/**
* Get the covariance matrix of unbound estimated parameters.
*
* @param problem estimation problem
* @return covariance matrix
* @exception EstimationException if the covariance matrix cannot be computed (singular problem)
*/
public double[][] getCovariances(EstimationProblem problem) throws EstimationException {
// set up the jacobian
updateJacobian();
// compute transpose(J).J, avoiding building big intermediate matrices
final int n = problem.getMeasurements().length;
final int m = problem.getUnboundParameters().length;
final int max = m * n;
double[][] jTj = new double[m][m];
for (int i = 0; i < m; ++i) {
for (int j = i; j < m; ++j) {
double sum = 0;
for (int k = 0; k < max; k += m) {
sum += jacobian[k + i] * jacobian[k + j];
}
jTj[i][j] = sum;
jTj[j][i] = sum;
}
}
try {
// compute the covariances matrix
RealMatrix inverse =
new LUDecompositionImpl(MatrixUtils.createRealMatrix(jTj)).getSolver().getInverse();
return inverse.getData();
} catch (InvalidMatrixException ime) {
throw new EstimationException("unable to compute covariances: singular problem");
}
}
public static RealMatrix randMatrix(int rows, int cols) {
RealMatrix res = MathFactory.createRealMatrix(rows, cols);
for (int i = 0; i < rows; i++) {
res.setRow(i, randVector(cols).getData());
}
return res;
}
/**
* Calculates the variance-covariance matrix of the regression parameters.
*
* <p>Var(b) = (X<sup>T</sup>X)<sup>-1</sup>
*
* <p>Uses QR decomposition to reduce (X<sup>T</sup>X)<sup>-1</sup> to
* (R<sup>T</sup>R)<sup>-1</sup>, with only the top p rows of R included, where p = the length of
* the beta vector.
*
* @return The beta variance-covariance matrix
*/
@Override
protected RealMatrix calculateBetaVariance() {
int p = X.getColumnDimension();
RealMatrix Raug = qr.getR().getSubMatrix(0, p - 1, 0, p - 1);
RealMatrix Rinv = new LUDecompositionImpl(Raug).getSolver().getInverse();
return Rinv.multiply(Rinv.transpose());
}
/**
* Throws IllegalArgumentException of the matrix does not have at least two columns and two rows
*
* @param matrix matrix to check for sufficiency
*/
private void checkSufficientData(final RealMatrix matrix) {
int nRows = matrix.getRowDimension();
int nCols = matrix.getColumnDimension();
if (nRows < 2 || nCols < 2) {
throw MathRuntimeException.createIllegalArgumentException(
"insufficient data: only {0} rows and {1} columns.", nRows, nCols);
}
}
@Override
public void step() {
stateHidden.setSubVector(0, weights0.operate(stateIn));
for (int i : series(h)) stateHidden.setEntry(i, activation.function(stateHidden.getEntry(i)));
stateOut.setSubVector(0, weights1.operate(stateHidden));
}
/**
* Calculates beta by GLS.
*
* <pre>
* b=(X' Omega^-1 X)^-1X'Omega^-1 y
* </pre>
*
* @return beta
*/
@Override
protected RealVector calculateBeta() {
RealMatrix OI = getOmegaInverse();
RealMatrix XT = X.transpose();
RealMatrix XTOIX = XT.multiply(OI).multiply(X);
RealMatrix inverse = new LUDecompositionImpl(XTOIX).getSolver().getInverse();
return inverse.multiply(XT).multiply(OI).operate(Y);
}
/**
* Returns a matrix of standard errors associated with the estimates in the correlation matrix.
* <br>
* <code>getCorrelationStandardErrors().getEntry(i,j)</code> is the standard error associated with
* <code>getCorrelationMatrix.getEntry(i,j)</code>
*
* <p>The formula used to compute the standard error is <br>
* <code>SE<sub>r</sub> = ((1 - r<sup>2</sup>) / (n - 2))<sup>1/2</sup></code> where <code>r
* </code> is the estimated correlation coefficient and <code>n</code> is the number of
* observations in the source dataset.
*
* @return matrix of correlation standard errors
*/
public RealMatrix getCorrelationStandardErrors() {
int nVars = correlationMatrix.getColumnDimension();
double[][] out = new double[nVars][nVars];
for (int i = 0; i < nVars; i++) {
for (int j = 0; j < nVars; j++) {
double r = correlationMatrix.getEntry(i, j);
out[i][j] = Math.sqrt((1 - r * r) / (nObs - 2));
}
}
return new BlockRealMatrix(out);
}
public void print(RealMatrix matrix, PrintWriter output, NumberFormat format, int width) {
for (int row = 0; row < matrix.getRowDimension(); row++) {
for (int column = 0; column < matrix.getColumnDimension(); column++) {
String number = format.format(matrix.getEntry(row, column));
int padding = Math.max(1, width - number.length());
for (int p = 0; p < padding; p++) output.print(' ');
output.print(number);
}
output.println();
}
}
/**
* Build a simple converter for correlated residuals with the specific weights.
