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fix: evaluate the tails from p and 1-p in stats/base/dists/cauchy/quantile #15778

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fix: evaluate the tails from p and 1-p in stats/base/dists/cauchy/quantile #15778

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7 changes: 7 additions & 0 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/lib/factory.js
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Original file line number Diff line number Diff line change
Expand Up @@ -68,6 +68,13 @@ function factory( x0, gamma ) {
if ( isnan( p ) || p < 0.0 || p > 1.0 ) {
return NaN;
}
// `p-0.5` rounds away the low-order bits of a `p` close to `0` (`tan(PI*(p-0.5)) = -1/tan(PI*p)`), so evaluate each tail from its own probability, which is exact (`1-p` for `p > 0.75`):
if ( p < 0.25 ) {
return x0 - ( gamma / tan( PI*p ) );
}
if ( p > 0.75 ) {
return x0 + ( gamma / tan( PI*( 1.0-p ) ) );
}
return x0 + ( gamma * tan( PI*( p-0.5 ) ) );
}
}
Expand Down
7 changes: 7 additions & 0 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/lib/main.js
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Expand Up @@ -83,6 +83,13 @@ function quantile( p, x0, gamma ) {
) {
return NaN;
}
// `p-0.5` rounds away the low-order bits of a `p` close to `0` (`tan(PI*(p-0.5)) = -1/tan(PI*p)`), so evaluate each tail from its own probability, which is exact (`1-p` for `p > 0.75`):
if ( p < 0.25 ) {
return x0 - ( gamma / tan( PI*p ) );
}
if ( p > 0.75 ) {
return x0 + ( gamma / tan( PI*( 1.0-p ) ) );
}
return x0 + ( gamma * tan( PI*( p-0.5 ) ) );
}

Expand Down
7 changes: 7 additions & 0 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/src/main.c
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Expand Up @@ -44,5 +44,12 @@ double stdlib_base_dists_cauchy_quantile( const double p, const double x0, const
) {
return 0.0/0.0; // NaN
}
// `p-0.5` rounds away the low-order bits of a `p` close to `0` (`tan(PI*(p-0.5)) = -1/tan(PI*p)`), so evaluate each tail from its own probability, which is exact (`1-p` for `p > 0.75`):
if ( p < 0.25 ) {
return x0 - ( gamma / stdlib_base_tan( STDLIB_CONSTANT_FLOAT64_PI * p ) );
}
if ( p > 0.75 ) {
return x0 + ( gamma / stdlib_base_tan( STDLIB_CONSTANT_FLOAT64_PI * ( 1.0 - p ) ) );
}
return x0 + gamma * stdlib_base_tan( STDLIB_CONSTANT_FLOAT64_PI * ( p - 0.5 ) );
}
1 change: 0 additions & 1 deletion lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/test/fixtures/julia/REQUIRE
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@@ -1,3 +1,2 @@
Distributions 0.23.8
julia 1.5
JSON 0.21
2 changes: 1 addition & 1 deletion ...ode_modules/@stdlib/stats/base/dists/cauchy/quantile/test/fixtures/julia/large_gamma.json
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2 changes: 1 addition & 1 deletion ...modules/@stdlib/stats/base/dists/cauchy/quantile/test/fixtures/julia/negative_median.json
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2 changes: 1 addition & 1 deletion ...modules/@stdlib/stats/base/dists/cauchy/quantile/test/fixtures/julia/positive_median.json
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14 changes: 10 additions & 4 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/test/fixtures/julia/runner.jl
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Expand Up @@ -16,14 +16,15 @@
# See the License for the specific language governing permissions and
# limitations under the License.

import Distributions: quantile, Cauchy
import JSON

"""
gen( p, x0, gamma, name )

Generate fixture data and write to file.

The quantile is evaluated in extended precision from the textbook formula `x0 + gamma*tan(pi*(p - 0.5))`, as forming `p - 0.5` in double precision rounds away the low-order bits of a `p` close to `0`.

