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Posted to commits@commons.apache.org by ah...@apache.org on 2021/12/14 08:50:27 UTC
[commons-statistics] 01/02: STATISTICS-39: Update Gamma distribution to use Number's gamma
This is an automated email from the ASF dual-hosted git repository.
aherbert pushed a commit to branch master
in repository https://gitbox.apache.org/repos/asf/commons-statistics.git
commit 9d3c178463dee38867427f871bc8d8c07556bf8b
Author: Alex Herbert <ah...@apache.org>
AuthorDate: Mon Dec 13 20:34:44 2021 +0000
STATISTICS-39: Update Gamma distribution to use Number's gamma
The PDF is computed using RegularizedGamma.P.derivative.
Additional test data added for shape 0.25, 0.5, 0.75.
Existing PDF test tolerances lowered where improvements have been made.
Reinstate disabled test of a very small shape parameter.
---
.../statistics/distribution/GammaDistribution.java | 151 +-
.../distribution/GammaDistributionTest.java | 79 +-
.../distribution/gamma-distribution-shape-0.25.csv | 1039 +++
.../distribution/gamma-distribution-shape-0.5.csv | 1295 ++++
.../distribution/gamma-distribution-shape-0.75.csv | 1167 ++++
.../distribution/gamma-distribution-shape-1.csv | 6400 +++++++++----------
.../distribution/gamma-distribution-shape-10.csv | 800 +--
.../distribution/gamma-distribution-shape-100.csv | 786 +--
.../distribution/gamma-distribution-shape-1000.csv | 6620 ++++++++++----------
.../distribution/gamma-distribution-shape-142.csv | 1520 ++---
.../distribution/gamma-distribution-shape-8.csv | 6400 +++++++++----------
.../statistics/distribution/gamma-distribution.mac | 94 +-
.../distribution/test.gamma.1.properties | 2 +-
13 files changed, 14898 insertions(+), 11455 deletions(-)
diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/GammaDistribution.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/GammaDistribution.java
index 8da73c2..63bf630 100644
--- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/GammaDistribution.java
+++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/GammaDistribution.java
@@ -16,7 +16,7 @@
*/
package org.apache.commons.statistics.distribution;
-import org.apache.commons.numbers.gamma.LanczosApproximation;
+import org.apache.commons.numbers.gamma.LogGamma;
import org.apache.commons.numbers.gamma.RegularizedGamma;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.AhrensDieterMarsagliaTsangGammaSampler;
@@ -29,65 +29,13 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
private static final double SUPPORT_LO = 0;
/** Support upper bound. */
private static final double SUPPORT_HI = Double.POSITIVE_INFINITY;
- /** Lanczos constant. */
- private static final double LANCZOS_G = LanczosApproximation.g();
+
/** The shape parameter. */
private final double shape;
/** The scale parameter. */
private final double scale;
- /**
- * The constant value of {@code shape + g + 0.5}, where {@code g} is the
- * Lanczos constant {@link LanczosApproximation#g()}.
- */
- private final double shiftedShape;
- /**
- * The constant value of
- * {@code shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape)},
- * where {@code L(shape)} is the Lanczos approximation returned by
- * {@link LanczosApproximation#value(double)}. This prefactor is used in
- * {@link #density(double)}, when no overflow occurs with the natural
- * calculation.
- */
- private final double densityPrefactor1;
- /**
- * The constant value of
- * {@code log(shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape))},
- * where {@code L(shape)} is the Lanczos approximation returned by
- * {@link LanczosApproximation#value(double)}. This prefactor is used in
- * {@link #logDensity(double)}, when no overflow occurs with the natural
- * calculation.
- */
- private final double logDensityPrefactor1;
- /**
- * The constant value of
- * {@code shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape)},
- * where {@code L(shape)} is the Lanczos approximation returned by
- * {@link LanczosApproximation#value(double)}. This prefactor is used in
- * {@link #density(double)}, when overflow occurs with the natural
- * calculation.
- */
- private final double densityPrefactor2;
- /**
- * The constant value of
- * {@code log(shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape))},
- * where {@code L(shape)} is the Lanczos approximation returned by
- * {@link LanczosApproximation#value(double)}. This prefactor is used in
- * {@link #logDensity(double)}, when overflow occurs with the natural
- * calculation.
