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/stats/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

Notes

  • In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/stats/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 );
}
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