source: branches/2925_AutoDiffForDynamicalModels/HeuristicLab.Problems.DynamicalSystemsModelling/3.3/Problem.cs @ 16152

#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;

namespace HeuristicLab.Problems.DynamicalSystemsModelling {
  public class Vector {
    public readonly static Vector Zero = new Vector(new double[0]);

    public static Vector operator +(Vector a, Vector b) {
      if (a == Zero) return b;
      if (b == Zero) return a;
      Debug.Assert(a.arr.Length == b.arr.Length);
      var res = new double[a.arr.Length];
      for (int i = 0; i < res.Length; i++)
        res[i] = a.arr[i] + b.arr[i];
      return new Vector(res);
    }
    public static Vector operator -(Vector a, Vector b) {
      if (b == Zero) return a;
      if (a == Zero) return -b;
      Debug.Assert(a.arr.Length == b.arr.Length);
      var res = new double[a.arr.Length];
      for (int i = 0; i < res.Length; i++)
        res[i] = a.arr[i] - b.arr[i];
      return new Vector(res);
    }
    public static Vector operator -(Vector v) {
      if (v == Zero) return Zero;
      for (int i = 0; i < v.arr.Length; i++)
        v.arr[i] = -v.arr[i];
      return v;
    }

    public static Vector operator *(double s, Vector v) {
      if (v == Zero) return Zero;
      if (s == 0.0) return Zero;
      var res = new double[v.arr.Length];
      for (int i = 0; i < res.Length; i++)
        res[i] = s * v.arr[i];
      return new Vector(res);
    }
    public static Vector operator *(Vector v, double s) {
      return s * v;
    }
    public static Vector operator /(double s, Vector v) {
      if (s == 0.0) return Zero;
      if (v == Zero) throw new ArgumentException("Division by zero vector");
      var res = new double[v.arr.Length];
      for (int i = 0; i < res.Length; i++)
        res[i] = 1.0 / v.arr[i];
      return new Vector(res);
    }
    public static Vector operator /(Vector v, double s) {
      return v * 1.0 / s;
    }


    private readonly double[] arr; // backing array;

    public Vector(double[] v) {
      this.arr = v;
    }

    public void CopyTo(double[] target) {
      Debug.Assert(arr.Length <= target.Length);
      Array.Copy(arr, target, arr.Length);
    }
  }

  [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";
    #endregion

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

    public IValueParameter<IRegressionProblemData> ProblemDataParameter {
      get { return (IValueParameter<IRegressionProblemData>)Parameters[ProblemDataParameterName]; }
    }
    public IValueParameter<ReadOnlyCheckedItemCollection<StringValue>> TargetVariablesParameter {
      get { return (IValueParameter<ReadOnlyCheckedItemCollection<StringValue>>)Parameters[TargetVariablesParameterName]; }
    }
    public IValueParameter<ReadOnlyCheckedItemCollection<StringValue>> FunctionSetParameter {
      get { return (IValueParameter<ReadOnlyCheckedItemCollection<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]; }
    }
    #endregion

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

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

    public ReadOnlyCheckedItemCollection<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; }
    }

    #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() {
      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 CheckedItemCollection<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<ReadOnlyCheckedItemCollection<StringValue>>(TargetVariablesParameterName, "Target variables (overrides setting in ProblemData)", targetVariables));
      Parameters.Add(new ValueParameter<ReadOnlyCheckedItemCollection<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)));

      RegisterEventHandlers();
      InitAllParameters();

      // TODO: do not clear selection of target variables when the input variables are changed
      // TODO: UI hangs when selecting / deselecting input variables because the encoding is updated on each item
    }


    public override double Evaluate(Individual individual, IRandom random) {
      var trees = individual.Values.Select(v => v.Value).OfType<ISymbolicExpressionTree>().ToArray(); // extract all trees from individual

      var problemData = ProblemData;
      var rows = ProblemData.TrainingIndices.ToArray();
      var targetVars = TargetVariables.CheckedItems.Select(i => i.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++;
      }

      var nodeIdx = new Dictionary<ISymbolicExpressionTreeNode, int>();

      foreach (var tree in trees) {
        foreach (var node in tree.Root.IterateNodesPrefix().Where(n => IsConstantNode(n))) {
          nodeIdx.Add(node, nodeIdx.Count);
        }
      }

      var theta = nodeIdx.Select(_ => random.NextDouble() * 2.0 - 1.0).ToArray(); // init params randomly from Unif(-1,1)

      double[] 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.minlbfgsoptimize(state, EvaluateObjectiveAndGradient, null,
          new object[] { trees, targetVars, problemData, nodeIdx, targetValues, rows, NumericIntegrationSteps, latentVariables }); //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) return double.MaxValue;
      }

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

      individual["OptTheta"] = new DoubleArray(optTheta); // write back optimized parameters so that we can use them in the Analysis method
      return optQuality;
    }

    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 nodeIdx = (Dictionary<ISymbolicExpressionTreeNode, int>)((object[])obj)[3];
      var targetValues = (double[,])((object[])obj)[4];
      var rows = (int[])((object[])obj)[5];
      var numericIntegrationSteps = (int)((object[])obj)[6];
      var latentVariables = (string[])((object[])obj)[7];

      var 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,
        rows,
        nodeIdx,                // TODO: is it Ok to use rows here ?
        x, numericIntegrationSteps).ToArray();


      // 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 => rows.Select(row => targetValues[row, c])) // colums 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) {
        for (int c = 0; c < y_pred.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(ReadOnlyItemList<ISymbolicExpressionTree>)));
      }

