R 中的回归:方法比较

在本文中,我提供了 R 中回归的各种示例。我比较了使用 R 的各种回归方法。我将介绍的方法是简单线性回归、多元线性回归、多项式回归、决策树回归和支持向量机回归。

R 中回归所需的包

我首先加载所需的包(除了 R 基础库):

library(dplyr)         # package for data wrangling
library(ggplot2)       # package for data visualization
library(rpart)         # package for decision tree regression analysis
library(randomForest)  # package for random forest regression analysis
library(e1071)         # package for support vector regression analysis
library(caTools)       # package for e.g. splitting into training and test sets

在 R 回归之前读取数据

现在,我读入了将用于在 R 中实施回归示例的数据。这些数据包含有关国内生产总值、私人投资总额、预期寿命、人口规模和研发投资的数据。

# reading in gross domestic product data
gdp_df <- read.csv("gross domestic product.csv",header=TRUE)

# reading in gross private investment data
investment_df <- read.csv("gross private investments.csv",header=TRUE)

# reading in life expectancy data
lifeexpectancy_df <- read.csv("life expectancy.csv",header=TRUE)

# reading in population data
population_df <- read.csv("population.csv",header=TRUE)

# reading in research and development data
rnd_df <- read.csv("research and development.csv",header=TRUE)

# joining all data into a final input data set
data_df <- gdp_df %>% inner_join(investment_df,by="DATE",fa) %>% 
  inner_join(lifeexpectancy_df,by="DATE") %>%
  inner_join(population_df,by="DATE") %>%
  inner_join(rnd_df,by="DATE")

# rename column headers
colnames(data_df) <-c("date",
                      "gdp",
                      "private_investment",
                      "life_expectancy",
                      "population",
                      "rnd")

# convert date entries from characters into date data types
data_df$date <- as.Date(data_df$date)

作为第一步,我想可视化数据。为了改进数据增长趋势的视觉比较,我对其进行了标准化。为此我定义了一个函数:

normalize_data <- function(x){
  return ((x-min(x))/(max(x)-min(x)))
}

使用该函数,我将 data_df 转换为标准化的 ggplot 友好数据框。我将其提供给 ggplot 并可视化数据:

dates <- c(data_df$date,
           data_df$date,
           data_df$date,
           data_df$date,
           data_df$date)

values <- c(normalize_data(data_df$gdp),
            normalize_data(data_df$private_investment),
            normalize_data(data_df$life_expectancy),
            normalize_data(data_df$population),
            normalize_data(data_df$rnd))

sources <- c(rep("gdp",times=nrow(data_df)),
             rep("private_investment",times=nrow(data_df)),
             rep("life_expectancy",times=nrow(data_df)),
             rep("population",times=nrow(data_df)),
             rep("rnd",times=nrow(data_df)))

# build ggplot-friendly data frame
ggdata_df <- as.data.frame(matrix(nrow=5*nrow(data_df),ncol=3))
colnames(ggdata_df) <- c("date","value","source")

# populating the ggplot-friendly data frame
ggdata_df$date <- as.Date(dates)
ggdata_df$value <- as.numeric(values)
ggdata_df$source <- sources

# visualize the normalized values (i.e. data set) with ggplot
ggplot(ggdata_df) + 
  geom_point(mapping = aes(x=dates,y=values,color=sources)) +
  ggtitle("Normalized data set values; time-series view") +
  xlab("Time") +
  ylab("Normalized observation values")
回归训练数据的数据概览

r 中的回归管道

在可视化感兴趣的数据集后,我在数据上测试了各种不同的回归模型。

对于 R 中回归的每种替代方法,过程始终相同:

  1. 将数据拆分为训练集和测试集
  2. 在训练集上训练预测器
  3. 根据训练集审查回归量的拟合优度
  4. 测试集上的测试回归器

R中的简单线性回归

我将从一个简单的线性回归模型开始我的 R 回归示例系列。我尝试根据数据集提供的其他观察数据来预测人口规模。

下面我实现了上面提到的管道(请注意,在这种情况下我不必规范化数据)。

由于人口规模将作为因变量,我现在有四个自变量可供选择。对于这个例子,我选择预期寿命(出生时)作为预测自变量。也就是说,对于我的 RI 回归示例,在这种情况下将得出人口规模与预期寿命之间的线性关系。

我的假设是,更高的预期寿命伴随着更高的人口规模(对于同一国家和地区)。

# randomly generate split
set.seed(123)
training_split <- sample.split(data_df$date,SplitRatio = 0.8)

# extract training and test sets
training_df <- subset(data_df, training_split)
test_df <- subset(data_df, !training_split)

