多元数据直观表示+线性回归分析

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1 多元数据直观表示

1.1 各省消费项目均值条形图

省份过多，各省的名称均不能全部显示

barplot(apply(data,1,mean))#按行做均值条形图

将横轴左边旋转90度，各省的名称均可显示

barplot(apply(data,1,mean),las=3)#按行做均值条形图

利用ggplot2包作图较为美观

data %>%
  mutate(Average_Consumption = rowMeans(select(., -1), na.rm = TRUE)) %>% 
  ggplot(aes(x = reorder(row.names(data), -Average_Consumption), y = Average_Consumption)) +
  geom_bar(stat = "identity", position = position_dodge(), colour = "black", fill = "steelblue") +
  labs(title = "各省消费项目均值条形图", x = "", y = "均值") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

1.2 各消费项目均值条形图

按消费项目做均值图条形图

barplot(apply(data,2,mean))#按列做均值图条形图

对不同项目的条形添加不同颜色

 barplot(apply(data,2,mean),col=1:8) #按列做彩色均值图条形图

去掉食品列后的数据按列做均值条形图

barplot(apply(data[,2:8],2,mean))

按消费项目做中位数条形图

barplot(apply(data,2,median))

利用ggplot作均值条形图

data %>% summarise(across(everything(), mean, na.rm = TRUE)) %>% 
  pivot_longer(cols = everything(), names_to = "Consumption_Type", values_to = "Average") %>% 
  mutate(
    Consumption_Type=factor(Consumption_Type,level=c('食品','衣着','设备','医疗','交通','教育','居住','杂项')),
  ) %>% 
  ggplot(aes(x = Consumption_Type, y = Average, fill = Consumption_Type)) +
  geom_bar(stat = "identity", position = position_dodge(), colour = "black") +
  theme_minimal() +
  labs(title = "各消费项目均值条形图", x = "类别", y = "均值",fill = "消费种类")

Warning: There was 1 warning in `summarise()`.
ℹ In argument: `across(everything(), mean, na.rm = TRUE)`.
Caused by warning:
! The `...` argument of `across()` is deprecated as of dplyr 1.1.0.
Supply arguments directly to `.fns` through an anonymous function instead.

  # Previously
  across(a:b, mean, na.rm = TRUE)

  # Now
  across(a:b, \(x) mean(x, na.rm = TRUE))

使各条形的颜色相同

data %>% summarise(across(everything(), mean, na.rm = TRUE)) %>% 
  pivot_longer(cols = everything(), names_to = "Consumption_Type", values_to = "Average") %>% 
  mutate(
    Consumption_Type=factor(Consumption_Type,level=c('食品','衣着','设备','医疗','交通','教育','居住','杂项')),
  ) %>% 
  ggplot(aes(x = Consumption_Type, y = Average)) +
  geom_bar(stat = "identity", position = position_dodge(), colour = "black", fill = "steelblue") +
  theme_minimal() +
  labs(title = "各消费项目均值条形图", x = "类别", y = "均值")

1.3 各消费项目箱线图

boxplot函数直接作箱线图，默认每个变量（列）作一个箱线，并将全部变量的箱线在同一个图中展示。

boxplot(data)#按列做箱线图

boxplot(data,horizontal=T,las=1)#箱线图中图形按水平放置

利用ggplot函数作箱线图，需要对数据转化为长结果数据

data %>% pivot_longer(cols = 1:8, names_to = "Consumption_Type", values_to = "Value") %>% 
  mutate(
    Consumption_Type=factor(Consumption_Type,level=c('食品','衣着','设备','医疗','交通','教育','居住','杂项')),
  ) %>% 
  ggplot(aes(x = Consumption_Type, y = Value)) +
  geom_boxplot() +
  labs(title = "各消费项目箱线图", x = "", y = "消费水平") +
  theme_minimal() #  + coord_flip()

