public void Test()
{
// Make a random list.
RandomHelper.Random = new Random(77);
List <VectorF> points = new List <VectorF>();
for (int i = 0; i < 10; i++)
{
var vector = new VectorF(4);
RandomHelper.Random.NextVectorF(vector, -1, 10);
points.Add(vector);
}
PrincipalComponentAnalysisF pca = new PrincipalComponentAnalysisF(points);
Assert.Greater(pca.Variances[0], pca.Variances[1]);
Assert.Greater(pca.Variances[1], pca.Variances[2]);
Assert.Greater(pca.Variances[2], pca.Variances[3]);
Assert.Greater(pca.Variances[3], 0);
Assert.IsTrue(pca.V.GetColumn(0).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(1).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(2).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(3).IsNumericallyNormalized);
// Compute covariance matrix and check if it is diagonal in the transformed space.
MatrixF cov = StatisticsHelper.ComputeCovarianceMatrix(points);
MatrixF transformedCov = pca.V.Transposed * cov * pca.V;
for (int row = 0; row < transformedCov.NumberOfRows; row++)
{
for (int column = 0; column < transformedCov.NumberOfColumns; column++)
{
if (row != column)
{
Assert.IsTrue(Numeric.IsZero(transformedCov[row, column]));
}
}
}
// The principal components must be Eigenvectors which means that multiplying with the covariance
// matrix does only change the length!
VectorF v0 = pca.V.GetColumn(0);
VectorF v0Result = cov * v0;
Assert.IsTrue(VectorF.AreNumericallyEqual(v0.Normalized, v0Result.Normalized));
VectorF v1 = pca.V.GetColumn(1);
VectorF v1Result = cov * v1;
Assert.IsTrue(VectorF.AreNumericallyEqual(v1.Normalized, v1Result.Normalized));
VectorF v2 = pca.V.GetColumn(2);
VectorF v2Result = cov * v2;
Assert.IsTrue(VectorF.AreNumericallyEqual(v2.Normalized, v2Result.Normalized));
VectorF v3 = pca.V.GetColumn(3);
VectorF v3Result = cov * v3;
Assert.IsTrue(VectorF.AreNumericallyEqual(v3.Normalized, v3Result.Normalized));
}
public void Test()
{
// Make a random list.
RandomHelper.Random = new Random(77);
List<VectorF> points = new List<VectorF>();
for (int i = 0; i < 10; i++)
{
var vector = new VectorF(4);
RandomHelper.Random.NextVectorF(vector, -1, 10);
points.Add(vector);
}
PrincipalComponentAnalysisF pca = new PrincipalComponentAnalysisF(points);
Assert.Greater(pca.Variances[0], pca.Variances[1]);
Assert.Greater(pca.Variances[1], pca.Variances[2]);
Assert.Greater(pca.Variances[2], pca.Variances[3]);
Assert.Greater(pca.Variances[3], 0);
Assert.IsTrue(pca.V.GetColumn(0).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(1).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(2).IsNumericallyNormalized);
Assert.IsTrue(pca.V.GetColumn(3).IsNumericallyNormalized);
// Compute covariance matrix and check if it is diagonal in the transformed space.
MatrixF cov = StatisticsHelper.ComputeCovarianceMatrix(points);
MatrixF transformedCov = pca.V.Transposed * cov * pca.V;
for (int row = 0; row < transformedCov.NumberOfRows; row++)
for (int column = 0; column < transformedCov.NumberOfColumns; column++)
if (row != column)
Assert.IsTrue(Numeric.IsZero(transformedCov[row, column]));
// The principal components must be Eigenvectors which means that multiplying with the covariance
// matrix does only change the length!
VectorF v0 = pca.V.GetColumn(0);
VectorF v0Result = cov * v0;
Assert.IsTrue(VectorF.AreNumericallyEqual(v0.Normalized, v0Result.Normalized));
VectorF v1 = pca.V.GetColumn(1);
VectorF v1Result = cov * v1;
Assert.IsTrue(VectorF.AreNumericallyEqual(v1.Normalized, v1Result.Normalized));
VectorF v2 = pca.V.GetColumn(2);
VectorF v2Result = cov * v2;
Assert.IsTrue(VectorF.AreNumericallyEqual(v2.Normalized, v2Result.Normalized));
VectorF v3 = pca.V.GetColumn(3);
VectorF v3Result = cov * v3;
Assert.IsTrue(VectorF.AreNumericallyEqual(v3.Normalized, v3Result.Normalized));
}
private void DoPca()
{
// Create random 2-dimensional vectors.
var points = new List <VectorF>();
for (int i = 0; i < 20; i++)
{
var x = RandomHelper.Random.NextFloat(-200, 200);
var y = RandomHelper.Random.NextFloat(-100, 100);
var randomVector = new VectorF(new float[] { x, y });
points.Add(randomVector);
}
// Draw a small cross for each point.
var debugRenderer = GraphicsScreen.DebugRenderer2D;
debugRenderer.Clear();
var center = new Vector3(640, 360, 0);
foreach (var point in points)
{
var x = point[0];
var y = point[1];
debugRenderer.DrawLine(
center + new Vector3(x - 10, y - 10, 0),
center + new Vector3(x + 10, y + 10, 0),
Color.Black,
true);
debugRenderer.DrawLine(
center + new Vector3(x + 10, y - 10, 0),
center + new Vector3(x - 10, y + 10, 0),
Color.Black,
true);
}
// Compute the average of the points.
var average = new Vector2F();
foreach (var point in points)
{
average += (Vector2F)point;
}
average /= points.Count;
// Compute the PCA of the point set.
var pca = new PrincipalComponentAnalysisF(points);
// Get the principal components.
// pca.V is a matrix where each column represents a principal component.
// The first column represents the first principal component, which can be loosely
// interpreted as the "direction of the widest spread".
Vector2F pc0 = (Vector2F)pca.V.GetColumn(0);
// The second column represents the second principal component, which is a vector
// orthogonal to the first.
Vector2F pc1 = (Vector2F)pca.V.GetColumn(1);
// pca.Variances contains the variances of the data points along the principal components.
// The square root of the variance is the standard deviation.
float standardDeviation0 = (float)Math.Sqrt(pca.Variances[0]);
float standardDeviation1 = (float)Math.Sqrt(pca.Variances[1]);
// Draw a line in the direction of the first principal component through the average point.
// The line length is proportional to the standard deviation with an arbitrary scaling.
debugRenderer.DrawLine(
center + new Vector3(average.X, average.Y, 0) - 3 * standardDeviation0 * new Vector3(pc0.X, pc0.Y, 0),
center + new Vector3(average.X, average.Y, 0) + 3 * standardDeviation0 * new Vector3(pc0.X, pc0.Y, 0),
Color.Black,
true);
// Draw a line in the direction of the second principal component through the average point.
debugRenderer.DrawLine(
center + new Vector3(average.X, average.Y, 0) - 3 * standardDeviation1 * new Vector3(pc1.X, pc1.Y, 0),
center + new Vector3(average.X, average.Y, 0) + 3 * standardDeviation1 * new Vector3(pc1.X, pc1.Y, 0),
Color.Black,
true);
}
