【原创】R语言多元线性回归特征降维：主成分回归和偏最小二乘案例分析报告附代码数据

R语言线性回归案例报告R初始指令安装“汽车”包：install.packages（“汽车”）加载库汽车加载汽车中的数据：数据（Salaries，package =“car”）查看您的办公桌上的数据（屏幕）：它所表示的薪水视图（帮助）和数据描述：help（薪水）变量的确切名称：名称（薪金）考虑谁是定量和定性的变量分散图虽然有些变量不是量化的，但相反，它们是绝对的，例如秩序是有序的，我们将要制作离散图考虑到图表，我们将运行变量之间的简单回归模型：“yrs.since.phd”“yrs.service”，但首先让我们来回顾一下变量之间的相关性。

因此，我们要确定假设正态性的相关系数考虑到这两个变量之间的相关性高，解释结果结果因变量yrs.since.phd的选择是正确的，请解释为什么编写表单的模型：y = intercept + oendiente * x解释截距和斜率假设检验根据测试结果，考虑到p值= 2e-16，是否拒绝了5％的显着性值的假设？斜率为零根据测试结果，考虑到p值= 2e-16，关于斜率的假设是否被拒绝了5％的显着性值？考虑到变量yrs.service在模型中是重要的模型和测试调整考虑到R的平方值为0.827，你认为该模型具有良好的线性拟合？解释调整的R平方值考虑到验证数据与模型拟合的测试由F统计得到：1894在1和395 DF，p值：<2.2e-16认为模型符合调整？ ##使用模型进行估计为变量x的以下值查找yrs.since.phd的估计值：Graficas del modeloValidación del modeloEn este caso se desea determinar los residuos(error)entre el modelo y lo observado y ver si los residuos cumplen con: 1. Tener media=0 2. Varianza= constante 3. Se distribuyen normal(0,constante) Los residuos se generan a continuación (por facilidad solo generamos los primeros 20 [1:20])Valores ajustados, es decir el valor de la variable y dad0 por el modelo (por facilidad solo generamos los primeros 20 [1:20])Prueba de normalidad, utilizamos la prueba QQ que permite deducir si los datos se ajustan a la normal (si los datos estan cerca de la línea)Validación del modeloEn este caso se desea determinar los residuos(error)entre el modelo y lo observado y ver si los residuos cumplen con: 1. Tener media=0 2. Varianza= constante 3. Se distribuyen normal(0,constante) Los residuos se generan a continuación (por facilidad solo generamos los primeros 20 [1:20])Valores ajustados, es decir el valor de la variable y dad0 por el modelo (por facilidad solo generamos los primeros 20 [1:20])Prueba de normalidad, utilizamos la prueba QQ que permite deducir si los datos se ajustan a la normal (si los datos estan cerca de la línea)condidera que los que estan a más de dos desviaciones estandar se distribuyen normal?。

r语言实现偏最小二乘法偏最小二乘法（Partial Least Squares Regression，简称PLSR）是一种多元统计分析方法，常用于建立预测模型。

在R语言中，我们可以使用plsr函数来实现偏最小二乘法。

在讲解具体实现之前，我们先来了解一下偏最小二乘法的原理。

偏最小二乘法是基于最小二乘法的一种改进方法，用于处理多重共线性问题。

在最小二乘法中，我们通过最小化预测值与真实值之间的平方误差来建立模型。

然而，在多重共线性存在的情况下，最小二乘法的结果可能会受到较大的误差影响。

偏最小二乘法通过将自变量和因变量进行正交变换，得到新的变量，从而消除了多重共线性的影响。

这些新的变量被称为偏最小二乘法的主成分，它们是原始变量的线性组合。

通过对主成分进行逐步回归分析，我们可以得到一个可靠的预测模型。

接下来，我们使用R语言中的plsr函数来实现偏最小二乘法。

首先，我们需要加载pls包，并读取我们的数据集。

假设我们的数据集包含了多个自变量和一个因变量。

```rlibrary(pls)data <- read.csv("data.csv")```然后，我们可以使用plsr函数来建立偏最小二乘法模型。

在plsr 函数中，我们需要指定自变量和因变量的列数，并选择主成分的数量。

