source: branches/2994-AutoDiffForIntervals/HeuristicLab.Algorithms.DataAnalysis/3.4/NonlinearRegression/NonlinearConstrainedRegression.cs @ 17456

#region License Information
/* HeuristicLab
 * Copyright (C) 2002-2019 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.Linq;
using System.Threading;
using HeuristicLab.Analysis;
using HeuristicLab.Common;
using HeuristicLab.Core;
using HeuristicLab.Data;
using HeuristicLab.Optimization;
using HeuristicLab.Parameters;
using HEAL.Attic;
using HeuristicLab.Problems.DataAnalysis;
using HeuristicLab.Problems.DataAnalysis.Symbolic;
using HeuristicLab.Problems.DataAnalysis.Symbolic.Regression;
using HeuristicLab.Random;
using HeuristicLab.Encodings.SymbolicExpressionTreeEncoding;

namespace HeuristicLab.Algorithms.DataAnalysis {
  /// <summary>
  /// Nonlinear regression data analysis algorithm.
  /// </summary>
  [Item("Nonlinear Regression with Constraints (NLR)", "Nonlinear regression (curve fitting) data analysis algorithm that supports interval constraints.")]
  [Creatable(CreatableAttribute.Categories.DataAnalysisRegression, Priority = 120)]
  [StorableType("B235DB6E-591F-4537-8D2F-C2D1232AAEFD")]
  public sealed class NonlinearConstrainedRegression : FixedDataAnalysisAlgorithm<IRegressionProblem> {
    private const string RegressionSolutionResultName = "Regression solution";
    private const string ModelStructureParameterName = "Model structure";
    private const string IterationsParameterName = "Iterations";
    private const string RestartsParameterName = "Restarts";
    private const string SetSeedRandomlyParameterName = "SetSeedRandomly";
    private const string SeedParameterName = "Seed";
    private const string InitParamsRandomlyParameterName = "InitializeParametersRandomly";
    private const string ApplyLinearScalingParameterName = "Apply linear scaling";

    public IFixedValueParameter<StringValue> ModelStructureParameter {
      get { return (IFixedValueParameter<StringValue>)Parameters[ModelStructureParameterName]; }
    }
    public IFixedValueParameter<IntValue> IterationsParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[IterationsParameterName]; }
    }

    public IFixedValueParameter<BoolValue> SetSeedRandomlyParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[SetSeedRandomlyParameterName]; }
    }

    public IFixedValueParameter<IntValue> SeedParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[SeedParameterName]; }
    }

    public IFixedValueParameter<IntValue> RestartsParameter {
      get { return (IFixedValueParameter<IntValue>)Parameters[RestartsParameterName]; }
    }

    public IFixedValueParameter<BoolValue> InitParametersRandomlyParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[InitParamsRandomlyParameterName]; }
    }

    public IFixedValueParameter<BoolValue> ApplyLinearScalingParameter {
      get { return (IFixedValueParameter<BoolValue>)Parameters[ApplyLinearScalingParameterName]; }
    }

    public string ModelStructure {
      get { return ModelStructureParameter.Value.Value; }
      set { ModelStructureParameter.Value.Value = value; }
    }

    public int Iterations {
      get { return IterationsParameter.Value.Value; }
      set { IterationsParameter.Value.Value = value; }
    }

    public int Restarts {
      get { return RestartsParameter.Value.Value; }
      set { RestartsParameter.Value.Value = value; }
    }

    public int Seed {
      get { return SeedParameter.Value.Value; }
      set { SeedParameter.Value.Value = value; }
    }

    public bool SetSeedRandomly {
      get { return SetSeedRandomlyParameter.Value.Value; }
      set { SetSeedRandomlyParameter.Value.Value = value; }
    }

    public bool InitializeParametersRandomly {
      get { return InitParametersRandomlyParameter.Value.Value; }
      set { InitParametersRandomlyParameter.Value.Value = value; }
    }

    public bool ApplyLinearScaling {
      get { return ApplyLinearScalingParameter.Value.Value; }
      set { ApplyLinearScalingParameter.Value.Value = value; }
    }

