Quantile
Quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. \(q\)-quantiles are values that partition a finite and ordered set of data into \(q\) subsets of (nearly) equal sizes. There are \(q-1\) of the \(q\)-quantiles, one for each integer \(k\) satisfying \(0<k<q\). The \(k\)-th \(q\)-quantile of a random variable is the value \(x\) such that the probability that a sample or the random variable will be less than \(x\) is at most \(\frac{k}{q}\) and the probability that a sample or the random variable will be more than \(x\) is at most \(\frac{q-k}{q}\).
Quantiles can also be applied to continuous distributions, providing a way to generalize rank statistics to continuous variables. When the cumulative distribution function of a random variable is know, the \(q\)-quantiles are the application of the quantile function to the values \(\{\frac{1}{q},\frac{2}{q},\dots,\frac{q-1}{q}\}\).
Code
In NM Dev, the class Quantile computes the quantiles for a data set. There are 9 different quantile definitions and implementations.
- INVERSE_OF_EMPIRICAL_CDF: the inverse of empirical distribution function
- INVERSE_OF_EMPIRICAL_CDF_WITH_AVERAGING_AT_DISCONTINUITIES: the inverse of empirical distribution function with averaging at discontinuities
- NEAREST_EVEN_ORDER_STATISTICS: the nearest even order statistic as in SAS
- LINEAR_INTERPOLATION_OF_EMPIRICAL_CDF: the linear interpolation of the empirical CDF
- MIDWAY_THROUGH_STEPS_OF_EMPIRICAL_CDF: a piecewise linear function where the knots are the values midway through the steps of the empirical CDF
- MINITAB_SPSS: the definition in Minitab and SPSS
- S: the definition in S
- APPROXIMATELY_MEDIAN_UNBIASED: the resulting quantile estimates are approximately median-unbiased regardless of the distribution of the sample
- APPROXIMATELY_UNBIASED_IF_DATA_IS_NORMAL: the resulting quantile estimates are approximately unbiased for the expected order statistics if the sample is normally distributed
// create an array of doubles for our dataset and quantiles
val values = doubleArrayOf(0.0, 1.0, 2.0, 3.0, 3.0, 3.0, 6.0, 7.0, 8.0, 9.0)
val qs = doubleArrayOf(1e-10, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8,
0.9, 0.95, 1.0)
// APPROXIMATELY_MEDIAN_UNBIASED
println("APPROXIMATELY_MEDIAN_UNBIASED")
// create Quantile object
val quantile1 = Quantile(values, Quantile.QuantileType.APPROXIMATELY_MEDIAN_UNBIASED)
println("Sample size: " + quantile1.N())
for (i in qs) {
println("Q(" + i + ") = " + quantile1.value(i))
}
println()
// "NEAREST_EVEN_ORDER_STATISTICS
println("NEAREST_EVEN_ORDER_STATISTICS")
// create Quantile object
val quantile2 = Quantile(values, Quantile.QuantileType.NEAREST_EVEN_ORDER_STATISTICS)
println("Sample size: " + quantile2.N())
for (i in qs) {
println("Q(" + i + ") = " + quantile2.value(i))
}
APPROXIMATELY_MEDIAN_UNBIASED
Sample size: 10
Q(1.0E-10) = 0.0
Q(0.1) = 0.3666666666666667
Q(0.15) = 0.8833333333333333
Q(0.2) = 1.4
Q(0.3) = 2.4333333333333336
Q(0.4) = 3.0
Q(0.5) = 3.0
Q(0.6) = 4.6
Q(0.7) = 6.566666666666666
Q(0.8) = 7.6
Q(0.9) = 8.633333333333333
Q(0.95) = 9.0
Q(1.0) = 9.0
NEAREST_EVEN_ORDER_STATISTICS
Sample size: 10
Q(1.0E-10) = 0.0
Q(0.1) = 0.0
Q(0.15) = 1.0
Q(0.2) = 1.0
Q(0.3) = 2.0
Q(0.4) = 3.0
Q(0.5) = 3.0
Q(0.6) = 3.0
Q(0.7) = 6.0
Q(0.8) = 7.0
Q(0.9) = 8.0
Q(0.95) = 9.0
Q(1.0) = 9.0