```rmodel <- plsr(Y ~ X1 + X2 + X3, data = data, ncomp = 2)```在上述代码中，Y代表因变量，X1、X2、X3代表自变量。

我们选择了2个主成分。

接下来，我们可以使用summary函数来查看模型的摘要信息。

```rsummary(model)```摘要信息中会显示出模型的R方值、标准误差以及每个主成分的贡献率。

我们还可以使用plot函数来绘制模型的各个主成分的贡献率图。

```rplot(model, ncomp = 2, asp = 1)```在图中，横轴代表主成分的数量，纵轴代表贡献率。

#第十章多元统计分析介绍#10.1 主成分分析与因子分析#10.1.1 主成分的简要定义与计算#10.1.2 主成分R 通用程序student<-data.frame(X1=c(148, 139, 160, 149, 159, 142, 153, 150, 151, 139, 140, 161, 158, 140, 137, 152, 149, 145, 160, 156,151, 147, 157, 147, 157, 151, 144, 141, 139, 148),X2=c(41, 34, 49, 36, 45, 31, 43, 43, 42, 31,29, 47, 49, 33, 31, 35, 47, 35, 47, 44,42, 38, 39, 30, 48, 36, 36, 30, 32, 38),X3=c(72, 71, 77, 67, 80, 66, 76, 77, 77, 68,64, 78, 78, 67, 66, 73, 82, 70, 74, 78,73, 73, 68, 65, 80, 74, 68, 67, 68, 70),X4=c(78, 76, 86, 79, 86, 76, 83, 79, 80, 74,74, 84, 83, 77, 73, 79, 79, 77, 87, 85,82, 78, 80, 75, 88, 80, 76, 76, 73, 78))student.pr<-princomp(student, cor=TRUE)summary(student.pr,loadings=TRUE)#10.1.3 因子分析的简要定义与计算#10.1.4 因子分析R 通用程序student<-read.table("e:/data/student.txt")names(student)=c("math", "phi", "chem", "lit", "his", "eng") fa<-factanal(student, factors=2)fa#10.2 判别分析#10.2.1 距离判别#10.2.2 Fisher 判别法#10.2.3 R 通用程序library(MASS)data(iris)attach(iris)names(iris)library(MASS)iris.lda <- lda(Species ~ Sepal.Length + Sepal.Width+ Petal.Length + Petal.Width)【原创】定制代写开发r/python/spss/matlab/WEKA/sas/sql/C++/stata/eviews数据挖掘和统计分析可视化调研报告程序等服务（附代码数据）,咨询：3025393450@有问题到淘宝找“大数据部落”就可以了iris.ldairis.pred=predict(iris.lda) $ classtable(iris.pred, Species)detach(iris)w <- read.table("e:/data/disc.txt")names(w)=c("group", "x1", "x2", "x3", "x4")library(MASS)z <- lda(group~x1+x2+x3+x4, data=w, prior=c(1, 1)/2) newdata<-rbind(c(8.85, 3.38, 5.17, 26.10), c(28.60, 2.40, 1.20, 127.0),c(20.70, 6.70, 7.60, 30.20), c(7.90, 2.40, 4.30, 33.20),c(3.19, 3.20, 1.43, 9.90), c(12.40, 5.10, 4.43, 24.60),c(16.80, 3.40, 2.31, 31.30), c(15.00, 2.70, 5.02, 64.00)) dimnames(newdata)<-list(NULL, c("x1", "x2", "x3", "x4")) newdata<-data.frame(newdata)predict(z, newdata=newdata)#10.3 聚类分析#10.3.1 基本思想#10.3.2 R通用程序x<-c(1, 2, 4.5, 6, 8)dim(x)<-c(5, 1)d<-dist(x)hc1<-hclust(d, "single")hc2<-hclust(d, "complete")hc3<-hclust(d, "median")hc4<-hclust(d, "ward")opar<-par(mfrow=c(2, 2))plot(hc1, hang=-1);plot(hc2, hang=-1)plot(hc3, hang=-1);plot(hc4, hang=-1)par(opar)data(iris);attach(iris)iris.hc<-hclust(dist(iris[,1:4]))plot(iris.hc, hang = -1)plclust(iris.hc,labels = FALSE, hang=-1)re<-rect.hclust(iris.hc,k=3)iris.id <- cutree(iris.hc,3)table(iris.id,Species)#10.4 典型相关分析#10.4.1 基本思想#10.4.2 R通用程序invest=read.table("e:/data/invest.txt")names(invest)=c("x1", "x2", "x3", "x4", "x5", "x6", "y1", "y2", "y3", "y4", "y5")ca<-cancor(invest[, 1:6], invest[, 7:11])ca#x10.5 对应分析#10.5.1 基本思想#10.5.2 R通用程序x.df=data.frame(HighlyFor=c(2, 6, 41, 72, 24), For =c(17, 65, 220, 224, 61),Against=c(17, 79, 327, 503, 300), HighlyAgainst=c(5, 6, 48, 47, 41))rownames(x.df)<-c("BelowPrimary", "Primary", "Secondary", "HighSchool","College")library(MASS)biplot(corresp(x.df, nf=2))。