*
* <p>The scalar objective function value is computed as:
*
* <pre>
* objective = y<sup>T</sup>y with y = scale×(observation-objective)
* </pre>
*
* <p>The array computed by the objective function, the observations array and the the scaling
* matrix must have consistent sizes or a {@link FunctionEvaluationException} will be triggered
* while computing the scalar objective.
*
* @param function vectorial residuals function to wrap
* @param observations observations to be compared to objective function to compute residuals
* @param scale scaling matrix
* @exception IllegalArgumentException if the observations vector and the scale matrix dimensions
* don't match (objective function dimension is checked only when the {@link #value(double[])}
* method is called)
*/
public LeastSquaresConverter(
final MultivariateVectorialFunction function,
final double[] observations,
final RealMatrix scale)
throws IllegalArgumentException {
if (observations.length != scale.getColumnDimension()) {
throw MathRuntimeException.createIllegalArgumentException(
"dimension mismatch {0} != {1}", observations.length, scale.getColumnDimension());
}
this.function = function;
this.observations = observations.clone();
this.weights = null;
this.scale = scale.copy();
}
public static String realMatToString(RealMatrix r) {
String res =
"RealMatrix("
+ r.getRowDimension()
+ "x"
+ r.getColumnDimension()
+ ") of type '"
+ r.getClass().getName()
+ "'\n";
for (int i = 0; i < r.getRowDimension(); i++) {
res += Arrays.toString(r.getRow(i)) + "\n";
}
return res;
}
@Override
public List<java.lang.Double> parameters() {
List<Double> parameters = new ArrayList<Double>((n + 1) * (h) + (h + 1) * n);
for (int i : series(h)) // to
for (int j : series(n + 1)) // from
parameters.add(weights0.getEntry(i, j));
for (int i : series(n)) // to
for (int j : series(h + 1)) // from
parameters.add(weights1.getEntry(i, j));
return parameters;
}
public static void main(String[] args) {
RealMatrix coefficients2 =
new Array2DRowRealMatrix(
new double[][] {
{0.0D, 1.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.857D, 0.0D, 0.054D, 0.018D, 0.0D, 0.071D, 0.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.0D, 1.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.857D, 0.0D, 0.054D, 0.018D, 0.0D, 0.071D, 0.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 1.0D, 0.0D, 0.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 1.0D, 0.0D, 0.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 1.0D, 0.0D, 0.0D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.6D, 0.4D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 1.0D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 1.0D, 0.0D, 0.0D, 1.0D},
{0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D, 0.0D}
},
false);
for (int i = 0; i < 11; i++) {
coefficients2.setEntry(i, i, -1d);
}
coefficients2 = coefficients2.transpose();
DecompositionSolver solver = new LUDecompositionImpl(coefficients2).getSolver();
System.out.println("1 method my Value :");
RealVector constants =
new ArrayRealVector(new double[] {-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, false);
RealVector solution = solver.solve(constants);
double[] data = solution.getData();
DecimalFormat df = new DecimalFormat();
df.setRoundingMode(RoundingMode.DOWN);
System.out.println("Корни уравнения:");
for (double dd : data) {
System.out.print(df.format(dd) + " ");
}
System.out.println();
System.out.println(
"Среднее число процессорных операций, выполняемых при одном прогоне алгоритма: "
+ operationsByProcess(data, arr));
System.out.println("Среднее число обращений к файлам:");
for (int i = 1; i < 4; i++) {
System.out.println(" Файл " + i + " : " + fileMiddleRequest(data, arr, i));
}
System.out.println("Среднее количество информации передаваемой при одном обращении к файлам:");
for (int i = 1; i < 4; i++) {
System.out.println(" Файл " + i + " : " + bitsPerFileTransfer(data, arr, i));
}
System.out.println(
"Сумма среднего числа обращений к основным операторам: " + operatorExecute(data, arr));
System.out.println("Средняя трудоемкость этапа: " + middleWork(data, arr));
}
/** {@inheritDoc} */
public double value(final double[] point) throws FunctionEvaluationException {
// compute residuals
final double[] residuals = function.value(point);
if (residuals.length != observations.length) {
throw new FunctionEvaluationException(
point, "dimension mismatch {0} != {1}", residuals.length, observations.length);
}
for (int i = 0; i < residuals.length; ++i) {
residuals[i] -= observations[i];
}
// compute sum of squares
double sumSquares = 0;
if (weights != null) {
for (int i = 0; i < residuals.length; ++i) {
final double ri = residuals[i];
sumSquares += weights[i] * ri * ri;
}
} else if (scale != null) {
for (final double yi : scale.operate(residuals)) {
sumSquares += yi * yi;
}
} else {
for (final double ri : residuals) {
sumSquares += ri * ri;
}
}
return sumSquares;
}
/**
* Returns a matrix of p-values associated with the (two-sided) null hypothesis that the
* corresponding correlation coefficient is zero.
