LambdaMethod.java
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package org.orekit.estimation.measurements.gnss;
import java.util.SortedSet;
import java.util.TreeSet;
import org.hipparchus.util.FastMath;
/** Decorrelation/reduction engine for LAMBDA method.
* <p>
* This class implements PJG Teunissen Least Square Ambiguity Decorrelation
* Adjustment (LAMBDA) method, as described in both the 1996 paper <a
* href="https://www.researchgate.net/publication/2790708_The_LAMBDA_method_for_integer_ambiguity_estimation_implementation_aspects">
* The LAMBDA method for integer ambiguity estimation: implementation aspects</a> by
* Paul de Jonge and Christian Tiberius and on the 2005 paper <a
* href="https://www.researchgate.net/publication/225518977_MLAMBDA_a_modified_LAMBDA_method_for_integer_least-squares_estimation">
* A modified LAMBDA method for integer least-squares estimation</a> by X.-W Chang, X. Yang
* and T. Zhou, Journal of Geodesy 79(9):552-565, DOI: 10.1007/s00190-005-0004-x
* </p>
* <p>
* It slightly departs on the original LAMBDA method as it does implement
* the following improvements proposed in the de Jonge and Tiberius 1996 paper
* that vastly speed up the search:
* </p>
* <ul>
* <li>alternate search starting from the middle and expanding outwards</li>
* <li>automatic shrinking of ellipsoid during the search</li>
* </ul>
* @see AmbiguitySolver
* @author Luc Maisonobe
* @since 10.0
*/
public class LambdaMethod extends AbstractLambdaMethod {
/** Margin factor to apply to estimated search limit parameter. */
private static final double CHI2_MARGIN_FACTOR = 1.1;
/** {@inheritDoc} */
@Override
protected void ltdlDecomposition() {
// get references to the diagonal and lower triangular parts
final double[] diag = getDiagReference();
final double[] low = getLowReference();
// perform Lᵀ.D.L decomposition
for (int k = diag.length - 1; k >= 0; --k) {
final double inv = 1.0 / diag[k];
for (int i = 0; i < k; ++i) {
final double lki = low[lIndex(k, i)];
for (int j = 0; j < i; ++j) {
low[lIndex(i, j)] -= lki * low[lIndex(k, j)] * inv;
}
diag[i] -= lki * lki * inv;
}
for (int j = 0; j < k; ++j) {
low[lIndex(k, j)] *= inv;
}
}
}
/** {@inheritDoc} */
@Override
protected void reduction() {
// get references to the diagonal and lower triangular parts
final double[] diag = getDiagReference();
final double[] low = getLowReference();
final int n = diag.length;
int kSaved = n - 2;
int k0 = kSaved;
while (k0 >= 0) {
final int k1 = k0 + 1;
if (k0 <= kSaved) {
for (int i = k0 + 1; i < n; ++i) {
integerGaussTransformation(i, k0);
}
}
final double lk1k0 = low[lIndex(k1, k0)];
final double delta = diag[k0] + lk1k0 * lk1k0 * diag[k1];
if (delta < diag[k1]) {
permutation(k0, delta);
kSaved = k0;
k0 = n - 2;
} else {
k0--;
}
}
}
/** {@inheritDoc} */
@Override
protected void discreteSearch() {
// get references to the diagonal part
final double[] diag = getDiagReference();
final int n = diag.length;
// estimate search domain limit
final long[] fixed = new long[n];
final double[] offsets = new double[n];
final double[] left = new double[n];
final double[] right = new double[n];
final int maxSolutions = getMaxSolution();
final double[] decorrelated = getDecorrelatedReference();
final AlternatingSampler[] samplers = new AlternatingSampler[n];
// set up top level sampling for last ambiguity
double chi2 = estimateChi2(fixed, offsets);
right[n - 1] = chi2 / diag[n - 1];
int index = n - 1;
while (index < n) {
if (index == -1) {
// all ambiguities have been fixed
final double squaredNorm = chi2 - diag[0] * (right[0] - left[0]);
addSolution(fixed, squaredNorm);
final int size = getSolutionsSize();
if (size >= maxSolutions) {
// shrink the ellipsoid
chi2 = squaredNorm;
right[n - 1] = chi2 / diag[n - 1];
for (int i = n - 1; i > 0; --i) {
samplers[i].setRadius(FastMath.sqrt(right[i]));
right[i - 1] = diag[i] / diag[i - 1] * (right[i] - left[i]);
}
samplers[0].setRadius(FastMath.sqrt(right[0]));
if (size > maxSolutions) {
removeSolution();
}
}
++index;
} else {
if (samplers[index] == null) {
// we start exploring a new ambiguity
samplers[index] = new AlternatingSampler(conditionalEstimate(index, offsets), FastMath.sqrt(right[index]));
} else {
// continue exploring the same ambiguity
samplers[index].generateNext();
}
if (samplers[index].inRange()) {
fixed[index] = samplers[index].getCurrent();
offsets[index] = fixed[index] - decorrelated[index];
final double delta = fixed[index] - samplers[index].getMidPoint();
left[index] = delta * delta;
if (index > 0) {
right[index - 1] = diag[index] / diag[index - 1] * (right[index] - left[index]);
}
// go down one level
--index;
} else {
// we have completed exploration of this ambiguity range
samplers[index] = null;
// go up one level
++index;
}
}
}
}
/** {@inheritDoc} */
@Override
protected void inverseDecomposition() {
// get references to the diagonal and lower triangular parts
final double[] diag = getDiagReference();
final double[] low = getLowReference();
final int n = diag.length;
// we rely on the following equation, where a low triangular
// matrix L of dimension n is split into sub-matrices of dimensions
// k, l and m with k + l + m = n
//
// [ A | | ] [ A⁻¹ | | ] [ Iₖ | | ]
// [ B | Iₗ | ] [ -BA⁻¹ | Iₗ | ] = [ | Iₗ | ]
// [ C | D | E ] [ E⁻¹ (DB - C) A⁻¹ | -E⁻¹D | E⁻¹ ] [ | | Iₘ ]
//
// considering we have already computed A⁻¹ (i.e. inverted rows 0 to k-1
// of L), and using l = 1 in the previous expression (i.e. the middle matrix
// is only one row), we see that elements 0 to k-1 of row k are given by -BA⁻¹
// and that element k is I₁ = 1. We can therefore invert L row by row and we
// obtain an inverse matrix L⁻¹ which is a low triangular matrix with unit
// diagonal. A⁻¹ is therefore also a low triangular matrix with unit diagonal.
