Laorie - 1 year ago 138
R Question

Building a Cumulative distribution function after Quantile Estimation

I have this data frame with two columns (Y and X).

With

`quantreg`
package i can estimate the quantiles of Y conditional on X.

I am unable to build CUMULATIVE density function of Y conditional on X. Can anybody 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")
``````

After this, how do I generate a Cumulative distribution function (Fx)?

This is my data.frame:

``````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")
``````

Can someone help me with this?

Thanks!

The cumulative distribution function for each `X` is the estimate of `Y` at each quantile `tau` with `X` held fixed. I'm not that experienced with quantile regression, but it seems like the output of `rq` isn't consistent in the sense that `Y` vs. `tau` for a given `X` is not necessarily monotonic.

For example, below is some code to get the cumulative distribution function for four values of `X`:

``````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
``````

Each row of `pred` is the cumulative distribution function for a given `X` value (where the `X` value is in the last column of `pred`). Below, we reshape this data frame to long format for plotting with ggplot.

``````# 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()
``````

You can see in the plot above that the cumulative distribution functions are not monotonic. It's easier to see what's happening if we instead plot the individual regression lines for each of the 101 quantiles. The cumulative distribution functions for each quantile are the vertical slices through the plot below. However, you can see that the regression lines for each quantile often cross each other, which is equivalent to the non-monontonicity (is that a word?) in the plot above.

As I said, I'm not that experienced with quantile regression, but maybe your data is too sparse to get stable slopes for each quantile. In particular, the single value at the left edge is problematic, but you won't get monotonic quantiles even if you remove that value.

``````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) +