source: branches/2925_AutoDiffForDynamicalModels/HeuristicLab.Problems.DynamicalSystemsModelling/3.3/Problem.cs @ 16398

#region License Information
/* HeuristicLab
 * Copyright (C) 2002-2018 Heuristic and Evolutionary Algorithms Laboratory (HEAL)
 *
 * This file is part of HeuristicLab.
 *
 * HeuristicLab 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.
 *
 * HeuristicLab 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 HeuristicLab. If not, see <http://www.gnu.org/licenses/>.
 */
#endregion

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using HeuristicLab.Analysis;
using HeuristicLab.Collections;
using HeuristicLab.Common;
using HeuristicLab.Core;
using HeuristicLab.Data;
using HeuristicLab.Encodings.SymbolicExpressionTreeEncoding;
using HeuristicLab.Optimization;
using HeuristicLab.Parameters;
using HeuristicLab.Persistence.Default.CompositeSerializers.Storable;
using HeuristicLab.Problems.DataAnalysis;
using HeuristicLab.Problems.DataAnalysis.Symbolic;
using HeuristicLab.Problems.Instances;
using Variable = HeuristicLab.Problems.DataAnalysis.Symbolic.Variable;

namespace HeuristicLab.Problems.DynamicalSystemsModelling {
  // Eine weitere Möglichkeit ist spline-smoothing der Daten (über Zeit) um damit für jede Zielvariable
  // einen bereinigten (Rauschen) Wert und die Ableitung dy/dt für alle Beobachtungen zu bekommen
  // danach kann man die Regression direkt für dy/dt anwenden (mit bereinigten y als inputs)
  [Item("Dynamical Systems Modelling Problem", "TODO")]
  [Creatable(CreatableAttribute.Categories.GeneticProgrammingProblems, Priority = 900)]
  [StorableClass]
  public sealed class Problem : SingleObjectiveBasicProblem<MultiEncoding>, IRegressionProblem, IProblemInstanceConsumer<IRegressionProblemData>, IProblemInstanceExporter<IRegressionProblemData> {
    #region parameter names
    private const string ProblemDataParameterName = "Data";
    private const string TargetVariablesParameterName = "Target variables";
    private const string FunctionSetParameterName = "Function set";
    private const string MaximumLengthParameterName = "Size limit";
    private const string MaximumParameterOptimizationIterationsParameterName = "Max. parameter optimization iterations";
    private const string NumberOfLatentVariablesParameterName = "Number of latent variables";
    private const string NumericIntegrationStepsParameterName = "Steps for numeric integration";
    private const string TrainingEpisodesParameterName = "Training episodes";
    private const string OptimizeParametersForEpisodesParameterName = "Optimize parameters for episodes";
    private const string OdeSolverParameterName = "ODE Solver";
    #endregion

    #region Parameter Properties
    IParameter IDataAnalysisProblem.ProblemDataParameter { get { return ProblemDataParameter; } }

    public IValueParameter<IRegressionProblemData> ProblemDataParameter {
      get { return (IValueParameter<IRegressionProblemData>)Parameters[ProblemDataParameterName]; }
    }
    public IValueParameter<ReadOnlyCheckedItemList<StringValue>> TargetVariablesParameter {
      get { return (IValueParameter<ReadOnlyCheckedItemList<StringValue>>)Parameters[TargetVariablesParameterName]; }
    }
    public IValueParameter<ReadOnlyCheckedItemList<StringValue>> FunctionSetParameter {
      get { return (IValueParameter<ReadOnlyCheckedItemList<StringValue>>)Parameters[FunctionSetParameterName]; }
    }
    public IFixedValueParameter<IntValue> MaximumLengthParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[MaximumLengthParameterName]; }
    }
    public IFixedValueParameter<IntValue> MaximumParameterOptimizationIterationsParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[MaximumParameterOptimizationIterationsParameterName]; }
    }
    public IFixedValueParameter<IntValue> NumberOfLatentVariablesParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[NumberOfLatentVariablesParameterName]; }
    }
    public IFixedValueParameter<IntValue> NumericIntegrationStepsParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[NumericIntegrationStepsParameterName]; }
    }
    public IValueParameter<ItemList<IntRange>> TrainingEpisodesParameter {
      get { return (IValueParameter<ItemList<IntRange>>)Parameters[TrainingEpisodesParameterName]; }
    }
    public IFixedValueParameter<BoolValue> OptimizeParametersForEpisodesParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[OptimizeParametersForEpisodesParameterName]; }
    }
    public IConstrainedValueParameter<StringValue> OdeSolverParameter {
      get { return (IConstrainedValueParameter<StringValue>)Parameters[OdeSolverParameterName]; }
    }
    #endregion

    #region Properties
    public IRegressionProblemData ProblemData {
      get { return ProblemDataParameter.Value; }
      set { ProblemDataParameter.Value = value; }
    }
    IDataAnalysisProblemData IDataAnalysisProblem.ProblemData { get { return ProblemData; } }

    public ReadOnlyCheckedItemList<StringValue> TargetVariables {
      get { return TargetVariablesParameter.Value; }
    }

    public ReadOnlyCheckedItemList<StringValue> FunctionSet {
      get { return FunctionSetParameter.Value; }
    }

    public int MaximumLength {
      get { return MaximumLengthParameter.Value.Value; }
    }
    public int MaximumParameterOptimizationIterations {
      get { return MaximumParameterOptimizationIterationsParameter.Value.Value; }
    }
    public int NumberOfLatentVariables {
      get { return NumberOfLatentVariablesParameter.Value.Value; }
    }
    public int NumericIntegrationSteps {
      get { return NumericIntegrationStepsParameter.Value.Value; }
    }
    public IEnumerable<IntRange> TrainingEpisodes {
      get { return TrainingEpisodesParameter.Value; }
    }
    public bool OptimizeParametersForEpisodes {
      get { return OptimizeParametersForEpisodesParameter.Value.Value; }
    }

    public string OdeSolver {
      get { return OdeSolverParameter.Value.Value; }
      set {
        var matchingValue = OdeSolverParameter.ValidValues.FirstOrDefault(v => v.Value == value);
        if (matchingValue == null) throw new ArgumentOutOfRangeException();
        else OdeSolverParameter.Value = matchingValue;
      }
    }

    #endregion

    public event EventHandler ProblemDataChanged;

    public override bool Maximization {
      get { return false; } // we minimize NMSE
    }

    #region item cloning and persistence
    // persistence
    [StorableConstructor]
    private Problem(bool deserializing) : base(deserializing) { }
    [StorableHook(HookType.AfterDeserialization)]
    private void AfterDeserialization() {
      if (!Parameters.ContainsKey(OptimizeParametersForEpisodesParameterName)) {
        Parameters.Add(new FixedValueParameter<BoolValue>(OptimizeParametersForEpisodesParameterName, "Flag to select if parameters should be optimized globally or for each episode individually.", new BoolValue(false)));
      }
      RegisterEventHandlers();
    }

