Logarithm of Cumulative Distribution Function

Evaluate the natural logarithm of the cumulative distribution function (CDF) for a raised cosine distribution.

The cumulative distribution function for a raised cosine random variable is

upper F left-parenthesis x semicolon mu comma s right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column for x less-than mu minus s 2nd Row 1st Column one-half left-bracket 1 plus StartFraction x minus mu Over s EndFraction plus StartFraction 1 Over pi EndFraction sine left-parenthesis StartFraction x minus mu Over s EndFraction pi right-parenthesis right-bracket 2nd Column for mu minus s less-than-or-equal-to x less-than-or-equal-to mu plus s 3rd Row 1st Column 1 2nd Column for x greater-than mu plus s EndLayout

where μ is the location parameter and s > 0 is the scale parameter.

Usage

var logcdf = require( '@stdlib/math/base/dists/cosine/logcdf' );

logcdf( x, mu, s )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a raised cosine distribution with parameters mu (location parameter) and s (scale parameter).

var y = logcdf( 2.0, 0.0, 3.0 );
// returns ~-0.029

y = logcdf( 0.0, 0.0, 1.0 );
// returns ~-0.693

y = logcdf( -1.0, 4.0, 2.0 );
// returns -Infinity

If provided NaN as any argument, the function returns NaN.

var y = logcdf( NaN, 0.0, 1.0 );
// returns NaN

y = logcdf( 0.0, NaN, 1.0 );
// returns NaN

y = logcdf( 0.0, 0.0, NaN );
// returns NaN

If provided s < 0, the function returns NaN.

var y = logcdf( 2.0, 0.0, -1.0 );
// returns NaN

If provided s = 0, the function evaluates the logarithm of the CDF for a degenerate distribution centered at mu.

var y = logcdf( 2.0, 8.0, 0.0 );
// returns -Infinity

y = logcdf( 8.0, 8.0, 0.0 );
// returns 0.0

y = logcdf( 10.0, 8.0, 0.0 );
// returns 0.0

logcdf.factory( mu, s )

Returns a function for evaluating the natural logarithm of the cumulative distribution function of a raised cosine distribution with parameters mu (location parameter) and s (scale parameter).

var mylogcdf = logcdf.factory( 10.0, 2.0 );

var y = mylogcdf( 10.0 );
// returns ~-0.693

y = mylogcdf( 8.0 );
// returns -Infinity

y = mylogcdf( 12.0 );
// returns 0.0

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/math/base/dists/cosine/logcdf' );

var mu;
var s;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 10.0;
    mu = randu() * 10.0;
    s = randu() * 10.0;
    y = logcdf( x, mu, s );
    console.log( 'x: %d, µ: %d, s: %d, ln(F(x;µ,s)): %d', x, mu, s, y );
}