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/*
* Java Information Dynamics Toolkit (JIDT)
* Copyright (C) 2012, Joseph T. Lizier
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
package infodynamics.measures.continuous.kraskov;
import java.util.Hashtable;
import java.util.Iterator;
import infodynamics.measures.continuous.ActiveInfoStorageCalculator;
import infodynamics.measures.continuous.ConditionalMutualInfoCalculatorMultiVariate;
import infodynamics.measures.continuous.TransferEntropyCalculator;
import infodynamics.measures.continuous.TransferEntropyCalculatorMultiVariate;
import infodynamics.measures.continuous.TransferEntropyCalculatorMultiVariateViaCondMutualInfo;
import infodynamics.measures.continuous.TransferEntropyCalculatorViaCondMutualInfo;
/**
* <p>Computes the differential transfer entropy (TE) between two multivariate
* <code>double[][]</code> time-series of observations
* (implementing {@link TransferEntropyCalculatorMultiVariate}),
* using Kraskov-Stoegbauer-Grassberger (KSG) estimation (see references below).
* This estimator is realised here by plugging in
* a {@link ConditionalMutualInfoCalculatorMultiVariateKraskov}
* as the calculator into the parent class {@link TransferEntropyCalculatorMultiVariateViaCondMutualInfo}.</p>
*
* <p>Crucially, the calculation is performed by examining
* neighbours in the full joint space (as specified by Frenzel and Pompe,
* and Gomez-Herrero et al.)
* rather than two MI calculators.</p>
*
* <p>Usage is as per the paradigm outlined for {@link TransferEntropyCalculatorMultiVariate},
* with:
* <ul>
* <li>The constructor step is either a simple call to {@link #TransferEntropyCalculatorMultiVariateKraskov()},
* or else specifies which KSG algorithm to implement via
* {@link #TransferEntropyCalculatorMultiVariateKraskov(String)};</li>
* <li>{@link #setProperty(String, String)} allowing properties defined for both
* {@link TransferEntropyCalculator#setProperty(String, String)} and
* {@link ConditionalMutualInfoCalculatorMultiVariateKraskov#setProperty(String, String)}
* as outlined
* in {@link TransferEntropyCalculatorViaCondMutualInfo#setProperty(String, String)});
* as well as for {@link #PROP_KRASKOV_ALG_NUM}.</li>
* <li>Computed values are in <b>nats</b>, not bits!</li>
* </ul>
* </p>
*
* <p><b>References:</b><br/>
* <ul>
* <li>T. Schreiber, <a href="http://dx.doi.org/10.1103/PhysRevLett.85.461">
* "Measuring information transfer"</a>,
* Physical Review Letters 85 (2) pp.461-464, 2000.</li>
* <li>J.T. Lizier, J. Heinzle, A. Horstmann, J.-D. Haynes, M. Prokopenko,
* <a href="http://dx.doi.org/10.1007/s10827-010-0271-2">
* "Multivariate information-theoretic measures reveal directed information
* structure and task relevant changes in fMRI connectivity"</a>,
* Journal of Computational Neuroscience, vol. 30, pp. 85-107, 2011.</li>
* <li>Frenzel and Pompe, <a href="http://dx.doi.org/10.1103/physrevlett.99.204101">
* "Partial Mutual Information for Coupling Analysis of Multivariate Time Series"</a>,
* Physical Review Letters, <b>99</b>, p. 204101+ (2007).</li>
* <li>G. Gomez-Herrero, W. Wu, K. Rutanen, M. C. Soriano, G. Pipa, and R. Vicente,
* <a href="http://arxiv.org/abs/1008.0539">
* "Assessing coupling dynamics from an ensemble of time series"</a>,
* arXiv:1008.0539 (2010).</li>
* <li>Kraskov, A., Stoegbauer, H., Grassberger, P.,
* <a href="http://dx.doi.org/10.1103/PhysRevE.69.066138">"Estimating mutual information"</a>,
* Physical Review E 69, (2004) 066138.</li>
* <li>J. T. Lizier, M. Prokopenko and A. Zomaya,
* <a href="http://dx.doi.org/10.1103/PhysRevE.77.026110">
* "Local information transfer as a spatiotemporal filter for complex systems"</a>
* Physical Review E 77, 026110, 2008.</li>
* </ul>
*
* @author Joseph Lizier (<a href="joseph.lizier at gmail.com">email</a>,
* <a href="http://lizier.me/joseph/">www</a>)
* @see TransferEntropyCalculatorMultiVariateViaCondMutualInfo
* @see TransferEntropyCalculatorMultiVariate
*
*/
public class TransferEntropyCalculatorMultiVariateKraskov
extends TransferEntropyCalculatorMultiVariateViaCondMutualInfo {
/**
* Class name for KSG conditional MI estimator via KSG algorithm 1
*/
public static final String COND_MI_CALCULATOR_KRASKOV1 = ConditionalMutualInfoCalculatorMultiVariateKraskov1.class.getName();
/**
* Class name for KSG conditional MI estimator via KSG algorithm 2
*/
public static final String COND_MI_CALCULATOR_KRASKOV2 = ConditionalMutualInfoCalculatorMultiVariateKraskov2.class.getName();
/**
* Property for setting which underlying Kraskov-Grassberger algorithm to use.
