/**
* @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 isPositiveInteger = require( '@stdlib/assert/is-positive-integer' ).isPrimitive;
var isArrayLike = require( '@stdlib/assert/is-array-like-object' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var Float64Array = require( '@stdlib/array/float64' );


// MAIN //

/**
* Returns an accumulator function which incrementally computes a moving arithmetic mean and unbiased sample variance.
*
* ## Method
*
* -   Let \\(W\\) be a window of \\(N\\) elements over which we want to compute an unbiased sample variance.
*
* -   The difference between the unbiased sample variance in a window \\(W_i\\) and the unbiased sample variance in a window \\(W_{i+1})\\) is given by
*
*     ```tex
*     \Delta s^2 = s_{i+1}^2 - s_{i}^2
*     ```
*
* -   If we multiply both sides by \\(N-1\\),
*
*     ```tex
*     (N-1)(\Delta s^2) = (N-1)s_{i+1}^2 - (N-1)s_{i}^2
*     ```
*
* -   If we substitute the definition of the unbiased sample variance having the form
*
*     ```tex
*     \begin{align*}
*     s^2 &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i - \bar{x})^2 \biggr) \\
*         &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} (x_i^2 - 2\bar{x}x_i + \bar{x}^2) \biggr) \\
*         &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2\bar{x} \sum_{i=1}^{N} x_i + \sum_{i=1}^{N} \bar{x}^2) \biggr) \\
*         &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - \frac{2N\bar{x}\sum_{i=1}^{N} x_i}{N} + N\bar{x}^2 \biggr) \\
*         &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - 2N\bar{x}^2 + N\bar{x}^2 \biggr) \\
*         &= \frac{1}{N-1} \biggl( \sum_{i=1}^{N} x_i^2 - N\bar{x}^2 \biggr)
*     \end{align*}
*     ```
*
*     we return
*
*     ```tex
*     (N-1)(\Delta s^2) = \biggl(\sum_{k=1}^N x_k^2 - N\bar{x}_{i+1}^2 \biggr) - \biggl(\sum_{k=0}^{N-1} x_k^2 - N\bar{x}_{i}^2 \biggr)
*     ```
*
* -   This can be further simplified by recognizing that subtracting the sums reduces to \\(x_N^2 - x_0^2\\); in which case,
*
*     ```tex
*     \begin{align*}
*     (N-1)(\Delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 + N\bar{x}_{i}^2 \\
*     &= x_N^2 - x_0^2 - N(\bar{x}_{i+1}^2 - \bar{x}_{i}^2) \\
*     &= x_N^2 - x_0^2 - N(\bar{x}_{i+1} - \bar{x}_{i})(\bar{x}_{i+1} + \bar{x}_{i})
*     \end{align*}
*     ```
*
* -   Recognizing that the difference of means can be expressed
*
*     ```tex
*     \bar{x}_{i+1} - \bar{x}_i = \frac{1}{N} \biggl( \sum_{k=1}^N x_k - \sum_{k=0}^{N-1} x_k \biggr) = \frac{x_N - x_0}{N}
*     ```
*
*     and substituting into the equation above
*
*     ```tex
*     (N-1)(\Delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
*     ```
*
* -   Rearranging terms gives us the update equation
*
*     ```tex
*     \begin{align*}
*     (N-1)(\Delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
*     &= (x_N - x_0)(x_N + x_0 - \bar{x}_{i+1} - \bar{x}_{i}) \\
*     &= (x_N - x_0)(x_N - \bar{x}_{i+1} + x_0 - \bar{x}_{i})
*     \end{align*}
*     ```
*
* @param {Collection} [out] - output array
* @param {PositiveInteger} window - window size
* @throws {TypeError} output argument must be array-like
* @throws {TypeError} window size must be a positive integer
* @returns {Function} accumulator function
*
* @example
* var accumulator = incrmmeanvar( 3 );
*
* var v = accumulator();
* // returns null
*
* v = accumulator( 2.0 );
* // returns [ 2.0, 0.0 ]
*
* v = accumulator( -5.0 );
* // returns [ -1.5, 24.5 ]
*
* v = accumulator( 3.0 );
* // returns [ 0.0, 19.0 ]
*
* v = accumulator( 5.0 );
* // returns [ 1.0, 28.0 ]
*
* v = accumulator();
* // returns [ 1.0, 28.0 ]
*/
function incrmmeanvar( out, window ) {
	var meanvar;
	var delta;
	var buf;
	var tmp;
	var M2;
	var mu;
	var d1;
	var d2;
	var W;
	var N;
	var n;
	var i;
	if ( arguments.length === 1 ) {
		meanvar = [ 0.0, 0.0 ];
		W = out;
	} else {
		if ( !isArrayLike( out ) ) {
			throw new TypeError( 'invalid argument. Output argument must be an array-like object. Value: `' + out + '`.' );
		}
		meanvar = out;
		W = window;
	}
	if ( !isPositiveInteger( W ) ) {
		throw new TypeError( 'invalid argument. Window size must be a positive integer. Value: `' + W + '`.' );
	}
	buf = new Float64Array( W );
	n = W - 1;
	M2 = 0.0;
	mu = 0.0;
	i = -1;
	N = 0;