- */
- private final double logDensityPrefactor2;
- /**
- * Lower bound on {@code y = x / scale} for the selection of the computation
- * method in {@link #density(double)}. For {@code y <= minY}, the natural
- * calculation overflows.
- */
- private final double minY;
- /**
- * Upper bound on {@code log(y)} ({@code y = x / scale}) for the selection
- * of the computation method in {@link #density(double)}. For
- * {@code log(y) >= maxLogY}, the natural calculation overflows.
- */
- private final double maxLogY;
+ /** Precomputed term for the log density: {@code -log(gamma(shape)) - log(scale)}. */
+ private final double minusLogGammaShapeMinusLogScale;
/**
* @param shape Shape parameter.
@@ -97,19 +45,7 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
double scale) {
this.shape = shape;
this.scale = scale;
- this.shiftedShape = shape + LANCZOS_G + 0.5;
- final double aux = Math.E / (2.0 * Math.PI * shiftedShape);
- this.densityPrefactor2 = shape * Math.sqrt(aux) / LanczosApproximation.value(shape);
- this.logDensityPrefactor2 = Math.log(shape) + 0.5 * Math.log(aux) -
- Math.log(LanczosApproximation.value(shape));
- this.densityPrefactor1 = this.densityPrefactor2 / scale *
- Math.pow(shiftedShape, -shape) *
- Math.exp(shape + LANCZOS_G);
- this.logDensityPrefactor1 = this.logDensityPrefactor2 - Math.log(scale) -
- Math.log(shiftedShape) * shape +
- shape + LANCZOS_G;
- this.minY = shape + LANCZOS_G - Math.log(Double.MAX_VALUE);
- this.maxLogY = Math.log(Double.MAX_VALUE) / (shape - 1.0);
+ this.minusLogGammaShapeMinusLogScale = -LogGamma.value(shape) - Math.log(scale);
}
/**
@@ -160,44 +96,6 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
*/
@Override
public double density(double x) {
- /* The present method must return the value of
- *
- * 1 a-1 - x 1 x a - x
- * ------------ x exp(---) = ---------- (-) exp(---)
- * Gamma(a) b^a b x Gamma(a) b b
- *
- * where a is the shape parameter, and b the scale parameter.
- * Substituting the Lanczos approximation of Gamma(a) leads to the
- * following expression of the density
- *
- * a e 1 y a
- * - sqrt(------------------) ---- (-----------) exp(a - y + g),
- * x 2 pi (a + g + 0.5) L(a) a + g + 0.5
- *
- * where y = x / b. The above formula is the "natural" computation, which
- * is implemented when no overflow is likely to occur. If overflow occurs
- * with the natural computation, the following identity is used. It is
- * based on the BOOST library
- * http://www.boost.org/doc/libs/1_35_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma.html
- * Formula (15) needs adaptations, which are detailed below.
- *
- * y a
- * (-----------) exp(a - y + g)
- * a + g + 0.5
- * y - a - g - 0.5 y (g + 0.5)
- * = exp(a log1pm(---------------) - ----------- + g),
- * a + g + 0.5 a + g + 0.5
- *
- * where log1pm(z) = log(1 + z) - z. Therefore, the value to be
- * returned is
- *
- * a e 1
- * - sqrt(------------------) ----
- * x 2 pi (a + g + 0.5) L(a)
- * y - a - g - 0.5 y (g + 0.5)
- * * exp(a log1pm(---------------) - ----------- + g).
- * a + g + 0.5 a + g + 0.5
- */
if (x <= SUPPORT_LO ||
x >= SUPPORT_HI) {
// Special case x=0
@@ -209,20 +107,7 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
return 0;
}
- final double y = x / scale;
- if ((y <= minY) || (Math.log(y) >= maxLogY)) {
- /*
- * Overflow.
- */
- final double aux1 = (y - shiftedShape) / shiftedShape;
- final double aux2 = shape * (Math.log1p(aux1) - aux1);
- final double aux3 = -y * (LANCZOS_G + 0.5) / shiftedShape + LANCZOS_G + aux2;
- return densityPrefactor2 / x * Math.exp(aux3);
- }
- /*
- * Natural calculation.