1.4 各消费项目星相图

星相图的每个角表示一个变量，可以得出，，北京，上海，广东，浙江四个地区的消费情况较为突出，其他地区的消费情况大致相同

stars(data)                #具有图例的360度星相图

stars(data,key.loc=c(17,7))                #具有图例的360度星相图

stars(data,full=F,key.loc=c(17,7))                #具有图例的180度星相图

stars(data,draw.segments=T,key.loc=c(17,7))#具有图例的360度彩色圆形星相图

stars(data,full=F,draw.segments=T,key.loc=c(17,7))#具有图例的180度彩色圆形星相图

1.5 各消费项目脸谱图

脸谱图较为生动形象，直观地表达样本之间的差异

library(aplpack) #加载aplpack包
aplpack::faces(data)

effect of variables:
 modified item       Var   
 "height of face   " "食品"
 "width of face    " "衣着"
 "structure of face" "设备"
 "height of mouth  " "医疗"
 "width of mouth   " "交通"
 "smiling          " "教育"
 "height of eyes   " "居住"
 "width of eyes    " "杂项"
 "height of hair   " "食品"
 "width of hair   "  "衣着"
 "style of hair   "  "设备"
 "height of nose  "  "医疗"
 "width of nose   "  "交通"
 "width of ear    "  "教育"
 "height of ear   "  "居住"

aplpack::faces(data[,2:8])#去掉第一个变量按每行7个做脸谱图

effect of variables:
 modified item       Var   
 "height of face   " "衣着"
 "width of face    " "设备"
 "structure of face" "医疗"
 "height of mouth  " "交通"
 "width of mouth   " "教育"
 "smiling          " "居住"
 "height of eyes   " "杂项"
 "width of eyes    " "衣着"
 "height of hair   " "设备"
 "width of hair   "  "医疗"
 "style of hair   "  "交通"
 "height of nose  "  "教育"
 "width of nose   "  "居住"
 "width of ear    "  "杂项"
 "height of ear   "  "衣着"

aplpack::faces(data[c(1,9,19,28,29,30),])#选择第1,9,19,28,29,30个观测的多元数据做脸谱图

effect of variables:
 modified item       Var   
 "height of face   " "食品"
 "width of face    " "衣着"
 "structure of face" "设备"
 "height of mouth  " "医疗"
 "width of mouth   " "交通"
 "smiling          " "教育"
 "height of eyes   " "居住"
 "width of eyes    " "杂项"
 "height of hair   " "食品"
 "width of hair   "  "衣着"
 "style of hair   "  "设备"
 "height of nose  "  "医疗"
 "width of nose   "  "交通"
 "width of ear    "  "教育"
 "height of ear   "  "居住"

library("TeachingDemos") #install.packages("TeachingDemos")
faces2(data,ncols=7) #TeachingDemos::faces(data)

1.6 各消费项目雷达图

ggplot2的扩展包ggiraphExtra能作雷达图

data[c(1,9,19,28,29,30),] %>% 
  mutate(省份=rownames(.)) %>% 
  ggRadar(aes(group = 省份))

1.7 各消费项目调和曲线图

source("msaR.R")#加自定义函数
msa.andrews(data)#绘制调和曲线图

msa.andrews(data[c(1,9,19,28,29,30),])

library("fmsb")
rddat=data[c(1,9,19,28,29,30),]
maxmin=rbind(apply(rddat,2,max),apply(rddat,2,min))
rddat=rbind(maxmin,rddat)
radarchart(rddat, axistype=2, pcol=topo.colors(6), plty=1, pdensity=seq(5,40,by=5), pangle=seq(0,150,by=30), pfcol=topo.colors(6))

#install.packages("andrews")
library(andrews)

Warning: package 'andrews' was built under R version 4.3.3

See the package vignette with `vignette("andrews")`

andrews(data,clr=5,ymax=6)

#选择第1,9,19,28,29,30个观测的多元数据做调和曲线图
andrews(data[c(1,9,19,28,29,30),],clr=5,ymax=6)

2 线性回归分析

## 一元线性回归2-1

每周加班工作时间(x)与签发新保单数目(y)呈明显正相关
每周加班工作时间(x)与签发新保单数目(y)的相关系数为0.95 。
利用每周加班工作时间(x)对签发新保单数目(y)作回归，回归方程为

\[ \widehat{y} =46.15+251.17\times x\]
随机误差\(\epsilon\)的标准差\(\sigma\)的估计值为127.06 5.x与y的决定系数

(R2=summary(model_fit)$r.sq)   #显示多元线性回归模型决定系数

[1] 0.9005

6.对回归方程做方差分析

anova(model_fit)#模型方差分析

Analysis of Variance Table

Response: y
          Df  Sum Sq Mean Sq F value  Pr(>F)    
x          1 1168713 1168713    72.4 2.8e-05 ***
Residuals  8  129147   16143                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

