1 | /// \ingroup newmat |
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2 | ///@{ |
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3 | |
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4 | /// \file newmatnl.h |
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5 | /// Header file for non-linear optimisation |
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6 | |
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7 | // Copyright (C) 1993,4,5: R B Davies |
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8 | |
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9 | #ifndef NEWMATNL_LIB |
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10 | #define NEWMATNL_LIB 0 |
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11 | |
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12 | #include "newmat.h" |
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13 | |
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14 | #ifdef use_namespace |
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15 | namespace NEWMAT { |
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16 | #endif |
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17 | |
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18 | |
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19 | |
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20 | /* |
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21 | |
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22 | This is a beginning of a series of classes for non-linear optimisation. |
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23 | |
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24 | At present there are two classes. FindMaximum2 is the basic optimisation |
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25 | strategy when one is doing an optimisation where one has first |
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26 | derivatives and estimates of the second derivatives. Class |
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27 | NonLinearLeastSquares is derived from FindMaximum2. This provides the |
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28 | functions that calculate function values and derivatives. |
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29 | |
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30 | A third class is now added. This is for doing maximum-likelihood when |
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31 | you have first derviatives and something like the Fisher Information |
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32 | matrix (eg the variance covariance matrix of the first derivatives or |
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33 | minus the second derivatives - this matrix is assumed to be positive |
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34 | definite). |
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35 | |
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36 | |
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37 | |
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38 | class FindMaximum2 |
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39 | |
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40 | Suppose T is the ColumnVector of parameters, F(T) the function we want |
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41 | to maximise, D(T) the ColumnVector of derivatives of F with respect to |
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42 | T, and S(T) the matrix of second derivatives. |
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43 | |
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44 | Then the basic iteration is given a value of T, update it to |
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45 | |
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46 | T - S.i() * D |
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47 | |
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48 | where .i() denotes inverse. |
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49 | |
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50 | If F was quadratic this would give exactly the right answer (except it |
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51 | might get a minimum rather than a maximum). Since F is not usually |
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52 | quadratic, the simple procedure would be to recalculate S and D with the |
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53 | new value of T and keep iterating until the process converges. This is |
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54 | known as the method of conjugate gradients. |
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55 | |
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56 | In practice, this method may not converge. FindMaximum2 considers an |
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57 | iteration of the form |
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58 | |
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59 | T - x * S.i() * D |
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60 | |
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61 | where x is a number. It tries x = 1 and uses the values of F and its |
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62 | slope with respect to x at x = 0 and x = 1 to fit a cubic in x. It then |
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63 | choses x to maximise the resulting function. This gives our new value of |
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64 | T. The program checks that the value of F is getting better and carries |
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65 | out a variety of strategies if it is not. |
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66 | |
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67 | The program also has a second strategy. If the successive values of T |
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68 | seem to be lying along a curve - eg we are following along a curved |
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69 | ridge, the program will try to fit this ridge and project along it. This |
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70 | does not work at present and is commented out. |
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71 | |
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72 | FindMaximum2 has three virtual functions which need to be over-ridden by |
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73 | a derived class. |
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74 | |
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75 | void Value(const ColumnVector& T, bool wg, Real& f, bool& oorg); |
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76 | |
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77 | T is the column vector of parameters. The function returns the value of |
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78 | the function to f, but may instead set oorg to true if the parameter |
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79 | values are not valid. If wg is true it may also calculate and store the |
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80 | second derivative information. |
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81 | |
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82 | bool NextPoint(ColumnVector& H, Real& d); |
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83 | |
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84 | Using the value of T provided in the previous call of Value, find the |
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85 | conjugate gradients adjustment to T, that is - S.i() * D. Also return |
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86 | |
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87 | d = D.t() * S.i() * D. |
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88 | |
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89 | NextPoint should return true if it considers that the process has |
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90 | converged (d very small) and false otherwise. The previous call of Value |
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91 | will have set wg to true, so that S will be available. |
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92 | |
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93 | Real LastDerivative(const ColumnVector& H); |
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94 | |
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95 | Return the scalar product of H and the vector of derivatives at the last |
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96 | value of T. |
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97 | |
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98 | The function Fit is the function that calls the iteration. |
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99 | |
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100 | void Fit(ColumnVector&, int); |
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101 | |
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102 | The arguments are the trial parameter values as a ColumnVector and the |
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103 | maximum number of iterations. The program calls a DataException if the |
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104 | initial parameters are not valid and a ConvergenceException if the |
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105 | process fails to converge. |
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106 | |
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107 | |
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108 | class NonLinearLeastSquares |