      // TODO extract common functionality from Evaluate and Analyze
      var bestIndividualAndQuality = this.GetBestIndividual(individuals, qualities);
      var optTheta = ((DoubleArray)bestIndividualAndQuality.Item1["OptTheta"]).ToArray(); // see evaluate
      var trees = bestIndividualAndQuality.Item1.Values.Select(v => v.Value).OfType<ISymbolicExpressionTree>().ToArray(); // extract all trees from individual
      var nodeIdx = new Dictionary<ISymbolicExpressionTreeNode, int>();


      foreach (var tree in trees) {
        foreach (var node in tree.Root.IterateNodesPrefix().Where(n => IsConstantNode(n))) {
          nodeIdx.Add(node, nodeIdx.Count);
        }
      }
      var problemData = ProblemData;
      var targetVars = TargetVariables.CheckedItems.Select(i => i.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>();
      var trainingRows = ProblemData.TrainingIndices.ToArray();
      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,
       trainingRows,
       nodeIdx,
       optTheta,
       NumericIntegrationSteps).ToArray();

      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, trainingPrediction.Select(arr => arr[colIdx].Item1).ToArray());
        trainingDataTable.Rows.Add(actualValuesRow);
        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,
       testRows,
       nodeIdx,
       optTheta,
       NumericIntegrationSteps).ToArray();

      for (int colIdx = 0; colIdx < targetVars.Length; colIdx++) {
        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);
      }

      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 modelList = new ItemList<ISymbolicExpressionTree>();
      foreach (var tree in trees) {
        var shownTree = (ISymbolicExpressionTree)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);
        }

        modelList.Add(shownTree);
      }
      results["Models"].Value = modelList.AsReadOnly();
      #endregion
    }


    #region interpretation
    private static IEnumerable<Tuple<double, Vector>[]> Integrate(
      ISymbolicExpressionTree[] trees, IDataset dataset, string[] inputVariables, string[] targetVariables, string[] latentVariables, IEnumerable<int> rows,
      Dictionary<ISymbolicExpressionTreeNode, int> nodeIdx, double[] parameterValues, int numericIntegrationSteps = 100) {

      int NUM_STEPS = numericIntegrationSteps ;
      double h = 1.0 / NUM_STEPS;

      // return first value as stored in the dataset
      yield return targetVariables
        .Select(targetVar => Tuple.Create(dataset.GetDoubleValue(targetVar, rows.First()), Vector.Zero))
        .ToArray();

      // 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
      foreach(var latentVar in latentVariables) {
        variableValues.Add(latentVar, Tuple.Create(0.0, Vector.Zero)); // we don't have observations for latent variables -> assume zero as starting value
      }
      var calculatedVariables = targetVariables.Concat(latentVariables); // TODO: must conincide with the order of trees in the encoding

      foreach (var t in rows.Skip(1)) {
        for (int step = 0; step < NUM_STEPS; step++) {
          var deltaValues = new Dictionary<string, Tuple<double, Vector>>();
          foreach (var tup in trees.Zip(calculatedVariables, Tuple.Create)) {
            var tree = tup.Item1;
            var targetVarName = tup.Item2;
            // skip programRoot and startSymbol
            var res = InterpretRec(tree.Root.GetSubtree(0).GetSubtree(0), variableValues, nodeIdx, parameterValues);
            deltaValues.Add(targetVarName, res);
          }

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

        // only return the target variables for calculation of errors
        yield return targetVariables
          .Select(targetVar => variableValues[targetVar])
          .ToArray();

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

    private static Tuple<double, Vector> InterpretRec(
      ISymbolicExpressionTreeNode node,
      Dictionary<string, Tuple<double, Vector>> variableValues,
      Dictionary<ISymbolicExpressionTreeNode, int> nodeIdx,
      double[] parameterValues
        ) {

      switch (node.Symbol.Name) {
        case "+": {
            var l = InterpretRec(node.GetSubtree(0), variableValues, nodeIdx, parameterValues); // TODO capture all parameters into a state type for interpretation
            var r = InterpretRec(node.GetSubtree(1), variableValues, nodeIdx, parameterValues);

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

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

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

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

            // 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'
                );
            }
          }
        default: {
            // distinguish other cases
            if (IsConstantNode(node)) {
              var vArr = new double[parameterValues.Length]; // backing array for vector
              vArr[nodeIdx[node]] = 1.0;
              var g = new Vector(vArr);
              return Tuple.Create(parameterValues[nodeIdx[node]], g);
            } else {
              // assume a variable name
              var varName = node.Symbol.Name;
              return variableValues[varName];
            }
          }
      }
    }
    #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<StringValue> e) {
      UpdateGrammarAndEncoding();
    }

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

    private void CheckedTargetVariablesChanged(object sender, CollectionItemsChangedEventArgs<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 ReadOnlyCheckedItemCollection<StringValue> CreateFunctionSet() {
      var l = new CheckedItemCollection<StringValue>();
      l.Add(new StringValue("+").AsReadOnly());
      l.Add(new StringValue("*").AsReadOnly());
      l.Add(new StringValue("%").AsReadOnly());
      l.Add(new StringValue("-").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 void UpdateTargetVariables() {
      var currentlySelectedVariables = TargetVariables.CheckedItems.Select(i => i.Value).ToArray();

      var newVariablesList = new CheckedItemCollection<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.Select(i => i.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)))
        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;
    }

    #endregion

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

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

  }
}