    // cloning 
    private Problem(Problem original, Cloner cloner)
      : base(original, cloner) {
      RegisterEventHandlers();
    }
    public override IDeepCloneable Clone(Cloner cloner) { return new Problem(this, cloner); }
    #endregion

    public Problem()
      : base() {
      var targetVariables = new CheckedItemList<StringValue>().AsReadOnly(); // HACK: it would be better to provide a new class derived from IDataAnalysisProblem
      var functions = CreateFunctionSet();
      Parameters.Add(new ValueParameter<IRegressionProblemData>(ProblemDataParameterName, "The data captured from the dynamical system. Use CSV import functionality to import data.", new RegressionProblemData()));
      Parameters.Add(new ValueParameter<ReadOnlyCheckedItemList<StringValue>>(TargetVariablesParameterName, "Target variables (overrides setting in ProblemData)", targetVariables));
      Parameters.Add(new ValueParameter<ReadOnlyCheckedItemList<StringValue>>(FunctionSetParameterName, "The list of allowed functions", functions));
      Parameters.Add(new FixedValueParameter<IntValue>(MaximumLengthParameterName, "The maximally allowed length of each expression. Set to a small value (5 - 25). Default = 10", new IntValue(10)));
      Parameters.Add(new FixedValueParameter<IntValue>(MaximumParameterOptimizationIterationsParameterName, "The maximum number of iterations for optimization of parameters (using L-BFGS). More iterations makes the algorithm slower, fewer iterations might prevent convergence in the optimization scheme. Default = 100", new IntValue(100)));
      Parameters.Add(new FixedValueParameter<IntValue>(NumberOfLatentVariablesParameterName, "Latent variables (unobserved variables) allow us to produce expressions which are integrated up and can be used in other expressions. They are handled similarly to target variables in forward simulation / integration. The difference to target variables is that there are no data to which the calculated values of latent variables are compared. Set to a small value (0 .. 5) as necessary (default = 0)", new IntValue(0)));
      Parameters.Add(new FixedValueParameter<IntValue>(NumericIntegrationStepsParameterName, "Number of steps in the numeric integration that are taken from one row to the next (set to 1 to 100). More steps makes the algorithm slower, less steps worsens the accuracy of the numeric integration scheme.", new IntValue(10)));
      Parameters.Add(new ValueParameter<ItemList<IntRange>>(TrainingEpisodesParameterName, "A list of ranges that should be used for training, each range represents an independent episode. This overrides the TrainingSet parameter in ProblemData.", new ItemList<IntRange>()));
      Parameters.Add(new FixedValueParameter<BoolValue>(OptimizeParametersForEpisodesParameterName, "Flag to select if parameters should be optimized globally or for each episode individually.", new BoolValue(false)));

      var solversStr = new string[] { "HeuristicLab", "CVODES" };
      var solvers = new ItemSet<StringValue>(
        solversStr.Select(s => new StringValue(s).AsReadOnly())
        );
      Parameters.Add(new ConstrainedValueParameter<StringValue>(OdeSolverParameterName, "The solver to use for solving the initial value ODE problems", solvers, solvers.First()));

      RegisterEventHandlers();
      InitAllParameters();

      // TODO: do not clear selection of target variables when the input variables are changed (keep selected target variables)
      // TODO: UI hangs when selecting / deselecting input variables because the encoding is updated on each item
      // TODO: use training range as default training episode
      // TODO: write back optimized parameters to solution?
      // TODO: optimization of starting values for latent variables in CVODES solver

    }

    public override double Evaluate(Individual individual, IRandom random) {
      var trees = individual.Values.Select(v => v.Value).OfType<ISymbolicExpressionTree>().ToArray(); // extract all trees from individual
      // write back optimized parameters to tree nodes instead of the separate OptTheta variable
      // retreive optimized parameters from nodes?

      if (OptimizeParametersForEpisodes) {
        int eIdx = 0;
        double totalNMSE = 0.0;
        int totalSize = 0;
        foreach (var episode in TrainingEpisodes) {
          double[] optTheta;
          double nmse;
          OptimizeForEpisodes(trees, random, new[] { episode }, out optTheta, out nmse);
          individual["OptTheta_" + eIdx] = new DoubleArray(optTheta); // write back optimized parameters so that we can use them in the Analysis method
          eIdx++;
          totalNMSE += nmse * episode.Size;
          totalSize += episode.Size;
        }
        return totalNMSE / totalSize;
      } else {
        double[] optTheta;
        double nmse;
        OptimizeForEpisodes(trees, random, TrainingEpisodes, out optTheta, out nmse);
        individual["OptTheta"] = new DoubleArray(optTheta); // write back optimized parameters so that we can use them in the Analysis method
        return nmse;
      }
    }

    private void OptimizeForEpisodes(
      ISymbolicExpressionTree[] trees,
      IRandom random,
      IEnumerable<IntRange> episodes,
      out double[] optTheta,
      out double nmse) {
      var rows = episodes.SelectMany(e => Enumerable.Range(e.Start, e.End - e.Start)).ToArray();
      var problemData = ProblemData;
      var targetVars = TargetVariables.CheckedItems.OrderBy(i => i.Index).Select(i => i.Value.Value).ToArray();
      var latentVariables = Enumerable.Range(1, NumberOfLatentVariables).Select(i => "λ" + i).ToArray(); // TODO: must coincide with the variables which are actually defined in the grammar and also for which we actually have trees
      var targetValues = new double[rows.Length, targetVars.Length];

      // collect values of all target variables
      var colIdx = 0;
      foreach (var targetVar in targetVars) {
        int rowIdx = 0;
        foreach (var value in problemData.Dataset.GetDoubleValues(targetVar, rows)) {
          targetValues[rowIdx, colIdx] = value;
          rowIdx++;
        }
        colIdx++;
      }

      // NOTE: the order of values in parameter matches prefix order of constant nodes in trees
      var paramNodes = new List<ISymbolicExpressionTreeNode>();
      foreach (var t in trees) {
        foreach (var n in t.IterateNodesPrefix()) {
          if (IsConstantNode(n))
            paramNodes.Add(n);
        }
      }
      // init params randomly
      // theta contains parameter values for trees and then the initial values for latent variables (a separate vector for each episode)
      // inital values for latent variables are also optimized
      var theta = new double[paramNodes.Count + latentVariables.Length * episodes.Count()];
      for (int i = 0; i < theta.Length; i++)
        theta[i] = random.NextDouble() * 2.0 - 1.0;

      optTheta = new double[0];
      if (theta.Length > 0) {
        alglib.minlbfgsstate state;
        alglib.minlbfgsreport report;
        alglib.minlbfgscreate(Math.Min(theta.Length, 5), theta, out state);
        alglib.minlbfgssetcond(state, 0.0, 0.0, 0.0, MaximumParameterOptimizationIterations);
        //alglib.minlbfgssetgradientcheck(state, 1e-6);
        alglib.minlbfgsoptimize(state, EvaluateObjectiveAndGradient, null,
          new object[] { trees, targetVars, problemData, targetValues, episodes.ToArray(), NumericIntegrationSteps, latentVariables, OdeSolver }); //TODO: create a type

        alglib.minlbfgsresults(state, out optTheta, out report);

        /*
         * 
         *         L-BFGS algorithm results

          INPUT PARAMETERS:
              State   -   algorithm state

          OUTPUT PARAMETERS:
              X       -   array[0..N-1], solution
              Rep     -   optimization report:
                          * Rep.TerminationType completetion code:
                              * -7    gradient verification failed.
                                      See MinLBFGSSetGradientCheck() for more information.
                              * -2    rounding errors prevent further improvement.
                                      X contains best point found.
                              * -1    incorrect parameters were specified
                              *  1    relative function improvement is no more than
                                      EpsF.
                              *  2    relative step is no more than EpsX.
                              *  4    gradient norm is no more than EpsG
                              *  5    MaxIts steps was taken
                              *  7    stopping conditions are too stringent,
                                      further improvement is impossible
                          * Rep.IterationsCount contains iterations count
                          * NFEV countains number of function calculations
         */
        if (report.terminationtype < 0) { nmse = 10E6; return; }
      }