R语言线性回归数据分析案例可视化报告从源下载数据集。

2.清理数据2.a放入数据列pimalm<-lm(class~npreg+glucose+bp+triceps+insulin+bmi+dia betes+age, data=pima)去除大p值的变量（p值> 0.005）Remove variables (insulin, age) with large p value (p value > 0.005) After the variables are dropped, the R-squared value remain about the same. This suggests the variables dropped do not have much effect on the model.Residual analysis shows almost straight line with distribution around zero. Due to this pattern, this model is not as robust.qqnorm(resid(pimalm), col="blue")qqline(resid(pimalm), col="red")The second dataset with much simpler variables. Although intuitively the variables both effect the output, the amount of effect by each variable is interesting. This dataset was examined to have a better sense of how multivariate regression will perform.allbacks.lm<-lm(weight~volume+area, data=allbacks) summary(allbacks.lm)qqnorm(resid(allbacks.lm), col="blue") qqline(resid(allbacks.lm), col="red")。

基于主成分分析的R语言在多元数据降维中的应用研究近年来，随着大数据时代的到来，我们面临的数据量越来越庞大。

在这样的背景下，如何从海量的数据中提取出有意义，且具有参考价值的信息，成为了数据科学和机器学习领域的主要挑战之一。

为了应对这一挑战，许多数据降维的方法被提出并得到了广泛应用。

在数据降维的方法中，主成分分析（Principal Component Analysis，简称PCA）是其中最具代表性的一种方法。

PCA通过线性变换将高维的特征空间映射到低维的特征空间中，从而实现数据降维。

其基本思想在于找到能够最大程度保留原始数据信息的特征轴，将其作为新的特征空间的主成分，从而达到数据降维的目的。

在R语言中，我们可以使用现有的函数库来实现主成分分析。

R语言是一门功能强大的数据分析和统计建模的编程语言，在数据科学和机器学习领域非常受欢迎。

下面，我们将详细介绍在多元数据降维中如何使用R语言进行基于主成分分析的应用研究。

首先，我们需要加载R语言中的相关函数库，以便使用主成分分析算法。

可以使用以下代码加载常用的函数库：```R# 加载常用函数库library(stats) # 统计函数库library(factoextra) # PCA可视化函数库```接下来，我们需要准备数据集，作为主成分分析的输入。

