[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r339354617
##
File path:
flink-ml-parent/flink-ml-lib/src/main/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussian.java
##
@@ -0,0 +1,148 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.BLAS;
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+import org.apache.flink.ml.common.linalg.SparseVector;
+import org.apache.flink.ml.common.linalg.Vector;
+
+import com.github.fommil.netlib.LAPACK;
+import org.netlib.util.intW;
+
+/**
+ * This class provides basic functionality for a Multivariate Gaussian
(Normal) Distribution.
+ */
+public class MultivariateGaussian {
+
+ private static final LAPACK LAPACK_INST = LAPACK.getInstance();
+ private static final double EPSILON;
+
+ static {
+ double eps = 1.0;
+ while ((1.0 + (eps / 2.0)) != 1.0) {
+ eps /= 2.0;
+ }
+ EPSILON = eps;
+ }
+
+ private final DenseVector mean;
+ private final DenseMatrix cov;
+
+ private DenseMatrix rootSigmaInv;
+ private double u;
+
+ // data buffers for computing pdf
+ private DenseVector delta;
+ private DenseVector v;
+
+ /**
+* The constructor.
+*
+* @param mean The mean vector of the distribution.
+* @param cov The covariance matrix of the distribution.
+*/
+ public MultivariateGaussian(DenseVector mean, DenseMatrix cov) {
+ this.mean = mean;
+ this.cov = cov;
+ this.delta = DenseVector.zeros(mean.size());
+ this.v = DenseVector.zeros(mean.size());
+ calculateCovarianceConstants();
+ }
+
+ /**
+* Returns density of this multivariate Gaussian at given point, x.
+*/
+ public double pdf(Vector x) {
+ return Math.exp(logpdf(x));
+ }
+
+ /**
+* Returns the log-density of this multivariate Gaussian at given
point, x.
+*/
+ public double logpdf(Vector x) {
+ int n = mean.size();
+ System.arraycopy(mean.getData(), 0, delta.getData(), 0, n);
+ BLAS.scal(-1.0, delta);
+ if (x instanceof DenseVector) {
+ BLAS.axpy(1., (DenseVector) x, delta);
+ } else if (x instanceof SparseVector) {
+ BLAS.axpy(1., (SparseVector) x, delta);
+ }
+ BLAS.gemv(1.0, rootSigmaInv, false, delta, 0., v);
+ return u - 0.5 * BLAS.dot(v, v);
+ }
+
+ /**
+* Compute distribution dependent constants.
+*
+* The probability density function is calculated as:
+* pdf(x) = (2*pi)^(-k/2)^ * det(sigma)^(-1/2)^ * exp((-1/2) * (x-mu).t
* inv(sigma) * (x-mu))
+*
+* Here we compute the following distribution dependent constants
that can be reused in each pdf computation:
+* A) u = log((2*pi)^(-k/2)^ * det(sigma)^(-1/2)^)
+* B) rootSigmaInv = sqrt(inv(sigma)) = D^(-1/2)^ * U.t
+*
+*
+* sigma = U * D * U.t
+* inv(sigma) = U * inv(D) * U.t = (D^{-1/2}^ * U.t).t *
(D^{-1/2}^ * U.t)
+* sqrt(inv(sigma)) = D^(-1/2)^ * U.t
+*
+*/
+ private void calculateCovarianceConstants() {
+ int n = this.mean.size();
+ int lwork = 3 * n - 1;
+ double[] matA = new double[n * n];
+ double[] work = new double[lwork];
+ double[] evs = new double[n];
+ intW info = new intW(0);
+
+ for (int i = 0; i < n; i++) {
+ System.arraycopy(cov.getData(), i * n, matA, i * n, i +
1);
+ }
+ LAPACK_INST.dsyev("V", "U", n, matA, n, evs, work, lwork, info);
+
+
[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r339354266
##
File path:
flink-ml-parent/flink-ml-lib/src/main/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussian.java
##
@@ -0,0 +1,148 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.BLAS;
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+import org.apache.flink.ml.common.linalg.SparseVector;
+import org.apache.flink.ml.common.linalg.Vector;
+
+import com.github.fommil.netlib.LAPACK;
+import org.netlib.util.intW;
+
+/**
+ * This class provides basic functionality for a Multivariate Gaussian
(Normal) Distribution.
