在分位數估計之后構建累積分布函數

Question

我有這個包含兩列（Y 和 X）的數據框。

使用quantreg包，我可以以 X 為條件估計 Y 的分位數。

我無法以 X 為條件構建 Y 的累積密度函數。有人可以幫助我嗎？

估計分位數：

library(quantreg)
taus=seq(0.1, 0.9, by = 0.01)
Quantis<-rq(data[,1] ~ data[,2],tau=taus,method="br")

在此之后，如何生成累積分布函數 (Fx)？

這是我的數據框：

structure(list(Y = c(NA, -1.793, -0.642, 1.189, -0.823, -1.715, 
1.623, 0.964, 0.395, -3.736, -0.47, 2.366, 0.634, -0.701, -1.692, 
0.155, 2.502, -2.292, 1.967, -2.326, -1.476, 1.464, 1.45, -0.797, 
1.27, 2.515, -0.765, 0.261, 0.423, 1.698, -2.734, 0.743, -2.39, 
0.365, 2.981, -1.185, -0.57, 2.638, -1.046, 1.931, 4.583, -1.276, 
1.075, 2.893, -1.602, 1.801, 2.405, -5.236, 2.214, 1.295, 1.438, 
-0.638, 0.716, 1.004, -1.328, -1.759, -1.315, 1.053, 1.958, -2.034, 
2.936, -0.078, -0.676, -2.312, -0.404, -4.091, -2.456, 0.984, 
-1.648, 0.517, 0.545, -3.406, -2.077, 4.263, -0.352, -1.107, 
-2.478, -0.718, 2.622, 1.611, -4.913, -2.117, -1.34, -4.006, 
-1.668, -1.934, 0.972, 3.572, -3.332, 1.094, -0.273, 1.078, -0.587, 
-1.25, -4.231, -0.439, 1.776, -2.077, 1.892, -1.069, 4.682, 1.665, 
1.793, -2.133, 1.651, -0.065, 2.277, 0.792, -3.469, 1.48, 0.958, 
-4.68, -2.909, 1.169, -0.941, -1.863, 1.814, -2.082, -3.087, 
0.505, -0.013, -0.12, -0.082, -1.944, 1.094, -1.418, -1.273, 
0.741, -1.001, -1.945, 1.026, 3.24, 0.131, -0.061, 0.086, 0.35, 
0.22, -0.704, 0.466, 8.255, 2.302, 9.819, 5.162, 6.51, -0.275, 
1.141, -0.56, -3.324, -8.456, -2.105, -0.666, 1.707, 1.886, -3.018, 
0.441, 1.612, 0.774, 5.122, 0.362, -0.903, 5.21, -2.927, -4.572, 
1.882, -2.5, -1.449, 2.627, -0.532, -2.279, -1.534, 1.459, -3.975, 
1.328, 2.491, -2.221, 0.811, 4.423, -3.55, 2.592, 1.196, -1.529, 
-1.222, -0.019, -1.62, 5.356, -1.885, 0.105, -1.366, -1.652, 
0.233, 0.523, -1.416, 2.495, 4.35, -0.033, -2.468, 2.623, -0.039, 
0.043, -2.015, -4.58, 0.793, -1.938, -1.105, 0.776, -1.953, 0.521, 
-1.276, 0.666, -1.919, 1.268, 1.646, 2.413, 1.323, 2.135, 0.435, 
3.747, -2.855, 4.021, -3.459, 0.705, -3.018, 0.779, 1.452, 1.523, 
-1.938, 2.564, 2.108, 3.832, 1.77, -3.087, -1.902, 0.644, 8.507
), X = c(0.056, 0.053, 0.033, 0.053, 0.062, 0.09, 0.11, 0.124, 
0.129, 0.129, 0.133, 0.155, 0.143, 0.155, 0.166, 0.151, 0.144, 
0.168, 0.171, 0.162, 0.168, 0.169, 0.117, 0.105, 0.075, 0.057, 
0.031, 0.038, 0.034, -0.016, -0.001, -0.031, -0.001, -0.004, 
-0.056, -0.016, 0.007, 0.015, -0.016, -0.016, -0.053, -0.059, 
-0.054, -0.048, -0.051, -0.052, -0.072, -0.063, 0.02, 0.034, 
0.043, 0.084, 0.092, 0.111, 0.131, 0.102, 0.167, 0.162, 0.167, 
0.187, 0.165, 0.179, 0.177, 0.192, 0.191, 0.183, 0.179, 0.176, 
0.19, 0.188, 0.215, 0.221, 0.203, 0.2, 0.191, 0.188, 0.19, 0.228, 
0.195, 0.204, 0.221, 0.218, 0.224, 0.233, 0.23, 0.258, 0.268, 
0.291, 0.275, 0.27, 0.276, 0.276, 0.248, 0.228, 0.223, 0.218, 
0.169, 0.188, 0.159, 0.156, 0.15, 0.117, 0.088, 0.068, 0.057, 
0.035, 0.021, 0.014, -0.005, -0.014, -0.029, -0.043, -0.046, 
-0.068, -0.073, -0.042, -0.04, -0.027, -0.018, -0.021, 0.002, 
0.002, 0.006, 0.015, 0.022, 0.039, 0.044, 0.055, 0.064, 0.096, 
0.093, 0.089, 0.173, 0.203, 0.216, 0.208, 0.225, 0.245, 0.23, 
0.218, -0.267, 0.193, -0.013, 0.087, 0.04, 0.012, -0.008, 0.004, 
0.01, 0.002, 0.008, 0.006, 0.013, 0.018, 0.019, 0.018, 0.021, 
0.024, 0.017, 0.015, -0.005, 0.002, 0.014, 0.021, 0.022, 0.022, 
0.02, 0.025, 0.021, 0.027, 0.034, 0.041, 0.04, 0.038, 0.033, 
0.034, 0.031, 0.029, 0.029, 0.029, 0.022, 0.021, 0.019, 0.021, 
0.016, 0.007, 0.002, 0.011, 0.01, 0.01, 0.003, 0.009, 0.015, 
0.018, 0.017, 0.021, 0.021, 0.021, 0.022, 0.023, 0.025, 0.022, 
0.022, 0.019, 0.02, 0.023, 0.022, 0.024, 0.022, 0.025, 0.025, 
0.022, 0.027, 0.024, 0.016, 0.024, 0.018, 0.024, 0.021, 0.021, 
0.021, 0.021, 0.022, 0.016, 0.015, 0.017, -0.017, -0.009, -0.003, 
-0.012, -0.009, -0.008, -0.024, -0.023)), .Names = c("Y", "X"
), row.names = c(NA, -234L), class = "data.frame")