* Will only be applied at the next initialisation.
*/
public final static String PROP_KRASKOV_ALG_NUM = "ALG_NUM";
/**
* Which Kraskov algorithm number we are using
*/
protected int kraskovAlgorithmNumber = 1;
protected boolean algChanged = false;
/**
* Storage for the properties ready to pass onto the underlying conditional MI calculators should they change
*/
protected Hashtable<String,String> props;
/**
* Property name for specifying which (if any) auto-embedding method to use.
* Valid values include {@link #AUTO_EMBED_METHOD_RAGWITZ}, {@link #AUTO_EMBED_METHOD_RAGWITZ_DEST_ONLY},
* {@link #AUTO_EMBED_METHOD_MAX_CORR_AIS}, {@link #AUTO_EMBED_METHOD_MAX_CORR_AIS_DEST_ONLY}
* {@link #AUTO_EMBED_METHOD_MAX_CORR_AIS_AND_TE}, and {@link #AUTO_EMBED_METHOD_NONE}.
* Defaults to {@link #AUTO_EMBED_METHOD_NONE}
*/
public static final String PROP_AUTO_EMBED_METHOD = "AUTO_EMBED_METHOD";
/**
* Valid value for the property {@link #PROP_AUTO_EMBED_METHOD} indicating that
* no auto embedding should be done (i.e. to use manually supplied parameters)
*/
public static final String AUTO_EMBED_METHOD_NONE = "NONE";
/**
* Valid value for the property {@link #PROP_AUTO_EMBED_METHOD} indicating that
* the Ragwitz optimisation technique should be used for automatic embedding
* for both source and destination time-series
*/
public static final String AUTO_EMBED_METHOD_RAGWITZ = "RAGWITZ";
/**
* Valid value for the property {@link #PROP_AUTO_EMBED_METHOD} indicating that
* the Ragwitz optimisation technique should be used for automatic embedding
* for the destination time-series only
*/
public static final String AUTO_EMBED_METHOD_RAGWITZ_DEST_ONLY = "RAGWITZ_DEST_ONLY";
/**
* Valid value for the property {@link #PROP_AUTO_EMBED_METHOD} indicating that
* the automatic embedding should be done by maximising the bias corrected
* AIS, for both source and destination time series
*/
public static final String AUTO_EMBED_METHOD_MAX_CORR_AIS = "MAX_CORR_AIS";
/**
* Valid value for the property {@link #PROP_AUTO_EMBED_METHOD} indicating that
* the automatic embedding should be done by maximising the bias corrected
* AIS for the target and subsequently maximising the TE over source embeddings,
* given a fixed source-target delay.