      // perform evaluation for optimal theta to get quality value
      double[] grad = new double[optTheta.Length];
      nmse = double.NaN;
      EvaluateObjectiveAndGradient(optTheta, ref nmse, grad,
        new object[] { trees, targetVars, problemData, targetValues, episodes.ToArray(), NumericIntegrationSteps, latentVariables, OdeSolver });
      if (double.IsNaN(nmse) || double.IsInfinity(nmse)) { nmse = 10E6; return; } // return a large value (TODO: be consistent by using NMSE)
    }

    private static void EvaluateObjectiveAndGradient(double[] x, ref double f, double[] grad, object obj) {
      var trees = (ISymbolicExpressionTree[])((object[])obj)[0];
      var targetVariables = (string[])((object[])obj)[1];
      var problemData = (IRegressionProblemData)((object[])obj)[2];
      var targetValues = (double[,])((object[])obj)[3];
      var episodes = (IntRange[])((object[])obj)[4];
      var numericIntegrationSteps = (int)((object[])obj)[5];
      var latentVariables = (string[])((object[])obj)[6];
      var odeSolver = (string)((object[])obj)[7];


      Tuple<double, Vector>[][] predicted = null; // one array of predictions for each episode
      // TODO: stop integration early for diverging solutions
      predicted = Integrate(
          trees,  // we assume trees contain expressions for the change of each target variable over time y'(t)
          problemData.Dataset,
          problemData.AllowedInputVariables.ToArray(),
          targetVariables,
          latentVariables,
          episodes,
          x,
          odeSolver,
          numericIntegrationSteps).ToArray();

      if (predicted.Length != targetValues.GetLength(0)) {
        f = double.MaxValue;
        Array.Clear(grad, 0, grad.Length);
        return;
      }

      // for normalized MSE = 1/variance(t) * MSE(t, pred)
      // TODO: Perf. (by standardization of target variables before evaluation of all trees)      
      var invVar = Enumerable.Range(0, targetVariables.Length)
        .Select(c => Enumerable.Range(0, targetValues.GetLength(0)).Select(row => targetValues[row, c])) // column vectors
        .Select(vec => vec.Variance())
        .Select(v => 1.0 / v)
        .ToArray();

      // objective function is NMSE
      f = 0.0;
      int n = predicted.Length;
      double invN = 1.0 / n;
      var g = Vector.Zero;
      int r = 0;
      foreach (var y_pred in predicted) {
        // y_pred contains the predicted values for target variables first and then predicted values for latent variables
        for (int c = 0; c < targetVariables.Length; c++) {

          var y_pred_f = y_pred[c].Item1;
          var y = targetValues[r, c];

          var res = (y - y_pred_f);
          var ressq = res * res;
          f += ressq * invN * invVar[c];
          g += -2.0 * res * y_pred[c].Item2 * invN * invVar[c];
        }
        r++;
      }

      g.CopyTo(grad);
    }

    public override void Analyze(Individual[] individuals, double[] qualities, ResultCollection results, IRandom random) {
      base.Analyze(individuals, qualities, results, random);

      if (!results.ContainsKey("Prediction (training)")) {
        results.Add(new Result("Prediction (training)", typeof(ReadOnlyItemList<DataTable>)));
      }
      if (!results.ContainsKey("Prediction (test)")) {
        results.Add(new Result("Prediction (test)", typeof(ReadOnlyItemList<DataTable>)));
      }
      if (!results.ContainsKey("Models")) {
        results.Add(new Result("Models", typeof(VariableCollection)));
      }
      if(!results.ContainsKey("SNMSE")) {
        results.Add(new Result("SNMSE", typeof(DoubleValue)));
      }

      var bestIndividualAndQuality = this.GetBestIndividual(individuals, qualities);
      var trees = bestIndividualAndQuality.Item1.Values.Select(v => v.Value).OfType<ISymbolicExpressionTree>().ToArray(); // extract all trees from individual

      results["SNMSE"].Value = new DoubleValue(bestIndividualAndQuality.Item2);

      var problemData = ProblemData;
      var targetVars = TargetVariables.CheckedItems.OrderBy(i => i.Index).Select(i => i.Value.Value).ToArray();
      var latentVariables = Enumerable.Range(1, NumberOfLatentVariables).Select(i => "λ" + i).ToArray(); // TODO: must coincide with the variables which are actually defined in the grammar and also for which we actually have trees

      var trainingList = new ItemList<DataTable>();

      if (OptimizeParametersForEpisodes) {
        var eIdx = 0;
        var trainingPredictions = new List<Tuple<double, Vector>[][]>();
        foreach (var episode in TrainingEpisodes) {
          var episodes = new[] { episode };
          var optTheta = ((DoubleArray)bestIndividualAndQuality.Item1["OptTheta_" + eIdx]).ToArray(); // see evaluate
          var trainingPrediction = Integrate(
                                   trees,  // we assume trees contain expressions for the change of each target variable over time y'(t)
                                   problemData.Dataset,
                                   problemData.AllowedInputVariables.ToArray(),
                                   targetVars,
                                   latentVariables,
                                   episodes,
                                   optTheta,
                                   OdeSolver,
                                   NumericIntegrationSteps).ToArray();
          trainingPredictions.Add(trainingPrediction);
          eIdx++;
        }

        // only for target values
        var trainingRows = TrainingEpisodes.SelectMany(e => Enumerable.Range(e.Start, e.End - e.Start));
        for (int colIdx = 0; colIdx < targetVars.Length; colIdx++) {
          var targetVar = targetVars[colIdx];
          var trainingDataTable = new DataTable(targetVar + " prediction (training)");
          var actualValuesRow = new DataRow(targetVar, "The values of " + targetVar, problemData.Dataset.GetDoubleValues(targetVar, trainingRows));
          var predictedValuesRow = new DataRow(targetVar + " pred.", "Predicted values for " + targetVar, trainingPredictions.SelectMany(arr => arr.Select(row => row[colIdx].Item1)).ToArray());
          trainingDataTable.Rows.Add(actualValuesRow);
          trainingDataTable.Rows.Add(predictedValuesRow);
          trainingList.Add(trainingDataTable);
        }
        results["Prediction (training)"].Value = trainingList.AsReadOnly();


        var models = new VariableCollection();

        foreach (var tup in targetVars.Zip(trees, Tuple.Create)) {
          var targetVarName = tup.Item1;
          var tree = tup.Item2;