7.对回归方程作残差图并做一些分析

plot(model_fit)#做散点图

8.根据\[ \widehat{y} =46.15+251.17\times x\]方程，带入y=1000,可以计算出需要的加班时间为3.798个小时

2.1 多元线性回归2-2

利用广告预算(x₁)和销售代理数目(x₂)对年销售额(y)作回归，回归方程为：

term estimate std.error statistic p.value

(Intercept) -22.74 30.69 -0.74 0.49

x1 0.15 0.11 1.33 0.24

x2 1.22 1.31 0.93 0.40

\[ \widehat{y} =-22.74+0.15\times x1+1.22\times x2\]
5%显著水平下，广告预算(x₁)和销售代理数目(x₂)的系数均不显著。
广告预算(x₁)与销售额(y)相关系数为0.5797；销售代理数目(x₂)与销售额(y)相关系数为0.4816 ；广告预算(x₁)和销售代理数目(x₂)与年销售额(y)的复相关系数为0.6586 。

term	estimate	std.error	statistic	p.value
(Intercept)	-22.74	30.69	-0.74	0.49
x1	0.15	0.11	1.33	0.24
x2	1.22	1.31	0.93	0.40

2.2 多元线性回归2-3

从回归方程中可知

term estimate std.error statistic p.value

(Intercept) -5213.1 12704.5 -0.41 0.70

GPA 8508.8 2721.6 3.13 0.03

年龄 181.6 283.5 0.64 0.55

\[ \widehat{起始工资} =-5213.12+8508.79\times GPA+181.58\times 年龄\]
含义

term	estimate	std.error	statistic	p.value
(Intercept)	-5213.1	12704.5	-0.41	0.70
GPA	8508.8	2721.6	3.13	0.03
年龄	181.6	283.5	0.64	0.55

summary(model_fit3)


Call:
lm(formula = 起始工资 ~ ., data = data4)

Residuals:
    1     2     3     4     5     6     7     8 
 1617   207  1282  -704 -2215  -770  -311   893 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)    -5213      12704   -0.41    0.699  
GPA             8509       2722    3.13    0.026 *
年龄             182        284    0.64    0.550  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1500 on 5 degrees of freedom
Multiple R-squared:  0.668, Adjusted R-squared:  0.535 
F-statistic: 5.02 on 2 and 5 DF,  p-value: 0.0637

当GPA=3,年龄=24，起始工资的预测值为2.4671^{4} 。

2.3 线性模型选择2-4

货运总量(y)、工业总产值(x₁)、农业总产值(x₂)、居民非商品支出(x₃)的相关系数矩阵为：

       y    x1    x2    x3
y  1.000 0.556 0.731 0.724
x1 0.556 1.000 0.113 0.398
x2 0.731 0.113 1.000 0.547
x3 0.724 0.398 0.547 1.000

散点图矩阵为：

回归方程为：

\[ \widehat{y} =-348.28+3.75\times x1+7.1\times x2+12.45\times x3\]
回归模型的R方为：0.8055 ，说明拟合优度好

回归模型F检验说明，t检验说明p值小于0.05，回归显著


Call:
lm(formula = y ~ ., data = data5)

Residuals:
   Min     1Q Median     3Q    Max 
-25.20 -17.03   2.63  11.68  33.23 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  -348.28     176.46   -1.97    0.096 .
x1              3.75       1.93    1.94    0.100  
x2              7.10       2.88    2.47    0.049 *
x3             12.45      10.57    1.18    0.284  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 23.4 on 6 degrees of freedom
Multiple R-squared:  0.806, Adjusted R-squared:  0.708 
F-statistic: 8.28 on 3 and 6 DF,  p-value: 0.0149

剔除不显著的后，回归模型为

$$ \widehat{y} =-348.28+3.75\times x1+7.1\times x2+12.45\times x3$$

逐步回归的选择的模型为y=-348.28+3.75x1+7.10x2+12.45x3

Start:  AIC=65.98
y ~ x1 + x2 + x3

       Df Sum of Sq  RSS  AIC
<none>              3297 66.0
- x3    1       762 4059 66.1
- x1    1      2072 5369 68.9
- x2    1      3340 6637 71.0


Call:
lm(formula = y ~ x1 + x2 + x3, data = data5)

Coefficients:
(Intercept)           x1           x2           x3  
    -348.28         3.75         7.10        12.45