*/
public static final String AUTO_EMBED_METHOD_MAX_CORR_AIS_AND_TE = "MAX_CORR_AIS_AND_TE";
/**
* Valid value for the property {@link #PROP_AUTO_EMBED_METHOD} indicating that
* the automatic embedding should be done by maximising the bias corrected
* AIS, for destination time series only
*/
public static final String AUTO_EMBED_METHOD_MAX_CORR_AIS_DEST_ONLY = "MAX_CORR_AIS_DEST_ONLY";
/**
* Internal variable tracking what type of auto embedding (if any)
* we are using
*/
protected String autoEmbeddingMethod = AUTO_EMBED_METHOD_NONE;
/**
* Property name for maximum embedding lengths (i.e. k for destination, and l for source if we're auto-embedding
* the source as well) for the auto-embedding search. Defaults to 1
*/
public static final String PROP_K_SEARCH_MAX = "AUTO_EMBED_K_SEARCH_MAX";
/**
* Internal variable for storing the maximum embedding length to search up to for
* automating the parameters.
*/
protected int k_search_max = 1;
/**
* Property name for maximum embedding delay (i.e. k_tau for destination, and l_tau for source if we're auto-embedding
* the source as well) for the auto-embedding search. Defaults to 1
*/
public static final String PROP_TAU_SEARCH_MAX = "AUTO_EMBED_TAU_SEARCH_MAX";
/**
* Internal variable for storing the maximum embedding delay to search up to for
* automating the parameters.
*/
protected int tau_search_max = 1;
/**
* Property name for the number of nearest neighbours to use for the auto-embedding search (Ragwitz criteria).
* Defaults to match the value in use for {@link MutualInfoCalculatorMultiVariateKraskov#PROP_K}
*/
public static final String PROP_RAGWITZ_NUM_NNS = "AUTO_EMBED_RAGWITZ_NUM_NNS";
/**
* Internal variable for storing the number of nearest neighbours to use for the
* auto embedding search (Ragwitz criteria)
*/
protected int ragwitz_num_nns = 1;
/**
* Internal variable to track whether the property {@link #PROP_RAGWITZ_NUM_NNS} has been
* set yet
*/
protected boolean ragwitz_num_nns_set = false;
/**
* Creates a new instance of the Kraskov-estimate style multivariate transfer entropy calculator
*
* Uses algorithm 1 by default, as per Gomez-Herro et al.
*
* @throws ClassNotFoundException
* @throws IllegalAccessException
* @throws InstantiationException
*
*/
public TransferEntropyCalculatorMultiVariateKraskov() throws InstantiationException, IllegalAccessException, ClassNotFoundException {
super(COND_MI_CALCULATOR_KRASKOV1);
kraskovAlgorithmNumber = 1;
props = new Hashtable<String,String>();
}
/**
* Creates a new instance of the Kraskov-Grassberger style transfer entropy calculator,
* with the supplied conditional MI calculator name
*
* @param calculatorName fully qualified name of the underlying MI class.
* Must be {@link #COND_MI_CALCULATOR_KRASKOV1} or {@link #COND_MI_CALCULATOR_KRASKOV2}
* @throws ClassNotFoundException
* @throws IllegalAccessException
* @throws InstantiationException
*
*/
public TransferEntropyCalculatorMultiVariateKraskov(String calculatorName) throws InstantiationException, IllegalAccessException, ClassNotFoundException {
super(calculatorName);
// Now check that it was one of our Kraskov-Grassberger calculators:
if (calculatorName.equalsIgnoreCase(COND_MI_CALCULATOR_KRASKOV1)) {
kraskovAlgorithmNumber = 1;
} else if (calculatorName.equalsIgnoreCase(COND_MI_CALCULATOR_KRASKOV2)) {
kraskovAlgorithmNumber = 2;
} else {
throw new ClassNotFoundException("Must be an underlying Kraskov-Grassberger conditional MI calculator");
}
props = new Hashtable<String,String>();
}
/* (non-Javadoc)
* @see infodynamics.measures.continuous.TransferEntropyCalculatorMultiVariateViaCondMutualInfo#initialise(int, int, int, int, int, int, int)
*/
@Override
public void initialise(int sourceDimensions, int destDimensions, int k, int k_tau, int l, int l_tau, int delay)
throws Exception {
if (algChanged) {
// The algorithm number was changed in a setProperties call:
String newCalcName = COND_MI_CALCULATOR_KRASKOV1;
if (kraskovAlgorithmNumber == 2) {
newCalcName = COND_MI_CALCULATOR_KRASKOV2;
}
@SuppressWarnings("unchecked")
Class<ConditionalMutualInfoCalculatorMultiVariate> condMiClass =
(Class<ConditionalMutualInfoCalculatorMultiVariate>) Class.forName(newCalcName);
ConditionalMutualInfoCalculatorMultiVariate newCondMiCalc = condMiClass.newInstance();
construct(newCondMiCalc);
// Set the properties for the Kraskov MI calculators (may pass in properties for our super class
// as well, but they should be ignored)
for (String key : props.keySet()) {
newCondMiCalc.setProperty(key, props.get(key));
}
algChanged = false;
}
super.initialise(sourceDimensions, destDimensions, k, k_tau, l, l_tau, delay);
}
/**
* Sets properties for the TE calculator.