# train predictor based on training set
predictor <- lm(formula = population ~ life_expectancy, data=training_df)

# print summary of simple linear regression
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy, data = training_df)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -15279  -6746   1131   4625  17762 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -840396.8    32055.5  -26.22   <2e-16 ***
## life_expectancy   14629.9      427.4   34.23   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8377 on 44 degrees of freedom
## Multiple R-squared:  0.9638, Adjusted R-squared:  0.963 
## F-statistic:  1172 on 1 and 44 DF,  p-value: < 2.2e-16

我使用 ggplot2 在 R 中可视化了这种回归的结果。在下面的代码中,我展示了如何这样做:

ggplot(training_df) + 
  geom_point(mapping=aes(x=life_expectancy,
                         y=population)) +
  geom_line(mapping=aes(x=life_expectancy,
                        y=predict(predictor,training_df)),color="red") +
  ggtitle("Population in dependence of life expectancy; linear regression (training)") +
  xlab("US life expectancy since birth [years]") +
  ylab("US population size [-]")
预期寿命导致的美国人口规模 R 的线性回归

我还想查看预测残差的直方图:

hist(predictor$residuals,
     main ="Histogram of model residuals",
     xlab="Model residuals [-]")

此外,我想看看预测变量观察大小的残差:

plot(x=training_df$life_expectancy,
     y=predictor$residuals, 
     main = "Model residuals in dependence of independent variable",
     xlab="US life expectancy [years]",
     ylab="Model residuals [-]")

这看起来有点系统性偏见。我也可以尝试在 qqnorm 图中展示这一点:

qqnorm(predictor$residuals)

训练完线性预测器后,我在测试集上测试预测性能:

# predict the test set values
predictions <- predict(predictor,test_df)

# visualize prediction accuracy
ggplot(test_df) + 
  geom_point(mapping=aes(x=life_expectancy,y=population)) +
  geom_line(mapping=aes(x=life_expectancy,predictions),color="red") +
  ggtitle("Population in dependence of life expectancy; linear regression (test)") +
  xlab("US life expectancy since birth [years]") +
  ylab("US population size [-]")
R中的简单线性回归

让我们看看是否可以使用多元线性回归方法获得更准确的预测。但首先,我们将整个数据集的简单线性回归预测存储在一个向量中:

predictions_slr <- predict(predictor,data_df)

多元线性回归

这次我在预测美国人口规模时首先考虑数据集中的所有变量:

# split data in training and test set
set.seed(123)
training_split <- sample.split(data_df$date, SplitRatio = 0.8)
training_df <- subset(data_df, training_split)
test_df <- subset(data_df, !training_split)

# train predictor with mutliple linear regression methodology, on training set
predictor <- lm(formula = population ~ gdp + 
                  private_investment +
                  life_expectancy +
                  rnd, training_df)

# summarize regression outcome 
summary(predictor)
## 
## Call:
## lm(formula = population ~ gdp + private_investment + life_expectancy + 
##     rnd, data = training_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10472.0  -1969.0    188.3   2421.8   7782.1 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        -3.202e+05  4.488e+04  -7.133 1.07e-08 ***
## gdp                 9.891e+00  2.970e+00   3.330  0.00184 ** 
## private_investment -3.272e+00  5.115e+00  -0.640  0.52598    
## life_expectancy     7.283e+03  6.318e+02  11.527 1.93e-14 ***
## rnd                -1.923e+02  8.065e+01  -2.385  0.02181 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3834 on 41 degrees of freedom
## Multiple R-squared:  0.9929, Adjusted R-squared:  0.9922 
## F-statistic:  1440 on 4 and 41 DF,  p-value: < 2.2e-16

与之前的简单线性回归相比,Adjusted R-squared 有所改善。私人投资量似乎对预测美国人口规模没有那么重要。因此,我采用逐步参数剔除法,在进行回归分析时剔除民间投资量作为自变量:

# regression without private investment volume as independent variable
predictor <- lm(formula=population~gdp + 
                  life_expectancy +
                  rnd, training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ gdp + life_expectancy + rnd, data = training_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10412.6  -2275.0    247.7   2209.6   7794.9 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -3.225e+05  4.442e+04  -7.259 6.20e-09 ***
## gdp              8.598e+00  2.160e+00   3.980 0.000267 ***
## life_expectancy  7.316e+03  6.252e+02  11.702 8.41e-15 ***
## rnd             -1.673e+02  7.006e+01  -2.388 0.021496 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3807 on 42 degrees of freedom
## Multiple R-squared:  0.9929, Adjusted R-squared:  0.9924 
## F-statistic:  1948 on 3 and 42 DF,  p-value: < 2.2e-16