*
* <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability that a random variable
* distributed as <code>t<sub>n-2</sub></code> takes a value with absolute value greater than or
* equal to <br>
* <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code>
*
* <p>The values in the matrix are sometimes referred to as the <i>significance</i> of the
* corresponding correlation coefficients.
*
* @return matrix of p-values
* @throws MathException if an error occurs estimating probabilities
*/
public RealMatrix getCorrelationPValues() throws MathException {
TDistribution tDistribution = new TDistributionImpl(nObs - 2);
int nVars = correlationMatrix.getColumnDimension();
double[][] out = new double[nVars][nVars];
for (int i = 0; i < nVars; i++) {
for (int j = 0; j < nVars; j++) {
if (i == j) {
out[i][j] = 0d;
} else {
double r = correlationMatrix.getEntry(i, j);
double t = Math.abs(r * Math.sqrt((nObs - 2) / (1 - r * r)));
out[i][j] = 2 * (1 - tDistribution.cumulativeProbability(t));
}
}
}
return new BlockRealMatrix(out);
}
/**
* Compute the "hat" matrix.
*
* <p>The hat matrix is defined in terms of the design matrix X by
* X(X<sup>T</sup>X)<sup>-1</sup>X<sup>T</sup>
*
* <p>The implementation here uses the QR decomposition to compute the hat matrix as Q
* I<sub>p</sub>Q<sup>T</sup> where I<sub>p</sub> is the p-dimensional identity matrix augmented
* by 0's. This computational formula is from "The Hat Matrix in Regression and ANOVA", David C.
* Hoaglin and Roy E. Welsch, <i>The American Statistician</i>, Vol. 32, No. 1 (Feb., 1978), pp.
* 17-22.
*
* @return the hat matrix
*/
public RealMatrix calculateHat() {
// Create augmented identity matrix
RealMatrix Q = qr.getQ();
final int p = qr.getR().getColumnDimension();
final int n = Q.getColumnDimension();
Array2DRowRealMatrix augI = new Array2DRowRealMatrix(n, n);
double[][] augIData = augI.getDataRef();
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (i == j && i < p) {
augIData[i][j] = 1d;
} else {
augIData[i][j] = 0d;
}
}
}
// Compute and return Hat matrix
return Q.multiply(augI).multiply(Q.transpose());
}
/**
* Modifies this map through a single backpropagation iteration using the given error values on
* the output nodes.
*
* @param error
*/
public void train(List<Double> error, double learningRate) {
RealVector eOut = new ArrayRealVector(error.size());
for (int i : series(error.size())) eOut.setEntry(i, error.get(i));
// * gHidden: delta for the non-bias nodes of the hidden layer
gHidden.setSubVector(0, stateHidden.getSubVector(0, n)); // optimize
for (int i : Series.series(gHidden.getDimension()))
gHidden.setEntry(i, activation.derivative(gHidden.getEntry(i)));
eHiddenL = weights1.transpose().operate(eOut);
eHidden.setSubVector(0, eHiddenL.getSubVector(0, h));
for (int i : series(h)) eHidden.setEntry(i, eHidden.getEntry(i) * gHidden.getEntry(i));
weights1Delta = MatrixTools.outer(eOut, stateHidden);
weights1Delta = weights1Delta.scalarMultiply(-1.0 * learningRate); // optimize
weights0Delta = MatrixTools.outer(eHidden, stateIn);
weights0Delta = weights0Delta.scalarMultiply(-1.0 * learningRate);
weights0 = weights0.add(weights0Delta);
weights1 = weights1.add(weights1Delta);
}
/**
* Derives a correlation matrix from a covariance matrix.
*
* <p>Uses the formula <br>
* <code>r(X,Y) = cov(X,Y)/s(X)s(Y)</code> where <code>r(·,·)</code> is the
* correlation coefficient and <code>s(·)</code> means standard deviation.
*
* @param covarianceMatrix the covariance matrix
* @return correlation matrix
*/
public RealMatrix covarianceToCorrelation(RealMatrix covarianceMatrix) {
int nVars = covarianceMatrix.getColumnDimension();
RealMatrix outMatrix = new BlockRealMatrix(nVars, nVars);
for (int i = 0; i < nVars; i++) {
double sigma = Math.sqrt(covarianceMatrix.getEntry(i, i));
outMatrix.setEntry(i, i, 1d);
for (int j = 0; j < i; j++) {
double entry =
covarianceMatrix.getEntry(i, j) / (sigma * Math.sqrt(covarianceMatrix.getEntry(j, j)));
outMatrix.setEntry(i, j, entry);
outMatrix.setEntry(j, i, entry);
}
}
return outMatrix;
}
/**
* Computes the correlation matrix for the columns of the input matrix.
