R 简明教程

R - Linear Regression

回归分析是一种非常广泛使用的统计工具，可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，其值是通过实验收集的。另一个变量称为响应变量，其值是从预测变量导出的。

Regression analysis is a very widely used statistical tool to establish a relationship model between two variables. One of these variable is called predictor variable whose value is gathered through experiments. The other variable is called response variable whose value is derived from the predictor variable.

在线性回归中，这两个变量通过一个方程相关联，其中这两个变量的指数（幂）为 1。在数学上，当作为图形绘制时，线性关系表示一条直线。任何变量的指数不等于 1 的非线性关系会形成一条曲线。

In Linear Regression these two variables are related through an equation, where exponent (power) of both these variables is 1. Mathematically a linear relationship represents a straight line when plotted as a graph. A non-linear relationship where the exponent of any variable is not equal to 1 creates a curve.

线性回归的通用数学方程为 -

The general mathematical equation for a linear regression is −

y = ax + b

以下是所用参数的描述 -

Following is the description of the parameters used −

  1. y is the response variable.

  2. x is the predictor variable.

  3. a and b are constants which are called the coefficients.

Steps to Establish a Regression

回归的一个简单示例是在知道身高时预测体重。为此，我们需要身高和体重之间的关系。

A simple example of regression is predicting weight of a person when his height is known. To do this we need to have the relationship between height and weight of a person.

创建此关系的步骤为 -

The steps to create the relationship is −

  1. Carry out the experiment of gathering a sample of observed values of height and corresponding weight.

  2. Create a relationship model using the lm() functions in R.

  3. Find the coefficients from the model created and create the mathematical equation using these

  4. Get a summary of the relationship model to know the average error in prediction. Also called residuals.

  5. To predict the weight of new persons, use the predict() function in R.

Input Data

以下是表示观测值的样本数据 -

Below is the sample data representing the observations −

# 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

lm() Function

此函数创建预测变量和响应变量之间的关系模型。

This function creates the relationship model between the predictor and the response variable.

Syntax

线性回归中 lm() 函数的基本语法是 -

The basic syntax for lm() function in linear regression is −

lm(formula,data)

以下是所用参数的描述 -

Following is the description of the parameters used −

  1. formula is a symbol presenting the relation between x and y.

  2. data is the vector on which the formula will be applied.

Create Relationship Model & get the Coefficients

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)

当我们执行上述代码时，会产生以下结果 -

When we execute the above code, it produces the following result −

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept)            x
   -38.4551          0.6746

Get the Summary of the Relationship

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))

当我们执行上述代码时，会产生以下结果 -

When we execute the above code, it produces the following result −

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

predict() Function

Syntax

预测中 predict() 的基本语法是 -

The basic syntax for predict() in linear regression is −

predict(object, newdata)

以下是所用参数的描述 -

Following is the description of the parameters used −

  1. object is the formula which is already created using the lm() function.

  2. newdata is the vector containing the new value for predictor variable.

Predict the weight of new persons

# 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)

当我们执行上述代码时，会产生以下结果 -

When we execute the above code, it produces the following result −

       1
76.22869

Visualize the Regression Graphically

# 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()

当我们执行上述代码时，会产生以下结果 -

When we execute the above code, it produces the following result −

linearregression