+ */
+public class MultivariateGaussian {
+
+ private static final LAPACK LAPACK_INST = LAPACK.getInstance();
+ private static final double EPSILON;
+
+ static {
+ double eps = 1.0;
+ while ((1.0 + (eps / 2.0)) != 1.0) {
+ eps /= 2.0;
+ }
+ EPSILON = eps;
+ }
+
+ private final DenseVector mean;
+ private final DenseMatrix cov;
+
+ private DenseMatrix rootSigmaInv;
+ private double u;
Review comment:
members can be `final`?
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[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r339354298
##
File path:
flink-ml-parent/flink-ml-lib/src/test/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussianTest.java
##
@@ -0,0 +1,57 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+
+import org.junit.Assert;
+import org.junit.Test;
+
+/**
+ * Test cases for MultivariateGaussian.
+ */
+public class MultivariateGaussianTest {
+ private static final double TOL = 1.0e-5;
+
Review comment:
add test for `Univariate` as well just in case?
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[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r339354650
##
File path:
flink-ml-parent/flink-ml-lib/src/main/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussian.java
##
@@ -0,0 +1,148 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.BLAS;
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+import org.apache.flink.ml.common.linalg.SparseVector;
+import org.apache.flink.ml.common.linalg.Vector;
+
+import com.github.fommil.netlib.LAPACK;
+import org.netlib.util.intW;
+
+/**
+ * This class provides basic functionality for a Multivariate Gaussian
(Normal) Distribution.
+ */
+public class MultivariateGaussian {
+
+ private static final LAPACK LAPACK_INST = LAPACK.getInstance();
+ private static final double EPSILON;
+
+ static {
+ double eps = 1.0;
+ while ((1.0 + (eps / 2.0)) != 1.0) {
+ eps /= 2.0;
+ }
+ EPSILON = eps;
+ }
+
+ private final DenseVector mean;
+ private final DenseMatrix cov;
+
+ private DenseMatrix rootSigmaInv;
+ private double u;
+
+ // data buffers for computing pdf
+ private DenseVector delta;
+ private DenseVector v;
+
+ /**
+* The constructor.
+*
+* @param mean The mean vector of the distribution.
+* @param cov The covariance matrix of the distribution.
+*/
+ public MultivariateGaussian(DenseVector mean, DenseMatrix cov) {
+ this.mean = mean;
+ this.cov = cov;
+ this.delta = DenseVector.zeros(mean.size());
+ this.v = DenseVector.zeros(mean.size());
+ calculateCovarianceConstants();
+ }
+
+ /**
+* Returns density of this multivariate Gaussian at given point, x.
+*/
+ public double pdf(Vector x) {
+ return Math.exp(logpdf(x));
+ }
+
+ /**
+* Returns the log-density of this multivariate Gaussian at given
point, x.
+*/
+ public double logpdf(Vector x) {
+ int n = mean.size();
+ System.arraycopy(mean.getData(), 0, delta.getData(), 0, n);
+ BLAS.scal(-1.0, delta);
+ if (x instanceof DenseVector) {
+ BLAS.axpy(1., (DenseVector) x, delta);
+ } else if (x instanceof SparseVector) {
+ BLAS.axpy(1., (SparseVector) x, delta);
+ }
+ BLAS.gemv(1.0, rootSigmaInv, false, delta, 0., v);
+ return u - 0.5 * BLAS.dot(v, v);
+ }
+
+ /**
+* Compute distribution dependent constants.
