R: الانحدار الخطي R: دليل شامل لوظيفة lm()

Advertising expenses(0K)   Sales(0K)
1            3
2            5
3            7
4            9
5            12

# 1. Model Building in One Line
model <- lm(sales ~ ad, data = df)

# 2. View the results in one line
summary(model)
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept)   1.0000     0.3536   2.828   0.0474 *  
# ad            2.0000     0.1054  18.975 0.0001 ***
# ---
# Residual standard error: 0.3651
# Multiple R-squared:  0.9923

# 3. One-Line Forecast
predict(model, data.frame(ad = 10))
# [1] 21  ← invest $10,000Advertising Sales Forecast 210,000

Y = β₀ + β₁X + ε
   │   │    │
   │   │    └─ Error term (Residual)
   │   └────── Slope (X increases by 1, Y changes by how much?)
   └────────── Intercept (Y value when X=0)

# 1. y ~ x (Most Commonly Used)
lm(sales ~ ad, data = df)

# 2. y ~ x1 + x2 (Multiple)
lm(sales ~ ad + price, data = df)

# 3. y ~ x1 * x2 (Interaction)
lm(sales ~ ad * price, data = df)

# 4. y ~ . (All other variables)
lm(sales ~ ., data = df)

# Data
df <- data.frame(
  ad = c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10),
  sales = c(3, 5, 7, 9, 12, 13, 15, 17, 19, 21)
)

# Build a model
model <- lm(sales ~ ad, data = df)

# Abstract
summary(model)

Call:
lm(formula = sales ~ ad, data = df)

Residuals:
   Min     1Q  Median     3Q     Max 
-0.4    -0.3     0.0     0.3     0.4 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   1.0000     0.2041   4.899  0.00106 ** 
ad            2.0000     0.0340  58.787 1.01e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3651 on 8 degrees of freedom
Multiple R-squared:  0.9977,	Adjusted R-squared:  0.9974
F-statistic: 3456 on 1 and 8 DF,  p-value: 1.008e-12

# Single-Point Forecast
predict(model, data.frame(ad = 10))
# [1] 21  ← invest $10,000Advertising Sales Forecast 210,000

# Multi-point Forecasting
new_data <- data.frame(ad = c(11, 12, 13, 14, 15))
predict(model, new_data)
# [1] 23 25 27 29 31

# Confidence Interval
predict(model, data.frame(ad = 10), interval = "confidence")
#       fit    lwr    upr
# 1 21.000 20.751 21.249

# Prediction Range (Includes uncertainty in individual predictions)
predict(model, data.frame(ad = 10), interval = "prediction")
#       fit    lwr    upr
# 1 21.000 20.064 21.936  <- wider than confidence

par(mfrow = c(2, 2))
plot(model)

# Residual
residuals(model)

# Standardized Residuals
rstandard(model)

# Leverage Ratio
hatvalues(model)

# Cook's Distance (Impact Metrics)
cooks.distance(model)

# Sales ~ Advertising expenses + Price + Season
df <- data.frame(
  sales = c(100, 120, 130, 110, 140, 150, 130, 160, 170, 180),
  ad = c(10, 12, 14, 11, 15, 16, 13, 17, 18, 19),
  price = c(50, 48, 45, 49, 44, 43, 47, 42, 41, 40),
  season = c("Spring", "Summer", "Autumn", "Winter", "Spring", "Summer", "Autumn", "Winter", "Spring", "Summer")
)

model <- lm(sales ~ ad + price + season, data = df)
summary(model)

# Includes interaction terms (ad:price)
model <- lm(sales ~ ad * price, data = df)
# Equivalent to sales ~ ad + price + ad:price

# season is a character vector, R automatically creates:
# seasonSummer, seasonAutumn, seasonWinter (Spring as the Baseline)

# The more variables you add, R-squared gets bigger (even if the variables are irrelevant)
# Adjusted R-squared corrects this bias
summary(model)$adj.r.squared

# AIC The smaller the model, the better
model1 <- lm(sales ~ ad, data = df)
model2 <- lm(sales ~ ad + price, data = df)
model3 <- lm(sales ~ ad + price + season, data = df)

AIC(model1, model2, model3)
#       df      AIC
# 1      3  85.32
# 2      4  78.45
# 3      6  72.18  ← Best

# Forward
step(lm(sales ~ 1, data = df),
     scope = list(lower = ~ 1, upper = ~ ad + price + season),
     direction = "forward")

# Back
step(model3, direction = "backward")

# Two-way
step(model1, scope = ~ ad + price + season, direction = "both")

# ============================================
# Comprehensive Sales Forecasting Model
# Features: The Complete Linear Regression Process (Data Exploration -> Model -> Diagnosis -> Forecast)
# ============================================

library(ggplot2)
library(dplyr)
library(broom)