* New property values are not guaranteed to take effect until the next call
* to an initialise method.
*
* <p>Valid property names, and what their
* values should represent, include:</p>
* <ul>
* <li>{@link #PROP_KRASKOV_ALG_NUM} -- which Kraskov algorithm number to use (1 or 2).</li>
* <li>Any properties accepted by {@link TransferEntropyCalculatorMultiVariateViaCondMutualInfo#setProperty(String, String)}</li>
* <li>Or properties accepted by the underlying
* {@link ConditionalMutualInfoCalculatorMultiVariateKraskov#setProperty(String, String)} implementation.</li>
* </ul>
* <p>One should set {@link ConditionalMutualInfoCalculatorMultiVariateKraskov#PROP_K} here, the number
* of neighbouring points one should count up to in determining the joint kernel size.</p>
* <p><b>Note:</b> further properties may be defined by child classes.</p>
*
* <p>Unknown property values are ignored.</p>
*
* @param propertyName name of the property
* @param propertyValue value of the property.
* @throws Exception if there is a problem with the supplied value).
*/
public void setProperty(String propertyName, String propertyValue)
throws Exception {
if (propertyName.equalsIgnoreCase(PROP_KRASKOV_ALG_NUM)) {
int previousAlgNumber = kraskovAlgorithmNumber;
kraskovAlgorithmNumber = Integer.parseInt(propertyValue);
if ((kraskovAlgorithmNumber != 1) && (kraskovAlgorithmNumber != 2)) {
throw new Exception("Kraskov algorithm number (" + kraskovAlgorithmNumber
+ ") must be either 1 or 2");
}
if (kraskovAlgorithmNumber != previousAlgNumber) {
algChanged = true;
}
if (debug) {
System.out.println(this.getClass().getSimpleName() + ": Set property " + propertyName +
" to " + propertyValue);
}
} else if (propertyName.equalsIgnoreCase(PROP_AUTO_EMBED_METHOD)) {
// New method set for determining the embedding parameters
autoEmbeddingMethod = propertyValue;
} else if (propertyName.equalsIgnoreCase(PROP_K_SEARCH_MAX)) {
// Set max embedding history length for auto determination of embedding
k_search_max = Integer.parseInt(propertyValue);
} else if (propertyName.equalsIgnoreCase(PROP_TAU_SEARCH_MAX)) {
// Set maximum embedding delay for auto determination of embedding
tau_search_max = Integer.parseInt(propertyValue);
} else if (propertyName.equalsIgnoreCase(PROP_RAGWITZ_NUM_NNS)) {
// Set the number of nearest neighbours to use in case of Ragwitz auto embedding:
ragwitz_num_nns = Integer.parseInt(propertyValue);
ragwitz_num_nns_set = true;
} else {
// Assume it was a property for the parent class or underlying conditional MI calculator
super.setProperty(propertyName, propertyValue);
props.put(propertyName, propertyValue); // This will keep properties for the super class as well as the cond MI calculator, but this is ok
}
}
@Override
public String getProperty(String propertyName) throws Exception {
if (propertyName.equalsIgnoreCase(PROP_KRASKOV_ALG_NUM)) {
return Integer.toString(kraskovAlgorithmNumber);
} else if (propertyName.equalsIgnoreCase(PROP_AUTO_EMBED_METHOD)) {
return autoEmbeddingMethod;
} else if (propertyName.equalsIgnoreCase(PROP_K_SEARCH_MAX)) {
return Integer.toString(k_search_max);
} else if (propertyName.equalsIgnoreCase(PROP_TAU_SEARCH_MAX)) {
return Integer.toString(tau_search_max);
} else if (propertyName.equalsIgnoreCase(PROP_RAGWITZ_NUM_NNS)) {
if (ragwitz_num_nns_set) {