- */
- return densityPrefactor1 * Math.exp(-y) * Math.pow(y, shape - 1);
+ return RegularizedGamma.P.derivative(shape, x / scale) / scale;
}
/** {@inheritDoc}
@@ -236,9 +121,6 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
*/
@Override
public double logDensity(double x) {
- /*
- * see the comment in {@link #density(double)} for computation details
- */
if (x <= SUPPORT_LO ||
x >= SUPPORT_HI) {
// Special case x=0
@@ -251,19 +133,15 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
}
final double y = x / scale;
- if ((y <= minY) || (Math.log(y) >= maxLogY)) {
- /*
- * Overflow.
- */
- final double aux1 = (y - shiftedShape) / shiftedShape;
- final double aux2 = shape * (Math.log1p(aux1) - aux1);
- final double aux3 = -y * (LANCZOS_G + 0.5) / shiftedShape + LANCZOS_G + aux2;
- return logDensityPrefactor2 - Math.log(x) + aux3;
+
+ // More accurate to log the density when it is finite.
+ // See NUMBERS-174: 'Log of the Gamma P Derivative'
+ final double p = RegularizedGamma.P.derivative(shape, y) / scale;
+ if (p <= Double.MAX_VALUE && p >= Double.MIN_NORMAL) {
+ return Math.log(p);
}
- /*
- * Natural calculation.
- */
- return logDensityPrefactor1 - y + Math.log(y) * (shape - 1);
+ // Use the log computation
+ return minusLogGammaShapeMinusLogScale - y + Math.log(y) * (shape - 1);
}
/**
@@ -287,7 +165,6 @@ public final class GammaDistribution extends AbstractContinuousDistribution {
} else if (x >= SUPPORT_HI) {
return 1;
}
-
return RegularizedGamma.P.value(shape, x / scale);
}
diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/GammaDistributionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/GammaDistributionTest.java
index 28b0d05..af06961 100644
--- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/GammaDistributionTest.java
+++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/GammaDistributionTest.java
@@ -24,7 +24,6 @@ import java.io.InputStreamReader;
import org.apache.commons.math3.stat.descriptive.SummaryStatistics;
import org.apache.commons.numbers.gamma.LanczosApproximation;
import org.junit.jupiter.api.Assertions;
-import org.junit.jupiter.api.Disabled;
import org.junit.jupiter.api.Test;
/**
@@ -67,37 +66,37 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
Assertions.assertEquals(4, dist.getVariance());
dist = GammaDistribution.of(1.1, 4.2);
- Assertions.assertEquals(1.1d * 4.2d, dist.getMean());
- Assertions.assertEquals(1.1d * 4.2d * 4.2d, dist.getVariance());
+ Assertions.assertEquals(1.1 * 4.2, dist.getMean());
+ Assertions.assertEquals(1.1 * 4.2 * 4.2, dist.getVariance());
}
@Test
void testAdditionalCumulativeProbability() {
- checkCumulativeProbability(-1.000, 4.0, 2.0, .0000);
- checkCumulativeProbability(15.501, 4.0, 2.0, .9499);
- checkCumulativeProbability(0.504, 4.0, 1.0, .0018);
- checkCumulativeProbability(10.011, 1.0, 2.0, .9933);
- checkCumulativeProbability(5.000, 2.0, 2.0, .7127);
+ checkCumulativeProbability(-1.000, 4.0, 2.0, 0.0);
+ checkCumulativeProbability(15.501, 4.0, 2.0, 0.94989465156755404);
+ checkCumulativeProbability(0.504, 4.0, 1.0, 0.0018026739713985257);
+ checkCumulativeProbability(10.011, 1.0, 2.0, 0.99329900998454213);
+ checkCumulativeProbability(5.000, 2.0, 2.0, 0.71270250481635422);
}
private void checkCumulativeProbability(double x, double a, double b, double expected) {
final GammaDistribution distribution = GammaDistribution.of(a, b);
final double actual = distribution.cumulativeProbability(x);
- Assertions.assertEquals(expected, actual, 10e-4, () -> "probability for " + x);
+ Assertions.assertEquals(expected, actual, expected * 1e-15, () -> "probability for " + x);
}
@Test
void testAdditionalInverseCumulativeProbability() {
- checkInverseCumulativeProbability(15.501, 4.0, 2.0, .9499);