在多元数据降维中，我们通常使用一个矩阵或数据框来表示多个变量的数据。

假设我们有一个数据集`data`，其中包含了`n`个样本和`p`个变量。

可以使用以下代码读取数据集：```R# 读取数据集data <- read.csv("data.csv") # 将文件名替换为实际的数据文件名```在数据准备完成后，我们可以使用`prcomp()`函数进行主成分分析。

该函数会返回一个PCA对象，其中包含了主成分分析的结果。

可以通过以下代码应用主成分分析：```R# 进行主成分分析pca <- prcomp(data, scale = TRUE) # 对数据进行标准化处理# 查看主成分分析结果print(pca)```在主成分分析完成后，我们可以使用`fviz_eig()`函数来可视化PCA的方差解释比例。

R语言数据分析回归研究案例：移民政策偏好是否有准确的刻板印象？数据重命名，重新编码，重组Group <chr> Count<dbl>Percent<dbl>6 476 56.00 5 179 21.062 60 7.063 54 6.354 46 5.41 1 27 3.18 0 8 0.94对Kirkegaard＆Bjerrekær2016的再分析确定用于本研究的32个国家的子集的总体准确性。

#降低样本的＃精确度GG_scatter(dk_fiscal, "mean_estimate", "dk_benefits_use",GG_scatter(dk_fiscal_sub, "mean_estimate", "dk_benefits_us e", case_names="Names")GG_scatter(dk_fiscal, "mean_estimate", "dk_fiscal", case_n ames="Names")#compare Muslim bias measures#can we make a bias measure that works without ratio scaleScore stereotype accuracy#add metric to main datad$stereotype_accuracy=indi_accuracy$pearson_rGG_save("figures/aggr_retest_stereotypes.png")GG_save("figures/aggregate_accuracy.png")GG_save("figures/aggregate_accuracy_no_SYR.png")Muslim bias in aggregate dataGG_save("figures/aggregate_muslim_bias.png")Immigrant preferences and stereotypesGG_save("figures/aggregate_muslim_bias_old_data.png") Immigrant preferences and stereotypesGG_save("figures/aggr_fiscal_net_opposition_no_SYR.png")GG_save("figures/aggr_stereotype_net_opposition.png")GG_save("figures/aggr_stereotype_net_opposition_no_SYR.pn g")lhs <chr>op<chr > rhs <chr> est <dbl> se <dbl> z <dbl> pvalue <dbl> net_opposition ~ mean_estimate_fiscal -4.4e-01 0.02303 -19.17 0.0e+00net_opposition~Muslim_frac 4.3e-02 0.05473 0.79 4.3e-01net_opposition~~net_opposition 6.9e-03 0.00175 3.94 8.3e-05dk_fiscal ~~ dk_fiscal 6.2e+03 0.00000 NA NAMuslim_frac~~Muslim_frac1.7e-01 0.0000NANAIndividual level modelsGG_scatter(example_muslim_bias, "Muslim", "resid", case_na mes="name")+#exclude Syria#distributiondescribe(d$Muslim_bias_r)%>%print()GG_save("figures/muslim_bias_dist.png")## `stat_bin()` using `bins = 30`. Pick better value with `GG_scatter(mediation_example, "Muslim", "resid", case_name s="name", repel_names=T)+scale_x_continuous("Muslim % in home country", labels=scal#stereotypes and preferencesmediation_model=plyr::ldply(seq_along_rows(d), function(rGG_denhist(mediation_model, "Muslim_resid_OLS", vline=medi an)## `stat_bin()` using `bins = 30`. Pick better value with `add to main datad$Muslim_preference=mediation_model$Muslim_resid_OLS Predictors of individual primary outcomes#party modelsrms::ols(stereotype_accuracy~party_vote, data=d)GG_group_means(d, "Muslim_bias_r", "party_vote")+ theme(axis.text.x=element_text(angle=-30, hjust=0))GG_group_means(d, "Muslim_preference", "party_vote")+#party agreement cors wtd.cors(d_parties)。

