newmatnl.h

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00001 //$$ newmatnl.h           definition file for non-linear optimisation
00002 
00003 // Copyright (C) 1993,4,5: R B Davies
00004 
00005 #ifndef NEWMATNL_LIB
00006 #define NEWMATNL_LIB 0
00007 
00008 #include "newmat.h"
00009 
00010 #ifdef use_namespace
00011 namespace NEWMAT {
00012 #endif
00013 
00014 
00015 
00016 /*
00017 
00018 This is a beginning of a series of classes for non-linear optimisation.
00019 
00020 At present there are two classes. FindMaximum2 is the basic optimisation
00021 strategy when one is doing an optimisation where one has first
00022 derivatives and estimates of the second derivatives. Class
00023 NonLinearLeastSquares is derived from FindMaximum2. This provides the
00024 functions that calculate function values and derivatives.
00025 
00026 A third class is now added. This is for doing maximum-likelihood when
00027 you have first derviatives and something like the Fisher Information
00028 matrix (eg the variance covariance matrix of the first derivatives or
00029 minus the second derivatives - this matrix is assumed to be positive
00030 definite).
00031 
00032 
00033 
00034    class FindMaximum2
00035 
00036 Suppose T is the ColumnVector of parameters, F(T) the function we want
00037 to maximise, D(T) the ColumnVector of derivatives of F with respect to
00038 T, and S(T) the matrix of second derivatives.
00039 
00040 Then the basic iteration is given a value of T, update it to
00041 
00042            T - S.i() * D
00043 
00044 where .i() denotes inverse.
00045 
00046 If F was quadratic this would give exactly the right answer (except it
00047 might get a minimum rather than a maximum). Since F is not usually
00048 quadratic, the simple procedure would be to recalculate S and D with the
00049 new value of T and keep iterating until the process converges. This is
00050 known as the method of conjugate gradients.
00051 
00052 In practice, this method may not converge. FindMaximum2 considers an
00053 iteration of the form
00054 
00055            T - x * S.i() * D
00056 
00057 where x is a number. It tries x = 1 and uses the values of F and its
00058 slope with respect to x at x = 0 and x = 1 to fit a cubic in x. It then
00059 choses x to maximise the resulting function. This gives our new value of
00060 T. The program checks that the value of F is getting better and carries
00061 out a variety of strategies if it is not.
00062 
00063 The program also has a second strategy. If the successive values of T
00064 seem to be lying along a curve - eg we are following along a curved
00065 ridge, the program will try to fit this ridge and project along it. This
00066 does not work at present and is commented out.
00067 
00068 FindMaximum2 has three virtual functions which need to be over-ridden by
00069 a derived class.
00070 
00071    void Value(const ColumnVector& T, bool wg, Real& f, bool& oorg);
00072 
00073 T is the column vector of parameters. The function returns the value of
00074 the function to f, but may instead set oorg to true if the parameter
00075 values are not valid. If wg is true it may also calculate and store the
00076 second derivative information.
00077 
00078    bool NextPoint(ColumnVector& H, Real& d);
00079 
00080 Using the value of T provided in the previous call of Value, find the
00081 conjugate gradients adjustment to T, that is - S.i() * D. Also return
00082 
00083            d = D.t() * S.i() * D.
00084 
00085 NextPoint should return true if it considers that the process has
00086 converged (d very small) and false otherwise. The previous call of Value
00087 will have set wg to true, so that S will be available.
00088 
00089    Real LastDerivative(const ColumnVector& H);
00090 
00091 Return the scalar product of H and the vector of derivatives at the last
00092 value of T.
00093 
00094 The function Fit is the function that calls the iteration.
00095 
00096    void Fit(ColumnVector&, int);
00097 
00098 The arguments are the trial parameter values as a ColumnVector and the
00099 maximum number of iterations. The program calls a DataException if the
00100 initial parameters are not valid and a ConvergenceException if the
00101 process fails to converge.
00102 
00103 
00104    class NonLinearLeastSquares
00105 
00106 This class is derived from FindMaximum2 and carries out a non-linear
00107 least squares fit. It uses a QR decomposition to carry out the
00108 operations required by FindMaximum2.
