public void PrincipalComponentAnalysis()
        {
            int D = 3;
            int N = 10;

            // construct a sample
            Random             rng    = new Random(1);
            MultivariateSample sample = new MultivariateSample(D);

            for (int i = 0; i < N; i++)
            {
                double x = 1.0 * rng.NextDouble() - 1.0;
                double y = 4.0 * rng.NextDouble() - 2.0;
                double z = 9.0 * rng.NextDouble() - 3.0;
                sample.Add(x, y, z);
            }

            // get its column means
            RowVector mu = new RowVector(D);

            for (int i = 0; i < D; i++)
            {
                mu[i] = sample.Column(i).Mean;
            }

            // get total variance
            double tVariance = GetTotalVariance(sample);

            Console.WriteLine(tVariance);

            // do a principal component analysis
            PrincipalComponentAnalysis pca = sample.PrincipalComponentAnalysis();

            Assert.IsTrue(pca.Dimension == sample.Dimension);
            Assert.IsTrue(pca.Count == sample.Count);

            // check that the PCs behave as expected
            for (int i = 0; i < pca.Dimension; i++)
            {
                PrincipalComponent pc = pca.Component(i);
                Assert.IsTrue(pc.Index == i);
                Assert.IsTrue(pc.Analysis == pca);
                Assert.IsTrue(TestUtilities.IsNearlyEqual(pc.Weight * pc.NormalizedVector(), pc.ScaledVector()));
                Assert.IsTrue((0.0 <= pc.VarianceFraction) && (pc.VarianceFraction <= 1.0));
                if (i == 0)
                {
                    Assert.IsTrue(pc.VarianceFraction == pc.CumulativeVarianceFraction);
                }
                else
                {
                    PrincipalComponent ppc = pca.Component(i - 1);
                    Assert.IsTrue(pc.VarianceFraction <= ppc.VarianceFraction);
                    Assert.IsTrue(TestUtilities.IsNearlyEqual(ppc.CumulativeVarianceFraction + pc.VarianceFraction, pc.CumulativeVarianceFraction));
                }
            }

            // express the sample in terms of principal components
            MultivariateSample csample = pca.TransformedSample();

            // check that the explained variances are as claimed
            for (int rD = 1; rD <= D; rD++)
            {
                MultivariateSample rSample = new MultivariateSample(D);
                foreach (double[] cEntry in csample)
                {
                    RowVector x = mu.Copy();
                    for (int i = 0; i < rD; i++)
                    {
                        PrincipalComponent pc = pca.Component(i);
                        x += (cEntry[i] * pc.Weight) * pc.NormalizedVector();
                    }
                    rSample.Add(x);
                }
                double rVariance = GetTotalVariance(rSample);
                Console.WriteLine("{0} {1}", rD, rVariance);
                Assert.IsTrue(TestUtilities.IsNearlyEqual(rVariance / tVariance, pca.Component(rD - 1).CumulativeVarianceFraction));
            }
        }
        public void PrincipalComponentAnalysis()
        {
            int D = 3;
            int N = 10;

            // construct a sample
            Random             rng    = new Random(1);
            MultivariateSample sample = new MultivariateSample(D);

            for (int i = 0; i < N; i++)
            {
                double x = 1.0 * rng.NextDouble() - 1.0;
                double y = 4.0 * rng.NextDouble() - 2.0;
                double z = 9.0 * rng.NextDouble() - 3.0;
                sample.Add(x, y, z);
            }

            // get its column means
            RowVector mu = new RowVector(D);

            for (int i = 0; i < D; i++)
            {
                mu[i] = sample.Column(i).Mean;
            }

            // get total variance
            double tVariance = GetTotalVariance(sample);

            Console.WriteLine(tVariance);

            // do a principal component analysis
            PrincipalComponentAnalysis pca = sample.PrincipalComponentAnalysis();

            Assert.IsTrue(pca.Dimension == sample.Dimension);
            Assert.IsTrue(pca.Count == sample.Count);

            // check that the PCs behave as expected
            Assert.IsTrue(pca.Components.Count == pca.Dimension);
            for (int i = 0; i < pca.Dimension; i++)
            {
                PrincipalComponent pc = pca.Components[i];
                Assert.IsTrue(pc.Index == i);
                Assert.IsTrue(pc.Analysis == pca);
                Assert.IsTrue(TestUtilities.IsNearlyEqual(pc.Weight * pc.NormalizedVector, pc.ScaledVector()));
                Assert.IsTrue(pca.MinimumDimension(pc.CumulativeVarianceFraction) == i + 1);
            }

            // Check enumerator, and verify that variance fractions behave as expected.
            int    count      = 0;
            double cumulative = 0.0;
            double previous   = Double.PositiveInfinity;

            foreach (PrincipalComponent pc in pca.Components)
            {
                Assert.IsTrue(pc.Index == count);
                count++;
                Assert.IsTrue((0.0 <= pc.VarianceFraction) && (pc.VarianceFraction <= 1.0));
                Assert.IsTrue(pc.VarianceFraction <= previous);
                previous    = pc.VarianceFraction;
                cumulative += pc.VarianceFraction;
                Assert.IsTrue(TestUtilities.IsNearlyEqual(cumulative, pc.CumulativeVarianceFraction));
            }
            Assert.IsTrue(count == pca.Components.Count);

            // express the sample in terms of principal components
            MultivariateSample csample = pca.TransformedSample();

            // check that the explained variances are as claimed
            for (int rD = 1; rD <= D; rD++)
            {
                MultivariateSample rSample = new MultivariateSample(D);
                foreach (double[] cEntry in csample)
                {
                    RowVector x = mu.Copy();
                    for (int i = 0; i < rD; i++)
                    {
                        PrincipalComponent pc = pca.Components[i];
                        x += (cEntry[i] * pc.Weight) * pc.NormalizedVector;
                    }
                    rSample.Add(x);
                }
                double rVariance = GetTotalVariance(rSample);
                Console.WriteLine("{0} {1}", rD, rVariance);
                Assert.IsTrue(TestUtilities.IsNearlyEqual(rVariance / tVariance, pca.Components[rD - 1].CumulativeVarianceFraction));
            }
        }