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109 | |
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110 | This class is derived from FindMaximum2 and carries out a non-linear |
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111 | least squares fit. It uses a QR decomposition to carry out the |
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112 | operations required by FindMaximum2. |
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113 | |
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114 | A prototype class R1_Col_I_D is provided. The user needs to derive a |
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115 | class from this which includes functions the predicted value of each |
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116 | observation its derivatives. An object from this class has to be |
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117 | provided to class NonLinearLeastSquares. |
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118 | |
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119 | Suppose we observe n normal random variables with the same unknown |
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120 | variance and such the i-th one has expected value given by f(i,P) |
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121 | where P is a column vector of unknown parameters and f is a known |
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122 | function. We wish to estimate P. |
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123 | |
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124 | First derive a class from R1_Col_I_D and override Real operator()(int i) |
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125 | to give the value of the function f in terms of i and the ColumnVector |
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126 | para defined in class R1_CoL_I_D. Also override ReturnMatrix |
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127 | Derivatives() to give the derivates of f at para and the value of i |
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128 | used in the preceeding call to operator(). Return the result as a |
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129 | RowVector. Construct an object from this class. Suppose in what follows |
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130 | it is called pred. |
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131 | |
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132 | Now constuct a NonLinearLeastSquaresObject accessing pred and optionally |
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133 | an iteration limit and an accuracy critierion. |
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134 | |
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135 | NonLinearLeastSquares NLLS(pred, 1000, 0.0001); |
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136 | |
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137 | The accuracy critierion should be somewhat less than one and 0.0001 is |
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138 | about the smallest sensible value. |
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139 | |
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140 | Define a ColumnVector P containing a guess at the value of the unknown |
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141 | parameter, and a ColumnVector Y containing the unknown data. Call |
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142 | |
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143 | NLLS.Fit(Y,P); |
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144 | |
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145 | If the process converges, P will contain the estimates of the unknown |
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146 | parameters. If it does not converge an exception will be generated. |
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147 | |
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148 | The following member functions can be called after you have done a fit. |
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149 | |
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150 | Real ResidualVariance() const; |
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151 | |
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152 | The estimate of the variance of the observations. |
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153 | |
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154 | void GetResiduals(ColumnVector& Z) const; |
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155 | |
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156 | The residuals of the individual observations. |
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157 | |
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158 | void GetStandardErrors(ColumnVector&); |
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159 | |
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160 | The standard errors of the observations. |
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161 | |
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162 | void GetCorrelations(SymmetricMatrix&); |
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163 | |
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164 | The correlations of the observations. |
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165 | |
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166 | void GetHatDiagonal(DiagonalMatrix&) const; |
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167 | |
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168 | Forms a diagonal matrix of values between 0 and 1. If the i-th value is |
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169 | larger than, say 0.2, then the i-th data value could have an undue |
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170 | influence on your estimates. |
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171 | |
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172 | |
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173 | */ |
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174 | |
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175 | class FindMaximum2 |
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176 | { |
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177 | virtual void Value(const ColumnVector&, bool, Real&, bool&) = 0; |
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178 | virtual bool NextPoint(ColumnVector&, Real&) = 0; |
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179 | virtual Real LastDerivative(const ColumnVector&) = 0; |
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180 | public: |
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181 | void Fit(ColumnVector&, int); |
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182 | virtual ~FindMaximum2() {} // to keep gnu happy |
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183 | }; |
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184 | |
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185 | class R1_Col_I_D |
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186 | { |
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187 | // The prototype for a Real function of a ColumnVector and an |
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188 | // integer. |
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189 | // You need to derive your function from this one and put in your |
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190 | // function for operator() and Derivatives() at least. |
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191 | // You may also want to set up a constructor to enter in additional |
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192 | // parameter values (that will not vary during the solve). |
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193 | |
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194 | protected: |
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195 | ColumnVector para; // Current x value |
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196 | |
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197 | public: |
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198 | virtual bool IsValid() { return true; } |
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199 | // is the current x value OK |
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200 | virtual Real operator()(int i) = 0; // i-th function value at current para |
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201 | virtual void Set(const ColumnVector& X) { para = X; } |
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202 | // set current para |
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203 | bool IsValid(const ColumnVector& X) |
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204 | { Set(X); return IsValid(); } |
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205 | // set para, check OK |
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206 | Real operator()(int i, const ColumnVector& X) |
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207 | { Set(X); return operator()(i); } |
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208 | // set para, return value |
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209 | virtual ReturnMatrix Derivatives() = 0; |
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210 | // return derivatives as RowVector |
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211 | virtual ~R1_Col_I_D() {} // to keep gnu happy |
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212 | }; |
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213 | |
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214 | |
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215 | class NonLinearLeastSquares : public FindMaximum2 |
