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* @license Apache-2.0
*
* Copyright (c) 2018 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' );
// MAIN //
/**
* Returns an accumulator function which incrementally computes a corrected sample excess kurtosis.
*
* ## Method
*
* The algorithm computes the sample excess kurtosis using the formula for `G_2` in [Joanes and Gill 1998][@joanes:1998]. In contrast to alternatives for calculating a sample kurtosis, `G_2` is an unbiased estimator under normality.
*
* ## 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
*
* @returns {Function} accumulator function
*
* @example
* var accumulator = incrkurtosis();
*
* 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
*/
function incrkurtosis() {
var deltaN2;
var deltaN;
var delta;
var term1;
var mean;
var tmp;
var g2;
var M2;
var M3;
var M4;
var N;
deltaN2 = 0.0;
deltaN = 0.0;
delta = 0.0;
term1 = 0.0;
mean = 0.0;
M2 = 0.0;
M3 = 0.0;
M4 = 0.0;
N = 0;
return accumulator;
/**
* If provided a value, the accumulator function returns an updated corrected sample excess kurtosis. If not provided a value, the accumulator function returns the current corrected sample excess kurtosis.
*
* @private
* @param {number} [x] - new value
* @returns {(number|null)} corrected sample excess kurtosis
*/
function accumulator( x ) {
if ( arguments.length === 0 ) {
if ( N < 4 ) {
return ( isnan( M4 ) ) ? NaN : null;
}
// Calculate the population excess kurtosis:
g2 = (( N*M4 ) / ( M2*M2 )) - 3.0;
// Return the corrected sample excess kurtosis:
return (N-1) / ( (N-2)*(N-3) ) * ( ((N+1)*g2) + 6.0 );
}
N += 1;
delta = x - mean;
deltaN = delta / N;
deltaN2 = deltaN * deltaN;
term1 = delta * deltaN * (N-1);
tmp = term1 * deltaN2 * ((N*N) - (3*N) + 3);
tmp += 6.0 * deltaN2 * M2;
tmp -= 4.0 * deltaN * M3;
M4 += tmp;
tmp = term1 * deltaN * (N-2);
tmp -= 3.0 * deltaN * M2;
M3 += tmp;
M2 += term1;
mean += deltaN;
if ( N < 4 ) {
return ( isnan( M4 ) ) ? NaN : null;
}
// Calculate the population excess kurtosis:
g2 = (N*M4 / ( M2*M2 )) - 3.0;
// Return the corrected sample excess kurtosis:
return (N-1) / ( (N-2)*(N-3) ) * ( ((N+1)*g2) + 6.0 );
}
}
// EXPORTS //
module.exports = incrkurtosis;
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