A quantile is a value which divides a frequency distribution such that there is a given proportion of data values below the quantile. For example, the median of a dataset is the

0.5

quantile because half the values are less than or equal to it; and the

0.25

quantile is the

25

th percentile.

nag_double_quantiles (g01amc) uses a modified version of Singleton's ‘median-of-three’ Quicksort algorithm (Singleton (1969)) to determine specified quantiles of a vector of real values. The input vector is partially sorted, as far as is required to compute desired quantiles; for a single quantile, this is much faster than sorting the entire vector. Where necessary, linear interpolation is also carried out to return the values of quantiles which lie between two data points.

4

References

Singleton R C (1969) An efficient algorithm for sorting with minimal storage: Algorithm 347 Comm. ACM 12 185–187

5

Arguments

1: $n$ – IntegerInput

On entry: the number of elements in the input vector rv.

Constraint:

n > 0

2: $rv [n]$ – doubleInput/Output

On entry: the vector whose quantiles are to be determined.

On exit: the order of the elements in rv is not, in general, preserved.

3: $nq$ – IntegerInput

On entry: the number of quantiles requested.

Constraint:

nq > 0

4: $q [nq]$ – const doubleInput

On entry: the quantiles to be calculated, in ascending order. Note that these must be between

0.0

and

1.0

, with

0.0

returning the smallest element and

1.0

the largest.

Constraints:

$0.0 \leq q [i - 1] \leq 1.0$ , for $i = 1, 2, \dots, nq$ ;
$q [i - 1] \leq q [i]$ , for $i = 1, 2, \dots, nq - 1$ .

5: $qv [nq]$ – doubleOutput

On exit:

qv [i - 1]

contains the quantile specified by the value provided in

q [i - 1]

, or an interpolated value if the quantile falls between two data values.

6: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .

On entry, $nq = 〈value〉$ .
Constraint: $nq > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_Q_NOT_ASCENDING: On entry, q was not in ascending order.
NE_Q_OUT_OF_RANGE: On entry, an element of q was less than $0.0$ or greater than $1.0$ .
NE_STACK_OVERFLOW: Internal error. Please contact NAG.

7

Accuracy

Not applicable.

8

Parallelism and Performance

nag_double_quantiles (g01amc) is not threaded in any implementation.

9

Further Comments

The average time taken by nag_double_quantiles (g01amc) is approximately proportional to

n \times (1 + \log (nq))

. The worst case time is proportional to

n^{2}

but this is extremely unlikely to occur.

10

Example

This example computes a list of quantiles from an array of doubles and an array of point values.

NAG Library Function Document

nag_double_quantiles (g01amc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description