*
* @param matrix matrix with columns representing variables to correlate
* @return correlation matrix
*/
public RealMatrix computeCorrelationMatrix(RealMatrix matrix) {
int nVars = matrix.getColumnDimension();
RealMatrix outMatrix = new BlockRealMatrix(nVars, nVars);
for (int i = 0; i < nVars; i++) {
for (int j = 0; j < i; j++) {
double corr = correlation(matrix.getColumn(i), matrix.getColumn(j));
outMatrix.setEntry(i, j, corr);
outMatrix.setEntry(j, i, corr);
}
outMatrix.setEntry(i, i, 1d);
}
return outMatrix;
}
/** {@inheritDoc} */
public double[] estimateRegressionParameters() {
RealMatrix b = calculateBeta();
return b.getColumn(0);
}
/**
* Create a PearsonsCorrelation from a RealMatrix whose columns represent variables to be
* correlated.
*
* @param matrix matrix with columns representing variables to correlate
*/
public PearsonsCorrelation(RealMatrix matrix) {
checkSufficientData(matrix);
nObs = matrix.getRowDimension();
correlationMatrix = computeCorrelationMatrix(matrix);
}
/** {@inheritDoc} */
public double[] estimateResiduals() {
RealMatrix b = calculateBeta();
RealMatrix e = Y.subtract(X.multiply(b));
return e.getColumn(0);
}
/**
* Calculates the residuals of multiple linear regression in matrix notation.
*
* <pre>
* u = y - X * b
* </pre>
*
* @return The residuals [n,1] matrix
*/
protected RealMatrix calculateResiduals() {
RealMatrix b = calculateBeta();
return Y.subtract(X.multiply(b));
}
/**
* Solve an estimation problem using a least squares criterion.
*
* <p>This method set the unbound parameters of the given problem starting from their current
* values through several iterations. At each step, the unbound parameters are changed in order to
* minimize a weighted least square criterion based on the measurements of the problem.
*
* <p>The iterations are stopped either when the criterion goes below a physical threshold under
* which improvement are considered useless or when the algorithm is unable to improve it (even if
* it is still high). The first condition that is met stops the iterations. If the convergence it
* not reached before the maximum number of iterations, an {@link EstimationException} is thrown.
*
* @param problem estimation problem to solve
* @exception EstimationException if the problem cannot be solved
* @see EstimationProblem
*/
@Override
public void estimate(EstimationProblem problem) throws EstimationException {
initializeEstimate(problem);
// work matrices
double[] grad = new double[parameters.length];
ArrayRealVector bDecrement = new ArrayRealVector(parameters.length);
double[] bDecrementData = bDecrement.getDataRef();
RealMatrix wGradGradT = MatrixUtils.createRealMatrix(parameters.length, parameters.length);
// iterate until convergence is reached
double previous = Double.POSITIVE_INFINITY;
do {
// build the linear problem
incrementJacobianEvaluationsCounter();
RealVector b = new ArrayRealVector(parameters.length);
RealMatrix a = MatrixUtils.createRealMatrix(parameters.length, parameters.length);
for (int i = 0; i < measurements.length; ++i) {
if (!measurements[i].isIgnored()) {
double weight = measurements[i].getWeight();
double residual = measurements[i].getResidual();
// compute the normal equation
for (int j = 0; j < parameters.length; ++j) {
grad[j] = measurements[i].getPartial(parameters[j]);
bDecrementData[j] = weight * residual * grad[j];
}
// build the contribution matrix for measurement i
for (int k = 0; k < parameters.length; ++k) {
double gk = grad[k];
for (int l = 0; l < parameters.length; ++l) {
wGradGradT.setEntry(k, l, weight * gk * grad[l]);
}
}
// update the matrices
a = a.add(wGradGradT);
b = b.add(bDecrement);
}
}
try {
// solve the linearized least squares problem
RealVector dX = new LUDecompositionImpl(a).getSolver().solve(b);
// update the estimated parameters
for (int i = 0; i < parameters.length; ++i) {
parameters[i].setEstimate(parameters[i].getEstimate() + dX.getEntry(i));
}
} catch (InvalidMatrixException e) {
throw new EstimationException("unable to solve: singular problem");
}
previous = cost;
updateResidualsAndCost();
} while ((getCostEvaluations() < 2)
|| (Math.abs(previous - cost) > (cost * steadyStateThreshold)
&& (Math.abs(cost) > convergence)));
}