R语言线性回归案例数据分析可视化报告在本实验中，我们将查看来自所有30个职业棒球大联盟球队的数据，并检查一个赛季的得分与其他球员统计数据之间的线性关系。

我们的目标是通过图表和数字总结这些关系，以便找出哪个变量（如果有的话）可以帮助我们最好地预测一个赛季中球队的得分情况。

数据用变量at_bats绘制这种关系作为预测。

关系看起来是线性的吗？如果你知道一个团队的at_bats，你会习惯使用线性模型来预测运行次数吗？散点图.如果关系看起来是线性的，我们可以用相关系数来量化关系的强度。

.残差平方和回想一下我们描述单个变量分布的方式。

回想一下，我们讨论了中心，传播和形状等特征。

能够描述两个数值变量（例如上面的runand at_bats）的关系也是有用的。

从前面的练习中查看你的情节，描述这两个变量之间的关系。

确保讨论关系的形式，方向和强度以及任何不寻常的观察。

正如我们用均值和标准差来总结单个变量一样，我们可以通过找出最符合其关联的线来总结这两个变量之间的关系。

使用下面的交互功能来选择您认为通过点云的最佳工作的线路。

# Click two points to make a line.After running this command, you’ll be prompted to click two points on the plot to define a line. Once you’ve done that, the line you specified will be shown in black and the residuals in blue. Note that there are 30 residuals, one for each of the 30 observations. Recall that the residuals are the difference between the observed values and the values predicted by the line:e i=y i−y^i ei=yi−y^iThe most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, you can rerun the plot command and add the argument showSquares = TRUE.## Click two points to make a line.Note that the output from the plot_ss function provides you with the slope and intercept of your line as well as the sum of squares.Run the function several times. What was the smallest sum of squares that you got? How does it compare to your neighbors?Answer: The smallest sum of squares is 123721.9. It explains the dispersion from mean. The linear modelIt is rather cumbersome to try to get the correct least squares line, i.e. the line that minimizes the sum of squared residuals, through trial and error. Instead we can use the lm function in R to fit the linear model (a.k.a. regression line).The first argument in the function lm is a formula that takes the form y ~ x. Here it can be read that we want to make a linear model of runs as a function of at_bats. The second argument specifies that R should look in the mlb11 data frame to find the runs and at_bats variables.The output of lm is an object that contains all of the information we need about the linear model that was just fit. We can access this information using the summary function.Let’s consider this output piece by piece. First, the formula used to describe the model is shown at the top. After the formula you find the five-number summary of the residuals. The “Coefficients” table shown next is key; its first column displays the linear model’s y-intercept and the coefficient of at_bats. With this table, we can write down the least squares regression line for the linear model:y^=−2789.2429+0.6305∗atbats y^=−2789.2429+0.6305∗atbatsOne last piece of information we will discuss from the summary output is the MultipleR-squared, or more simply, R2R2. The R2R2value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 37.3% of the variability in runs is explained by at-bats.output, write the equation of the regression line. What does the slope tell us in thecontext of the relationship between success of a team and its home runs?Answer: homeruns has positive relationship with runs, which means 1 homeruns increase 1.835 times runs.Prediction and prediction errors Let’s create a scatterplot with the least squares line laid on top.The function abline plots a line based on its slope and intercept. Here, we used a shortcut by providing the model m1, which contains both parameter estimates. This line can be used to predict y y at any value of x x. When predictions are made for values of x x that are beyond the range of the observed data, it is referred to as extrapolation and is not usually recommended. However, predictions made within the range of the data are more reliable. They’re also used to compute the residuals.many runs would he or she predict for a team with 5,578 at-bats? Is this an overestimate or an underestimate, and by how much? In other words, what is the residual for thisprediction?Model diagnosticsTo assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.Linearity: You already checked if the relationship between runs and at-bats is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. at-bats. Recall that any code following a # is intended to be a comment that helps understand the code but is ignored by R.6.Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between runs and at-bats?Answer: the residuals has normal linearity of the relationship between runs ans at-bats, which mean is 0.Nearly normal residuals: To check this condition, we can look at a histogramor a normal probability plot of the residuals.7.Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?Answer: Yes.It’s nearly normal.Constant variability:1. Choose another traditional variable from mlb11 that you think might be a goodpredictor of runs. Produce a scatterplot of the two variables and fit a linear model. Ata glance, does there seem to be a linear relationship?Answer: Yes, the scatterplot shows they have a linear relationship..1.How does this relationship compare to the relationship between runs and at_bats?Use the R22 values from the two model summaries to compare. Does your variable seem to predict runs better than at_bats? How can you tell?1. Now that you can summarize the linear relationship between two variables, investigatethe relationships between runs and each of the other five traditional variables. Which variable best predicts runs? Support your conclusion using the graphical andnumerical methods we’ve discussed (for the sake of conciseness, only include output for the best variable, not all five).Answer: The new_obs is the best predicts runs since it has smallest Std. Error, which the points are on or very close to the line.1.Now examine the three newer variables. These are the statistics used by the author of Moneyball to predict a teams success. In general, are they more or less effective at predicting runs that the old variables? Explain using appropriate graphical andnumerical evidence. Of all ten variables we’ve analyzed, which seems to be the best predictor of runs? Using the limited (or not so limited) information you know about these baseball statistics, does your result make sense?Answer: ‘new_slug’ as 87.85% ,‘new_onbase’ as 77.85% ,and ‘new_obs’ as 68.84% are predicte better on ‘runs’ than old variables.1. Check the model diagnostics for the regression model with the variable you decidedwas the best predictor for runs.This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by the faculty and TAs of UCLA Statistics.。