return Integer.toString(ragwitz_num_nns);
} else {
return condMiCalc.getProperty(ConditionalMutualInfoCalculatorMultiVariateKraskov.PROP_K);
}
} else {
// Assume it was a property for the parent class or underlying conditional MI calculator
return super.getProperty(propertyName);
}
}
@Override
public void preFinaliseAddObservations() throws Exception {
// Automatically determine the embedding parameters for the given time series
if (autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_NONE)) {
return;
}
// Else we need to auto embed
// If we need to check which embedding method later:
if (autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_RAGWITZ) ||
autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_RAGWITZ_DEST_ONLY) ||
autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_MAX_CORR_AIS) ||
autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_MAX_CORR_AIS_DEST_ONLY) ||
autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_MAX_CORR_AIS_AND_TE)) {
// Use a Kraskov AIS calculator to embed both time-series individually:
ActiveInfoStorageCalculatorMultiVariateKraskov aisCalc = new ActiveInfoStorageCalculatorMultiVariateKraskov();
// Set the properties for the underlying MI Kraskov calculator here to match ours:
for (String key : props.keySet()) {
aisCalc.setProperty(key, props.get(key));
}
// Set the auto-embedding properties as we require:
if (autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_RAGWITZ) ||
autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_RAGWITZ_DEST_ONLY)) {
// We're doing Ragwitz auto-embedding
aisCalc.setProperty(ActiveInfoStorageCalculatorMultiVariateKraskov.PROP_AUTO_EMBED_METHOD,
ActiveInfoStorageCalculatorMultiVariateKraskov.AUTO_EMBED_METHOD_RAGWITZ);
// In case !ragwitz_num_nns_set and our condMiCalc has a different default number of
// kNNs for Kraskov search than miCalc, we had best supply the number directly here:
aisCalc.setProperty(ActiveInfoStorageCalculatorMultiVariateKraskov.PROP_RAGWITZ_NUM_NNS,
getProperty(PROP_RAGWITZ_NUM_NNS));
} else {
// We're doing max bias-corrected AIS embeding:
aisCalc.setProperty(ActiveInfoStorageCalculatorMultiVariateKraskov.PROP_AUTO_EMBED_METHOD,
ActiveInfoStorageCalculatorMultiVariateKraskov.AUTO_EMBED_METHOD_MAX_CORR_AIS);
}
aisCalc.setProperty(ActiveInfoStorageCalculatorMultiVariateKraskov.PROP_K_SEARCH_MAX,
Integer.toString(k_search_max));
aisCalc.setProperty(ActiveInfoStorageCalculatorMultiVariateKraskov.PROP_TAU_SEARCH_MAX,
Integer.toString(tau_search_max));
// Embed the destination:
if (debug) {
System.out.println("Starting embedding of destination:");
}
aisCalc.initialise(destDimensions);
aisCalc.startAddObservations();
if ((sourceDimensions == 1) && (destDimensions == 1)) {
for (double[] destination : vectorOfDestinationTimeSeries) {
aisCalc.addObservations(destination);
}
} else {
for (double[][] destination : vectorOfMultiVariateDestinationTimeSeries) {
aisCalc.addObservations(destination);
}
}
aisCalc.finaliseAddObservations();
// Set the auto-embedding parameters for the destination:
k = Integer.parseInt(aisCalc.getProperty(ActiveInfoStorageCalculator.K_PROP_NAME));
k_tau = Integer.parseInt(aisCalc.getProperty(ActiveInfoStorageCalculator.TAU_PROP_NAME));
if (debug) {
System.out.printf("Embedding parameters for destination set to k=%d,k_tau=%d\n",