          var origTreeVar = new HeuristicLab.Core.Variable(targetVarName + "(original)");
          origTreeVar.Value = (ISymbolicExpressionTree)tree.Clone();
          models.Add(origTreeVar);
        }
        results["Models"].Value = models;
      } else {
        var optTheta = ((DoubleArray)bestIndividualAndQuality.Item1["OptTheta"]).ToArray(); // see evaluate
        var trainingPrediction = Integrate(
                                   trees,  // we assume trees contain expressions for the change of each target variable over time dy/dt
                                   problemData.Dataset,
                                   problemData.AllowedInputVariables.ToArray(),
                                   targetVars,
                                   latentVariables,
                                   TrainingEpisodes,
                                   optTheta,
                                   OdeSolver,
                                   NumericIntegrationSteps).ToArray();
        // for target values and latent variables
        var trainingRows = TrainingEpisodes.SelectMany(e => Enumerable.Range(e.Start, e.End - e.Start));
        for (int colIdx = 0; colIdx < trees.Length; colIdx++) {
          // is target variable
          if (colIdx < targetVars.Length) {
            var targetVar = targetVars[colIdx];
            var trainingDataTable = new DataTable(targetVar + " prediction (training)");
            var actualValuesRow = new DataRow(targetVar, "The values of " + targetVar, problemData.Dataset.GetDoubleValues(targetVar, trainingRows));
            var predictedValuesRow = new DataRow(targetVar + " pred.", "Predicted values for " + targetVar, trainingPrediction.Select(arr => arr[colIdx].Item1).ToArray());
            trainingDataTable.Rows.Add(actualValuesRow);
            trainingDataTable.Rows.Add(predictedValuesRow);
            trainingList.Add(trainingDataTable);
          } else {
            var latentVar = latentVariables[colIdx - targetVars.Length];
            var trainingDataTable = new DataTable(latentVar + " prediction (training)");
            var predictedValuesRow = new DataRow(latentVar + " pred.", "Predicted values for " + latentVar, trainingPrediction.Select(arr => arr[colIdx].Item1).ToArray());
            var emptyRow = new DataRow(latentVar);
            trainingDataTable.Rows.Add(emptyRow);
            trainingDataTable.Rows.Add(predictedValuesRow);
            trainingList.Add(trainingDataTable);
          }
        }
        // TODO: DRY for training and test
        var testList = new ItemList<DataTable>();
        var testRows = ProblemData.TestIndices.ToArray();
        var testPrediction = Integrate(
         trees,  // we assume trees contain expressions for the change of each target variable over time y'(t)
         problemData.Dataset,
         problemData.AllowedInputVariables.ToArray(),
         targetVars,
         latentVariables,
         new IntRange[] { ProblemData.TestPartition },
         optTheta,
         OdeSolver,
         NumericIntegrationSteps).ToArray();

        for (int colIdx = 0; colIdx < trees.Length; colIdx++) {
          // is target variable
          if (colIdx < targetVars.Length) {
            var targetVar = targetVars[colIdx];
            var testDataTable = new DataTable(targetVar + " prediction (test)");
            var actualValuesRow = new DataRow(targetVar, "The values of " + targetVar, problemData.Dataset.GetDoubleValues(targetVar, testRows));
            var predictedValuesRow = new DataRow(targetVar + " pred.", "Predicted values for " + targetVar, testPrediction.Select(arr => arr[colIdx].Item1).ToArray());
            testDataTable.Rows.Add(actualValuesRow);
            testDataTable.Rows.Add(predictedValuesRow);
            testList.Add(testDataTable);

          } else {
            var latentVar = latentVariables[colIdx - targetVars.Length];
            var testDataTable = new DataTable(latentVar + " prediction (test)");
            var predictedValuesRow = new DataRow(latentVar + " pred.", "Predicted values for " + latentVar, testPrediction.Select(arr => arr[colIdx].Item1).ToArray());
            var emptyRow = new DataRow(latentVar);
            testDataTable.Rows.Add(emptyRow);
            testDataTable.Rows.Add(predictedValuesRow);
            testList.Add(testDataTable);
          }
        }

        results["Prediction (training)"].Value = trainingList.AsReadOnly();
        results["Prediction (test)"].Value = testList.AsReadOnly();
        #region simplification of models
        // TODO the dependency of HeuristicLab.Problems.DataAnalysis.Symbolic is not ideal
        var models = new VariableCollection();    // to store target var names and original version of tree

        int nextParIdx = 0;
        for (int idx = 0; idx < trees.Length; idx++) {
          var varName = string.Empty;
          if (idx < targetVars.Length) {
            varName = targetVars[idx];
          } else {
            varName = latentVariables[idx - targetVars.Length];
          }
          var tree = trees[idx];

          // when we reference HeuristicLab.Problems.DataAnalysis.Symbolic we can translate symbols
          var shownTree = new SymbolicExpressionTree(TranslateTreeNode(tree.Root, optTheta.ToArray(),
            ref nextParIdx));

          // var shownTree = (SymbolicExpressionTree)tree.Clone();
          // var constantsNodeOrig = tree.IterateNodesPrefix().Where(IsConstantNode);
          // var constantsNodeShown = shownTree.IterateNodesPrefix().Where(IsConstantNode);
          // 
          // foreach (var n in constantsNodeOrig.Zip(constantsNodeShown, (original, shown) => new { original, shown })) {
          //   double constantsVal = optTheta[nodeIdx[n.original]];
          // 
          //   ConstantTreeNode replacementNode = new ConstantTreeNode(new Constant()) { Value = constantsVal };
          // 
          //   var parentNode = n.shown.Parent;
          //   int replacementIndex = parentNode.IndexOfSubtree(n.shown);
          //   parentNode.RemoveSubtree(replacementIndex);
          //   parentNode.InsertSubtree(replacementIndex, replacementNode);
          // }

          var origTreeVar = new HeuristicLab.Core.Variable(varName + "(original)");
          origTreeVar.Value = (ISymbolicExpressionTree)tree.Clone();
          models.Add(origTreeVar);
          var simplifiedTreeVar = new HeuristicLab.Core.Variable(varName + "(simplified)");
          simplifiedTreeVar.Value = TreeSimplifier.Simplify(shownTree);
          models.Add(simplifiedTreeVar);

        }
        results["Models"].Value = models;
        #endregion
      }
    }


    #region interpretation

    // the following uses auto-diff to calculate the gradient w.r.t. the parameters forward in time.
    // this is basically the method described in Gronwall T. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 1919;20:292–296.

    // a comparison of three potential calculation methods for the gradient is given in:
    // Sengupta, B., Friston, K. J., & Penny, W. D. (2014). Efficient gradient computation for dynamical models. Neuroimage, 98(100), 521–527. http://doi.org/10.1016/j.neuroimage.2014.04.040
    // "Our comparison establishes that the adjoint method is computationally more efficient for numerical estimation of parametric gradients
    // for state-space models — both linear and non-linear, as in the case of a dynamical causal model (DCM)"

    // for a solver with the necessary features see: https://computation.llnl.gov/projects/sundials/cvodes

    private static IEnumerable<Tuple<double, Vector>[]> Integrate(
      ISymbolicExpressionTree[] trees, IDataset dataset,
      string[] inputVariables, string[] targetVariables, string[] latentVariables, IEnumerable<IntRange> episodes,
      double[] parameterValues,
      string odeSolver, int numericIntegrationSteps = 100) {

      // TODO: numericIntegrationSteps is only relevant for the HeuristicLab solver
      var episodeIdx = 0;
      foreach (var episode in episodes) {
        var rows = Enumerable.Range(episode.Start, episode.End - episode.Start);

        // integrate forward starting with known values for the target in t0

        var variableValues = new Dictionary<string, Tuple<double, Vector>>();
        var t0 = rows.First();
        foreach (var varName in inputVariables) {
          variableValues.Add(varName, Tuple.Create(dataset.GetDoubleValue(varName, t0), Vector.Zero));
        }
        foreach (var varName in targetVariables) {
          variableValues.Add(varName, Tuple.Create(dataset.GetDoubleValue(varName, t0), Vector.Zero));
        }
        // add value entries for latent variables which are also integrated
        // initial values are at the end of the parameter vector
        // separete initial values for each episode
        var initialValueIdx = parameterValues.Length - episodes.Count() * latentVariables.Length + episodeIdx * latentVariables.Length;
        foreach (var latentVar in latentVariables) {
          var arr = new double[parameterValues.Length]; // backing array
          arr[initialValueIdx] = 1.0;
          var g = new Vector(arr);
          variableValues.Add(latentVar,
            Tuple.Create(parameterValues[initialValueIdx], g)); // we don't have observations for latent variables therefore we optimize the initial value for each episode 
          initialValueIdx++;
        }

        var calculatedVariables = targetVariables.Concat(latentVariables).ToArray(); // TODO: must conincide with the order of trees in the encoding