+*
+* The probability density function is calculated as:
+* pdf(x) = (2*pi)^(-k/2)^ * det(sigma)^(-1/2)^ * exp((-1/2) * (x-mu).t
* inv(sigma) * (x-mu))
+*
+* Here we compute the following distribution dependent constants
that can be reused in each pdf computation:
+* A) u = log((2*pi)^(-k/2)^ * det(sigma)^(-1/2)^)
+* B) rootSigmaInv = sqrt(inv(sigma)) = D^(-1/2)^ * U.t
+*
+*
+* sigma = U * D * U.t
+* inv(sigma) = U * inv(D) * U.t = (D^{-1/2}^ * U.t).t *
(D^{-1/2}^ * U.t)
+* sqrt(inv(sigma)) = D^(-1/2)^ * U.t
+*
+*/
+ private void calculateCovarianceConstants() {
+ int n = this.mean.size();
+ int lwork = 3 * n - 1;
+ double[] matA = new double[n * n];
+ double[] work = new double[lwork];
+ double[] evs = new double[n];
+ intW info = new intW(0);
+
+ for (int i = 0; i < n; i++) {
+ System.arraycopy(cov.getData(), i * n, matA, i * n, i +
1);
+ }
+ LAPACK_INST.dsyev("V", "U", n, matA, n, evs, work, lwork, info);
+
+
[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r339354278
##
File path:
flink-ml-parent/flink-ml-lib/src/main/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussian.java
##
@@ -0,0 +1,148 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.BLAS;
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+import org.apache.flink.ml.common.linalg.SparseVector;
+import org.apache.flink.ml.common.linalg.Vector;
+
+import com.github.fommil.netlib.LAPACK;
+import org.netlib.util.intW;
+
+/**
+ * This class provides basic functionality for a Multivariate Gaussian
(Normal) Distribution.
+ */
+public class MultivariateGaussian {
+
+ private static final LAPACK LAPACK_INST = LAPACK.getInstance();
+ private static final double EPSILON;
+
+ static {
+ double eps = 1.0;
+ while ((1.0 + (eps / 2.0)) != 1.0) {
+ eps /= 2.0;
+ }
+ EPSILON = eps;
+ }
+
+ private final DenseVector mean;
+ private final DenseMatrix cov;
+
+ private DenseMatrix rootSigmaInv;
+ private double u;
+
+ // data buffers for computing pdf
+ private DenseVector delta;
+ private DenseVector v;
+
+ /**
+* The constructor.
+*
+* @param mean The mean vector of the distribution.
+* @param cov The covariance matrix of the distribution.
+*/
+ public MultivariateGaussian(DenseVector mean, DenseMatrix cov) {
+ this.mean = mean;
+ this.cov = cov;
+ this.delta = DenseVector.zeros(mean.size());
+ this.v = DenseVector.zeros(mean.size());
+ calculateCovarianceConstants();
+ }
+
+ /**
+* Returns density of this multivariate Gaussian at given point, x.
+*/
+ public double pdf(Vector x) {
+ return Math.exp(logpdf(x));
+ }
+
+ /**
+* Returns the log-density of this multivariate Gaussian at given
point, x.
+*/
+ public double logpdf(Vector x) {
+ int n = mean.size();
+ System.arraycopy(mean.getData(), 0, delta.getData(), 0, n);
+ BLAS.scal(-1.0, delta);
+ if (x instanceof DenseVector) {
+ BLAS.axpy(1., (DenseVector) x, delta);
+ } else if (x instanceof SparseVector) {
+ BLAS.axpy(1., (SparseVector) x, delta);
+ }
+ BLAS.gemv(1.0, rootSigmaInv, false, delta, 0., v);
+ return u - 0.5 * BLAS.dot(v, v);
+ }
+
+ /**
+* Compute distribution dependent constants.
+*
+* The probability density function is calculated as:
+* pdf(x) = (2*pi)^(-k/2)^ * det(sigma)^(-1/2)^ * exp((-1/2) * (x-mu).t
* inv(sigma) * (x-mu))
+*
+* Here we compute the following distribution dependent constants
that can be reused in each pdf computation:
+* A) u = log((2*pi)^(-k/2)^ * det(sigma)^(-1/2)^)
+* B) rootSigmaInv = sqrt(inv(sigma)) = D^(-1/2)^ * U.t
+*
+*
+* sigma = U * D * U.t
+* inv(sigma) = U * inv(D) * U.t = (D^{-1/2}^ * U.t).t *
(D^{-1/2}^ * U.t)
+* sqrt(inv(sigma)) = D^(-1/2)^ * U.t
+*
+*/
+ private void calculateCovarianceConstants() {
+ int n = this.mean.size();
+ int lwork = 3 * n - 1;
+ double[] matA = new double[n * n];
+ double[] work = new double[lwork];
+ double[] evs = new double[n];
+ intW info = new intW(0);
+
+ for (int i = 0; i < n; i++) {
+ System.arraycopy(cov.getData(), i * n, matA, i * n, i +
1);
+ }
Review comment:
`System.arraycopy(cov.getData(), 0, matA, 0, n*n);`?