# Arguments

* `p`: input value
Expand All @@ -41,9 +42,8 @@ julia> gen( p, x0, gamma, \"data.json\" );
```
"""
function gen( p, x0, gamma, name )
z = Array{Float64}( undef, length(p) );
for i in eachindex(p)
z[ i ] = quantile( Cauchy( x0[i], gamma[i] ), p[i] );
z = setprecision( BigFloat, 2048 ) do
Float64.( BigFloat.( x0 ) .+ ( BigFloat.( gamma ) .* tan.( BigFloat( pi ) .* ( BigFloat.( p ) .- 0.5 ) ) ) )
end

# Store data to be written to file as a collection:
Expand Down Expand Up @@ -87,3 +87,9 @@ p = rand( 1000 );
x0 = rand( 1000 ) .* 1.0;
gamma = rand( 1000 ) .* 50.0;
gen( p, x0, gamma, "large_gamma.json" );

# Tails (`p` and `1-p` log-spaced; `x0 = 0`, so that the result does not also measure the rounding of `x0 + ...`):
p = vcat( exp10.( range( -300.0, stop = -1.0, length = 500 ) ), 1.0 .- exp10.( range( -16.0, stop = -1.0, length = 500 ) ) );
x0 = zeros( 1000 );
gamma = ( rand( 1000 ) .* 20.0 ) .+ 0.1;
gen( p, x0, gamma, "tails.json" );
1 change: 1 addition & 0 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/test/fixtures/julia/tails.json
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84 changes: 29 additions & 55 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/test/test.factory.js
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Original file line number Diff line number Diff line change
Expand Up @@ -22,10 +22,9 @@

var tape = require( 'tape' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var abs = require( '@stdlib/math/base/special/abs' );
var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var EPS = require( '@stdlib/constants/float64/eps' );
var factory = require( './../lib/factory.js' );


Expand All @@ -34,6 +33,7 @@ var factory = require( './../lib/factory.js' );
var largeGamma = require( './fixtures/julia/large_gamma.json' );
var negativeMedian = require( './fixtures/julia/negative_median.json' );
var positiveMedian = require( './fixtures/julia/positive_median.json' );
var tails = require( './fixtures/julia/tails.json' );


// TESTS //
Expand Down Expand Up @@ -131,36 +131,11 @@ tape( 'if provided a nonpositive `gamma`, the created function always returns `N
tape( 'the created function evaluates the quantile function at `p` given parameters `x0` and `gamma` (large `gamma`)', function test( t ) {
var expected;
var quantile;
var delta;
var gamma;
var tol;
var x0;
var i;
var p;
var y;
var i;

/*
* Higher tolerance than EPS because Julia gives slightly different results for
* |x| ~<= 3*pi/4:
*
* Example 1:
* x = -1.35646279095478;
* Julia (tan): -4.593961172862999
* stdlib (tan): -4.593961172863
* Mathematica: -4.59396117286300026311049650877442413097818001966176559315
*
* Example 2:
* x = 1.4710248292410089
* Julia (tan): 9.989623320530624
* stdlib (tan): 9.989623320530626
* Mathematica: 9.989623320530629158499137574831736702146195199133529403233
*
* Example 3:
* x = 1.528545878614728
* Julia (tan): 23.654302824341386
* stdlib (tan): 23.65430282434139
* Mathematica: 23.65430282434144042648214719732782590575979471046811610915...
*/

expected = largeGamma.expected;
p = largeGamma.p;
Expand All @@ -169,27 +144,19 @@ tape( 'the created function evaluates the quantile function at `p` given paramet
for ( i = 0; i < p.length; i++ ) {
quantile = factory( x0[i], gamma[i] );
y = quantile( p[i] );
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'p: '+p[i]+', x0: '+x0[i]+', gamma: '+gamma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 50.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. p: '+p[ i ]+'. x0: '+x0[i]+'. gamma: '+gamma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 28 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the created function evaluates the quantile function at `p` given `x0` and `gamma` (`x0 > 0`)', function test( t ) {
var expected;
var quantile;
var delta;
var gamma;
var tol;
var x0;
var i;
var p;
var y;
var i;

expected = positiveMedian.expected;
p = positiveMedian.p;
Expand All @@ -198,27 +165,19 @@ tape( 'the created function evaluates the quantile function at `p` given `x0` an
for ( i = 0; i < p.length; i++ ) {
quantile = factory( x0[i], gamma[i] );
y = quantile( p[i] );
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'p: '+p[i]+', x0: '+x0[i]+', gamma: '+gamma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 125.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. p: '+p[ i ]+'. x0: '+x0[i]+'. gamma: '+gamma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 162 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the created function evaluates the quantile function at `p` given `x0` and `gamma` (`x0 < 0`)', function test( t ) {
var expected;
var quantile;
var delta;
var gamma;
var tol;
var x0;
var i;
var p;
var y;
var i;

expected = negativeMedian.expected;
p = negativeMedian.p;
Expand All @@ -227,13 +186,28 @@ tape( 'the created function evaluates the quantile function at `p` given `x0` an
for ( i = 0; i < p.length; i++ ) {
quantile = factory( x0[i], gamma[i] );
y = quantile( p[i] );
