quantile

Quantile

Introduction

Given the non-NaN numbers $a_1,a_2,\ldots,a_n$ sorted in ascending order. Then, $a_i$ corresponds to the quantile $(i-0.5)/n$, denoted by $q_i$. For example, for a sorted sequence of 5 numbers, they correspond to $q_1=0.1$, $q_2=0.3$, $q_3=0.5$, $q_4=0.7$ and $q_5=0.9$, respectively. In general, the quantile $q$, where $0\leq q\leq 1$, is given by interpolation:

Usage

x = quantile(A, q)

• q should be a vector of real numbers lying in $[0,1]$.
• When A is a scalar, x is a vector with length length(q). Each element of x is equal to A
• When A is a vector, x is a vector with length length(q) where x(i) contains the quantile q(i) of A.
• When A is a matrix or multi-dimensional array, it takes each vector of A along its first non-singleton dimension, and computes the vector's quantiles. The vector collapses or expands into a vector of length length(q) storing the quantiles.
• size(x,k) == length(q) holds, where k is the first non-singleton dimension of A, whereas size(x,i) == size(A,i) holds for all i not equal to k.
• It gives the same results as prctile(A, q * 100).

x = quantile(A, p, 'all')

• q should be a vector of real numbers lying in $[0,1]$.
• The output x is a vector, where x(i) stores the q(i) quantile of the data set containing all numbers in A.
• x and q have the same lengths.
• It gives the same results as prctile(A, q * 100, 'all').

x = quantile(A, p, dim)

• q should be a vector of real numbers lying in $[0,1]$.
• It takes each vector of A along its dim-th dimension and computes the vector's quantiles. The vector collapses or expands into a vector of lengthlength(q) storing the quantiles.
• size(x,dim) == length(q) hold, whereas size(x,i) == size(A,i) holds for all i not equal to dim.
• It gives the same results as prctile(A, q * 100, dim).

x = quantile(A, p, dimVec)

• q should be a vector of real numbers lying in $[0,1]$.
• It takes all elements contained in each of the spaces spanning across dimensions dimVec(1), dimVec(2), ..., dimVec(end) and compute the elements' quantiles.
• It gives the same results as prctile(A, q * 100, dimVec)

Discussion

When computing a quantile of a set of numbers, all NaN values are removed from the set before performing the computation. If the set is empty after the removal of NaN values, an NaN will be returned as the output.

Examples

• In this example, quantile takes each column (first non-singleton dimension) of a and then computes the column's quantiles. Since q has 4 numbers, each column of x also has length 4.

Input

a=rand(2,3,4)
x=quantile(a,[10 20 30 40] * 0.01)

Output

a(:, :, 1) = 1e-1 ×
 2.0711   9.1147   7.5377
 8.2067   9.4177   0.8654

a(:, :, 2) = 1e-1 ×
 8.2799   7.3872   1.4412
 0.4373   9.6350   7.7508

a(:, :, 3) = 1e-1 ×
 6.6587   4.9619   2.3629
 1.8065   2.5954   2.8663

a(:, :, 4) = 1e-1 ×
 3.7219   3.0862   7.1284
 3.6086   6.9071   6.5927

x(:, :, 1) = 1e-1 ×
 2.0711   9.1147   0.8654
 2.0711   9.1147   0.8654
 2.6847   9.1450   1.5327
 3.9118   9.2056   2.8671

x(:, :, 2) = 1e-1 ×
 0.4373   7.3872   1.4412
 0.4373   7.3872   1.4412
 1.2216   7.6120   2.0721
 2.7901   8.0616   3.3341

x(:, :, 3) = 1e-1 ×
 1.8065   2.5954   2.3629
 1.8065   2.5954   2.3629
 2.2917   2.8321   2.4132
 3.2621   3.3054   2.5139

x(:, :, 4) = 1e-1 ×
 3.6086   3.0862   6.5927
 3.6086   3.0862   6.5927
 3.6200   3.4683   6.6463
 3.6426   4.2325   6.7534

• In this example, quantile takes each row (2nd dimension) of a and then computes the row's quantiles. Since q has 4 numbers, each row of x also has length 4.

Input

a=rand(2,3,4)
x=quantile(a,[10 20 30 40] * 0.01,2)

Output

a(:, :, 1) = 1e-1 ×
 5.4991   5.6710   8.9774
 5.7987   5.5879   4.6090

a(:, :, 2) = 1e-1 ×
 7.5114   8.4010   8.3770
 0.1907   2.0860   4.2590

a(:, :, 3) = 1e-1 ×
 4.0253   6.8010   1.8290
 2.8301   0.9264   4.0676

a(:, :, 4) = 1e-1 ×
 5.7757   7.7034   8.7879
 9.4666   8.2765   8.8977

x(:, :, 1) = 1e-1 ×
 5.4991   5.5163   5.5679   5.6195
 4.6090   4.7069   5.0006   5.2942

x(:, :, 2) = 1e-1 ×
 7.5114   7.5980   7.8577   8.1174
 0.1907   0.3802   0.9488   1.5174

x(:, :, 3) = 1e-1 ×
 1.8290   2.0487   2.7075   3.3664
 0.9264   1.1168   1.6879   2.2590

x(:, :, 4) = 1e-1 ×
 5.7757   5.9685   6.5468   7.1251
 8.2765   8.3386   8.5250   8.7113

• In this example, quantile takes all numbers in a and then computes their quantiles. Since q has 4 numbers, the array a is collapsed into a vector of length 4.

Input

a=rand(2,3,4)
x=quantile(a,[10 20 30 40] * 0.01,'all')

Output

a(:, :, 1) = 1e-1 ×
 6.0797   1.1510   9.2457
 0.5870   5.0227   1.6294

a(:, :, 2) = 1e-1 ×
 0.6660   0.5263   3.3619
 4.9154   3.0374   3.7822

a(:, :, 3) = 1e-1 ×
 1.7551   2.4141   5.7546
 1.4208   5.8595   5.1658

a(:, :, 4) = 1e-1 ×
 7.5946   1.6099   4.2575
 0.4456   7.2544   3.8506

x = 1e-1 × 
 0.5809
 1.2320
 1.6236
 2.4764

• In this example, quantile takes all numbers in a plane spanning the 1st (row) and 2nd (column) dimensions and then computes the numbers' quantiles. The array a has 4 such planes, therefore the output x has length 4 along the direction of the 3rd dimension (pointing into the screen). Since q has 5 numbers, each of the planes collapses into a vector of 5 numbers.

Input

a=rand(2,3,4)
x=quantile(a,[10 20 30 40 50] * 0.01,[1 2])

Output

a(:, :, 1) = 1e-1 ×
 8.1641   0.9487   7.6470
 8.2029   5.4701   8.9867

a(:, :, 2) = 1e-1 ×
 0.0313   1.3611   5.7518
 6.6171   0.6510   7.4514

a(:, :, 3) = 1e-1 ×
 3.5090   9.4290   1.7631
 6.1744   1.7446   8.1828

a(:, :, 4) = 1e-1 ×
 6.0778   2.1679   0.1072
 1.1016   7.9887   2.6365

x(:, :, 1) = 1e-1 ×
 1.4009
 4.1137
 6.1232
 7.4293
 7.9056

x(:, :, 2) = 1e-1 ×
 0.0933
 0.4651
 0.8640
 1.2901
 3.5564

x(:, :, 3) = 1e-1 ×
 1.7465
 1.7576
 2.2869
 3.3344
 4.8417

x(:, :, 4) = 1e-1 ×
 0.2066
 0.8033
 1.4215
 2.0613
 2.4022