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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_quantiles_stream_arbitrary (g01ap)

## Purpose

nag_stat_quantiles_stream_arbitrary (g01ap) finds approximate quantiles from a large arbitrary-sized data stream using an out-of-core algorithm.

## Syntax

[ind, np, qv, rcomm, icomm, ifail] = g01ap(ind, rv, eps, q, rcomm, icomm, 'nb', nb, 'nq', nq, 'lrcomm', lrcomm, 'licomm', licomm)
[ind, np, qv, rcomm, icomm, ifail] = nag_stat_quantiles_stream_arbitrary(ind, rv, eps, q, rcomm, icomm, 'nb', nb, 'nq', nq, 'lrcomm', lrcomm, 'licomm', licomm)

## Description

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$0.5$ quantile because half the values are less than or equal to it.
nag_stat_quantiles_stream_arbitrary (g01ap) uses a slightly modified version of an algorithm described in a paper by Zhang and Wang (2007) to determine ε$\epsilon$-approximate quantiles of a large arbitrary-sized data stream of real values, where ε$\epsilon$ is a user-defined approximation factor. Let m$m$ denote the number of data elements processed so far then, given any quantile q[0.0,1.0]$q\in \left[0.0,1.0\right]$, an ε$\epsilon$-approximate quantile is defined as an element in the data stream whose rank falls within [(qε)m,(q + ε)m] $\left[\left(q-\epsilon \right)m,\left(q+\epsilon \right)m\right]$. In case of more than one ε$\epsilon$-approximate quantile being available, the one closest to qm$qm$ is used.

## References

Zhang Q and Wang W (2007) A fast algorithm for approximate quantiles in high speed data streams Proceedings of the 19th International Conference on Scientific and Statistical Database Management IEEE Computer Society 29

## Parameters

### Compulsory Input Parameters

1:     ind – int64int32nag_int scalar
On initial entry: must be set to 0$0$.
Indicates the action required in the current call to nag_stat_quantiles_stream_arbitrary (g01ap).
ind = 0${\mathbf{ind}}=0$
Initialize the communication arrays and attempt to process the first nb values from the data stream. eps, rv and nb must be set and licomm must be at least 10$10$.
ind = 1${\mathbf{ind}}=1$
Attempt to process the next block of nb values from the data stream. The calling program must update rv and (if required) nb, and re-enter nag_stat_quantiles_stream_arbitrary (g01ap) with all other parameters unchanged.
ind = 2${\mathbf{ind}}=2$
Continue calculation following the reallocation of either or both of the communication arrays rcomm and icomm.
ind = 3${\mathbf{ind}}=3$
Calculate the nq ε$\epsilon$-approximate quantiles specified in q. The calling program must set q and nq and re-enter nag_stat_quantiles_stream_arbitrary (g01ap) with all other parameters unchanged. This option can be chosen only when npexp(1.0) / eps${\mathbf{np}}\ge ⌈\mathrm{exp}\left(1.0\right)/{\mathbf{eps}}⌉$.
Constraint: ind = 0${\mathbf{ind}}=0$, 1$1$, 2$2$ or 3$3$.
2:     rv( : $:$) – double array
Note: the dimension of the array rv must be at least nb${\mathbf{nb}}$ if ind = 0${\mathbf{ind}}=0$, 1$1$ or 2$2$.
If ind = 0${\mathbf{ind}}=0$, 1$1$ or 2$2$, the vector containing the current block of data, otherwise rv is not referenced.
3:     eps – double scalar
Approximation factor ε$\epsilon$.
Constraint: eps > 0.0 ​ and ​eps1.0${\mathbf{eps}}>0.0\text{​ and ​}{\mathbf{eps}}\le 1.0$.
4:     q( : $:$) – double array
Note: the dimension of the array q must be at least nq${\mathbf{nq}}$ if ind = 3${\mathbf{ind}}=3$.
If ind = 3${\mathbf{ind}}=3$, the quantiles to be calculated, otherwise q is not referenced. Note that q(i) = 0.0${\mathbf{q}}\left(i\right)=0.0$, corresponds to the minimum value and q(i) = 1.0${\mathbf{q}}\left(i\right)=1.0$ to the maximum value.
Constraint: if ind = 3${\mathbf{ind}}=3$, 0.0q(i)1.0$0.0\le {\mathbf{q}}\left(\mathit{i}\right)\le 1.0$, for i = 1,2,,nq$\mathit{i}=1,2,\dots ,{\mathbf{nq}}$.
5:     rcomm(lrcomm) – double array
lrcomm, the dimension of the array, must satisfy the constraint
• if ind = 0${\mathbf{ind}}=0$, lrcomm1${\mathbf{lrcomm}}\ge 1$;
• otherwise lrcommicomm(1)${\mathbf{lrcomm}}\ge {\mathbf{icomm}}\left(1\right)$.
If ind = 1${\mathbf{ind}}=1$ or 2$2$ then the first l$l$ elements of rcomm as supplied to nag_stat_quantiles_stream_arbitrary (g01ap) must be identical to the first l$l$ elements of rcomm returned from the last call to nag_stat_quantiles_stream_arbitrary (g01ap), where l$l$ is the value of lrcomm used in the last call. In other words, the contents of rcomm must not be altered between calls to this function. If rcomm needs to be reallocated then its contents must be preserved. If ind = 0${\mathbf{ind}}=0$ then rcomm need not be set.
6:     icomm(licomm) – int64int32nag_int array
licomm, the dimension of the array, must satisfy the constraint
• if ind = 0${\mathbf{ind}}=0$, licomm10${\mathbf{licomm}}\ge 10$;
• otherwise licommicomm(2)${\mathbf{licomm}}\ge {\mathbf{icomm}}\left(2\right)$.
If ind = 1${\mathbf{ind}}=1$ or 2$2$ then the first l$l$ elements of icomm as supplied to nag_stat_quantiles_stream_arbitrary (g01ap) must be identical to the first l$l$ elements of icomm returned from the last call to nag_stat_quantiles_stream_arbitrary (g01ap), where l$l$ is the value of licomm used in the last call. In other words, the contents of icomm must not be altered between calls to this function. If icomm needs to be reallocated then its contents must be preserved. If ind = 0${\mathbf{ind}}=0$ then icomm need not be set.

