1 | /*! |
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2 | \file |
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3 | \brief Bayesian Filtering for linear Gaussian models (Kalman Filter) and extensions |
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4 | \author Vaclav Smidl. |
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5 | |
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6 | ----------------------------------- |
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7 | BDM++ - C++ library for Bayesian Decision Making under Uncertainty |
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8 | |
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9 | Using IT++ for numerical operations |
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10 | ----------------------------------- |
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11 | */ |
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12 | |
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13 | #ifndef KF_H |
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14 | #define KF_H |
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15 | |
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16 | |
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17 | #include "../math/functions.h" |
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18 | #include "../stat/exp_family.h" |
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19 | #include "../math/chmat.h" |
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20 | #include "../base/user_info.h" |
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21 | |
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22 | namespace bdm { |
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23 | |
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24 | /*! |
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25 | * \brief Basic Kalman filter with full matrices (education purpose only)! Will be deleted soon! |
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26 | */ |
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27 | |
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28 | class KalmanFull { |
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29 | protected: |
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30 | int dimx, dimy, dimu; |
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31 | mat A, B, C, D, R, Q; |
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32 | |
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33 | //cache |
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34 | mat _Pp, _Ry, _iRy, _K; |
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35 | public: |
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36 | //posterior |
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37 | //! Mean value of the posterior density |
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38 | vec mu; |
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39 | //! Variance of the posterior density |
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40 | mat P; |
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41 | |
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42 | bool evalll; |
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43 | double ll; |
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44 | public: |
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45 | //! Full constructor |
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46 | KalmanFull ( mat A, mat B, mat C, mat D, mat R, mat Q, mat P0, vec mu0 ); |
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47 | //! Here dt = [yt;ut] of appropriate dimensions |
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48 | void bayes ( const vec &dt ); |
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49 | //! print elements of KF |
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50 | friend std::ostream &operator<< ( std::ostream &os, const KalmanFull &kf ); |
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51 | //! For EKFfull; |
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52 | KalmanFull() {}; |
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53 | }; |
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54 | |
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55 | |
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56 | /*! |
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57 | * \brief Kalman filter with covariance matrices in square root form. |
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58 | |
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59 | Parameter evolution model:\f[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \f] |
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60 | Observation model: \f[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \f] |
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61 | Where $e_t$ and $w_t$ are independent vectors Normal(0,1)-distributed disturbances. |
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62 | */ |
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63 | template<class sq_T> |
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64 | |
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65 | class Kalman : public BM { |
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66 | protected: |
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67 | //! Indetifier of output rv |
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68 | RV rvy; |
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69 | //! Indetifier of exogeneous rv |
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70 | RV rvu; |
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71 | //! cache of rv.count() |
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72 | int dimx; |
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73 | //! cache of rvy.count() |
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74 | int dimy; |
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75 | //! cache of rvu.count() |
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76 | int dimu; |
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77 | //! Matrix A |
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78 | mat A; |
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79 | //! Matrix B |
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80 | mat B; |
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81 | //! Matrix C |
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82 | mat C; |
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83 | //! Matrix D |
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84 | mat D; |
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85 | //! Matrix Q in square-root form |
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86 | sq_T Q; |
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87 | //! Matrix R in square-root form |
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88 | sq_T R; |
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89 | |
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90 | //!posterior density on $x_t$ |
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91 | enorm<sq_T> est; |
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92 | //!preditive density on $y_t$ |
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93 | enorm<sq_T> fy; |
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94 | |
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95 | //! placeholder for Kalman gain |
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96 | mat _K; |
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97 | //! cache of fy.mu |
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98 | vec& _yp; |
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99 | //! cache of fy.R |
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100 | sq_T& _Ry; |
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101 | //!cache of est.mu |
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102 | vec& _mu; |
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103 | //!cache of est.R |
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104 | sq_T& _P; |