消除私人投资量改善了调整后的 R 平方(略微)。但是,我仍然有兴趣查看残差的分布。

首先,残差直方图:

hist(predictor$residuals)

接下来,残差的 qqnorm 图:

qqnorm(predictor$residuals)

我现在通过预测测试集总体值来评估模型性能:

# predict test set population values
predictions <- predict(predictor,test_df)

# visualize model performance vs test set, along timeline
ggplot(test_df) + 
  geom_point(mapping=aes(x=date,y=population)) +
  geom_line(mapping=aes(x=date,y=predictions),color="red") +
  ggtitle("Multiple linear regression prediction of US population") +
  xlab("Date") +
  ylab("US population size [-]")
R中的多元线性回归预测我们的人口规模

在继续进行多项式回归之前,我将整个数据集的预测存储在一个向量中:

predictions_mlr <- predict(predictor,data_df)

R中的多项式回归

考虑到向后排除后的多元线性回归结果,预期寿命似乎对预测人口规模很有帮助,毕竟。

因此,我尝试在 R 中进行多项式回归,根据美国出生时的预期寿命预测美国人口规模。

我的方法是添加多项式项,只要这样可以改进调整后的 R 平方:

# split in training and test set
set.seed(123)
training_split <- sample.split(data_df$date,SplitRatio = 0.8)
training_df <- subset(data_df, training_split)
test_df <- subset(data_df, !training_split)

# train predictor on first order term only, i.e. simple linear regression
predictor <- lm(formula = population ~ life_expectancy,training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy, data = training_df)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -15279  -6746   1131   4625  17762 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -840396.8    32055.5  -26.22   <2e-16 ***
## life_expectancy   14629.9      427.4   34.23   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8377 on 44 degrees of freedom
## Multiple R-squared:  0.9638, Adjusted R-squared:  0.963 
## F-statistic:  1172 on 1 and 44 DF,  p-value: < 2.2e-16

我现在问自己:我可以通过添加一个二阶多项式项来改进调整后的 R 平方吗?让我测试一下:

# polynomial two term regression
training_df$LE2 <- training_df$life_expectancy^2
predictor <- lm(formula = population ~ life_expectancy + LE2, training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy + LE2, data = training_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10785.9  -4081.8   -715.1   4186.9   8913.4 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     4014088.8   604959.5   6.635 4.35e-08 ***
## life_expectancy -116186.0    16295.0  -7.130 8.34e-09 ***
## LE2                 879.9      109.6   8.029 4.31e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5360 on 43 degrees of freedom
## Multiple R-squared:  0.9855, Adjusted R-squared:  0.9848 
## F-statistic:  1463 on 2 and 43 DF,  p-value: < 2.2e-16

是的。改进了调整后的 R 平方。如果我添加第三个术语会怎样?

# polynomial three term regression
training_df$LE3 <- training_df$life_expectancy^3
predictor <- lm(formula = population ~ life_expectancy + LE2 + LE3, training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy + LE2 + LE3, data = training_df)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -10908  -3739   -862   4079   9125 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)
## (Intercept)      1.196e+07  2.240e+07   0.534    0.596
## life_expectancy -4.367e+05  9.037e+05  -0.483    0.631
## LE2              5.187e+03  1.214e+04   0.427    0.671
## LE3             -1.927e+01  5.432e+01  -0.355    0.725
## 
## Residual standard error: 5415 on 42 degrees of freedom
## Multiple R-squared:  0.9856, Adjusted R-squared:  0.9845 
## F-statistic: 955.5 on 3 and 42 DF,  p-value: < 2.2e-16

添加第三项并没有改善调整后的 R 平方。如果我在 R 中的回归中添加第四项会发生什么?