- checkInverseCumulativeProbability(0.504, 4.0, 1.0, .0018);
- checkInverseCumulativeProbability(10.011, 1.0, 2.0, .9933);
- checkInverseCumulativeProbability(5.000, 2.0, 2.0, .7127);
+ checkInverseCumulativeProbability(15.501, 4.0, 2.0, 0.94989465156755404);
+ checkInverseCumulativeProbability(0.504, 4.0, 1.0, 0.0018026739713985257);
+ checkInverseCumulativeProbability(10.011, 1.0, 2.0, 0.99329900998454213);
+ checkInverseCumulativeProbability(5.000, 2.0, 2.0, 0.71270250481635422);
}
private void checkInverseCumulativeProbability(double expected, double a, double b, double p) {
final GammaDistribution distribution = GammaDistribution.of(a, b);
final double actual = distribution.inverseCumulativeProbability(p);
- Assertions.assertEquals(expected, actual, 10e-4, () -> "critical value for " + p);
+ Assertions.assertEquals(expected, actual, expected * 1e-10, () -> "critical value for " + p);
}
@Test
@@ -122,19 +121,13 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
// To force overflow condition
// R2.5: print(dgamma(x, shape=1000, rate=100), digits=10)
checkDensity(1000, 100, x, new double[]{0.000000000e+00, 0.000000000e+00, 0.000000000e+00, 0.000000000e+00, 0.000000000e+00, 3.304830256e-84});
-
- // TODO - This requires more test cases that evaluate points when the density
- // switches the computation.
}
/**
- * Test a shape below 1. This captures a known failure condition.
- * The conditions to detect overflow are incorrect and the computation
- * switches to an inaccurate method that computes pdf == 0 when it should
- * be very large.
+ * Test a shape far below 1. Support for very small shape parameters
+ * was fixed in STATISTICS-39.
*/
@Test
- @Disabled("Implementation fails for small x")
void testShapeBelow1() {
// R2.5: print(dgamma(x, shape=0.05, rate=1), digits=20)
final double[] x = new double[]{1e-100, 1e-10, 1e-5, 0.1};
@@ -147,7 +140,7 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
private void checkDensity(double alpha, double rate, double[] x, double[] expected) {
final GammaDistribution dist = GammaDistribution.of(alpha, 1 / rate);
for (int i = 0; i < x.length; i++) {
- Assertions.assertEquals(expected[i], dist.density(x[i]), Math.abs(expected[i]) * 1e-5);
+ Assertions.assertEquals(expected[i], dist.density(x[i]), Math.abs(expected[i]) * 1e-9);
}
}
@@ -171,15 +164,19 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
checkLogDensity(0.1, 4, x, new double[]{-inf, 10.319872287, -3.490250753, -6.114083216, -10.737915678, -23.562577337});
// R2.5: print(dgamma(x, shape=.1, rate=1, log=TRUE), digits=10)
checkLogDensity(0.1, 1, x, new double[]{-inf, 10.181245850, -2.128880189, -3.252712652, -4.876545114, -8.701206773});
- // To force overflow condition
+ // To force overflow condition to pdf=zero
// R2.5: print(dgamma(x, shape=1000, rate=100, log=TRUE), digits=10)
checkLogDensity(1000, 100, x, new double[]{-inf, -15101.7453846, -2042.5042706, -1400.0502372, -807.5962038, -192.2217627});
+ // To force overflow condition to pdf=infinity
+ final double[] x1 = {1e-315, 1e-320, 1e-323};
+ // scipy.stats gamma(1e-2).logpdf(x1)
+ checkLogDensity(0.01, 1, x1, new double[]{713.46168137365419, 724.85948860402209, 731.70997561537104});
}
private void checkLogDensity(double alpha, double rate, double[] x, double[] expected) {
final GammaDistribution dist = GammaDistribution.of(alpha, 1 / rate);
for (int i = 0; i < x.length; i++) {
- Assertions.assertEquals(expected[i], dist.logDensity(x[i]), Math.abs(expected[i]) * 1e-5);
+ Assertions.assertEquals(expected[i], dist.logDensity(x[i]), Math.abs(expected[i]) * 1e-9);
}
}
@@ -213,15 +210,6 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
Math.exp(-x / scale) / Math.exp(logGamma(shape));
}
- // TODO
- // Test against using Commons Numbers LogGamma.
