From 9497342ef7333eed7cc490ca93a5291a1ffd5c81 Mon Sep 17 00:00:00 2001 From: Xiangyun Huang Date: Sun, 8 Oct 2023 14:31:41 +0800 Subject: [PATCH] =?UTF-8?q?=E5=88=86=E5=B1=82=E6=AD=A3=E6=80=81=E6=A8=A1?= =?UTF-8?q?=E5=9E=8B=E4=B8=8E=20Stan=20(#116)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit * 正态总体方差齐性检验方法的适用范围 * 制作概率分布的分位数表 * 区域数据分析 * 分层正态模型 * Fix Duplicate chunk label * 对比结果 * 函数 mcmc_pairs 替代函数 pairs * 补充参考文献 * tweak * summary 函数 * Fix typo * 添加更多的说明 * tweak table * 相比于 nlme 包,lme4 包和 blme 包提供更多严格的检查 --- _quarto.yml | 2 + analyze-areal-data.qmd | 137 ++++++++ common-statistical-tests.qmd | 87 ++++- data/scotland/scotland.dbf | Bin 0 -> 6817 bytes data/scotland/scotland.shp | Bin 0 -> 60748 bytes data/scotland/scotland.shx | Bin 0 -> 548 bytes hierarchical-normal-models.qmd | 572 +++++++++++++++++++++++++++++++++ references.bib | 107 +++++- regression-and-correlation.qmd | 4 +- 9 files changed, 901 insertions(+), 8 deletions(-) create mode 100644 analyze-areal-data.qmd create mode 100755 data/scotland/scotland.dbf create mode 100755 data/scotland/scotland.shp create mode 100755 data/scotland/scotland.shx create mode 100644 hierarchical-normal-models.qmd 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 0000000000000000000000000000000000000000..44ee8812295e4aacf2f78c3513105de55ecec488 GIT binary patch literal 6817 zcmZ`-%Wm676m?M)?I!J_yRP^HCiv82Y0FL;#CC$XK$Zb5Q5GSZ3@FNu|Gwwkk!0P= z(JUlt=4j5`_Z|NF{r7(+XJ=;@zx4PVj{7g)KVG@%8NYrrkh_c9>)n*BzccImKV$li zOMm~V0aw+g@2WY;W&IDmeD?WaQ+Chg;?(-}&mZNye_OZv|CkrcrQ1{Xx97hp`l{b9 z%KiTK|2J*(xC6NJ|D_N3E{KyL2=2H2s_g2bS>Q*2xiC^gW*wA|IWghHLgEpYCSf!> zUI>CL9A(EdQ4put;I62;hP$~Dr+!XCIrtrivEB$%5je+THa=beI-Hv0#ZzqXv3V}L zrrd0}BmO+~a}vtI?>x+pw+3AhjD<5%Qc=!vs-Y2eds#Hg<@x1yzA8jqzitCUF;a3) zLM0A4MYrQHH@Z#2RLwxQI9za9F3P5zukgqbkNt8I%0ceax*ggxnrba!^iqhTiBPr<#WR);XdeiLt>&^SIsFl6xkxms6lv<}u^wggr;$SQ#3- z4SXo4C9by3qI5EnGmpHS0_8e6r$add96uk0nc+;s#E2d<*zT@__to-18s}b4fdVZq 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