        // return first value as stored in the dataset
        yield return calculatedVariables
          .Select(calcVarName => variableValues[calcVarName])
          .ToArray();

        var prevT = rows.First(); // TODO: here we should use a variable for t if it is available. Right now we assume equidistant measurements.
        foreach (var t in rows.Skip(1)) {
          if (odeSolver == "HeuristicLab")
            IntegrateHL(trees, calculatedVariables, variableValues, parameterValues, numericIntegrationSteps);
          else if (odeSolver == "CVODES")
            IntegrateCVODES(trees, calculatedVariables, variableValues, parameterValues, t - prevT);
          else throw new InvalidOperationException("Unknown ODE solver " + odeSolver);
          prevT = t;

          // This check doesn't work with the HeuristicLab integrator if there are input variables 
          //if (variableValues.Count == targetVariables.Length) {
          // only return the target variables for calculation of errors
          var res = calculatedVariables
            .Select(targetVar => variableValues[targetVar])
            .ToArray();
          if (res.Any(ri => double.IsNaN(ri.Item1) || double.IsInfinity(ri.Item1))) yield break;
          yield return res;
          //} else {
          //  yield break; // stop early on problems
          //}


          // update for next time step
          foreach (var varName in inputVariables) {
            variableValues[varName] = Tuple.Create(dataset.GetDoubleValue(varName, t), Vector.Zero);
          }
        }
        episodeIdx++;
      }
    }

    #region CVODES


    /// <summary>
    ///  Here we use CVODES to solve the ODE. Forward sensitivities are used to calculate the gradient for parameter optimization
    /// </summary>
    /// <param name="trees">Each equation in the ODE represented as a tree</param>
    /// <param name="calculatedVariables">The names of the calculated variables</param>
    /// <param name="variableValues">The start values of the calculated variables as well as their sensitivites over parameters</param>
    /// <param name="parameterValues">The current parameter values</param>
    /// <param name="t">The time t up to which we need to integrate.</param>
    private static void IntegrateCVODES(
      ISymbolicExpressionTree[] trees, // f(y,p) in tree representation
      string[] calculatedVariables, // names of elements of y
      Dictionary<string, Tuple<double, Vector>> variableValues,  //  y (intput and output) input: y(t0), output: y(t0+t)
      double[] parameterValues, // p
      double t // duration t for which we want to integrate
      ) {

      // the RHS of the ODE
      // dy/dt = f(y_t,x_t,p)
      CVODES.CVRhsFunc f = CreateOdeRhs(trees, calculatedVariables, parameterValues);
      // the Jacobian ∂f/∂y
      CVODES.CVDlsJacFunc jac = CreateJac(trees, calculatedVariables, parameterValues);

      // the RHS for the forward sensitivities (∂f/∂y)s_i(t) + ∂f/∂p_i
      CVODES.CVSensRhsFn sensF = CreateSensitivityRhs(trees, calculatedVariables, parameterValues);

      // setup solver
      int numberOfEquations = trees.Length;
      IntPtr y = IntPtr.Zero;
      IntPtr cvode_mem = IntPtr.Zero;
      IntPtr A = IntPtr.Zero;
      IntPtr yS0 = IntPtr.Zero;
      IntPtr linearSolver = IntPtr.Zero;
      var ns = parameterValues.Length; // number of parameters

      try {
        y = CVODES.N_VNew_Serial(numberOfEquations);
        // init y to current values of variables
        // y must be initialized before calling CVodeInit
        for (int i = 0; i < calculatedVariables.Length; i++) {
          CVODES.NV_Set_Ith_S(y, i, variableValues[calculatedVariables[i]].Item1);
        }

        cvode_mem = CVODES.CVodeCreate(CVODES.MultistepMethod.CV_ADAMS, CVODES.NonlinearSolverIteration.CV_FUNCTIONAL);

        var flag = CVODES.CVodeInit(cvode_mem, f, 0.0, y);
        Debug.Assert(CVODES.CV_SUCCESS == flag);

        double relTol = 1.0e-2;
        double absTol = 1.0;
        flag = CVODES.CVodeSStolerances(cvode_mem, relTol, absTol);  // TODO: probably need to adjust absTol per variable
        Debug.Assert(CVODES.CV_SUCCESS == flag);

        A = CVODES.SUNDenseMatrix(numberOfEquations, numberOfEquations);
        Debug.Assert(A != IntPtr.Zero);

        linearSolver = CVODES.SUNDenseLinearSolver(y, A);
        Debug.Assert(linearSolver != IntPtr.Zero);

        flag = CVODES.CVDlsSetLinearSolver(cvode_mem, linearSolver, A);
        Debug.Assert(CVODES.CV_SUCCESS == flag);

        flag = CVODES.CVDlsSetJacFn(cvode_mem, jac);
        Debug.Assert(CVODES.CV_SUCCESS == flag);

        yS0 = CVODES.N_VCloneVectorArray_Serial(ns, y); // clone the output vector for each parameter
        unsafe {
          // set to initial sensitivities supplied by caller
          for (int pIdx = 0; pIdx < ns; pIdx++) {
            var yS0_i = *((IntPtr*)yS0.ToPointer() + pIdx);
            for (var varIdx = 0; varIdx < calculatedVariables.Length; varIdx++) {
              CVODES.NV_Set_Ith_S(yS0_i, varIdx, variableValues[calculatedVariables[varIdx]].Item2[pIdx]);
            }
          }
        }

        flag = CVODES.CVodeSensInit(cvode_mem, ns, CVODES.CV_SIMULTANEOUS, sensF, yS0);
        Debug.Assert(CVODES.CV_SUCCESS == flag);

        flag = CVODES.CVodeSensEEtolerances(cvode_mem);
        Debug.Assert(CVODES.CV_SUCCESS == flag);

        // make one forward integration step
        double tout = 0.0; // first output time
        flag = CVODES.CVode(cvode_mem, t, y, ref tout, CVODES.CV_NORMAL);
        if (flag == CVODES.CV_SUCCESS) {
          Debug.Assert(t == tout);

          // get sensitivities
          flag = CVODES.CVodeGetSens(cvode_mem, ref tout, yS0);
          Debug.Assert(CVODES.CV_SUCCESS == flag);

          // update variableValues based on integration results
          for (int varIdx = 0; varIdx < calculatedVariables.Length; varIdx++) {
            var yi = CVODES.NV_Get_Ith_S(y, varIdx);
            var gArr = new double[parameterValues.Length];
            for (var pIdx = 0; pIdx < parameterValues.Length; pIdx++) {
              unsafe {
                var yS0_pi = *((IntPtr*)yS0.ToPointer() + pIdx);
                gArr[pIdx] = CVODES.NV_Get_Ith_S(yS0_pi, varIdx);
              }
            }
            variableValues[calculatedVariables[varIdx]] = Tuple.Create(yi, new Vector(gArr));
          }
        } else {
          variableValues.Clear();   // indicate problems by not returning new values
        }