    [StorableConstructor]
    private NonlinearConstrainedRegression(StorableConstructorFlag _) : base(_) { }
    private NonlinearConstrainedRegression(NonlinearConstrainedRegression original, Cloner cloner)
      : base(original, cloner) {
    }
    public NonlinearConstrainedRegression()
      : base() {
      Problem = new RegressionProblem();
      Parameters.Add(new FixedValueParameter<StringValue>(ModelStructureParameterName, "The function for which the parameters must be fit (only numeric constants are tuned).", new StringValue("1.0 * x*x + 0.0")));
      Parameters.Add(new FixedValueParameter<IntValue>(IterationsParameterName, "The maximum number of iterations for constants optimization.", new IntValue(200)));
      Parameters.Add(new FixedValueParameter<IntValue>(RestartsParameterName, "The number of independent random restarts (>0)", new IntValue(10)));
      Parameters.Add(new FixedValueParameter<IntValue>(SeedParameterName, "The PRNG seed value.", new IntValue()));
      Parameters.Add(new FixedValueParameter<BoolValue>(SetSeedRandomlyParameterName, "Switch to determine if the random number seed should be initialized randomly.", new BoolValue(true)));
      Parameters.Add(new FixedValueParameter<BoolValue>(InitParamsRandomlyParameterName, "Switch to determine if the real-valued model parameters should be initialized randomly in each restart.", new BoolValue(false)));
      Parameters.Add(new FixedValueParameter<BoolValue>(ApplyLinearScalingParameterName, "Switch to determine if linear scaling terms should be added to the model", new BoolValue(true)));

      SetParameterHiddenState();

      InitParametersRandomlyParameter.Value.ValueChanged += (sender, args) => {
        SetParameterHiddenState();
      };
    }

    private void SetParameterHiddenState() {
      var hide = !InitializeParametersRandomly;
      RestartsParameter.Hidden = hide;
      SeedParameter.Hidden = hide;
      SetSeedRandomlyParameter.Hidden = hide;
    }

    [StorableHook(HookType.AfterDeserialization)]
    private void AfterDeserialization() {
      SetParameterHiddenState();
      InitParametersRandomlyParameter.Value.ValueChanged += (sender, args) => {
        SetParameterHiddenState();
      };
    }

    public override IDeepCloneable Clone(Cloner cloner) {
      return new NonlinearConstrainedRegression(this, cloner);
    }

    #region nonlinear regression
    protected override void Run(CancellationToken cancellationToken) {
      IRegressionSolution bestSolution = null;
      if (InitializeParametersRandomly) {
        var qualityTable = new DataTable("RMSE table");
        qualityTable.VisualProperties.YAxisLogScale = true;
        var trainRMSERow = new DataRow("RMSE (train)");
        trainRMSERow.VisualProperties.ChartType = DataRowVisualProperties.DataRowChartType.Points;
        var testRMSERow = new DataRow("RMSE test");
        testRMSERow.VisualProperties.ChartType = DataRowVisualProperties.DataRowChartType.Points;

        qualityTable.Rows.Add(trainRMSERow);
        qualityTable.Rows.Add(testRMSERow);
        Results.Add(new Result(qualityTable.Name, qualityTable.Name + " for all restarts", qualityTable));
        if (SetSeedRandomly) Seed = RandomSeedGenerator.GetSeed();
        var rand = new MersenneTwister((uint)Seed);
        bestSolution = CreateRegressionSolution((RegressionProblemData)Problem.ProblemData, ModelStructure, Iterations, ApplyLinearScaling, rand);
        trainRMSERow.Values.Add(bestSolution.TrainingRootMeanSquaredError);
        testRMSERow.Values.Add(bestSolution.TestRootMeanSquaredError);
        for (int r = 0; r < Restarts; r++) {
          var solution = CreateRegressionSolution((RegressionProblemData)Problem.ProblemData, ModelStructure, Iterations, ApplyLinearScaling, rand);
          trainRMSERow.Values.Add(solution.TrainingRootMeanSquaredError);
          testRMSERow.Values.Add(solution.TestRootMeanSquaredError);
          if (solution.TrainingRootMeanSquaredError < bestSolution.TrainingRootMeanSquaredError) {
            bestSolution = solution;
          }
        }
      } else {
        bestSolution = CreateRegressionSolution((RegressionProblemData)Problem.ProblemData, ModelStructure, Iterations, ApplyLinearScaling);
      }