// This property is used in the loops below to speed up the computation of -BA⁻¹
final double[] row = new double[n - 1];
diag[0] = 1.0 / diag[0];
for (int k = 1; k < n; ++k) {
// compute row k of lower triangular part, by computing -BA⁻¹
final int iK = lIndex(k, 0);
System.arraycopy(low, iK, row, 0, k);
for (int j = 0; j < k; ++j) {
double sum = row[j];
for (int l = j + 1; l < k; ++l) {
sum += row[l] * low[lIndex(l, j)];
}
low[iK + j] = -sum;
}
// diagonal part
diag[k] = 1.0 / diag[k];
}
}
/** Compute a safe estimate of search limit parameter χ².
* <p>
* The estimate is based on section 4.11 in de Jonge and Tiberius 1996,
* computing χ² such that it includes at least a few preset grid points
* </p>
* @param fixed placeholder for test fixed ambiguities
* @param offsets placeholder for offsets between fixed ambiguities and float ambiguities
* @return safe estimate of search limit parameter χ²
*/
private double estimateChi2(final long[] fixed, final double[] offsets) {
// get references to the diagonal part
final double[] diag = getDiagReference();
final int n = diag.length;
// maximum number of solutions seeked
final int maxSolutions = getMaxSolution();
// get references to the decorrelated ambiguities
final double[] decorrelated = getDecorrelatedReference();
// initialize test points, assuming ambiguities have been fully decorrelated
final AlternatingSampler[] samplers = new AlternatingSampler[n];
for (int i = 0; i < n; ++i) {
samplers[i] = new AlternatingSampler(decorrelated[i], 0.0);
}
final SortedSet<Double> squaredNorms = new TreeSet<>();
// first test point at center
for (int i = 0; i < n; ++i) {
fixed[i] = samplers[i].getCurrent();
}
squaredNorms.add(squaredNorm(fixed, offsets));
while (squaredNorms.size() < maxSolutions) {
// add a series of grid points, each shifted from center along a different axis
final int remaining = FastMath.min(n, maxSolutions - squaredNorms.size());
for (int i = 0; i < remaining; ++i) {
final long saved = fixed[i];
samplers[i].generateNext();
fixed[i] = samplers[i].getCurrent();
squaredNorms.add(squaredNorm(fixed, offsets));
fixed[i] = saved;
}
}
// select a limit ensuring at least the needed number of grid points are in the search domain
int count = 0;
for (final double s : squaredNorms) {
if (++count == maxSolutions) {
return s * CHI2_MARGIN_FACTOR;
}
}
// never reached
return Double.NaN;
}
/** Compute squared norm of a set of fixed ambiguities.
* @param fixed fixed ambiguities
* @param offsets placeholder for offsets between fixed ambiguities and float ambiguities
* @return squared norm of a set of fixed ambiguities
*/
private double squaredNorm(final long[] fixed, final double[] offsets) {
// get references to the lower triangular part and the decorrelated ambiguities
final double[] diag = getDiagReference();
final double[] decorrelated = getDecorrelatedReference();
double n2 = 0;
for (int i = diag.length - 1; i >= 0; --i) {
offsets[i] = fixed[i] - decorrelated[i];
final double delta = fixed[i] - conditionalEstimate(i, offsets);
n2 += diag[i] * delta * delta;
}
return n2;
}
/** Compute conditional estimate of an ambiguity.
* <p>
* This corresponds to equation 4.4 in de Jonge and Tiberius 1996
* </p>
* @param i index of the ambiguity
* @param offsets offsets between already fixed ambiguities and float ambiguities
* @return conditional estimate of ambiguity â<sub>i|i+1...n</sub>
*/
private double conditionalEstimate(final int i, final double[] offsets) {
// get references to the diagonal and lower triangular parts
final double[] diag = getDiagReference();
final double[] low = getLowReference();
// get references to the decorrelated ambiguities
final double[] decorrelated = getDecorrelatedReference();
double conditional = decorrelated[i];
for (int j = i + 1; j < diag.length; ++j) {
conditional -= low[lIndex(j, i)] * offsets[j];
}
return conditional;
}
}