        // cleanup all allocated objects
      } finally {
        if (y != IntPtr.Zero) CVODES.N_VDestroy_Serial(y);
        if (cvode_mem != IntPtr.Zero) CVODES.CVodeFree(ref cvode_mem);
        if (linearSolver != IntPtr.Zero) CVODES.SUNLinSolFree(linearSolver);
        if (A != IntPtr.Zero) CVODES.SUNMatDestroy(A);
        if (yS0 != IntPtr.Zero) CVODES.N_VDestroyVectorArray_Serial(yS0, ns);
      }
    }


    private static CVODES.CVRhsFunc CreateOdeRhs(
      ISymbolicExpressionTree[] trees,
      string[] calculatedVariables,
      double[] parameterValues) {
      // we don't need to calculate a gradient here 
      return (double t,
              IntPtr y, // N_Vector, current value of y (input)
              IntPtr ydot, // N_Vector (calculated value of y' (output)
              IntPtr user_data // optional user data, (unused here)
              ) => {
                // TODO: perf
                var nodeValues = new Dictionary<ISymbolicExpressionTreeNode, Tuple<double, Vector>>();

                int pIdx = 0;
                foreach (var tree in trees) {
                  foreach (var n in tree.IterateNodesPrefix()) {
                    if (IsConstantNode(n)) {
                      nodeValues.Add(n, Tuple.Create(parameterValues[pIdx], Vector.Zero)); // here we do not need a gradient
                      pIdx++;
                    } else if (n.SubtreeCount == 0) {
                      // for variables and latent variables get the value from variableValues
                      var varName = n.Symbol.Name;
                      var varIdx = Array.IndexOf(calculatedVariables, varName); // TODO: perf!
                      if (varIdx < 0) throw new InvalidProgramException();
                      var y_i = CVODES.NV_Get_Ith_S(y, (long)varIdx);
                      nodeValues.Add(n, Tuple.Create(y_i, Vector.Zero)); // no gradient needed
                    }
                  }
                }
                for (int i = 0; i < trees.Length; i++) {
                  var tree = trees[i];
                  var res_i = InterpretRec(tree.Root.GetSubtree(0).GetSubtree(0), nodeValues);
                  CVODES.NV_Set_Ith_S(ydot, i, res_i.Item1);
                }
                return 0;
              };
    }

    private static CVODES.CVDlsJacFunc CreateJac(
      ISymbolicExpressionTree[] trees,
      string[] calculatedVariables,
      double[] parameterValues) {

      return (
        double t, // current time (input)
        IntPtr y, // N_Vector, current value of y (input)
        IntPtr fy, // N_Vector, current value of f (input)
        IntPtr Jac, // SUNMatrix ∂f/∂y (output, rows i contains are ∂f_i/∂y vector)
        IntPtr user_data, // optional (unused here)
        IntPtr tmp1, // N_Vector, optional (unused here)
        IntPtr tmp2, // N_Vector, optional (unused here)
        IntPtr tmp3 // N_Vector, optional (unused here)
      ) => {
        // here we need to calculate partial derivatives for the calculated variables y
        var nodeValues = new Dictionary<ISymbolicExpressionTreeNode, Tuple<double, Vector>>();
        int pIdx = 0;
        foreach (var tree in trees) {
          foreach (var n in tree.IterateNodesPrefix()) {
            if (IsConstantNode(n)) {
              nodeValues.Add(n, Tuple.Create(parameterValues[pIdx], Vector.Zero)); // here we need a gradient over y which is zero for parameters
              pIdx++;
            } else if (n.SubtreeCount == 0) {
              // for variables and latent variables we use values supplied in y and init gradient vectors accordingly
              var varName = n.Symbol.Name;
              var varIdx = Array.IndexOf(calculatedVariables, varName); // TODO: perf!
              if (varIdx < 0) throw new InvalidProgramException();

              var y_i = CVODES.NV_Get_Ith_S(y, (long)varIdx);
              var gArr = new double[CVODES.NV_LENGTH_S(y)]; // backing array
              gArr[varIdx] = 1.0;
              var g = new Vector(gArr);
              nodeValues.Add(n, Tuple.Create(y_i, g));
            }
          }
        }

        for (int i = 0; i < trees.Length; i++) {
          var tree = trees[i];
          var res = InterpretRec(tree.Root.GetSubtree(0).GetSubtree(0), nodeValues);
          var g = res.Item2;
          for (int j = 0; j < calculatedVariables.Length; j++) {
            CVODES.SUNDenseMatrix_Set(Jac, i, j, g[j]);
          }
        }
        return 0; // on success
      };
    }


    // to calculate sensitivities RHS for all equations at once
    // must compute (∂f/∂y)s_i(t) + ∂f/∂p_i and store in ySdot.
    // Index i refers to parameters, dimensionality of matrix and vectors is number of equations
    private static CVODES.CVSensRhsFn CreateSensitivityRhs(ISymbolicExpressionTree[] trees, string[] calculatedVariables, double[] parameterValues) {
      return (
              int Ns, // number of parameters
              double t, // current time
              IntPtr y, // N_Vector y(t) (input)
              IntPtr ydot, // N_Vector dy/dt(t) (input)
              IntPtr yS, // N_Vector*, one vector for each parameter (input)
              IntPtr ySdot, // N_Vector*, one vector for each parameter (output)
              IntPtr user_data, // optional (unused here)
              IntPtr tmp1, // N_Vector, optional (unused here)
              IntPtr tmp2 // N_Vector, optional (unused here)
        ) => {
          // here we need to calculate partial derivatives for the calculated variables y as well as for the parameters
          var nodeValues = new Dictionary<ISymbolicExpressionTreeNode, Tuple<double, Vector>>();
          var d = calculatedVariables.Length + parameterValues.Length; // dimensionality of gradient
          // first collect variable values
          foreach (var tree in trees) {
            foreach (var n in tree.IterateNodesPrefix()) {
              if (IsVariableNode(n)) {
                // for variables and latent variables we use values supplied in y and init gradient vectors accordingly
                var varName = n.Symbol.Name;
                var varIdx = Array.IndexOf(calculatedVariables, varName); // TODO: perf!
                if (varIdx < 0) throw new InvalidProgramException();

                var y_i = CVODES.NV_Get_Ith_S(y, (long)varIdx);
                var gArr = new double[d]; // backing array
                gArr[varIdx] = 1.0;
                var g = new Vector(gArr);
                nodeValues.Add(n, Tuple.Create(y_i, g));
              }
            }
          }
          // then collect constants
          int pIdx = 0;
          foreach (var tree in trees) {
            foreach (var n in tree.IterateNodesPrefix()) {
              if (IsConstantNode(n)) {
                var gArr = new double[d];
                gArr[calculatedVariables.Length + pIdx] = 1.0;
                var g = new Vector(gArr);
                nodeValues.Add(n, Tuple.Create(parameterValues[pIdx], g));
                pIdx++;
              }
            }
          }
          // gradient vector is [∂f/∂y_1, ∂f/∂y_2, ... ∂f/∂yN, ∂f/∂p_1 ... ∂f/∂p_K]


          for (pIdx = 0; pIdx < Ns; pIdx++) {
            unsafe {
              var sDot_pi = *((IntPtr*)ySdot.ToPointer() + pIdx);
              CVODES.N_VConst_Serial(0.0, sDot_pi);
            }
          }

          for (int i = 0; i < trees.Length; i++) {
            var tree = trees[i];
            var res = InterpretRec(tree.Root.GetSubtree(0).GetSubtree(0), nodeValues);
            var g = res.Item2;