      Results.Add(new Result(RegressionSolutionResultName, "The nonlinear regression solution.", bestSolution));
      Results.Add(new Result("Root mean square error (train)", "The root of the mean of squared errors of the regression solution on the training set.", new DoubleValue(bestSolution.TrainingRootMeanSquaredError)));
      Results.Add(new Result("Root mean square error (test)", "The root of the mean of squared errors of the regression solution on the test set.", new DoubleValue(bestSolution.TestRootMeanSquaredError)));

    }

    /// <summary>
    /// Fits a model to the data by optimizing the numeric constants.
    /// Model is specified as infix expression containing variable names and numbers. 
    /// The starting point for the numeric constants is initialized randomly if a random number generator is specified (~N(0,1)). Otherwise the user specified constants are
    /// used as a starting point. 
    /// </summary>-
    /// <param name="problemData">Training and test data</param>
    /// <param name="modelStructure">The function as infix expression</param>
    /// <param name="maxIterations">Number of constant optimization iterations (using Levenberg-Marquardt algorithm)</param>
    /// <param name="random">Optional random number generator for random initialization of numeric constants.</param>
    /// <returns></returns>
    public static ISymbolicRegressionSolution CreateRegressionSolution(RegressionProblemData problemData, string modelStructure, int maxIterations, bool applyLinearScaling, IRandom rand = null) {
      var parser = new InfixExpressionParser();
      var tree = parser.Parse(modelStructure);
      // parser handles double and string variables equally by creating a VariableTreeNode
      // post-process to replace VariableTreeNodes by FactorVariableTreeNodes for all string variables
      var factorSymbol = new FactorVariable();
      factorSymbol.VariableNames =
        problemData.AllowedInputVariables.Where(name => problemData.Dataset.VariableHasType<string>(name));
      factorSymbol.AllVariableNames = factorSymbol.VariableNames;
      factorSymbol.VariableValues =
        factorSymbol.VariableNames.Select(name =>
        new KeyValuePair<string, Dictionary<string, int>>(name,
        problemData.Dataset.GetReadOnlyStringValues(name).Distinct()
        .Select((n, i) => Tuple.Create(n, i))
        .ToDictionary(tup => tup.Item1, tup => tup.Item2)));

      foreach (var parent in tree.IterateNodesPrefix().ToArray()) {
        for (int i = 0; i < parent.SubtreeCount; i++) {
          var varChild = parent.GetSubtree(i) as VariableTreeNode;
          var factorVarChild = parent.GetSubtree(i) as FactorVariableTreeNode;
          if (varChild != null && factorSymbol.VariableNames.Contains(varChild.VariableName)) {
            parent.RemoveSubtree(i);
            var factorTreeNode = (FactorVariableTreeNode)factorSymbol.CreateTreeNode();
            factorTreeNode.VariableName = varChild.VariableName;
            factorTreeNode.Weights =
              factorTreeNode.Symbol.GetVariableValues(factorTreeNode.VariableName).Select(_ => 1.0).ToArray();
            // weight = 1.0 for each value
            parent.InsertSubtree(i, factorTreeNode);
          } else if (factorVarChild != null && factorSymbol.VariableNames.Contains(factorVarChild.VariableName)) {
            if (factorSymbol.GetVariableValues(factorVarChild.VariableName).Count() != factorVarChild.Weights.Length)
              throw new ArgumentException(
                string.Format("Factor variable {0} needs exactly {1} weights",
                factorVarChild.VariableName,
                factorSymbol.GetVariableValues(factorVarChild.VariableName).Count()));
            parent.RemoveSubtree(i);
            var factorTreeNode = (FactorVariableTreeNode)factorSymbol.CreateTreeNode();
            factorTreeNode.VariableName = factorVarChild.VariableName;
            factorTreeNode.Weights = factorVarChild.Weights;
            parent.InsertSubtree(i, factorTreeNode);
          }
        }
      }

