Build Accumulated Density Probability using R

Question

I have this two-column date frame (Y and X)

With the package quantreg I can estimate the amounts of Y given x.

Done this I’m not managing to build the CUMULATIVE Y conditional density function. Could someone help me?

Estimating the amounts:

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

Now, how do I generate my accumulated density function?

Follows the dataframe:

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

The cumulative distribution function, in its case, is a function of x. Therefore, there is not only a cumulative density function, but one for each possible x.

With the following code you can view some accumulated distributions, for some svalores of x. This problem soon becomes complicated, since you are usually adjusting quantile regression to more than one variable, unlike your example.

library(tidyverse)
df <- data.frame(X = c(0, 0.3))
bind_cols(
  df, 
  predict(Quantis, df) %>% as.data.frame()              
) %>%
  gather(quantil, valor, starts_with("tau")) %>%
  mutate(quantil = parse_number(quantil)) %>%
  ggplot(aes(x = valor, y = quantil, colour = as.factor(X))) + 
  geom_line() + 
  labs(y = "F(y)", x = "y", colour = "X")

These are the survival functions estimated by your quantile regression model. Quantile regression assumes no distribution of the response variable, so there is no closed formula for the cumulative distribution. You could take a distribution to Y and estimate a parametric model.

For example:

Ajuste uma regressão linear normal para `Y` dado X:

> reg <- lm(Y ~ X, data = dados)
> summary(reg)

Call:
lm(formula = Y ~ X, data = dados)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6213 -1.7343  0.0677  1.4585  9.9230 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.1800     0.2017   0.892    0.373
X            -1.4715     1.7462  -0.843    0.400

Residual standard error: 2.461 on 231 degrees of freedom
  (1 observation deleted due to missingness)
Multiple R-squared:  0.003065,  Adjusted R-squared:  -0.001251 
F-statistic: 0.7102 on 1 and 231 DF,  p-value: 0.4003

> sigma(reg)
[1] 2.461144

Now the cumulative distribution function is given by:

source: Wikipedia

The cumulative density function in your example would be given by the formula above, replacing u for 0.1800 - 1.4715*x and sigma for 2.461144.

Estimates of the cumulative distribution function using this model would be:

mu <- function(x){
  0.1800 - 1.4715*x
}
bind_rows(
  data.frame(
    X = 0, 
    quantil = seq(0,1,by = 0.01),
    valor = qnorm(seq(0,1,by = 0.01), mean = mu(0), sd = sigma(reg))
    ),
  data.frame(
    X = 0.3, 
    quantil = seq(0,1,by = 0.01),
    valor = qnorm(seq(0,1,by = 0.01), mean = mu(0.3), sd = sigma(reg))
  )
) %>%
  ggplot(aes(x = valor, y = quantil, colour = as.factor(X))) + 
  geom_line() + 
  labs(y = "F(y)", x = "y", colour = "X")

The function qnorm of R is exactly the cumulative distribution function of Normal.