            // update ySdot = (∂f/∂y)s_i(t) + ∂f/∂p_i

            for (pIdx = 0; pIdx < Ns; pIdx++) {
              unsafe {
                var sDot_pi = *((IntPtr*)ySdot.ToPointer() + pIdx);
                var s_pi = *((IntPtr*)yS.ToPointer() + pIdx);

                var v = CVODES.NV_Get_Ith_S(sDot_pi, i);
                // (∂f/∂y)s_i(t)
                var p = 0.0;
                for (int yIdx = 0; yIdx < calculatedVariables.Length; yIdx++) {
                  p += g[yIdx] * CVODES.NV_Get_Ith_S(s_pi, yIdx);
                }
                // + ∂f/∂p_i
                CVODES.NV_Set_Ith_S(sDot_pi, i, v + p + g[calculatedVariables.Length + pIdx]);
              }
            }

          }
          return 0; // on success
        };
    }
    #endregion

    private static void IntegrateHL(
      ISymbolicExpressionTree[] trees,
      string[] calculatedVariables, // names of integrated variables
      Dictionary<string, Tuple<double, Vector>> variableValues, //y (intput and output) input: y(t0), output: y(t0+1)
      double[] parameterValues,
      int numericIntegrationSteps) {

      var nodeValues = new Dictionary<ISymbolicExpressionTreeNode, Tuple<double, Vector>>();

      // the nodeValues for parameters are constant over time
      // TODO: this needs to be done only once for each iteration in gradient descent (whenever parameter values change)
      // NOTE: the order must be the same as above (prefix order for nodes)
      int pIdx = 0;
      foreach (var tree in trees) {
        foreach (var node in tree.Root.IterateNodesPrefix()) {
          if (IsConstantNode(node)) {
            var gArr = new double[parameterValues.Length]; // backing array
            gArr[pIdx] = 1.0;
            var g = new Vector(gArr);
            nodeValues.Add(node, new Tuple<double, Vector>(parameterValues[pIdx], g));
            pIdx++;
          } else if (node.SubtreeCount == 0) {
            // for (latent) variables get the values from variableValues
            var varName = node.Symbol.Name;
            nodeValues.Add(node, variableValues[varName]);
          }
        }
      }


      double h = 1.0 / numericIntegrationSteps;
      for (int step = 0; step < numericIntegrationSteps; step++) {
        var deltaValues = new Dictionary<string, Tuple<double, Vector>>();
        for (int i = 0; i < trees.Length; i++) {
          var tree = trees[i];
          var targetVarName = calculatedVariables[i];

          // Root.GetSubtree(0).GetSubtree(0) skips programRoot and startSymbol
          var res = InterpretRec(tree.Root.GetSubtree(0).GetSubtree(0), nodeValues);
          deltaValues.Add(targetVarName, res);
        }

        // update variableValues for next step, trapezoid integration
        foreach (var kvp in deltaValues) {
          var oldVal = variableValues[kvp.Key];
          var newVal = Tuple.Create(
            oldVal.Item1 + h * kvp.Value.Item1,
            oldVal.Item2 + h * kvp.Value.Item2
          );
          variableValues[kvp.Key] = newVal;
        }


        foreach(var node in nodeValues.Keys.ToArray()) {
          if(node.SubtreeCount == 0 && !IsConstantNode(node)) {
            // update values for (latent) variables
            var varName = node.Symbol.Name;
            nodeValues[node] = variableValues[varName];
          }
        }
      }
    }

    private static Tuple<double, Vector> InterpretRec(
      ISymbolicExpressionTreeNode node,
      Dictionary<ISymbolicExpressionTreeNode, Tuple<double, Vector>> nodeValues      // contains value and gradient vector for a node (variables and constants only)
        ) {

      switch (node.Symbol.Name) {
        case "+": {
            var l = InterpretRec(node.GetSubtree(0), nodeValues);
            var r = InterpretRec(node.GetSubtree(1), nodeValues);

            return Tuple.Create(l.Item1 + r.Item1, l.Item2 + r.Item2);
          }
        case "*": {
            var l = InterpretRec(node.GetSubtree(0), nodeValues);
            var r = InterpretRec(node.GetSubtree(1), nodeValues);

            return Tuple.Create(l.Item1 * r.Item1, l.Item2 * r.Item1 + l.Item1 * r.Item2);
          }

        case "-": {
            var l = InterpretRec(node.GetSubtree(0), nodeValues);
            var r = InterpretRec(node.GetSubtree(1), nodeValues);

            return Tuple.Create(l.Item1 - r.Item1, l.Item2 - r.Item2);
          }
        case "%": {
            var l = InterpretRec(node.GetSubtree(0), nodeValues);
            var r = InterpretRec(node.GetSubtree(1), nodeValues);

            // protected division
            if (r.Item1.IsAlmost(0.0)) {
              return Tuple.Create(0.0, Vector.Zero);
            } else {
              return Tuple.Create(
                l.Item1 / r.Item1,
                l.Item1 * -1.0 / (r.Item1 * r.Item1) * r.Item2 + 1.0 / r.Item1 * l.Item2 // (f/g)' = f * (1/g)' + 1/g * f' = f * -1/g² * g' + 1/g * f'
                );
            }
          }
        case "sin": {
            var x = InterpretRec(node.GetSubtree(0), nodeValues);
            return Tuple.Create(
              Math.Sin(x.Item1),
              Vector.Cos(x.Item2) * x.Item2
            );
          }
        case "cos": {
            var x = InterpretRec(node.GetSubtree(0), nodeValues);
            return Tuple.Create(
              Math.Cos(x.Item1),
              -Vector.Sin(x.Item2) * x.Item2
            );
          }
        case "sqr": {
            var x = InterpretRec(node.GetSubtree(0), nodeValues);
            return Tuple.Create(
              x.Item1 * x.Item1,
              2.0 * x.Item1 * x.Item2
            );
          }
        default: {
            return nodeValues[node];  // value and gradient for constants and variables must be set by the caller
          }
      }
    }
    #endregion

    #region events
    /*
     * Dependencies between parameters:
     * 
     * ProblemData
     *    |
     *    V
     * TargetVariables   FunctionSet    MaximumLength    NumberOfLatentVariables
     *               |   |                 |                   |
     *               V   V                 |                   |
     *             Grammar <---------------+-------------------
     *                |
     *                V
     *            Encoding
     */
    private void RegisterEventHandlers() {
      ProblemDataParameter.ValueChanged += ProblemDataParameter_ValueChanged;
      if (ProblemDataParameter.Value != null) ProblemDataParameter.Value.Changed += ProblemData_Changed;

      TargetVariablesParameter.ValueChanged += TargetVariablesParameter_ValueChanged;
      if (TargetVariablesParameter.Value != null) TargetVariablesParameter.Value.CheckedItemsChanged += CheckedTargetVariablesChanged;

      FunctionSetParameter.ValueChanged += FunctionSetParameter_ValueChanged;
      if (FunctionSetParameter.Value != null) FunctionSetParameter.Value.CheckedItemsChanged += CheckedFunctionsChanged;

      MaximumLengthParameter.Value.ValueChanged += MaximumLengthChanged;