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'p: '+p[i]+', x0: '+x0[i]+', gamma: '+gamma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 50.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. p: '+p[ i ]+'. x0: '+x0[i]+'. gamma: '+gamma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 326 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the created function evaluates the quantile function at `p` in the tails (`p` or `1-p` close to `0`)', function test( t ) {
var expected;
var quantile;
var gamma;
var x0;
var i;
var p;
var y;

expected = tails.expected;
p = tails.p;
x0 = tails.x0;
gamma = tails.gamma;
for ( i = 0; i < p.length; i++ ) {
quantile = factory( x0[i], gamma[i] );
y = quantile( p[i] );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
}
t.end();
});
82 changes: 27 additions & 55 deletions lib/node_modules/@stdlib/stats/base/dists/cauchy/quantile/test/test.native.js
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Original file line number Diff line number Diff line change
Expand Up @@ -24,17 +24,17 @@ var resolve = require( 'path' ).resolve;
var tape = require( 'tape' );
var tryRequire = require( '@stdlib/utils/try-require' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var abs = require( '@stdlib/math/base/special/abs' );
var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var EPS = require( '@stdlib/constants/float64/eps' );


// FIXTURES //

var largeGamma = require( './fixtures/julia/large_gamma.json' );
var negativeMedian = require( './fixtures/julia/negative_median.json' );
var positiveMedian = require( './fixtures/julia/positive_median.json' );
var tails = require( './fixtures/julia/tails.json' );


// VARIABLES //
Expand Down Expand Up @@ -104,103 +104,75 @@ tape( 'if provided a nonpositive `gamma`, the function always returns `NaN`', op
tape( 'the function evaluates the quantile function at `p` given `x0` and `gamma` (large `gamma`)', opts, function test( t ) {
var expected;
var gamma;
var delta;
var tol;
var x0;
var i;
var p;
var y;
var i;

/*
* Higher tolerance than EPS because Julia gives slightly different results for
* |x| ~<= 3*pi/4:
*
* Example 1:
* x = -1.35646279095478;
* Julia (tan): -4.593961172862999
* stdlib (tan): -4.593961172863
* Mathematica: -4.59396117286300026311049650877442413097818001966176559315
*
* Example 2:
* x = 1.4710248292410089
* Julia (tan): 9.989623320530624
* stdlib (tan): 9.989623320530626
* Mathematica: 9.989623320530629158499137574831736702146195199133529403233
*
* Example 3:
* x = 1.528545878614728
* Julia (tan): 23.654302824341386
* stdlib (tan): 23.65430282434139
* Mathematica: 23.65430282434144042648214719732782590575979471046811610915...
*/

expected = largeGamma.expected;
p = largeGamma.p;
x0 = largeGamma.x0;
gamma = largeGamma.gamma;
for ( i = 0; i < p.length; i++ ) {
y = quantile( p[i], x0[i], gamma[i] );
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'p: '+p[i]+', x0:'+x0[i]+', gamma: '+gamma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 50.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. p: '+p[ i ]+'. x0: '+x0[i]+'. gamma: '+gamma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 28 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the function evaluates the quantile function at `p` given `x0` and `gamma` (`x0 > 0`)', opts, function test( t ) {
var expected;
var delta;
var gamma;
var tol;
var x0;
var i;
var p;
var y;
var i;

expected = positiveMedian.expected;
p = positiveMedian.p;
x0 = positiveMedian.x0;
gamma = positiveMedian.gamma;
for ( i = 0; i < p.length; i++ ) {
y = quantile( p[i], x0[i], gamma[i] );
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'p: '+p[i]+', x0:'+x0[i]+', gamma: '+gamma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 125.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. p: '+p[ i ]+'. x0: '+x0[i]+'. gamma: '+gamma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 162 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the function evaluates the quantile function at `p` given `x0` and `gamma` (`x0 < 0`)', opts, function test( t ) {
var expected;
var delta;
var gamma;
var tol;
var x0;
var i;
var p;
var y;
var i;

expected = negativeMedian.expected;
p = negativeMedian.p;
x0 = negativeMedian.x0;
gamma = negativeMedian.gamma;
for ( i = 0; i < p.length; i++ ) {
y = quantile( p[i], x0[i], gamma[i] );
if ( y === expected[i] ) {
t.strictEqual( y, expected[i], 'p: '+p[i]+', x0:'+x0[i]+', gamma: '+gamma[i]+', y: '+y+', expected: '+expected[i] );
} else {
delta = abs( y - expected[ i ] );
tol = 90.0 * EPS * abs( expected[ i ] );
t.ok( delta <= tol, 'within tolerance. p: '+p[ i ]+'. x0: '+x0[i]+'. gamma: '+gamma[i]+'. y: '+y+'. E: '+expected[ i ]+'. Δ: '+delta+'. tol: '+tol+'.' );
}
t.strictEqual( isAlmostSameValue( y, expected[ i ], 326 ), true, 'returns expected value' );
}
t.end();
});

tape( 'the function evaluates the quantile function at `p` in the tails (`p` or `1-p` close to `0`)', opts, function test( t ) {
var expected;
var gamma;
var x0;
var i;
var p;
var y;

expected = tails.expected;
p = tails.p;
x0 = tails.x0;
gamma = tails.gamma;
for ( i = 0; i < p.length; i++ ) {
y = quantile( p[i], x0[i], gamma[i] );
t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
}
t.end();
});
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