[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r337847736
##
File path:
flink-ml-parent/flink-ml-lib/src/main/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussian.java
##
@@ -0,0 +1,138 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.BLAS;
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+import org.apache.flink.ml.common.linalg.SparseVector;
+import org.apache.flink.ml.common.linalg.Vector;
+
+import com.github.fommil.netlib.LAPACK;
+import org.netlib.util.intW;
+
+/**
+ * This class provides basic functionality for a Multivariate Gaussian
(Normal) Distribution.
+ */
+public class MultivariateGaussian {
+
+ private static final LAPACK LAPACK_INST = LAPACK.getInstance();
+ private static final double EPSILON;
+
+ static {
+ double eps = 1.0;
+ while ((1.0 + (eps / 2.0)) != 1.0) {
+ eps /= 2.0;
+ }
+ EPSILON = eps;
+ }
+
+ private final DenseVector mean;
+ private final DenseMatrix cov;
+
+ private DenseMatrix rootSigmaInv;
+ private double u;
+
+ // data buffers for computing pdf
+ private DenseVector delta;
+ private DenseVector v;
+
+ /**
+* The constructor.
+*
+* @param mean The mean vector of the distribution.
+* @param cov The covariance matrix of the distribution.
+*/
+ public MultivariateGaussian(DenseVector mean, DenseMatrix cov) {
+ this.mean = mean;
+ this.cov = cov;
+ this.delta = DenseVector.zeros(mean.size());
+ this.v = DenseVector.zeros(mean.size());
+ calculateCovarianceConstants();
+ }
+
+ /**
+* Returns density of this multivariate Gaussian at given point, x.
+*/
+ public double pdf(Vector x) {
+ return Math.exp(logpdf(x));
+ }
+
+ /**
+* Returns the log-density of this multivariate Gaussian at given
point, x.
+*/
+ public double logpdf(Vector x) {
+ int n = mean.size();
+ System.arraycopy(mean.getData(), 0, delta.getData(), 0, n);
+ BLAS.scal(-1.0, delta);
+ if (x instanceof DenseVector) {
+ BLAS.axpy(1., (DenseVector) x, delta);
+ } else if (x instanceof SparseVector) {
+ BLAS.axpy(1., (SparseVector) x, delta);
+ }
+ BLAS.gemv(1.0, rootSigmaInv, false, delta, 0., v);
+ return u - 0.5 * BLAS.dot(v, v);
+ }
+
+ /**
+* Compute distribution dependent constants.
+*
+* rootSigmaInv = D^(-1/2)^ * U.t, where sigma = U * D * U.t .
Review comment:
much more informatively:
```
*
* sigma = U * D * U.t
* inv(Sigma) = U * inv(D) * U.t = = (D^{-1/2}^ * U.t).t * (D^{-1/2}^ *
U.t)
* sqrt(inv(Sigma)) = D^(-1/2)^ * U.t
*
```
since the word `rootSigmaInv` is not intuitively understandable to me.
(especially "root" doesn't mean square-root)
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[GitHub] [flink] walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add the class for multivariate Gaussian Distribution.
walterddr commented on a change in pull request #9733: [FLINK-14154][ml] Add
the class for multivariate Gaussian Distribution.
URL: https://github.com/apache/flink/pull/9733#discussion_r337846841
##
File path:
flink-ml-parent/flink-ml-lib/src/test/java/org/apache/flink/ml/common/statistics/basicstatistic/MultivariateGaussianTest.java
##
@@ -0,0 +1,57 @@
+/*
+ * 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.
+ */
+
+package org.apache.flink.ml.common.statistics.basicstatistic;
+
+import org.apache.flink.ml.common.linalg.DenseMatrix;
+import org.apache.flink.ml.common.linalg.DenseVector;
+
+import org.junit.Assert;
+import org.junit.Test;
+
+/**
+ * Test cases for MultivariateGaussian.
+ */
+public class MultivariateGaussianTest {
+ private static final double TOL = 1.0e-5;
Review comment:
Is there a reason why we use a fixed `TOL` while during the actual execution
we use the limited bi-section to determine the actual TOL for the system?
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