### Optional Input Parameters

1:     nb – int64int32nag_int scalar
Default: The dimension of the array rv.
If ind = 0${\mathbf{ind}}=0$, 1$1$ or 2$2$, the size of the current block of data. The size of blocks of data in array rv can vary; therefore nb can change between calls to nag_stat_quantiles_stream_arbitrary (g01ap).
Constraint: if ind = 0${\mathbf{ind}}=0$, 1$1$ or 2$2$, nb > 0${\mathbf{nb}}>0$.
2:     nq – int64int32nag_int scalar
Default: The dimension of the array q.
If ind = 3${\mathbf{ind}}=3$, the number of quantiles requested, otherwise nq is not referenced.
Constraint: if ind = 3${\mathbf{ind}}=3$, nq > 0${\mathbf{nq}}>0$.
3:     lrcomm – int64int32nag_int scalar
Default: The dimension of the array rcomm.
The dimension of the array rcomm as declared in the (sub)program from which nag_stat_quantiles_stream_arbitrary (g01ap) is called.
Constraints:
• if ind = 0${\mathbf{ind}}=0$, lrcomm1${\mathbf{lrcomm}}\ge 1$;
• otherwise lrcommicomm(1)${\mathbf{lrcomm}}\ge {\mathbf{icomm}}\left(1\right)$.
4:     licomm – int64int32nag_int scalar
Default: The dimension of the array icomm.
The dimension of the array icomm as declared in the (sub)program from which nag_stat_quantiles_stream_arbitrary (g01ap) is called.
Constraints:
• if ind = 0${\mathbf{ind}}=0$, licomm10${\mathbf{licomm}}\ge 10$;
• otherwise licommicomm(2)${\mathbf{licomm}}\ge {\mathbf{icomm}}\left(2\right)$.

None.

