compute-gammaincinv

gammaincinv

Inverse incomplete gamma function.

Computes the inverse of the lower incomplete gamma function

Specifically, for given p and a it finds the x such that p = P(x, a).

The function can also be used to invert the upper incomplete gamma function, which is defined as follows:

Again, for given p and a the function returns the x which satisfies p = Q(x, a).

Installation

$ npm install compute-gammaincinv

For use in the browser, use browserify.

Usage

var gammaincinv = require( 'compute-gammaincinv' );

gammaincinv( p, a[, opts] )

Inverts element-wise the regularized incomplete gamma function. p can be a number, array, typed array or matrix. a has to be either an array or matrix of equal dimensions as p or a single number. The function returns either an array with the same length as the p array, a matrix with the same dimensions as the p matrix or a single number. Contrary to the more commonly used definitoon, in this implementation the first argument is p and the second argument is the scale factor a.

var matrix = require( 'dstructs-matrix' ),
	data,
	mat,
	out,
	i;

out = gammaincinv( 0.5, 20 );
// returns ~19.668

out = gammaincinv( 1e-4, 1.0001 );
// returns ~0.0001

data = [ 0.1, 0.2, 0.3 ];
out = gammaincinv( 0.9, data );
// returns [ ~0.266, ~0.605, ~0.885 ]

out = gammaincinv( data, 2 );
// returns [ ~0.532, ~0.824, ~1.097 ]

data = new Float32Array( [0.1,0.2,0.3] );
out = gammaincinv( data, 2 );
// returns Float64Array( [~0.532,~0.824,~1.097] )

data = new Float32Array( 6 );
for ( i = 0; i < 6; i++ ) {
	data[ i ] = i / 6;
}
mat = matrix( data, [3,2], 'float32' );
/*
	[    0  1/6
	   2/6  3/6
	   4/6  5/6 ]
*/

out = gammaincinv( mat, 4 );
/*
	[  0      ~6.972
	  ~8.394  ~9.669
	  ~11.067 ~12.987 ]
*/

The function accepts the following options:

• accessor: accessor function for accessing array values.
• dtype: output typed array or matrix data type. Default: float64.
• copy: boolean indicating if the function should return a new data structure. Default: true.
• path: deepget/deepset key path.
• sep: deepget/deepset key path separator. Default: '.'.
• tail:string indicating whether to invert the 'lower' or 'upper' regularized incomplete gamma function. Default: 'lower'.

By default, the function inverts the lower regularized incomplete gamma function, P(x,a). To invert the upper function instead, i.e. Q(x,a), set the tail option to 'upper'.

var l, u, bool;

l = gammaincinv( 0.6, 2 )
// returns ~2.022

u = gammaincinv( 0.6, 2, {
	'tail': 'upper'
});
// returns ~1.376

For object arrays, provide an accessor function for accessing array values.

data = [
	{'x':0.1},
	{'x':0.2},
	{'x':0.3},
	{'x':0.4},
	{'x':0.5},
	{'x':0.6},
	{'x':0.7},
	{'x':0.8}
	{'x':0.9}
];

function getValue( d, i ) {
	return d.x;
}

var out = gammaincinv( data, 10, {
	'accessor': getValue
});
// returns [ ~6.221, ~7.289, ~8.133, ~8.904, ~9.669, ~10.476, ~11.388, ~12.519, ~14.206 ]

When inverting the incomplete gamma function for values between two object arrays, provide an accessor function which accepts 3 arguments.

var data = [
	['beep', 0.1],
	['boop', 0.2],
	['bip', 0.3],
	['bap', 0.4],
	['baz', 0.5]
];

var arr = [
	{'x': 1},
	{'x': 2},
	{'x': 3},
	{'x': 4},
	{'x': 5}
];

function getValue( d, i, j ) {
	if ( j === 0 ) {
		return d[ 1 ];
	}
	return d.x;
}

var out = gammaincinv( data, arr, {
	'accessor': getValue
});
// returns [ ~0.105, ~0.824, ~1.914, ~3.211, ~4.671 ]

Note: j corresponds to the input array index, where j=0 is the index for the first input array and j=1 is the index for the second input array.

To deepset an object array, provide a key path and, optionally, a key path separator.

var data = [
	{'x':[0,0.1]},
	{'x':[1,0.2]},
	{'x':[2,0.3]},
	{'x':[3,0.4]},
	{'x':[4,0.5]}
];

var out = gammaincinv( data, 4, 'x|1', '|' );
/*
	[
		{'x':[0,~1.745]},
		{'x':[1,~2.297]},
		{'x':[2,~2.764]},
		{'x':[3,~3.211},
		{'x':[4,~3.672]}
	]
*/

var bool = ( data === out );
// returns true

By default, when provided a typed array or matrix, the output data structure is float64 in order to preserve precision. To specify a different data type, set the dtype option (see matrix for a list of acceptable data types).

var data, out;

data = new Float64Array( [ 0.1, 0.2, 0.3 ] );

out = gammaincinv( data, 5, {
	'dtype': 'int32',
});
// returns Int32Array( [2,3,3] )

// Works for plain arrays, as well...
out = gammaincinv( [0.1, 0.2, 0.3 ], 5, {
	'dtype': 'uint8'
});
// returns Uint8Array( [2,3,3] )

