Build Accumulated Density Probability using R

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

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

Done, I can not construct the conditional Y-CUMP density function. Could someone help me?

Estimating the quantiles: 

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 cumulative density function?

The dataframe follows: 

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")
    
asked by anonymous 08.10.2016 / 21:51

1 answer

2

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

With the following code you can see some cumulative distributions, for some values of x . This problem soon gets tricky, since you're usually adjusting the quantile regression to more than one variable, different from 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")

Thesearethesurvivalfunctionsestimatedbytheirquantileregressionmodel.Thequantileregressionassumesnodistributionoftheresponsevariable,sothereisnoclosedformulaforthecumulativedistribution.YoucouldassumeadistributiontoYandestimateaparametricmodel.

Forexample:

Ajusteumaregressãolinearnormalpara'Y'dadoX:>reg<-lm(Y~X,data=dados)>summary(reg)Call:lm(formula=Y~X,data=dados)Residuals:Min1QMedian3QMax-8.6213-1.73430.06771.45859.9230Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)0.18000.20170.8920.373X-1.47151.7462-0.8430.400Residualstandarderror:2.461on231degreesoffreedom(1observationdeletedduetomissingness)MultipleR-squared:0.003065,AdjustedR-squared:-0.001251F-statistic:0.7102on1and231DF,p-value:0.4003>sigma(reg)[1]2.461144

Nowthecumulativedistributionfunctionisgivenby:

source: Wikipedia

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

    
10.10.2016 / 15:08
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