# polynomial four term regression
training_df$LE4 <- training_df$life_expectancy^4
predictor <- lm(formula = population ~ life_expectancy + LE2 + LE3 + LE4, training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy + LE2 + LE3 + LE4, 
##     data = training_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11422.7  -2990.7     -4.8   2420.7  11496.0 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -2.345e+09  6.041e+08  -3.882 0.000369 ***
## life_expectancy  1.264e+08  3.250e+07   3.889 0.000362 ***
## LE2             -2.553e+06  6.555e+05  -3.895 0.000355 ***
## LE3              2.290e+04  5.873e+03   3.900 0.000350 ***
## LE4             -7.697e+01  1.972e+01  -3.903 0.000346 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4680 on 41 degrees of freedom
## Multiple R-squared:  0.9895, Adjusted R-squared:  0.9884 
## F-statistic: 963.4 on 4 and 41 DF,  p-value: < 2.2e-16

添加第四项改进了 R 中的回归,因为调整后的 R 平方增加了。我继续添加项,添加第五个和第六个回归项。

# polynomial six term regression
training_df$LE5 <- training_df$life_expectancy^5
training_df$LE6 <- training_df$life_expectancy^6
predictor <- lm(formula = population ~ life_expectancy + LE2 + LE3 + 
                  LE4 + LE5 + LE6, training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy + LE2 + LE3 + LE4 + 
##     LE5 + LE6, data = training_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12579.2  -2669.2   -306.2   1781.2  12719.2 
## 
## Coefficients: (1 not defined because of singularities)
##                   Estimate Std. Error t value Pr(>|t|)  
## (Intercept)     -3.842e+10  1.696e+10  -2.265   0.0290 *
## life_expectancy  2.459e+09  1.097e+09   2.243   0.0305 *
## LE2             -6.145e+07  2.768e+07  -2.220   0.0322 *
## LE3              7.276e+05  3.312e+05   2.197   0.0339 *
## LE4             -3.632e+03  1.671e+03  -2.174   0.0357 *
## LE5                     NA         NA      NA       NA  
## LE6              4.285e-02  2.013e-02   2.128   0.0395 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4491 on 40 degrees of freedom
## Multiple R-squared:  0.9905, Adjusted R-squared:  0.9894 
## F-statistic: 837.9 on 5 and 40 DF,  p-value: < 2.2e-16

添加第五个和第六个回归项改进了调整后的 R 平方。我决定继续添加第七和第八项:

# polynomial seven term regression
training_df$LE7 <- training_df$life_expectancy^7
training_df$LE8 <- training_df$life_expectancy^8
predictor <- lm(formula = population ~ life_expectancy + LE2 + LE3 + LE4 + 
                  LE5 + LE6 + LE7 + LE8, training_df)

# review model performance
summary(predictor)
## 
## Call:
## lm(formula = population ~ life_expectancy + LE2 + LE3 + LE4 + 
##     LE5 + LE6 + LE7 + LE8, data = training_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12579.2  -2669.2   -306.2   1781.2  12719.2 
## 
## Coefficients: (3 not defined because of singularities)
##                   Estimate Std. Error t value Pr(>|t|)  
## (Intercept)     -3.842e+10  1.696e+10  -2.265   0.0290 *
## life_expectancy  2.459e+09  1.097e+09   2.243   0.0305 *
## LE2             -6.145e+07  2.768e+07  -2.220   0.0322 *
## LE3              7.276e+05  3.312e+05   2.197   0.0339 *
## LE4             -3.632e+03  1.671e+03  -2.174   0.0357 *
## LE5                     NA         NA      NA       NA  
## LE6              4.285e-02  2.013e-02   2.128   0.0395 *
## LE7                     NA         NA      NA       NA  
## LE8                     NA         NA      NA       NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4491 on 40 degrees of freedom
## Multiple R-squared:  0.9905, Adjusted R-squared:  0.9894 
## F-statistic: 837.9 on 5 and 40 DF,  p-value: < 2.2e-16

调整后的 R 平方没有改善。我决定在这里停止多项式回归并可视化回归结果:

ggplot(training_df) + 
  geom_point(mapping=aes(x=life_expectancy,y=population)) +
  geom_line(mapping=aes(x=life_expectancy,y=predict(predictor,training_df)),
            color="red") +
  ggtitle("Trained polynomial regression model; degree 6") + 
  xlab("US life expectancy at birth [years]") + 
  ylab("US population size [-]")
R中的六次多项式回归

现在我针对测试集测试多项式模型:

# add terms to test data frame
test_df$LE2 <- test_df$life_expectancy^2
test_df$LE3 <- test_df$life_expectancy^3
test_df$LE4 <- test_df$life_expectancy^4
test_df$LE5 <- test_df$life_expectancy^5
test_df$LE6 <- test_df$life_expectancy^6
test_df$LE7 <- test_df$life_expectancy^7
test_df$LE8 <- test_df$life_expectancy^8