      // var interpreter = new SymbolicDataAnalysisExpressionTreeLinearInterpreter();
      // 
      // SymbolicRegressionConstantOptimizationEvaluator.OptimizeConstants(interpreter, tree, problemData, problemData.TrainingIndices,
      //   applyLinearScaling: applyLinearScaling, maxIterations: maxIterations,
      //   updateVariableWeights: false, updateConstantsInTree: true);


      var intervals = problemData.IntervalConstraints;
      var constraintsParser = new IntervalConstraintsParser();
      var constraints = constraintsParser.Parse(intervals.Value);
      var dataIntervals = problemData.VariableRanges.VariableIntervals;

      // convert constants to variables named theta...
      var treeForDerivation = ReplaceConstWithVar(tree, out List<string> thetaNames, out List<double> thetaValues);

      // create trees for relevant derivatives
      Dictionary<string, ISymbolicExpressionTree> derivatives = new Dictionary<string, ISymbolicExpressionTree>();
      var allThetaNodes = thetaNames.Select(_ => new List<ConstantTreeNode>()).ToArray();
      var constraintTrees = new List<ISymbolicExpressionTree>();
      foreach (var constraint in constraints) {
        if (constraint.IsDerivation) {
          var df = DerivativeCalculator.Derive(treeForDerivation, constraint.Variable);

          // alglib requires constraint expressions of the form c(x) <= 0
          // -> we make two expressions, one for the lower bound and one for the upper bound

          if (constraint.Interval.UpperBound < double.PositiveInfinity) {
            var df_smaller_upper = Subtract((ISymbolicExpressionTree)df.Clone(), CreateConstant(constraint.Interval.UpperBound));
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(df_smaller_upper, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
          if (constraint.Interval.LowerBound > double.NegativeInfinity) {
            var df_larger_lower = Subtract(CreateConstant(constraint.Interval.LowerBound), (ISymbolicExpressionTree)df.Clone());
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(df_larger_lower, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
        } else {
          if (constraint.Interval.UpperBound < double.PositiveInfinity) {
            var f_smaller_upper = Subtract((ISymbolicExpressionTree)treeForDerivation.Clone(), CreateConstant(constraint.Interval.UpperBound));
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(f_smaller_upper, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
          if (constraint.Interval.LowerBound > double.NegativeInfinity) {
            var f_larger_lower = Subtract(CreateConstant(constraint.Interval.LowerBound), (ISymbolicExpressionTree)treeForDerivation.Clone());
            // convert variables named theta back to constants
            var df_prepared = ReplaceVarWithConst(f_larger_lower, thetaNames, thetaValues, allThetaNodes);
            constraintTrees.Add(df_prepared);
          }
        }
      }

      var preparedTree = ReplaceVarWithConst(treeForDerivation, thetaNames, thetaValues, allThetaNodes);

      // initialize constants randomly
      if (rand != null) {
        for (int i = 0; i < allThetaNodes.Length; i++) {
          double f = Math.Exp(NormalDistributedRandom.NextDouble(rand, 0, 1));
          double scale = rand.NextDouble() < 0.5 ? -1 : 1;
          thetaValues[i] = scale * thetaValues[i] * f;
          foreach (var constNode in allThetaNodes[i]) constNode.Value = thetaValues[i];
        }
      }

      void UpdateThetaValues(double[] theta) {
        for (int i = 0; i < theta.Length; ++i) {
          foreach (var constNode in allThetaNodes[i]) constNode.Value = theta[i];
        }
      }

      // define the callback used by the alglib optimizer
      // the x argument for this callback represents our theta
      void calculate_jacobian(double[] x, double[] fi, double[,] jac, object obj) {
        UpdateThetaValues(x);

        var autoDiffEval = new VectorAutoDiffEvaluator();
        autoDiffEval.Evaluate(preparedTree, problemData.Dataset, problemData.TrainingIndices.ToArray(),
          GetParameterNodes(preparedTree, allThetaNodes), out double[] fi_eval, out double[,] jac_eval);
        var target = problemData.TargetVariableTrainingValues.ToArray();