R语言实现偏最小二乘回归法partialleastsquares（PLS）回归原文链接：/?p=8652偏最小二乘回归是一种回归形式。

当使用pls时，新的线性组合有助于解释模型中的自变量和因变量。

在本文中，我们将使用pls在“ Mroz”数据集中预测“收入”。

library(pls);library(Ecdat)data("Mroz")str(Mroz)## 'data.frame': 753 obs. of 18 variables:## $ work : Factor w/ 2 levels "yes","no": 2 2 2 22 2 2 2 22 ...## $ hoursw : int 1610 16561980 4561568 20321440 1 0201458 1600 ...## $ child6 : int 1 0 1 01 0 0 000 ...## $ child618 : int 0 2 3 32 0 2 022 ...## $ agew : int 32 30 35 3431 54 37 544839 ...## $ educw : int 12 12 12 1214 12 16 121212 ...## $ hearnw : num 3.35 1.394.55 1.14.59 ...## $ wagew : num 2.65 2.654.04 3.253.6 4.75.95 9.980 4.15 ...## $ hoursh : int 2708 23103072 19202000 10402670 4 1201995 2100 ...## $ ageh : int 34 30 40 5332 57 37 535243 ...## $ educh : int 12 9 12 1012 11 12 8412 ...## $ wageh : num 4.03 8.443.58 3.5410 ...## $ income : int 16310 2180021040 730027300 19495 21152 1890020405 20425 ...## $ educwm : int 12 7 12 712 14 14 377 ...## $ educwf : int 7 7 7 714 7 7 377 ...## $ unemprate : num 5 11 5 59.5 7.55 5 3 5 ...## $ city : Factor w/ 2 levels "no","yes": 1 2 1 12 2 1 11 1 ...## $ experience: int 14 5 15 67 33 11 352421 ...首先，我们必须通过将数据分为训练和测试集来准备数据。