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216 | { |
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217 | // these replace the corresponding functions in FindMaximum2 |
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218 | void Value(const ColumnVector&, bool, Real&, bool&); |
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219 | bool NextPoint(ColumnVector&, Real&); |
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220 | Real LastDerivative(const ColumnVector&); |
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221 | |
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222 | Matrix X; // the things we need to do the |
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223 | ColumnVector Y; // QR triangularisation |
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224 | UpperTriangularMatrix U; // see the write-up in newmata.txt |
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225 | ColumnVector M; |
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226 | Real errorvar, criterion; |
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227 | int n_obs, n_param; |
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228 | const ColumnVector* DataPointer; |
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229 | RowVector Derivs; |
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230 | SymmetricMatrix Covariance; |
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231 | DiagonalMatrix SE; |
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232 | R1_Col_I_D& Pred; // Reference to predictor object |
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233 | int Lim; // maximum number of iterations |
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234 | |
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235 | public: |
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236 | NonLinearLeastSquares(R1_Col_I_D& pred, int lim=1000, Real crit=0.0001) |
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237 | : criterion(crit), Pred(pred), Lim(lim) {} |
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238 | void Fit(const ColumnVector&, ColumnVector&); |
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239 | Real ResidualVariance() const { return errorvar; } |
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240 | void GetResiduals(ColumnVector& Z) const { Z = Y; } |
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241 | void GetStandardErrors(ColumnVector&); |
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242 | void GetCorrelations(SymmetricMatrix&); |
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243 | void GetHatDiagonal(DiagonalMatrix&) const; |
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244 | |
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245 | private: |
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246 | void MakeCovariance(); |
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247 | }; |
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248 | |
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249 | |
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250 | // The next class is the prototype class for calculating the |
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251 | // log-likelihood. |
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252 | // I assume first derivatives are available and something like the |
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253 | // Fisher Information or variance/covariance matrix of the first |
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254 | // derivatives or minus the matrix of second derivatives is |
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255 | // available. This matrix must be positive definite. |
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256 | |
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257 | class LL_D_FI |
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258 | { |
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259 | protected: |
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260 | ColumnVector para; // current parameter values |
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261 | bool wg; // true if FI matrix wanted |
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262 | |
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263 | public: |
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264 | virtual void Set(const ColumnVector& X) { para = X; } |
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265 | // set parameter values |
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266 | virtual void WG(bool wgx) { wg = wgx; } |
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267 | // set wg |
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268 | |
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269 | virtual bool IsValid() { return true; } |
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270 | // return true is para is OK |
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271 | bool IsValid(const ColumnVector& X, bool wgx=true) |
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272 | { Set(X); WG(wgx); return IsValid(); } |
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273 | |
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274 | virtual Real LogLikelihood() = 0; // return the loglikelihhod |
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275 | Real LogLikelihood(const ColumnVector& X, bool wgx=true) |
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276 | { Set(X); WG(wgx); return LogLikelihood(); } |
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277 | |
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278 | virtual ReturnMatrix Derivatives() = 0; |
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279 | // column vector of derivatives |
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280 | virtual ReturnMatrix FI() = 0; // Fisher Information matrix |
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281 | virtual ~LL_D_FI() {} // to keep gnu happy |
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282 | }; |
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283 | |
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284 | // This is the class for doing the maximum likelihood estimation |
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285 | |
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286 | class MLE_D_FI : public FindMaximum2 |
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287 | { |
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288 | // these replace the corresponding functions in FindMaximum2 |
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289 | void Value(const ColumnVector&, bool, Real&, bool&); |
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290 | bool NextPoint(ColumnVector&, Real&); |
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291 | Real LastDerivative(const ColumnVector&); |
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292 | |
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293 | // the things we need for the analysis |
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294 | LL_D_FI& LL; // reference to log-likelihood |
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295 | int Lim; // maximum number of iterations |
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296 | Real Criterion; // convergence criterion |
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297 | ColumnVector Derivs; // for the derivatives |
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298 | LowerTriangularMatrix LT; // Cholesky decomposition of FI |
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299 | SymmetricMatrix Covariance; |
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300 | DiagonalMatrix SE; |
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301 | |
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302 | public: |
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303 | MLE_D_FI(LL_D_FI& ll, int lim=1000, Real criterion=0.0001) |
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304 | : LL(ll), Lim(lim), Criterion(criterion) {} |
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305 | void Fit(ColumnVector& Parameters); |
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306 | void GetStandardErrors(ColumnVector&); |
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307 | void GetCorrelations(SymmetricMatrix&); |
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308 | |
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309 | private: |
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310 | void MakeCovariance(); |
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311 | }; |
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312 | |
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313 | |
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314 | #ifdef use_namespace |
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315 | } |
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316 | #endif |
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317 | |
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318 | |
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319 | |
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320 | #endif |
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321 | |
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322 | // body file: newmatnl.cpp |
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323 | |
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324 | |
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325 | |
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326 | ///@} |
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327 | |
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