Answer 1

每個X的累積分布函數是每個分位數tau上Y的估計值，其中X保持固定。 我對分位數回歸沒有經驗，但似乎rq的輸出不一致，因為給定X Y與tau不一定是單調的。

例如，下面是一些代碼來獲取X四個值的累積分布函數：

library(quantreg)
library(reshape2)
library(ggplot2)

taus=seq(0,1,0.01)
Quantis<-rq(Y ~ X, tau=taus, method="br", data=data)

# Get predictions of Y for four X values for all taus
X = seq(min(data$X), max(data$X), length.out=4)
pred = as.data.frame(predict(Quantis, newdata=data.frame(X)))
pred$X = X

pred每一行都是給定X值的累積分布函數（其中X值位於pred的最后一列）。 下面，我們將此數據框重塑為長格式，以便使用 ggplot 進行繪圖。

# Reshape predictions to long format
pred = melt(pred, id.var="X", variable.name="tau", value.name="Y")
pred$tau = as.numeric(gsub("tau= (.*)", "\\1", pred$tau))

# Plot predictions
ggplot(pred, aes(x=Y, y=tau, color=factor(X))) +
  geom_step() + theme_bw()

您可以在上圖中看到累積分布函數不是單調的。 如果我們為 101 個分位數（即tau的 101 個值）中的每一個繪制單獨的回歸線，則更容易看出發生了什么。 Y在每個X處的累積分布函數是通過下圖的垂直切片。 但是，您可以看到每個分位數的回歸線經常相互交叉，這相當於上圖中的非單調性（這是一個詞嗎？）。

為了相互一致，分位數回歸線不應相互交叉。 正如我所說，我對分位數回歸沒有那么豐富的經驗，但也許您的數據太稀疏而無法為每個分位數獲得穩定的斜率。 特別是，左邊緣的單個值有問題，但即使刪除該值，也不會得到單調分位數。

ggplot(pred, aes(x=X, y=Y, colour=tau, group=factor(tau))) +
  geom_point(data=data, aes(X,Y), inherit.aes=FALSE) +
  geom_line(show.legend=FALSE, alpha=0.7) +
  scale_colour_gradient2(low="red", mid="yellow", high="blue", midpoint=0.5) +
  theme_bw()