在R中导出LDA/QDA划分线方程并同图绘制的方法咨询

下面提供基于Base R和ggplot2的自定义方案，同时推导并导出LDA（线性）和QDA（二次）的划分线方程。

Base R 实现（自定义性强） #

核心思路是生成覆盖数据范围的网格点，计算每个点的后验概率，再绘制概率等于0.5的轮廓线（即划分边界）。

library(MASS)

# 原始数据与模型构建
set.seed(123)
Ng <- 100 
group.a.x <- rnorm(n = Ng, mean = 2, sd = 3)
group.a.y <- rnorm(n = Ng, mean = 2, sd = 3)
group.b.x <- rnorm(n = Ng, mean = 11, sd = 3)
group.b.y <- rnorm(n = Ng, mean = 11, sd = 3)
group.a <- data.frame(x = group.a.x, y = group.a.y, group = "A")
group.b <- data.frame(x = group.b.x, y = group.b.y, group = "B")
my.xy <- rbind(group.a, group.b)
mdlLDA <- lda(group ~ x + y, data = my.xy)
mdlQDA <- qda(group ~ x + y, data = my.xy)

# 1. 创建密集网格数据
x_range <- range(my.xy$x)
y_range <- range(my.xy$y)
grid <- expand.grid(
  x = seq(x_range[1], x_range[2], length.out = 200),
  y = seq(y_range[1], y_range[2], length.out = 200)
)

# 2. 计算每个网格点的后验概率
grid$lda_prob <- predict(mdlLDA, newdata = grid)$posterior[, "A"]
grid$qda_prob <- predict(mdlQDA, newdata = grid)$posterior[, "A"]

# 3. 绘制图形
plot(my.xy$x, my.xy$y, col = my.xy$group, pch = 16, 
     xlab = "x", ylab = "y", main = "LDA vs QDA 划分线对比")
# 添加LDA划分线（A类后验概率=0.5的边界）
contour(grid$x, grid$y, matrix(grid$lda_prob, 200, 200), 
        levels = 0.5, add = TRUE, col = "red", lwd = 2, lty = 1)
# 添加QDA划分线
contour(grid$x, grid$y, matrix(grid$qda_prob, 200, 200), 
        levels = 0.5, add = TRUE, col = "blue", lwd = 2, lty = 2)
# 添加图例
legend("topleft", legend = c("LDA 划分线", "QDA 划分线", "Group A", "Group B"),
       col = c("red", "blue", "black", "red"), lwd = c(2,2,NA,NA), 
       pch = c(NA,NA,16,16), lty = c(1,2,NA,NA))

ggplot2 实现（简洁易自定义） #

利用geom_contour直接绘制概率边界，配合ggplot的主题系统轻松调整样式：

library(ggplot2)
library(tidyr)

# 延续上面的grid数据，转换为长格式
grid_long <- pivot_longer(grid, cols = c(lda_prob, qda_prob), 
                          names_to = "model", values_to = "prob")

# 可视化
ggplot() +
  geom_point(data = my.xy, aes(x = x, y = y, color = group), size = 2) +
  geom_contour(data = grid_long, aes(x = x, y = y, z = prob, color = model, linetype = model),
               breaks = 0.5, linewidth = 1.2) +
  scale_color_manual(values = c("A" = "black", "B" = "red", "lda_prob" = "darkred", "qda_prob" = "darkblue")) +
  scale_linetype_manual(values = c("lda_prob" = 1, "qda_prob" = 2)) +
  labs(x = "x", y = "y", title = "LDA vs QDA 划分线对比", color = "类别/模型", linetype = "模型") +
  theme_minimal()

导出划分线方程 #

划分线的本质是两类后验概率相等的边界，我们可以通过模型参数直接推导方程：

LDA 线性划分线方程 #

LDA假设两类共享协方差矩阵，决策边界是直线，令两类判别函数相等即可推导：

# 提取LDA参数
mu_A <- mdlLDA$means[1,]
mu_B <- mdlLDA$means[2,]
cov_mat <- cov(my.xy[,1:2])
inv_cov <- solve(cov_mat)
pi_A <- sum(my.xy$group == "A")/nrow(my.xy)
pi_B <- 1 - pi_A

# 计算线性方程系数
coef_x <- inv_cov[1,1]*(mu_A[1] - mu_B[1]) + inv_cov[1,2]*(mu_A[2] - mu_B[2])
coef_y <- inv_cov[2,1]*(mu_A[1] - mu_B[1]) + inv_cov[2,2]*(mu_A[2] - mu_B[2])
intercept <- -0.5*(t(mu_A) %*% inv_cov %*% mu_A - t(mu_B) %*% inv_cov %*% mu_B) + log(pi_A/pi_B)

# 输出方程
cat("LDA划分线方程：", round(coef_x,4), "*x +", round(coef_y,4), "*y +", round(intercept,4), "= 0\n")
cat("整理为y关于x的表达式：y =", round(-coef_x/coef_y,4), "*x +", round(-intercept/coef_y,4), "\n")

QDA 二次划分线方程 #

QDA允许两类有独立协方差矩阵，决策边界是二次曲线，推导如下：

# 提取QDA参数
cov_A <- cov(my.xy[my.xy$group == "A",1:2])
cov_B <- cov(my.xy[my.xy$group == "B",1:2])
inv_cov_A <- solve(cov_A)
inv_cov_B <- solve(cov_B)

# 计算二次项、一次项和常数项系数
coef_x2 <- 0.5*(inv_cov_B[1,1] - inv_cov_A[1,1])
coef_xy <- (inv_cov_B[1,2] - inv_cov_A[1,2])
coef_y2 <- 0.5*(inv_cov_B[2,2] - inv_cov_A[2,2])
coef_x <- (inv_cov_A %*% mu_A)[1] - (inv_cov_B %*% mu_B)[1]
coef_y <- (inv_cov_A %*% mu_A)[2] - (inv_cov_B %*% mu_B)[2]
intercept <- -0.5*(t(mu_A) %*% inv_cov_A %*% mu_A - t(mu_B) %*% inv_cov_B %*% mu_B) - 
  0.5*log(det(cov_A)/det(cov_B)) + log(pi_A/pi_B)

# 输出方程
cat("QDA划分线方程：", round(coef_x2,4), "*x² +", round(coef_xy,4), "*xy +", round(coef_y2,4), "*y² +",
    round(coef_x,4), "*x +", round(coef_y,4), "*y +", round(intercept,4), "= 0\n")

内容的提问来源于stack exchange，提问作者Alex