Probability Density Function

Cauchy distribution probability density function (PDF).

The probability density function (PDF) for a Cauchy random variable is

where gamma > 0 is the scale parameter and x0 is the location parameter.

Installation

$ npm install distributions-cauchy-pdf

For use in the browser, use browserify.

Usage

var pdf = require( 'distributions-cauchy-pdf' );

pdf( x[, options] )

Evaluates the probability density function (PDF) for the Cauchy distribution. x may be either a number, an array, a typed array, or a matrix.

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

out = pdf( 1 );
// returns ~0.159

out = pdf( -1 );
// returns ~0.159

x = [ 0, 0.5, 1, 1.5, 2, 2.5 ];
out = pdf( x );
// returns [ ~0.318, ~0.255, ~0.159, ~0.0979, ~0.0637, ~0.0439 ]

x = new Float32Array( x );
out = pdf( x );
// returns Float64Array( [~0.318,~0.255,~0.159,~0.0979,~0.0637,~0.0439] )

x = new Float32Array( 6 );
for ( i = 0; i < 6; i++ ) {
	x[ i ] = i * 0.5;
}
mat = matrix( x, [3,2], 'float32' );
/*
	[ 0  0.5
	  1  1.5
	  2  2.5 ]
*/

out = pdf( mat );
/*
	[ ~0.318 ~0.255
	  ~0.159 ~0.0979
	  ~0.0637 ~0.0439 ]
*/

The function accepts the following options:

• gamma: scale parameter. Default: 1.
• x0: location parameter. Default: 0.
• 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: '.'.

A Cauchy distribution is a function of two parameters: gamma > 0(scale parameter) and x0(location parameter). By default, gamma is equal to 1 and x0 is equal to 0. To adjust either parameter, set the corresponding option.

var x = [ 0, 0.5, 1, 1.5, 2, 2.5 ];

var out = pdf( x, {
	'gamma': 0.1,
	'x0': 1,
});
// returns [ ~0.0315, ~0.122, ~3.18, ~0.122, ~0.0315, ~0.0141 ]

For non-numeric arrays, provide an accessor function for accessing array values.

var data = [
	[0,0],
	[1,0.5],
	[2,1],
	[3,1.5],
	[4,2],
	[5,2.5]
];

function getValue( d, i ) {
	return d[ 1 ];
}

var out = pdf( data, {
	'accessor': getValue
});
// returns [ ~0.318, ~0.255, ~0.159, ~0.0979, ~0.0637, ~0.0439 ]

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

var data = [
	{'x':[0,0]},
	{'x':[1,0.5]},
	{'x':[2,1]},
	{'x':[3,1.5]},
	{'x':[4,2]},
	{'x':[5,2.5]}
];

var out = pdf( data, {
	'path': 'x/1',
	'sep': '/'
});
/*
	[
		{'x':[0,~0.318]},
		{'x':[1,~0.255]},
		{'x':[2,~0.159]},
		{'x':[3,~0.0979]},
		{'x':[4,0.0637]},
		{'x':[5,~0.0439]}
	]
*/

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 x, out;

x = new Int8Array( [0,1,2,3,4] );

out = pdf( x, {
	'gamma': 0.1,
	'x0': 2,
	'dtype': 'int32'
});
// returns Int32Array( [0,0,3,0,0] )

// Works for plain arrays, as well...
out = pdf( [0,0.5,1,1.5,2], {
	'gamma': 0.1,
	'x0': 2,
	'dtype': 'uint8'
});
// returns Uint8Array( [0,0,3,0,0] )

By default, the function returns a new data structure. To mutate the input data structure (e.g., when input values can be discarded or when optimizing memory usage), set the copy option to false.

var bool,
	mat,
	out,
	x,
	i;

x = [ 0, 0.5, 1, 1.5, 2 ];

out = pdf( x, {
	'copy': false
});
// returns [ ~0.318, ~0.255, ~0.159, ~0.0979, ~0.0637, ~0.0439 ]

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

x = new Float32Array( 6 );
for ( i = 0; i < 6; i++ ) {
	x[ i ] = i * 0.5;
}
mat = matrix( x, [3,2], 'float32' );
/*
	[ 0  0.5
	  1  1.5
	  2  2.5 ]
*/

out = pdf( mat, {
	'copy': false
});
/*
	[ ~0.318 ~0.255
	  ~0.159 ~0.0979
	  ~0.0637 ~0.0439 ]
*/

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

Notes

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

  var data, out;
  
  out = pdf( null );
  // returns NaN
  
  out = pdf( true );
  // returns NaN
  
  out = pdf( {'a':'b'} );
  // returns NaN
  
  out = pdf( [ true, null, [] ] );
  // returns [ NaN, NaN, NaN ]
  
  function getValue( d, i ) {
  	return d.x;
  }
  data = [
  	{'x':true},
  	{'x':[]},
  	{'x':{}},
  	{'x':null}
  ];
  
  out = pdf( data, {
  	'accessor': getValue
  });
  // returns [ NaN, NaN, NaN, NaN ]
  
  out = pdf( data, {
  	'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 = pdf( [ true, null, [] ], {
  	'dtype': 'int8'
  });
  // returns Int8Array( [0,0,0] );

Examples

var pdf = require( 'distributions-cauchy-pdf' ),
	matrix = require( 'dstructs-matrix' );

var data,
	mat,
	out,
	tmp,
	i;

// Plain arrays...
data = new Array( 10 );
for ( i = 0; i < data.length; i++ ) {
	data[ i ] = i * 0.5;
}
out = pdf( data );

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

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

// Typed arrays...
data = new Float32Array( 10 );
for ( i = 0; i < data.length; i++ ) {
	data[ i ] = i * 0.5;
}
out = pdf( data );

// Matrices...
mat = matrix( data, [5,2], 'int32' );
out = pdf( mat );

// Matrices (custom output data type)...
out = pdf( mat, {
	'dtype': 'uint8'
});

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

License

MIT license.

Copyright

Copyright © 2015. The Compute.io Authors.