        // calc sum of squared errors and gradient
        var sse = 0.0;
        var g = new double[x.Length];
        for (int i = 0; i < target.Length; i++) {
          var res = target[i] - fi_eval[i];
          sse += res * res;
          for (int j = 0; j < g.Length; j++) {
            g[j] += -2.0 * res * jac_eval[i, j];
          }
        }

        fi[0] = sse;
        for (int j = 0; j < x.Length; j++) { jac[0, j] = g[j]; }

        var intervalEvaluator = new IntervalEvaluator();
        for (int i = 0; i < constraintTrees.Count; i++) {
          var interval = intervalEvaluator.Evaluate(constraintTrees[i], dataIntervals, GetParameterNodes(constraintTrees[i], allThetaNodes),
            out double[] lowerGradient, out double[] upperGradient);

          // we transformed this to a constraint c(x) <= 0, so only the upper bound is relevant for us
          fi[i + 1] = interval.UpperBound;
          for (int j = 0; j < x.Length; j++) {
            jac[i + 1, j] = upperGradient[j];
          }
        }
      }

      // prepare alglib
      alglib.minnlcstate state;
      alglib.minnlcreport rep;
      var x0 = thetaValues.ToArray();

      alglib.minnlccreate(x0.Length, x0, out state);
      double epsx = 1e-6;
      int maxits = 0;
      alglib.minnlcsetalgoslp(state);
      alglib.minnlcsetcond(state, 0, maxits);
      var s = Enumerable.Repeat(1d, x0.Length).ToArray();  // scale is set to unit scale
      alglib.minnlcsetscale(state, s);

      // set boundary constraints
      // var boundaryLower = Enumerable.Repeat(-10d, n).ToArray();
      // var boundaryUpper = Enumerable.Repeat(10d, n).ToArray();
      // alglib.minnlcsetbc(state, boundaryLower, boundaryUpper);
      // set non-linear constraints: 0 equality constraints, 1 inequality constraint
      alglib.minnlcsetnlc(state, 0, constraintTrees.Count);

      alglib.minnlcoptimize(state, calculate_jacobian, null, null);
      alglib.minnlcresults(state, out double[] xOpt, out rep);

      var interpreter = new SymbolicDataAnalysisExpressionTreeLinearInterpreter();
      UpdateThetaValues(xOpt);
      var model = new SymbolicRegressionModel(problemData.TargetVariable, (ISymbolicExpressionTree)preparedTree.Clone(), (ISymbolicDataAnalysisExpressionTreeInterpreter)interpreter.Clone());
      if (applyLinearScaling)
        model.Scale(problemData);

      SymbolicRegressionSolution solution = new SymbolicRegressionSolution(model, (IRegressionProblemData)problemData.Clone());
      solution.Model.Name = "Regression Model";
      solution.Name = "Regression Solution";
      return solution;
    }

    private static ISymbolicExpressionTreeNode[] GetParameterNodes(ISymbolicExpressionTree tree, List<ConstantTreeNode>[] allNodes) {
      // TODO better solution necessary
      var treeConstNodes = tree.IterateNodesPostfix().OfType<ConstantTreeNode>().ToArray();
      var paramNodes = new ISymbolicExpressionTreeNode[allNodes.Length];
      for (int i = 0; i < paramNodes.Length; i++) {
        paramNodes[i] = allNodes[i].SingleOrDefault(n => treeConstNodes.Contains(n));
      }
      return paramNodes;
    }