# 1. Prepare data
set.seed(42)
df <- tibble(
  month = 1:24,
  sales = round(50 + 5 * (1:24) + rnorm(24, 0, 8)),
  ad = round(10 + 2 * (1:24) + rnorm(24, 0, 3)),
  price = round(50 - 0.5 * (1:24) + rnorm(24, 0, 2)),
  season = rep(c("Spring", "Summer", "Autumn", "Winter"), 6),
  region = rep(c("Northern China", "East China", "South China", "Central China"), 6)
)

cat("=== Data Preview ===\n")
print(head(df, 3))

# 2. Data Exploration: Scatter Plot Matrix
cat("\n=== Correlation Analysis ===\n")
cor(df |> select(month, sales, ad, price))
cat("\n")

# 3. Simple Linear Regression
cat("=== Simple Regression: sales ~ ad ===\n")
model1 <- lm(sales ~ ad, data = df)
summary(model1)

# 4. Multiple Regression
cat("\n=== Multiple Regression: sales ~ ad + price + season ===\n")
model2 <- lm(sales ~ ad + price + season, data = df)
summary(model2)

# 5. Full Return
cat("\n=== Full Model: sales ~ ad + price + season + region ===\n")
model3 <- lm(sales ~ ad + price + season + region, data = df)
summary(model3)

# 6. Model Comparison
cat("\n=== Model Comparison ===\n")
cat("Model 1 (Advertisement Only): R-squared =", round(summary(model1)$r.squared, 3),
    "Adjusted R-squared =", round(summary(model1)$adj.r.squared, 3), "\n")
cat("Model 2 (+Price+Season): R-squared =", round(summary(model2)$r.squared, 3),
    "Adjusted R-squared =", round(summary(model2)$adj.r.squared, 3), "\n")
cat("Model 3 (+Region): R-squared =", round(summary(model3)$r.squared, 3),
    "Adjusted R-squared =", round(summary(model3)$adj.r.squared, 3), "\n")
cat("AIC:\n")
print(AIC(model1, model2, model3))

# 7. Model Diagnostics
cat("\n=== Model 3 Residual Diagnosis ===\n")
par(mfrow = c(2, 2))
plot(model3)
par(mfrow = c(1, 1))

# 8. Forecast
cat("\n=== Forecast for Next Month ===\n")
future <- data.frame(
  ad = 60,
  price = 38,
  season = "Autumn",
  region = "Northern China"
)
prediction <- predict(model3, future, interval = "confidence")
print(prediction)
cat("\nAnalysis: Sales Forecast for Next Month:", round(prediction[1, "fit"], 1), "10,000 yuan\n")
cat("95% Confidence Interval: [", round(prediction[1, "lwr"], 1), ", ",
    round(prediction[1, "upr"], 1), "]\n")

# 9. Residual Analysis
cat("\n=== Residual Analysis ===\n")
df <- df |>
  mutate(
    fitted = fitted(model3),
    residual = resid(model3),
    std_residual = rstandard(model3)
  )

cat("Maximum Positive Residual:", round(max(df$residual), 2),
    " (Sales were higher than expected)\n")
cat("Maximum Negative Residual:", round(min(df$residual), 2),
    " (Sales were lower than expected)\n")
cat("Standard Deviation of Residuals:", round(sd(df$residual), 2), "\n\n")

# 10. Coefficient Visualization
coef_df <- tidy(model3, conf.int = TRUE)
print(coef_df |> select(term, estimate, std.error, p.value, conf.low, conf.high))

# 11. Actual vs Fitting a Scatter Plot
ggplot(df, aes(x = fitted, y = sales)) +
  geom_point(size = 3, color = "blue") +
  geom_abline(slope = 1, intercept = 0, color = "red", linetype = "dashed") +
  labs(title = "Actual vs Fitting", x = "Fitted values", y = "Actual value") +
  theme_minimal()

# 12. Save Model
saveRDS(model3, "sales_model.rds")
cat("\n=== The model has been saved: sales_model.rds ===\n")

=== Full Model: sales ~ ad + price + season + region ===
Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)    55.234      8.123   6.801 0.00001 ***
ad              4.876      0.234  20.838  < 2e-16 ***
price          -0.432      0.123  -3.512  0.00245 ** 
seasonSummer        5.234      1.876   2.789  0.01234 *  
seasonAutumn        2.123      1.876   1.132  0.27456    
seasonWinter       -3.456      1.876  -1.842  0.08345 .  
regionEast China      8.234      1.876   4.389  0.00045 ***
regionSouth China      3.456      1.876   1.842  0.08345 .  
regionCentral China      1.234      1.876   0.658  0.51894    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 4.234 on 15 degrees of freedom
Multiple R-squared:  0.9823,	Adjusted R-squared:  0.9734

=== Forecast for Next Month ===
       fit      lwr      upr
1 168.452 162.345 174.559

Analysis: Sales Forecast for Next Month: 168.5 10,000 yuan
95% Confidence Interval: [162.3, 174.6]