By default, the function returns a new data structure. To mutate the input data structure, set the copy option to false.

var data,
	bool,
	mat,
	out,
	i;

data = [ 0.1, 0.2, 0.3 ];

out = gammaincinv( data, 2, {
	'copy': false
});
// returns  [ ~0.532, ~0.824, ~1.097 ]

bool = ( data === out );
// returns true

data = new Float32Array( 6 );
for ( i = 0; i < 6; i++ ) {
	data[ i ] = i / 6;
}
mat = matrix( data, [3,2], 'float32' );
/*
	[  0   1/6
	  2/6  3/6
	  4/6  5/6 ]
*/

out = gammaincinv( mat, 4, {
	'copy': false
});
/*
	[  0      ~6.972
	  ~8.394  ~9.669
	  ~11.067 ~12.987 ]
*/

bool = ( mat === out );
// returns true

Notes

• If an element is not a numeric value, the returned value is NaN.

  var data, out;
  
  out = gammaincinv( null, 1 );
  // returns NaN
  
  out = gammaincinv( true, 1 );
  // returns NaN
  
  out = gammaincinv( {'a':'b'}, 1 );
  // returns NaN
  
  out = gammaincinv( [ true, null, [] ], 1 );
  // returns [ NaN, NaN, NaN ]
  
  function getValue( d, i ) {
  	return d.x;
  }
  data = [
  	{'x':true},
  	{'x':[]},
  	{'x':{}},
  	{'x':null}
  ];
  
  out = gammaincinv( data, 1, {
  	'accessor': getValue
  });
  // returns [ NaN, NaN, NaN, NaN ]
  
  out = gammaincinv( data, 1, {
  	'path': 'x'
  });
  /*
  	[
  		{'x':NaN},
  		{'x':NaN},
  		{'x':NaN,
  		{'x':NaN}
  	]
  */

• Be careful when providing a data structure which contains non-numeric elements and specifying an integer output data type, as NaN values are cast to 0.

  var out = gammaincinv( [ true, null, [] ], 1, {
  	'dtype': 'int8'
  });
  // returns Int8Array( [0,0,0] );

• When calling the function with a numeric value as the first argument and a matrix or array as the second argument, only the dtype option is applicable.

  	// Valid:
  	var out = gammaincinv( 1, [ 1, 2, 3 ], {
  		'dtype': 'int8'
  	});
  	// returns Int8Array( [0,0,0] )
  
  	// Not valid:
  	var out = gammaincinv( 0.5, [ 1, 2, 3 ], {
  		'copy': false
  	});
  	// throws an error

Implementation

The code used to calculate the inverse incomplete gamma function has been translated from the Fortran module GammaCHI by Amparo Gil, Javier Segura and Nico M. Temme. It uses different methods of computation depending on the values of the input values: Taylor, asymptotic expansions and high-order Newton methods.

References

1. A. Gil, J. Segura and N.M. Temme, GammaCHI: a package for the inversion and computation of the gamma and chi-square distribution functions (central and noncentral). Computer Physics Commun
2. A. Gil, J. Segura and N.M. Temme. Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios. SIAM J Sci Comput. (2012) 34(6), A2965-A2981

Examples

var matrix = require( 'dstructs-matrix' ),
	gammaincinv = require( 'compute-gammaincinv' );

var data,
	mat,
	out,
	tmp,
	i;

// Plain arrays...
data = new Array( 100 );
for ( i = 0; i < data.length; i++ ) {
	data[ i ] = Math.random();
}
out = gammaincinv( data, 1 );

// Object arrays (accessors)...
function getValue( d ) {
	return d.x;
}
for ( i = 0; i < data.length; i++ ) {
	data[ i ] = {
		'x': data[ i ]
	};
}
out = gammaincinv( data, 1, {
	'accessor': getValue
});

// Deep set arrays...
for ( i = 0; i < data.length; i++ ) {
	data[ i ] = {
		'x': [ i, data[ i ].x ]
	};
}
out = gammaincinv( data, 1, {
	'path': 'x/1',
	'sep': '/'
});

// Typed arrays...
data = new Float64Array( 100 );
for ( i = 0; i < data.length; i++ ) {
	data[ i ] = Math.random();
}
tmp = gammaincinv( data, 1 );
out = '';
for ( i = 0; i < data.length; i++ ) {
	out += tmp[ i ];
	if ( i < data.length-1 ) {
		out += ',';
	}
}

// Matrices...
mat = matrix( data, [10,10], 'float64' );
out = gammaincinv( mat, 1 );

// Matrices (custom output data type)...
out = gammaincinv( mat, 1, {
	'dtype': 'float32'
});

To run the example code from the top-level application directory,

$ node ./examples/index.js

Tests

Unit

Unit tests use the Mocha test framework with Chai assertions. To run the tests, execute the following command in the top-level application directory:

$ make test

All new feature development should have corresponding unit tests to validate correct functionality.

Test Coverage

This repository uses Istanbul as its code coverage tool. To generate a test coverage report, execute the following command in the top-level application directory:

$ make test-cov

Istanbul creates a ./reports/coverage directory. To access an HTML version of the report,

$ make view-cov

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License

MIT license.

Copyright

Copyright © <%= year %>. The Compute.io Authors.