Simple_Linear_Regression_R.rmd

---
title: "簡單線性迴歸 (Simple Linear Regression)"
author: "JianKai Wang (https://sophia.ddns.net/)"
date: "2018年3月14日"
output: html_document
---

## 準備資料

底下模擬鱷魚的長度與重量資料。

```{r}
alligator = data.frame(
  lnLength = c(3.87, 3.61, 4.33, 3.43, 3.81, 3.83, 3.46, 3.76, 3.50, 3.58, 4.19, 3.78, 3.71, 3.73, 3.78),
  lnWeight = c(4.87, 3.93, 6.46, 3.33, 4.38, 4.70, 3.50, 4.50, 3.58, 3.64, 5.90, 4.43, 4.38, 4.42, 4.25)
)

plot(
  alligator$lnLength, 
  alligator$lnWeight,
  xlab = "Snout vent length (inches) on log scale",
  ylab = "Weight (pounds) on log scale",
  main = "Alligators in Central Florida"
)
```

## 建立簡單線性迴歸模型

```r
# the prototype of lm
# subset: 選擇特徵的向量
# weights: 特徵的加權值
# lm(formula, data, subset, weights, na.action, ...)
```

```{r}
set.seed(123)
training_idx <- sample(1:nrow(alligator), 10)
training_data <- alligator[training_idx,]
testing_data <- alligator[-training_idx,]
model = lm(lnWeight ~ lnLength, data = training_data)
summary(model)
```

透過函式 **summary** 可以列出線性迴歸的相關資訊，包含 F-test, $R^2$ 值, 與殘差分析等。

## 繪圖檢視迴歸結果

```{r}
plot(
  training_data$lnLength, 
  training_data$lnWeight,
  xlab = "Snout vent length (inches) on log scale",
  ylab = "Weight (pounds) on log scale",
  main = "Alligators in Central Florida"
)
abline(model)
```

## 自建立方程式並用於迴歸預測

```{r}
# 取得迴歸係數
beta = coef(model)
beta
```

由上可得迴歸方程式為

$$y=3.431098*x-8.476067$$

```{r}
# 建立迴歸方程式
y1 <- function(x1) {
  return(x1 * beta[2] + beta[1])
}

# 建立測試資料
new_length_data <- matrix(c(4.0, 4.1), nrow=1)
new_weight_data <- apply(new_length_data, 1, y1)

# 將新產生資料進行繪圖
plot(
  training_data$lnLength, 
  training_data$lnWeight,
  xlab = "Snout vent length (inches) on log scale",
  ylab = "Weight (pounds) on log scale",
  main = "Alligators in Central Florida"
)
abline(model)
abline(v = 4.0, col="red", lty=2)
abline(v = 4.1, col="red", lty=2)
points(new_length_data, new_weight_data, col="red")
```

### 透過內建函式預測結果

```{r}
lm.pred <- predict(model, testing_data)
data.frame(true=testing_data$lnWeight, pred=lm.pred)
```

### 計算殘差

```{r}
# type 為殘差計算方式，有 working, response, deviance, pearson, partial 等
residuals(model, type="working")
```