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* @license Apache-2.0
*
* Copyright (c) 2026 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var isPositiveInteger = require( '@stdlib/assert/is-positive-integer' ).isPrimitive;
var Float64Array = require( '@stdlib/array/float64' );
var format = require( '@stdlib/string/format' );
// MAIN //
/**
* Returns an accumulator function which incrementally computes a moving corrected sample excess kurtosis.
*
* ## Method
*
* The algorithm maintains running central moment accumulators `M2`, `M3`,
* and `M4` using Welford's online algorithm. When the window is full, each
* new datum triggers an O(1) downdate (removing the oldest value) followed
* by an O(1) update (adding the newest value).
*
* ### Welford update (adding value `x`, going from `n-1` to `n` values)
*
* ```text
* delta = x - mean_prev
* deltaN = delta / n
* term1 = delta * deltaN * (n - 1)
* M4 += term1 * deltaN^2 * (n^2 - 3n + 3) + 6*deltaN^2*M2 - 4*deltaN*M3
* M3 += term1 * deltaN * (n - 2) - 3*deltaN*M2
* M2 += term1
* mean += deltaN
* ```
*
* ### Welford downdate (removing value `xo`, going from `n` to `n-1` values)
*
* ```text
* mean_new = (n*mean - xo) / (n - 1)
* delta = xo - mean_new
* deltaN = delta / n
* term1 = delta * deltaN * (n - 1)
* M2_new = M2 - term1
* M3_new = M3 - term1*deltaN*(n-2) + 3*deltaN*M2_new
* M4_new = M4 - term1*deltaN^2*(n^2-3n+3) - 6*deltaN^2*M2_new + 4*deltaN*M3_new
* ```
*
* The corrected sample excess kurtosis is
*
* ```text
* G_2 = (n-1)/((n-2)*(n-3)) * ((n+1)*g2 + 6)
* ```
*
* where `g2 = n*M4/M2^2 - 3`.
*
* ## Notes
*
* - The kurtosis is only defined for a window containing at least 4 values.
* For fewer values the accumulator returns `null`.
*
* - Input values are **not** type-checked. If a `NaN` enters the window,
* the accumulator returns `NaN` until the `NaN` leaves the window, at
* which point the central moments are recomputed from the circular buffer.
*
* ## References
*
* - Joanes, D. N., and C. A. Gill. 1998. "Comparing measures of sample
* skewness and kurtosis." _Journal of the Royal Statistical Society:
* Series D (The Statistician)_ 47 (1). Blackwell Publishers Ltd: 183-89.
* doi:[10.1111/1467-9884.00122][@joanes:1998].
*
* [@joanes:1998]: http://dx.doi.org/10.1111/1467-9884.00122
*
* @param {PositiveInteger} W - window size
* @throws {TypeError} must provide a positive integer
* @returns {Function} accumulator function
*
* @example
* var accumulator = mkurtosis( 4 );
*
* var kurtosis = accumulator();
* // returns null
*
* kurtosis = accumulator( 2.0 );
* // returns null
*
* kurtosis = accumulator( 2.0 );
* // returns null
*
* kurtosis = accumulator( -4.0 );
* // returns null
*
* kurtosis = accumulator( -4.0 );
* // returns -6.0
*
* kurtosis = accumulator( 3.0 );
* // returns -5.652200677131425
*/
function mkurtosis( W ) {
var deltaN2;
var deltaN;
var delta;
var term1;
var mean;
var nans;
var buf;
var M2;
var M3;
var M4;
var g2;
var N;
var n;
var i;
if ( !isPositiveInteger( W ) ) {
throw new TypeError( format( 'invalid argument. Must provide a positive integer. Value: `%s`.', W ) );
}
buf = new Float64Array( W );
mean = 0.0;
M2 = 0.0;
M3 = 0.0;
M4 = 0.0;
N = 0;
i = -1;
nans = 0;
return accumulator;
/**
* If provided a value, returns an updated moving corrected sample excess kurtosis; otherwise, returns the current moving corrected sample excess kurtosis.