多元线性回归模型中，如果所有特征一起上，容易造成过拟合使测试数据误差方差过大；因此减少不必要的特征，简化模型是减小方差的一个重要步骤。

除了直接对特征筛选，来也可以进行特征压缩，减少某些不重要的特征系数，系数压缩趋近于0就可以认为舍弃该特征。

岭回归（Ridge Regression）和Lasso回归是在普通最小二乘线性回归的基础上加上正则项以对参数进行压缩惩罚。

首先，对于普通的最小二乘线性回归，它的代价函数是：线性回归RSS通过拟合系数β来使RSS最小。

方法很简单，求偏导利用线性代数解方程组即可。

根据线性代数的理论可知，只要样本量合适，它就存在唯一解，也就是该模型的最优解。

这么做尽管使RSS达到了最小，它还是把所有的特征看作同样重要的程度来求解，并没有做任何特征选择，因此存在过拟合的可能。

岭回归在OLS回归模型的RSS上加上了惩罚项（l2范数），这样代价函数就成为：岭回归的代价函数λ是一个非负的调节参数，可以看到：当λ=0时，此时它与RSS一致，没有起到任何惩罚作用；当λ -> ∞时，它的惩罚项也就是无穷大，而为了使代价函数最小，只能压缩系数β趋近于0。

但是因为λ不可能为无穷大，二次项求偏导时总会保留变量本身，所以事实上它也不可能真正地将某个特征压缩为0。

尽管系数较小可以有效减小方差，但依然留着一大长串特征会使模型不便于解释。

这是岭回归的缺点。

lasso回归的正项则就把二次项改成了一次绝对值（l1范数），具体为：lasso回归的代价函数一次项求导可以抹去变量本身，因此lasso回归的系数可以为0。

这样可以起来真正的特征筛选效果。

无论对于岭回归还是lasso回归，本质都是通过调节λ来实现模型误差vs方差的平衡调整。

训练构建岭回归模型cv.out的图如下：cv.out 当log(λ)为5.+时（log(bestlam)=5.3），MSE最小。

训练构建lasso回归模型。

偏最小二乘法代码讲解偏最小二乘法（Partial Least Squares，简称PLS）是一种多元回归分析方法，它通过同时考虑自变量和因变量之间的相关性，将多个自变量线性组合成少数几个综合变量，然后再将这些综合变量与因变量进行回归分析。

以下是使用Python实现偏最小二乘法的代码示例：```pythonimport numpy as npdef pls_regression(X, Y, num_components):# 数据标准化X_mean = np.mean(X, axis=0)X_std = np.std(X, axis=0)X = (X - X_mean) / X_stdY_mean = np.mean(Y)Y_std = np.std(Y)Y = (Y - Y_mean) / Y_std# 初始化权重矩阵和系数矩阵W = np.zeros((X.shape[1], num_components))C = np.zeros((Y.shape[1], num_components))T = np.zeros((X.shape[0], num_components))P = np.zeros((X.shape[1], num_components))U = np.zeros((Y.shape[0], num_components))B = np.zeros((num_components, num_components))B_orth = np.zeros((num_components, num_components))for i in range(num_components):# 计算回归系数W[:, i] = np.dot(X.T, Y)W[:, i] = W[:, i] / np.linalg.norm(W[:, i])C[:, i] = np.dot(Y.T, W[:, i])P[:, i] = np.dot(X, W[:, i])T[:, i] = np.dot(X, P[:, i])U[:, i] = np.dot(Y, C[:, i])# 计算回归系数矩阵B[i, i] = np.dot(T[:, i].T, U[:, i]) / np.dot(T[:, i].T, T[:, i])Y = Y - np.dot(T[:, i].reshape(-1, 1), C[:, i].reshape(1, -1))# 计算正交化的回归系数矩阵if i > 0:B_orth[i, i-1] = np.linalg.norm(T[:, i-1]) / np.linalg.norm(T[:, i]) T[:, i] = T[:, i] - np.dot(T[:, i-1], B_orth[i, i-1])P[:, i] = P[:, i] - np.dot(P[:, i-1], B_orth[i, i-1])C[:, i] = np.dot(Y.T, T[:, i]) / np.dot(T[:, i].T, T[:, i])# 计算预测值Y_pred = np.dot(T, B.T) * Y_std + Y_meanreturn Y_pred```使用该代码，我们可以对给定的自变量X和因变量Y进行偏最小二乘回归分析。