlation.qmd
@@ -161,8 +161,8 @@ str(state_x77)
 ```{r}
 #| label: fig-state-x77-scatter
 #| fig-cap: "预期寿命与人均收入的关系图"
-#| fig-width: 4.5
-#| fig-height: 3.5
+#| fig-width: 6
+#| fig-height: 4.5
 #| code-fold: true
 #| echo: !expr knitr::is_html_output()
 #| fig-showtext: true