00109 
00110 A prototype class R1_Col_I_D is provided. The user needs to derive a
00111 class from this which includes functions the predicted value of each
00112 observation its derivatives. An object from this class has to be
00113 provided to class NonLinearLeastSquares.
00114 
00115 Suppose we observe n normal random variables with the same unknown
00116 variance and such the i-th one has expected value given by f(i,P)
00117 where P is a column vector of unknown parameters and f is a known
00118 function. We wish to estimate P.
00119 
00120 First derive a class from R1_Col_I_D and override Real operator()(int i)
00121 to give the value of the function f in terms of i and the ColumnVector
00122 para defined in class R1_CoL_I_D. Also override ReturnMatrix
00123 Derivatives() to give the derivates of f at para and the value of i
00124 used in the preceeding call to operator(). Return the result as a
00125 RowVector. Construct an object from this class. Suppose in what follows
00126 it is called pred.
00127 
00128 Now constuct a NonLinearLeastSquaresObject accessing pred and optionally
00129 an iteration limit and an accuracy critierion.
00130 
00131    NonLinearLeastSquares NLLS(pred, 1000, 0.0001);
00132 
00133 The accuracy critierion should be somewhat less than one and 0.0001 is
00134 about the smallest sensible value.
00135 
00136 Define a ColumnVector P containing a guess at the value of the unknown
00137 parameter, and a ColumnVector Y containing the unknown data. Call
00138 
00139    NLLS.Fit(Y,P);
00140 
00141 If the process converges, P will contain the estimates of the unknown
00142 parameters. If it does not converge an exception will be generated.
00143 
00144 The following member functions can be called after you have done a fit.
00145 
00146 Real ResidualVariance() const;
00147 
00148 The estimate of the variance of the observations.
00149 
00150 void GetResiduals(ColumnVector& Z) const;
00151 
00152 The residuals of the individual observations.
00153 
00154 void GetStandardErrors(ColumnVector&);
00155 
00156 The standard errors of the observations.
00157 
00158 void GetCorrelations(SymmetricMatrix&);
00159 
00160 The correlations of the observations.
00161 
00162 void GetHatDiagonal(DiagonalMatrix&) const;
00163 
00164 Forms a diagonal matrix of values between 0 and 1. If the i-th value is
00165 larger than, say 0.2, then the i-th data value could have an undue
00166 influence on your estimates.
00167 
00168 
00169 */
00170 
00171 class FindMaximum2
00172 {
00173    virtual void Value(const ColumnVector&, bool, Real&, bool&) = 0;
00174    virtual bool NextPoint(ColumnVector&, Real&) = 0;
00175    virtual Real LastDerivative(const ColumnVector&) = 0;
00176 public:
00177    void Fit(ColumnVector&, int);
00178    virtual ~FindMaximum2() {}            // to keep gnu happy
00179 };
00180 
00181 class R1_Col_I_D
00182 {
00183    // The prototype for a Real function of a ColumnVector and an
00184    // integer.
00185    // You need to derive your function from this one and put in your
00186    // function for operator() and Derivatives() at least.
00187    // You may also want to set up a constructor to enter in additional
00188    // parameter values (that will not vary during the solve).