      NumberOfLatentVariablesParameter.Value.ValueChanged += NumLatentVariablesChanged;
    }

    private void NumLatentVariablesChanged(object sender, EventArgs e) {
      UpdateGrammarAndEncoding();
    }

    private void MaximumLengthChanged(object sender, EventArgs e) {
      UpdateGrammarAndEncoding();
    }

    private void FunctionSetParameter_ValueChanged(object sender, EventArgs e) {
      FunctionSetParameter.Value.CheckedItemsChanged += CheckedFunctionsChanged;
    }

    private void CheckedFunctionsChanged(object sender, CollectionItemsChangedEventArgs<IndexedItem<StringValue>> e) {
      UpdateGrammarAndEncoding();
    }

    private void TargetVariablesParameter_ValueChanged(object sender, EventArgs e) {
      TargetVariablesParameter.Value.CheckedItemsChanged += CheckedTargetVariablesChanged;
    }

    private void CheckedTargetVariablesChanged(object sender, CollectionItemsChangedEventArgs<IndexedItem<StringValue>> e) {
      UpdateGrammarAndEncoding();
    }

    private void ProblemDataParameter_ValueChanged(object sender, EventArgs e) {
      ProblemDataParameter.Value.Changed += ProblemData_Changed;
      OnProblemDataChanged();
      OnReset();
    }

    private void ProblemData_Changed(object sender, EventArgs e) {
      OnProblemDataChanged();
      OnReset();
    }

    private void OnProblemDataChanged() {
      UpdateTargetVariables();        // implicitly updates other dependent parameters
      var handler = ProblemDataChanged;
      if (handler != null) handler(this, EventArgs.Empty);
    }

    #endregion

    #region  helper

    private void InitAllParameters() {
      UpdateTargetVariables(); // implicitly updates the grammar and the encoding      
    }

    private ReadOnlyCheckedItemList<StringValue> CreateFunctionSet() {
      var l = new CheckedItemList<StringValue>();
      l.Add(new StringValue("+").AsReadOnly());
      l.Add(new StringValue("*").AsReadOnly());
      l.Add(new StringValue("%").AsReadOnly());
      l.Add(new StringValue("-").AsReadOnly());
      l.Add(new StringValue("sin").AsReadOnly());
      l.Add(new StringValue("cos").AsReadOnly());
      l.Add(new StringValue("sqr").AsReadOnly());
      return l.AsReadOnly();
    }

    private static bool IsConstantNode(ISymbolicExpressionTreeNode n) {
      return n.Symbol.Name.StartsWith("θ");
    }
    private static bool IsLatentVariableNode(ISymbolicExpressionTreeNode n) {
      return n.Symbol.Name.StartsWith("λ");
    }
    private static bool IsVariableNode(ISymbolicExpressionTreeNode n) {
      return (n.SubtreeCount == 0) && !IsConstantNode(n) && !IsLatentVariableNode(n);
    }


    private void UpdateTargetVariables() {
      var currentlySelectedVariables = TargetVariables.CheckedItems
        .OrderBy(i => i.Index)
        .Select(i => i.Value.Value)
        .ToArray();

      var newVariablesList = new CheckedItemList<StringValue>(ProblemData.Dataset.VariableNames.Select(str => new StringValue(str).AsReadOnly()).ToArray()).AsReadOnly();
      var matchingItems = newVariablesList.Where(item => currentlySelectedVariables.Contains(item.Value)).ToArray();
      foreach (var matchingItem in matchingItems) {
        newVariablesList.SetItemCheckedState(matchingItem, true);
      }
      TargetVariablesParameter.Value = newVariablesList;
    }

    private void UpdateGrammarAndEncoding() {
      var encoding = new MultiEncoding();
      var g = CreateGrammar();
      foreach (var targetVar in TargetVariables.CheckedItems) {
        encoding = encoding.Add(new SymbolicExpressionTreeEncoding(targetVar + "_tree", g, MaximumLength, MaximumLength)); // only limit by length
      }
      for (int i = 1; i <= NumberOfLatentVariables; i++) {
        encoding = encoding.Add(new SymbolicExpressionTreeEncoding("λ" + i + "_tree", g, MaximumLength, MaximumLength));
      }
      Encoding = encoding;
    }

    private ISymbolicExpressionGrammar CreateGrammar() {
      // whenever ProblemData is changed we create a new grammar with the necessary symbols
      var g = new SimpleSymbolicExpressionGrammar();
      g.AddSymbols(FunctionSet.CheckedItems.OrderBy(i => i.Index).Select(i => i.Value.Value).ToArray(), 2, 2);

      // TODO
      //g.AddSymbols(new[] {
      //  "exp",
      //  "log", // log( <expr> ) // TODO: init a theta to ensure the value is always positive
      //  "exp_minus" // exp((-1) * <expr>
      //}, 1, 1);

      foreach (var variableName in ProblemData.AllowedInputVariables.Union(TargetVariables.CheckedItems.Select(i => i.Value.Value)))
        g.AddTerminalSymbol(variableName);

      // generate symbols for numeric parameters for which the value is optimized using AutoDiff
      // we generate multiple symbols to balance the probability for selecting a numeric parameter in the generation of random trees
      var numericConstantsFactor = 2.0;
      for (int i = 0; i < numericConstantsFactor * (ProblemData.AllowedInputVariables.Count() + TargetVariables.CheckedItems.Count()); i++) {
        g.AddTerminalSymbol("θ" + i); // numeric parameter for which the value is optimized using AutoDiff
      }

      // generate symbols for latent variables
      for (int i = 1; i <= NumberOfLatentVariables; i++) {
        g.AddTerminalSymbol("λ" + i); // numeric parameter for which the value is optimized using AutoDiff
      }

      return g;
    }


    private ISymbolicExpressionTreeNode TranslateTreeNode(ISymbolicExpressionTreeNode n, double[] parameterValues, ref int nextParIdx) {
      ISymbolicExpressionTreeNode translatedNode = null;
      if (n.Symbol is StartSymbol) {
        translatedNode = new StartSymbol().CreateTreeNode();
      } else if (n.Symbol is ProgramRootSymbol) {
        translatedNode = new ProgramRootSymbol().CreateTreeNode();
      } else if (n.Symbol.Name == "+") {
        translatedNode = new Addition().CreateTreeNode();
      } else if (n.Symbol.Name == "-") {
        translatedNode = new Subtraction().CreateTreeNode();
      } else if (n.Symbol.Name == "*") {
        translatedNode = new Multiplication().CreateTreeNode();
      } else if (n.Symbol.Name == "%") {
        translatedNode = new Division().CreateTreeNode();
      } else if (n.Symbol.Name == "sin") {
        translatedNode = new Sine().CreateTreeNode();
      } else if (n.Symbol.Name == "cos") {
        translatedNode = new Cosine().CreateTreeNode();
      } else if (n.Symbol.Name == "sqr") {
        translatedNode = new Square().CreateTreeNode();
      } else if (IsConstantNode(n)) {
        var constNode = (ConstantTreeNode)new Constant().CreateTreeNode();
        constNode.Value = parameterValues[nextParIdx];
        nextParIdx++;
        translatedNode = constNode;
      } else {
        // assume a variable name
        var varName = n.Symbol.Name;
        var varNode = (VariableTreeNode)new Variable().CreateTreeNode();
        varNode.Weight = 1.0;
        varNode.VariableName = varName;
        translatedNode = varNode;
      }
      foreach (var child in n.Subtrees) {
        translatedNode.AddSubtree(TranslateTreeNode(child, parameterValues, ref nextParIdx));
      }
      return translatedNode;
    }
    #endregion

    #region Import & Export
    public void Load(IRegressionProblemData data) {
      Name = data.Name;
      Description = data.Description;
      ProblemData = data;
    }

    public IRegressionProblemData Export() {
      return ProblemData;
    }
    #endregion

  }
}