	return accumulator;

	/**
	* If provided a value, the accumulator function returns updated accumulated values. If not provided a value, the accumulator function returns the current accumulated values.
	*
	* @private
	* @param {number} [x] - input value
	* @returns {(ArrayLikeObject|null)} output array or null
	*/
	function accumulator( x ) {
		var k;
		var v;
		if ( arguments.length === 0 ) {
			if ( N === 0 ) {
				return null;
			}
			meanvar[ 0 ] = mu;
			if ( N === 1 ) {
				if ( isnan( M2 ) ) {
					meanvar[ 1 ] = NaN;
				} else {
					meanvar[ 1 ] = 0.0;
				}
			} else if ( N < W ) {
				meanvar[ 1 ] = M2 / (N-1);
			} else {
				meanvar[ 1 ] = M2 / n;
			}
			return meanvar;
		}
		// Update the index for managing the circular buffer:
		i = (i+1) % W;

		// Case: incoming value is NaN, the sliding second moment is automatically NaN...
		if ( isnan( x ) ) {
			N = W; // explicitly set to avoid `N < W` branch
			mu = NaN;
			M2 = NaN;
		}
		// Case: initial window...
		else if ( N < W ) {
			buf[ i ] = x; // update buffer
			N += 1;
			delta = x - mu;
			mu += delta / N;
			M2 += delta * (x - mu);

			meanvar[ 0 ] = mu;
			if ( N === 1 ) {
				meanvar[ 1 ] = 0.0;
			} else {
				meanvar[ 1 ] = M2 / (N-1);
			}
			return meanvar;
		}
		// Case: N = W = 1
		else if ( N === 1 ) {
			mu = x;
			M2 = 0.0;
			meanvar[ 0 ] = x;
			meanvar[ 1 ] = 0.0;
			return meanvar;
		}
		// Case: outgoing value is NaN, and, thus, we need to compute the accumulated values...
		else if ( isnan( buf[ i ] ) ) {
			N = 1;
			mu = x;
			M2 = 0.0;
			for ( k = 0; k < W; k++ ) {
				if ( k !== i ) {
					v = buf[ k ];
					if ( isnan( v ) ) {
						N = W; // explicitly set to avoid `N < W` branch
						mu = NaN;
						M2 = NaN;
						break; // second moment is automatically NaN, so no need to continue
					}
					N += 1;
					delta = v - mu;
					mu += delta / N;
					M2 += delta * (v - mu);
				}
			}
		}
		// Case: neither the current second moment nor the incoming value are NaN, so we need to update the accumulated values...
		else if ( isnan( M2 ) === false ) {
			tmp = buf[ i ];
			delta = x - tmp;
			d1 = tmp - mu;
			mu += delta / W;
			d2 = x - mu;
			M2 += delta * (d1 + d2);
		}
		// Case: the current second moment is NaN, so nothing to do until the buffer no longer contains NaN values...
		buf[ i ] = x;

		meanvar[ 0 ] = mu;
		meanvar[ 1 ] = M2 / n;
		return meanvar;
	}
}


// EXPORTS //

module.exports = incrmmeanvar;