    #endregion

    #region helper
    private static ISymbolicExpressionTree ReplaceVarWithConst(ISymbolicExpressionTree tree, List<string> thetaNames, List<double> thetaValues, List<ConstantTreeNode>[] thetaNodes) {
      var copy = (ISymbolicExpressionTree)tree.Clone();
      var nodes = copy.IterateNodesPostfix().ToList();
      for (int i = 0; i < nodes.Count; i++) {
        var n = nodes[i] as VariableTreeNode;
        if (n != null) {
          var thetaIdx = thetaNames.IndexOf(n.VariableName);
          if (thetaIdx >= 0) {
            var parent = n.Parent;
            if(thetaNodes[thetaIdx].Any()) {
              // HACKY: REUSE CONSTANT TREE NODE IN SEVERAL TREES
              // we use this trick to allow autodiff over thetas when thetas occurr multiple times in the tree (e.g. in derived trees)
              var constNode = thetaNodes[thetaIdx].First();
              var childIdx = parent.IndexOfSubtree(n);
              parent.RemoveSubtree(childIdx);
              parent.InsertSubtree(childIdx, constNode);
            } else {
              var constNode = (ConstantTreeNode)CreateConstant(thetaValues[thetaIdx]);
              var childIdx = parent.IndexOfSubtree(n);
              parent.RemoveSubtree(childIdx);
              parent.InsertSubtree(childIdx, constNode);
              thetaNodes[thetaIdx].Add(constNode);
            }
          }
        }
      }
      return copy;
    }

    private static ISymbolicExpressionTree ReplaceConstWithVar(ISymbolicExpressionTree tree, out List<string> thetaNames, out List<double> thetaValues) {
      thetaNames = new List<string>();
      thetaValues = new List<double>();
      var copy = (ISymbolicExpressionTree)tree.Clone();
      var nodes = copy.IterateNodesPostfix().ToList();

      int n = 1;
      for (int i = 0; i < nodes.Count; ++i) {
        var node = nodes[i];
        /*if (node is VariableTreeNode variableTreeNode) {
          var thetaVar = (VariableTreeNode)new Problems.DataAnalysis.Symbolic.Variable().CreateTreeNode();
          thetaVar.Weight = 1;
          thetaVar.VariableName = $"θ{n++}";

          thetaNames.Add(thetaVar.VariableName);
          thetaValues.Add(variableTreeNode.Weight);
          variableTreeNode.Weight = 1; // set to unit weight

          var parent = variableTreeNode.Parent;
          var prod = MakeNode<Multiplication>(thetaVar, variableTreeNode);
          if (parent != null) {
            var index = parent.IndexOfSubtree(variableTreeNode);
            parent.RemoveSubtree(index);
            parent.InsertSubtree(index, prod);
          }
        } else*/ if (node is ConstantTreeNode constantTreeNode) {
          var thetaVar = (VariableTreeNode)new Problems.DataAnalysis.Symbolic.Variable().CreateTreeNode();
          thetaVar.Weight = 1;
          thetaVar.VariableName = $"θ{n++}";

          thetaNames.Add(thetaVar.VariableName);
          thetaValues.Add(constantTreeNode.Value);

          var parent = constantTreeNode.Parent;
          if (parent != null) {
            var index = constantTreeNode.Parent.IndexOfSubtree(constantTreeNode);
            parent.RemoveSubtree(index);
            parent.InsertSubtree(index, thetaVar);
          }
        }
      }
      return copy;
    }

    private static ISymbolicExpressionTreeNode CreateConstant(double value) {
      var constantNode = (ConstantTreeNode)new Constant().CreateTreeNode();
      constantNode.Value = value;
      return constantNode;
    }

    private static ISymbolicExpressionTree Subtract(ISymbolicExpressionTree t, ISymbolicExpressionTreeNode b) {
      var sub = MakeNode<Subtraction>(t.Root.GetSubtree(0).GetSubtree(0), b);
      t.Root.GetSubtree(0).RemoveSubtree(0);
      t.Root.GetSubtree(0).InsertSubtree(0, sub);
      return t;
    }
    private static ISymbolicExpressionTree Subtract(ISymbolicExpressionTreeNode b, ISymbolicExpressionTree t) {
      var sub = MakeNode<Subtraction>(b, t.Root.GetSubtree(0).GetSubtree(0));
      t.Root.GetSubtree(0).RemoveSubtree(0);
      t.Root.GetSubtree(0).InsertSubtree(0, sub);
      return t;
    }

    private static ISymbolicExpressionTreeNode MakeNode<T>(params ISymbolicExpressionTreeNode[] fs) where T : ISymbol, new() {
      var node = new T().CreateTreeNode();
      foreach (var f in fs) node.AddSubtree(f);
      return node;
    }
    #endregion
  }
}