00189 
00190 protected:
00191    ColumnVector para;                 // Current x value
00192 
00193 public:
00194    virtual bool IsValid() { return true; }
00195                                        // is the current x value OK
00196    virtual Real operator()(int i) = 0; // i-th function value at current para
00197    virtual void Set(const ColumnVector& X) { para = X; }
00198                                        // set current para
00199    bool IsValid(const ColumnVector& X)
00200       { Set(X); return IsValid(); }
00201                                        // set para, check OK
00202    Real operator()(int i, const ColumnVector& X)
00203       { Set(X); return operator()(i); }
00204                                        // set para, return value
00205    virtual ReturnMatrix Derivatives() = 0;
00206                                        // return derivatives as RowVector
00207    virtual ~R1_Col_I_D() {}            // to keep gnu happy
00208 };
00209 
00210 
00211 class NonLinearLeastSquares : public FindMaximum2
00212 {
00213    // these replace the corresponding functions in FindMaximum2
00214    void Value(const ColumnVector&, bool, Real&, bool&);
00215    bool NextPoint(ColumnVector&, Real&);
00216    Real LastDerivative(const ColumnVector&);
00217 
00218    Matrix X;                         // the things we need to do the
00219    ColumnVector Y;                   // QR triangularisation
00220    UpperTriangularMatrix U;          // see the write-up in newmata.txt
00221    ColumnVector M;
00222    Real errorvar, criterion;
00223    int n_obs, n_param;
00224    const ColumnVector* DataPointer;
00225    RowVector Derivs;
00226    SymmetricMatrix Covariance;
00227    DiagonalMatrix SE;
00228    R1_Col_I_D& Pred;                 // Reference to predictor object
00229    int Lim;                          // maximum number of iterations
00230 
00231 public:
00232    NonLinearLeastSquares(R1_Col_I_D& pred, int lim=1000, Real crit=0.0001)
00233       : criterion(crit), Pred(pred), Lim(lim) {}
00234    void Fit(const ColumnVector&, ColumnVector&);
00235    Real ResidualVariance() const { return errorvar; }
00236    void GetResiduals(ColumnVector& Z) const { Z = Y; }
00237    void GetStandardErrors(ColumnVector&);
00238    void GetCorrelations(SymmetricMatrix&);
00239    void GetHatDiagonal(DiagonalMatrix&) const;
00240 
00241 private:
00242    void MakeCovariance();
00243 };
00244 
00245 
00246 // The next class is the prototype class for calculating the
00247 // log-likelihood.
00248 // I assume first derivatives are available and something like the 
00249 // Fisher Information or variance/covariance matrix of the first
00250 // derivatives or minus the matrix of second derivatives is
00251 // available. This matrix must be positive definite.
00252 
00253 class LL_D_FI
00254 {
00255 protected:
00256    ColumnVector para;                  // current parameter values
00257    bool wg;                         // true if FI matrix wanted
00258 
00259 public:
00260    virtual void Set(const ColumnVector& X) { para = X; }
00261                                        // set parameter values
00262    virtual void WG(bool wgx) { wg = wgx; }
00263                                        // set wg
00264 
00265    virtual bool IsValid() { return true; }
00266                                        // return true is para is OK
00267    bool IsValid(const ColumnVector& X, bool wgx=true)
00268       { Set(X); WG(wgx); return IsValid(); }
00269 
00270    virtual Real LogLikelihood() = 0;   // return the loglikelihhod
00271    Real LogLikelihood(const ColumnVector& X, bool wgx=true)
00272       { Set(X); WG(wgx); return LogLikelihood(); }
00273 
00274    virtual ReturnMatrix Derivatives() = 0;
00275                                        // column vector of derivatives
00276    virtual ReturnMatrix FI() = 0;      // Fisher Information matrix
00277    virtual ~LL_D_FI() {}               // to keep gnu happy
00278 };
00279 
00280 // This is the class for doing the maximum likelihood estimation
00281 
00282 class MLE_D_FI : public FindMaximum2
00283 {
00284    // these replace the corresponding functions in FindMaximum2
00285    void Value(const ColumnVector&, bool, Real&, bool&);
00286    bool NextPoint(ColumnVector&, Real&);
00287    Real LastDerivative(const ColumnVector&);
00288 
00289    // the things we need for the analysis
00290    LL_D_FI& LL;                        // reference to log-likelihood
00291    int Lim;                            // maximum number of iterations
00292    Real Criterion;                     // convergence criterion
00293    ColumnVector Derivs;                // for the derivatives
00294    LowerTriangularMatrix LT;           // Cholesky decomposition of FI
00295    SymmetricMatrix Covariance;
00296    DiagonalMatrix SE;
00297 
00298 public:
00299    MLE_D_FI(LL_D_FI& ll, int lim=1000, Real criterion=0.0001)
00300       : LL(ll), Lim(lim), Criterion(criterion) {}
00301    void Fit(ColumnVector& Parameters);
00302    void GetStandardErrors(ColumnVector&);
00303    void GetCorrelations(SymmetricMatrix&);
00304 
00305 private:
00306    void MakeCovariance();
00307 };
00308 
00309 
00310 #ifdef use_namespace
00311 }
00312 #endif
00313 
00314 
00315 
00316 #endif
00317 
00318 // body file: newmatnl.cpp
00319 
00320 
00321 
00322 

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