# visualize predictor performance on test set
predictions <- predict(predictor,test_df)
ggplot(test_df) + 
  geom_point(mapping=aes(x=life_expectancy,y=population)) +
  geom_line(mapping=aes(x=life_expectancy,y=predictions), color = "red") +
  ggtitle("Polynomial regression on test data; degree 6") + 
  xlab("US life expectancy at birth [years]") + 
  ylab("US population size [-]")
R中的多项式回归

在继续进行决策树回归之前,我将整个数据集的预测存储在一个向量中:

data_df$LE2 <- data_df$life_expectancy^2
data_df$LE3 <- data_df$life_expectancy^3
data_df$LE4 <- data_df$life_expectancy^4
data_df$LE5 <- data_df$life_expectancy^5
data_df$LE6 <- data_df$life_expectancy^6
data_df$LE7 <- data_df$life_expectancy^7
data_df$LE8 <- data_df$life_expectancy^8
predictions_poly <- predict(predictor,data_df)

R 中的决策树回归

另一种回归方法是决策树回归。在这种类型的回归中,解决方案空间被划分为子集,在训练集中测量的平均观察值构成了子集的预测。

在下面的部分中,我在 R 中实现了决策树回归:

# split in training and test set
set.seed(123)
training_split <- sample.split(data_df$date,SplitRatio = 0.8)
training_df <- subset(data_df, training_split)
test_df <- subset(data_df, !training_split)

# conduct decision tree regression
predictor <- rpart(formula = population ~ gdp +
                     private_investment +
                     life_expectancy + 
                     population + 
                     rnd,
                   data = training_df)

# generate predictions from test set
predictions <- predict(predictor,test_df)
residuals <- test_df$population - predictions

#  visualize prediction accuracy along the timeline
ggplot(test_df) + 
  geom_point(mapping = aes(x=date,y=population)) + 
  geom_point(mapping = aes(x=date,y=predictions), color = "red") + 
  ggtitle("Decision tree regression performance on test set (red: predictions)") + 
  xlab("Date") +
  ylab("US population size [-]")
R 中的决策树回归

在继续支持向量回归之前,我将对整个数据集的预测存储在另一个向量中:

predictions_dtr <- predict(predictor,data_df)

支持向量回归

最后,我进行支持向量回归。我只提供多元线性回归发现在预测美国人口规模方面具有统计显着性的变量:

# split in training and test set
set.seed(123)
training_split <- sample.split(data_df$date,SplitRatio = 0.8)
training_df <- subset(data_df, training_split)
test_df <- subset(data_df, !training_split)

# construct predictor
predictor <- svm(formula = population ~ gdp + 
                   life_expectancy + 
                   rnd,
                 data = training_df,
                 type = "eps-regression")

# visualize prediction accuracy on test set along time line
ggplot(test_df) + 
  geom_point(mapping = aes(x=date, y = population)) + 
  geom_line(mapping = aes(x=date, y = predict(predictor,test_df)), color = "red") + 
  ggtitle("Support vector regression (eps methode) on test set") + 
  xlab("Date") + 
  ylab("US population size [-]")
R中的支持向量回归

与其他回归部分一样,我将整个数据集的预测作为解空间存储到一个单独的向量中:

predictions_svr <- predict(predictor,data_df)

模型预测的比较

使用具有存储预测值的向量,我们可以比较时间轴上的不同预测模型:

# construct ggplot friendly data frame
ggdata_df <- as.data.frame(matrix(nrow=nrow(data_df)*6,ncol=3))
colnames(ggdata_df) <- c("date","value","source")
ggdata_df$date <- rep(data_df$date,times=6)
ggdata_df$value <- c(predictions_slr,
                     predictions_mlr,
                     predictions_poly,
                     predictions_dtr,
                     predictions_svr,
                     data_df$population)
ggdata_df$source <- c(rep("simple linear", times=nrow(data_df)),
                      rep("multiple linear", times=nrow(data_df)),
                      rep("polynomial", times=nrow(data_df)),
                      rep("decision tree", times=nrow(data_df)),
                      rep("support vector", times=nrow(data_df)),
                      rep("original observations", times=nrow(data_df)))

# visualize
ggplot(ggdata_df) + 
  geom_point(mapping = aes(x=date, y=value, color=source)) +
  ggtitle("Comparison of model predictions against real observations values") + 
  xlab("Date") +
  ylab("US population size [-]")
比较 R 中的各种回归方法

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