k, k_tau);
}
if (autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_RAGWITZ) ||
autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_MAX_CORR_AIS)) {
// Embed the source also:
if (debug) {
System.out.println("Starting embedding of source:");
}
aisCalc.initialise(sourceDimensions);
aisCalc.startAddObservations();
if ((sourceDimensions == 1) && (destDimensions == 1)) {
for (double[] source : vectorOfSourceTimeSeries) {
aisCalc.addObservations(source);
}
} else {
for (double[][] source : vectorOfMultiVariateSourceTimeSeries) {
aisCalc.addObservations(source);
}
}
aisCalc.finaliseAddObservations();
// Set the auto-embedding parameters for the source:
l = Integer.parseInt(aisCalc.getProperty(ActiveInfoStorageCalculator.K_PROP_NAME));
l_tau = Integer.parseInt(aisCalc.getProperty(ActiveInfoStorageCalculator.TAU_PROP_NAME));
if (debug) {
System.out.printf("Embedding parameters for source set to l=%d,l_tau=%d\n",
l, l_tau);
}
} else if (autoEmbeddingMethod.equalsIgnoreCase(AUTO_EMBED_METHOD_MAX_CORR_AIS_AND_TE)) {
if (debug) {
System.out.println("Starting embedding of source:");
}
// Instantiate a new calculator to optimize the embedding parameters
TransferEntropyCalculatorMultiVariateKraskov teEmbeddingCalc =
new TransferEntropyCalculatorMultiVariateKraskov();
// Set all properties of the current calculator except embedding method
for (String key : props.keySet()) {
teEmbeddingCalc.setProperty(key, props.get(key));
}
teEmbeddingCalc.setProperty(PROP_AUTO_EMBED_METHOD, AUTO_EMBED_METHOD_NONE);
double bestTE = Double.NEGATIVE_INFINITY;
int l_candidate_best = 1;
int l_tau_candidate_best = 1;
// Iterate over all possible embeddings
for (int l_candidate = 1; l_candidate <= k_search_max; l_candidate++) {
for (int l_tau_candidate = 1; l_tau_candidate <= tau_search_max; l_tau_candidate++) {
teEmbeddingCalc.initialise(sourceDimensions, destDimensions, k, k_tau, l_candidate, l_tau_candidate, delay);
teEmbeddingCalc.startAddObservations();
if ((sourceDimensions == 1) && (destDimensions == 1)) {
Iterator<double[]> destIterator = vectorOfDestinationTimeSeries.iterator();
for (double[] source : vectorOfSourceTimeSeries) {
double[] dest = destIterator.next();
teEmbeddingCalc.addObservations(source, dest);
}
} else {
Iterator<double[][]> destIterator = vectorOfMultiVariateDestinationTimeSeries.iterator();
for (double[][] source : vectorOfMultiVariateSourceTimeSeries) {
double[][] dest = destIterator.next();
teEmbeddingCalc.addObservations(source, dest);
}
}
teEmbeddingCalc.finaliseAddObservations();
double thisTE = teEmbeddingCalc.computeAverageLocalOfObservations();
if (debug) {
System.out.printf("TE for l=%d, l_tau=%d is %.3f\n",
l_candidate, l_tau_candidate, thisTE);
}
if (thisTE > bestTE) {
// This parameter setting is the best so far:
bestTE = thisTE;
l_candidate_best = l_candidate;
l_tau_candidate_best = l_tau_candidate;
}
if (l_candidate == 1) {
// tau is irrelevant, so no point testing other values
break;
}
}
}
l = l_candidate_best;
l_tau = l_tau_candidate_best;
if (debug) {
System.out.printf("Embedding parameters for source set to l=%d,l_tau=%d\n",
l, l_tau);
}
}
// Now that embedding parameters are finalised:
startTimeForFirstDestEmbedding =
computeStartTimeForFirstDestEmbedding(k, k_tau, l, l_tau, delay);
}
}
}
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