- // The implementation for LogGamma has improved since Math-849 due to the use
- // of a LogGamma1P implementation when x < 8.
- // Compute logpdf(x) =
- // (shape - 1) * Math.log(x) - x / scale - LogGamma.value(shape) - shape * Math.log(scale)
- // This is the method used in scipy stats.
- // The result is used with exp to compute the pdf.
-
/**
* MATH-753: large values of x or shape parameter cause density(double) to
* overflow. Reference data is generated with the Maxima script
@@ -336,13 +324,28 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
}
@Test
+ void testMath753Shape025() {
+ doTestMath753(0.25, 1.0, 1.0, 0.0, 0.0, "gamma-distribution-shape-0.25.csv");
+ }
+
+ @Test
+ void testMath753Shape05() {
+ doTestMath753(0.5, 1.0, 1.0, 0.0, 0.0, "gamma-distribution-shape-0.5.csv");
+ }
+
+ @Test
+ void testMath753Shape075() {
+ doTestMath753(0.75, 1.0, 1.0, 0.0, 0.0, "gamma-distribution-shape-0.75.csv");
+ }
+
+ @Test
void testMath753Shape1() {
- doTestMath753(1.0, 1.5, 0.5, 0.0, 0.0, "gamma-distribution-shape-1.csv");
+ doTestMath753(1.0, 1.0, 0.5, 0.0, 0.0, "gamma-distribution-shape-1.csv");
}
@Test
void testMath753Shape8() {
- doTestMath753(8.0, 1.5, 1.0, 0.0, 0.0, "gamma-distribution-shape-8.csv");
+ doTestMath753(8.0, 1.0, 1.0, 0.0, 0.0, "gamma-distribution-shape-8.csv");
}
@Test
@@ -357,11 +360,11 @@ class GammaDistributionTest extends BaseContinuousDistributionTest {
@Test
void testMath753Shape142() {
- doTestMath753(142.0, 3.3, 1.6, 40.0, 40.0, "gamma-distribution-shape-142.csv");
+ doTestMath753(142.0, 1.5, 1.0, 40.0, 40.0, "gamma-distribution-shape-142.csv");
}
@Test
void testMath753Shape1000() {
- doTestMath753(1000.0, 1.0, 1.0, 160.0, 230.0, "gamma-distribution-shape-1000.csv");
+ doTestMath753(1000.0, 1.0, 1.0, 160.0, 200.0, "gamma-distribution-shape-1000.csv");
}
}
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.25.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.25.csv
new file mode 100644
index 0000000..f99859e
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.25.csv
@@ -0,0 +1,1039 @@
+#
+# Licensed to the Apache Software Foundation (ASF) under one or more
+# contributor license agreements. See the NOTICE file distributed with
+# this work for additional information regarding copyright ownership.
+# The ASF licenses this file to You under the Apache License, Version 2.0
+# (the "License"); you may not use this file except in compliance with
+# the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
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diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.5.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.5.csv
new file mode 100644
index 0000000..83611f9
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.5.csv
@@ -0,0 +1,1295 @@
+#
+# Licensed to the Apache Software Foundation (ASF) under one or more
+# contributor license agreements. See the NOTICE file distributed with
+# this work for additional information regarding copyright ownership.
+# The ASF licenses this file to You under the Apache License, Version 2.0
+# (the "License"); you may not use this file except in compliance with
+# the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
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diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.75.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.75.csv
new file mode 100644
index 0000000..e8c1ac1
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-0.75.csv
@@ -0,0 +1,1167 @@
+#
+# Licensed to the Apache Software Foundation (ASF) under one or more
+# contributor license agreements. See the NOTICE file distributed with
+# this work for additional information regarding copyright ownership.
+# The ASF licenses this file to You under the Apache License, Version 2.0
+# (the "License"); you may not use this file except in compliance with
+# the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
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diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-1.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-1.csv
index af00c10..fa1f072 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-1.csv
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/gamma-distribution-shape-1.csv
@@ -13,3203 +13,3203 @@
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
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