*
* @private
* @param {number} [x] - new value
* @returns {(number|null)} corrected sample excess kurtosis or null
*/
function accumulator( x ) {
var xo;
if ( arguments.length === 0 ) {
if ( nans > 0 ) {
return NaN;
}
n = ( N < W ) ? N : W;
if ( n < 4 ) {
return null;
}
g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
}
// Advance the circular buffer index:
i = ( i + 1 ) % W;
if ( N < W ) {
// Fill phase: window is not yet full.
buf[ i ] = x;
N += 1;
if ( isnan( x ) ) {
nans += 1;
}
if ( nans > 0 ) {
return NaN;
}
// Welford update (add x, going from N-1 to N):
n = N;
delta = x - mean;
deltaN = delta / n;
deltaN2 = deltaN * deltaN;
term1 = delta * deltaN * ( n - 1 );
M4 += term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
M4 += 6.0 * deltaN2 * M2;
M4 -= 4.0 * deltaN * M3;
M3 += term1 * deltaN * ( n - 2 );
M3 -= 3.0 * deltaN * M2;
M2 += term1;
mean += deltaN;
if ( n < 4 ) {
return null;
}
g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
}
// Slide phase: window is full (N >= W).
xo = buf[ i ];
buf[ i ] = x;
// Track NaN count changes:
if ( isnan( xo ) ) {
nans -= 1;
}
if ( isnan( x ) ) {
nans += 1;
}
// If a NaN just left the window and none remain, recompute from scratch
// because NaN corrupted the running moment state:
if ( nans === 0 && isnan( xo ) ) {
recompute();
} else if ( nans > 0 ) {
// NaN still present or just entered; skip moment update:
return NaN;
} else {
// Normal case: downdate xo, then update x.
n = W;
// Downdate: remove xo from (n, mean, M2, M3, M4).
mean = ( ( n * mean ) - xo ) / ( n - 1 );
// Recover delta/deltaN as they were when xo was originally added:
delta = xo - mean;
deltaN = delta / n;
deltaN2 = deltaN * deltaN;
term1 = delta * deltaN * ( n - 1 );
M2 -= term1;
M3 -= term1 * deltaN * ( n - 2 );
M3 += 3.0 * deltaN * M2;
M4 -= term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
M4 -= 6.0 * deltaN2 * M2;
M4 += 4.0 * deltaN * M3;
// Update: add x to (n-1) values to recover a full window of n:
delta = x - mean;
deltaN = delta / n;
deltaN2 = deltaN * deltaN;
term1 = delta * deltaN * ( n - 1 );
M4 += term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
M4 += 6.0 * deltaN2 * M2;
M4 -= 4.0 * deltaN * M3;
M3 += term1 * deltaN * ( n - 2 );
M3 -= 3.0 * deltaN * M2;
M2 += term1;
mean += deltaN;
}
n = W;
if ( n < 4 ) {
return null;
}
g2 = ( n * M4 / ( M2 * M2 ) ) - 3.0;
return ( (n-1) / ( (n-2)*(n-3) ) ) * ( ( (n+1)*g2 ) + 6.0 );
}
/**
* Recomputes central moments M2, M3, M4 and the mean from the circular buffer.
*
* @private
*/
function recompute() {
var v;
var k;
mean = 0.0;
M2 = 0.0;
M3 = 0.0;
M4 = 0.0;
// Iterate over buffer from oldest (index (i+1)%W) to newest (index i):
for ( k = 1; k <= W; k++ ) {
v = buf[ ( i + k ) % W ];
// Welford update for position k (1-indexed):
n = k;
delta = v - mean;
deltaN = delta / n;
deltaN2 = deltaN * deltaN;
term1 = delta * deltaN * ( n - 1 );
M4 += term1 * deltaN2 * ( ( n*n ) - ( 3*n ) + 3 );
M4 += 6.0 * deltaN2 * M2;
M4 -= 4.0 * deltaN * M3;
M3 += term1 * deltaN * ( n - 2 );
M3 -= 3.0 * deltaN * M2;
M2 += term1;
mean += deltaN;
}
}
}
// EXPORTS //
module.exports = mkurtosis;
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