回归分析是一种非常广泛使用的统计工具，用于建立两个变量之间的关系模型。 这些变量之一称为预测变量，其值通过实验收集。 另一个变量称为响应变量，其值从预测变量派生。
在线性回归中，这两个变量通过方程相关，其中这两个变量的指数（幂）为1.数学上，线性关系表示当绘制为曲线图时的直线。 任何变量的指数不等于1的非线性关系将创建一条曲线。
线性回归的一般数学方程为 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">y = ax + b
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以下是所使用的参数的描述 -
y是响应变量。
x是预测变量。
a和b被称为系数常数。
建立回归的步骤
回归的简单例子是当人的身高已知时预测人的体重。 为了做到这一点，我们需要有一个人的身高和体重之间的关系。
创建关系的步骤是 -
进行收集高度和相应重量的观测值的样本的实验。
使用R语言中的lm()函数创建关系模型。
从创建的模型中找到系数，并使用这些创建数学方程
获得关系模型的摘要以了解预测中的平均误差。 也称为残差。
为了预测新人的体重，使用R中的predict()函数。
输入数据
下面是代表观察的样本数据 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"># Values of height
151, 174, 138, 186, 128, 136, 179, 163, 152, 131
Values of weight.
63, 81, 56, 91, 47, 57, 76, 72, 62, 48
</pre>
LM()函数
此函数创建预测变量和响应变量之间的关系模型。
语法
线性回归中lm()函数的基本语法是 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">lm(formula,data)
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以下是所使用的参数的说明 -
公式是表示x和y之间的关系的符号。
数据是应用公式的向量。
创建关系模型并获取系数
<pre class="prettyprint notranslate tryit" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
Apply the lm() function.
relation <- lm(y~x)
print(relation)
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当我们执行上面的代码，它产生以下结果 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">Call:
lm(formula = y ~ x)
Coefficients:
(Intercept) x
-38.4551 0.6746
</pre>
获取相关的摘要
<pre class="prettyprint notranslate tryit" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
Apply the lm() function.
relation <- lm(y~x)
print(summary(relation))
</pre>
当我们执行上面的代码，它产生以下结果 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-6.3002 -1.6629 0.0412 1.8944 3.9775
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -38.45509 8.04901 -4.778 0.00139 **
x 0.67461 0.05191 12.997 1.16e-06 ***
Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.253 on 8 degrees of freedom
Multiple R-squared: 0.9548, Adjusted R-squared: 0.9491
F-statistic: 168.9 on 1 and 8 DF, p-value: 1.164e-06
</pre>
predict()函数
语法
线性回归中的predict（）的基本语法是 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">predict(object, newdata)
</pre>
以下是所使用的参数的描述 -
object是已使用lm()函数创建的公式。
newdata是包含预测变量的新值的向量。
预测新人的体重
<pre class="prettyprint notranslate tryit" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"># The predictor vector.
x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
The resposne vector.
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
Apply the lm() function.
relation <- lm(y~x)
Find weight of a person with height 170.
a <- data.frame(x = 170)
result <- predict(relation,a)
print(result)
</pre>
当我们执行上面的代码，它产生以下结果 -
<pre class="result notranslate" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"> 1
76.22869
</pre>
以图形方式可视化回归
<pre class="prettyprint notranslate tryit" style="margin: 15px 0px; padding: 10px 5px; position: relative; width: auto; max-width: 700px; box-sizing: border-box; display: block; line-height: 1.7; background: rgb(239, 239, 239); border-radius: 3px; font-size: 14px; font-family: Consolas, "Courier New", Courier, monospace; overflow-x: auto; border: 1px solid rgb(221, 221, 221); word-wrap: break-word !important; white-space: pre-wrap !important; color: rgb(0, 0, 0); font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"># Create the predictor and response variable.
x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
relation <- lm(y~x)
Give the chart file a name.
png(file = "linearregression.png")
Plot the chart.
plot(y,x,col = "blue",main = "Height & Weight Regression",
abline(lm(x~y)),cex = 1.3,pch = 16,xlab = "Weight in Kg",ylab = "Height in cm")
Save the file.
dev.off()
</pre>
当我们执行上面的代码，它产生以下结果 -