### Output Parameters

1:     ind – int64int32nag_int scalar
Indicates output from the call.
ind = 1${\mathbf{ind}}=1$
nag_stat_quantiles_stream_arbitrary (g01ap) has processed np data points and expects to be called again with additional data.
ind = 2${\mathbf{ind}}=2$
Either one or more of the communication arrays rcomm and icomm is too small. The new minimum lengths of rcomm and icomm have been returned in icomm(1)${\mathbf{icomm}}\left(1\right)$ and icomm(2)${\mathbf{icomm}}\left(2\right)$ respectively. If the new minimum length is greater than the current length then the corresponding communication array needs to be reallocated, its contents preserved and nag_stat_quantiles_stream_arbitrary (g01ap) called again with all other parameters unchanged.
If there is more data to be processed, it is recommended that lrcomm and licomm are made significantly bigger than the minimum to limit the number of reallocations.
ind = 3${\mathbf{ind}}=3$
nag_stat_quantiles_stream_arbitrary (g01ap) has returned the requested ε$\epsilon$-approximate quantiles in qv. These quantiles are based on np data points.
2:     np – int64int32nag_int scalar
m$m$, the number of elements processed so far.
3:     qv( : $:$) – double array
Note: the dimension of the array qv must be at least nq${\mathbf{nq}}$ if ind = 3${\mathbf{ind}}=3$.
If ind = 3${\mathbf{ind}}=3$, qv(i)${\mathbf{qv}}\left(i\right)$ contains the ε$\epsilon$-approximate quantiles specified by the value provided in q(i)${\mathbf{q}}\left(i\right)$.
4:     rcomm(lrcomm) – double array
rcomm holds information required by subsequent calls to nag_stat_quantiles_stream_arbitrary (g01ap)
5:     icomm(licomm) – int64int32nag_int array
icomm(1)${\mathbf{icomm}}\left(1\right)$ holds the minimum required length for rcomm and icomm(2)${\mathbf{icomm}}\left(2\right)$ holds the minimum required length for icomm. The remaining elements of icomm are used for communication between subsequent calls to nag_stat_quantiles_stream_arbitrary (g01ap).
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).
As an out-of-core function nag_stat_quantiles_stream_arbitrary (g01ap) will only perform certain parameter checks when a data checkpoint (including completion of data input) is signaled. As such it will usually be inappropriate to halt program execution when an error is detected since any errors may be subsequently resolved without losing any processing already carried out. Therefore setting ifail to a value of 1​ or ​1$-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value 1$1$ is recommended. When the value 1​ or ​1$-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: ind = 0${\mathbf{ind}}=0$, 1$1$, 2$2$ or 3$3$.
ifail = 2${\mathbf{ifail}}=2$
Constraint: 0.0 < eps1.0$0.0<{\mathbf{eps}}\le 1.0$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: if ind = 0${\mathbf{ind}}=0$, 1$1$ or 2$2$ then nb > 0${\mathbf{nb}}>0$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: licomm10${\mathbf{licomm}}\ge 10$.
ifail = 5${\mathbf{ifail}}=5$
Constraint: lrcomm1${\mathbf{lrcomm}}\ge 1$.
ifail = 6${\mathbf{ifail}}=6$
The contents of icomm have been altered between calls to this function.
ifail = 7${\mathbf{ifail}}=7$
The contents of rcomm have been altered between calls to this function.
ifail = 8${\mathbf{ifail}}=8$
Number of data elements streamed, _$_$ is not sufficient for a quantile query when .
Supply more data or reprocess the data with a higher eps value.
ifail = 9${\mathbf{ifail}}=9$
Constraint: if ind = 3${\mathbf{ind}}=3$ then nq > 0${\mathbf{nq}}>0$.
ifail = 10${\mathbf{ifail}}=10$
Constraint: if ind = 3${\mathbf{ind}}=3$ then 0.0q(i)1.0$0.0\le {\mathbf{q}}\left(i\right)\le 1.0$ for all i$i$.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

The average time taken by nag_stat_quantiles_stream_arbitrary (g01ap) scales as np log(1 / εlog(εnp)) ${\mathbf{np}}\mathrm{log}\left(1/\epsilon \mathrm{log}\left(\epsilon {\mathbf{np}}\right)\right)$.
It is not possible to determine in advance the final size of the communication arrays rcomm and icomm without knowing the size of the dataset. However, if a rough size (n$n$) is known, the speed of the computation can be increased if the sizes of the communication arrays are not smaller than
 lrcomm = (log2(n × eps + 1.0) − 2) × ⌈1.0 / eps⌉ + 1 + x + 2 × min (x, ⌈x / 2.0⌉ + 1 ) × y + 1 licomm = (log2(n × eps + 1.0) − 2) × (2 × (⌈1.0 / eps⌉ + 1) + 1) + 2 × (x + 2 × min (x, ⌈x / 2.0⌉ + 1 ) × y) + y + 11
$lrcomm = ( log2( n×eps+1.0 ) - 2 ) × ⌈ 1.0/eps ⌉ +1+x+ 2× min(x, ⌈ x/2.0 ⌉ +1 ) × y +1 licomm = ( log2( n×eps+1.0 ) - 2 ) × ( 2 × ( ⌈1.0/eps⌉ +1 ) + 1 ) + 2 × ( x+2× min(x, ⌈x/2.0⌉ +1 ) × y ) + y + 11$
where
 x = max (1,⌊log (eps × n) / eps ⌋) y = log2(n / x + 1.0) + 1 .
$x= max(1,⌊ log⁡ (eps×n) / eps ⌋) y = log2(n/x+1.0) +1 .$

## Example

```function nag_stat_quantiles_stream_arbitrary_example
n = 0;
eps = 0.2;
q = [0.25; 0.5; 1.0];
nb = 20;
% For this example we are using a string as the source of data.
datastream = strcat('34.01 57.95 44.88 22.04 28.84 4.43 0.32 20.82 20.53 ', ...
' 13.08  7.99 54.03 23.21 26.73 39.72  0.97 39.05 38.78 19.38 51.34 24.08 ', ...
' 12.41 58.11 35.90 40.38 27.41 19.80  6.02 45.33 36.34 43.14 53.84 39.49 ', ...
'  9.04 36.74 58.72 59.95 15.41 33.05 39.54 33.24 58.67 54.12 39.48 43.73 ', ...
' 24.15 55.72  8.87 40.47 46.18 20.36  6.95 36.86 49.24 56.83 43.87 29.86 ', ...
' 22.49 25.29 33.17');
rcomm = zeros(100, 1);
icomm = zeros(400, 1, 'int64');
ind = int64(0);
repeat = true;
pos    = 0;
while (repeat)
if (ind==0 || ind==1)
[rv, new_pos] = textscan(datastream(pos+1:end), '%f', nb);
pos = pos+new_pos;

nd = numel(rv{1});
if nd == 0
break;
elseif nd < nb
nb = nd;
repeat = false;
elseif pos == numel(datastream)
repeat = false;
end
n = n+nb;
end

[ind, np, qv, rcomm, icomm, ifail] = ...
nag_stat_quantiles_stream_arbitrary(ind, rv{1}, eps, q, rcomm, icomm);

if ifail ~= 0
break;
end

% If ind=2, the communication arrays are too small, so extend them
% and call the routine again with the same rv
if ind == 2
if numel(rcomm) < icomm(1)
rcomm(icomm(1)) = 0;
end
if numel(icomm) < icomm(2)
icomm(icomm(2)) = 0;
end
end
end

if ifail == 0
% Call NAG routine again to calculate quantiles specified in vector q
ind = int64(3);
[ind, np, qv, rcomm, icomm, ifail] = ...
nag_stat_quantiles_stream_arbitrary(ind, 0, eps, q, rcomm, icomm);
else
fprintf('\nAn error occurred processing the observations\n');
fprintf('ifail = %d\n', ifail);
end

% Print the results
if ifail == 0
fprintf('\nInput Data:\n %d observations\n eps = %5.2f\n\n', n, eps);
fprintf('Quantile     Result\n');
for i = 1:numel(q)
fprintf('%7.2f     %7.2f\n', q(i), qv(i));
end
else
fprintf('\nAn error occurred calculating the quantiles, ifail = %d\n', ifail);
end
```
```

Input Data:
60 observations
eps =  0.20

Quantile     Result
0.25       22.49
0.50       39.54
1.00       59.95

```
```function g01ap_example
n = 0;
eps = 0.2;
q = [0.25; 0.5; 1.0];
nb = 20;
% For this example we are using a string as the source of data.
datastream = strcat('34.01 57.95 44.88 22.04 28.84 4.43 0.32 20.82 20.53 ', ...
' 13.08  7.99 54.03 23.21 26.73 39.72  0.97 39.05 38.78 19.38 51.34 24.08 ', ...
' 12.41 58.11 35.90 40.38 27.41 19.80  6.02 45.33 36.34 43.14 53.84 39.49 ', ...
'  9.04 36.74 58.72 59.95 15.41 33.05 39.54 33.24 58.67 54.12 39.48 43.73 ', ...
' 24.15 55.72  8.87 40.47 46.18 20.36  6.95 36.86 49.24 56.83 43.87 29.86 ', ...
' 22.49 25.29 33.17');
rcomm = zeros(100, 1);
icomm = zeros(400, 1, 'int64');
ind = int64(0);
repeat = true;
pos    = 0;
while (repeat)
if (ind==0 || ind==1)
[rv, new_pos] = textscan(datastream(pos+1:end), '%f', nb);
pos = pos+new_pos;

nd = numel(rv{1});
if nd == 0
break;
elseif nd < nb
nb = nd;
repeat = false;
elseif pos == numel(datastream)
repeat = false;
end
n = n+nb;
end

[ind, np, qv, rcomm, icomm, ifail] = g01ap(ind, rv{1}, eps, q, rcomm, icomm);

if ifail ~= 0
break;
end

% If ind=2, the communication arrays are too small, so extend them
% and call the routine again with the same rv
if ind == 2
if numel(rcomm) < icomm(1)
rcomm(icomm(1)) = 0;
end
if numel(icomm) < icomm(2)
icomm(icomm(2)) = 0;
end
end
end

if ifail == 0
% Call NAG routine again to calculate quantiles specified in vector q
ind = int64(3);
[ind, np, qv, rcomm, icomm, ifail] = g01ap(ind, 0, eps, q, rcomm, icomm);
else
fprintf('\nAn error occurred processing the observations\n');
fprintf('ifail = %d\n', ifail);
end

% Print the results
if ifail == 0
fprintf('\nInput Data:\n %d observations\n eps = %5.2f\n\n', n, eps);
fprintf('Quantile     Result\n');
for i = 1:numel(q)
fprintf('%7.2f     %7.2f\n', q(i), qv(i));
end
else
fprintf('\nAn error occurred calculating the quantiles, ifail = %d\n', ifail);
end
```
```

Input Data:
60 observations
eps =  0.20

Quantile     Result
0.25       22.49
0.50       39.54
1.00       59.95

```