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105 | |
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106 | public: |
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107 | //! Default constructor |
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108 | Kalman ( ); |
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109 | //! Copy constructor |
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110 | Kalman ( const Kalman<sq_T> &K0 ); |
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111 | //! Set parameters with check of relevance |
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112 | void set_parameters ( const mat &A0, const mat &B0, const mat &C0, const mat &D0, const sq_T &Q0, const sq_T &R0 ); |
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113 | //! Set estimate values, used e.g. in initialization. |
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114 | void set_est ( const vec &mu0, const sq_T &P0 ) { |
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115 | sq_T pom ( dimy ); |
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116 | est.set_parameters ( mu0, P0 ); |
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117 | P0.mult_sym ( C, pom ); |
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118 | fy.set_parameters ( C*mu0, pom ); |
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119 | }; |
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120 | |
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121 | //! Here dt = [yt;ut] of appropriate dimensions |
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122 | void bayes ( const vec &dt ); |
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123 | //!access function |
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124 | const epdf& posterior() const { |
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125 | return est; |
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126 | } |
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127 | const enorm<sq_T>* _e() const { |
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128 | return &est; |
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129 | } |
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130 | //!access function |
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131 | mat& __K() { |
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132 | return _K; |
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133 | } |
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134 | //!access function |
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135 | vec _dP() { |
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136 | return _P->getD(); |
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137 | } |
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138 | }; |
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139 | |
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140 | /*! \brief Kalman filter in square root form |
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141 | |
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142 | Trivial example: |
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143 | \include kalman_simple.cpp |
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144 | |
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145 | */ |
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146 | class KalmanCh : public Kalman<chmat> { |
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147 | protected: |
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148 | //! pre array (triangular matrix) |
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149 | mat preA; |
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150 | //! post array (triangular matrix) |
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151 | mat postA; |
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152 | |
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153 | public: |
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154 | //! copy constructor |
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155 | BM* _copy_() const { |
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156 | KalmanCh* K = new KalmanCh; |
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157 | K->set_parameters ( A, B, C, D, Q, R ); |
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158 | K->set_statistics ( _mu, _P ); |
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159 | return K; |
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160 | } |
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161 | //! Set parameters with check of relevance |
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162 | void set_parameters ( const mat &A0, const mat &B0, const mat &C0, const mat &D0, const chmat &Q0, const chmat &R0 ); |
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163 | void set_statistics ( const vec &mu0, const chmat &P0 ) { |
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164 | est.set_parameters ( mu0, P0 ); |
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165 | }; |
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166 | |
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167 | |
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168 | /*!\brief Here dt = [yt;ut] of appropriate dimensions |
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169 | |
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170 | The following equality hold::\f[ |
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171 | \left[\begin{array}{cc} |
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172 | R^{0.5}\\ |
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173 | P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ |
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174 | & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} |
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175 | R_{y}^{0.5} & KA'\\ |
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176 | & P_{t+1|t}^{0.5}\\ |
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177 | \\\end{array}\right]\f] |
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178 | |
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179 | Thus this object evaluates only predictors! Not filtering densities. |
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180 | */ |
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181 | void bayes ( const vec &dt ); |
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182 | }; |
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183 | |
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184 | /*! |
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185 | \brief Extended Kalman Filter in full matrices |
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186 | |
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187 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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188 | */ |
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189 | class EKFfull : public KalmanFull, public BM { |
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190 | protected: |
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191 | //! Internal Model f(x,u) |
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192 | shared_ptr<diffbifn> pfxu; |
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193 | |
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194 | //! Observation Model h(x,u) |
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195 | shared_ptr<diffbifn> phxu; |
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196 | |
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197 | enorm<fsqmat> E; |
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198 | public: |
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199 | //! Default constructor |
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200 | EKFfull ( ); |
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201 | |
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202 | //! Set nonlinear functions for mean values and covariance matrices. |
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203 | void set_parameters ( const shared_ptr<diffbifn> &pfxu, const shared_ptr<diffbifn> &phxu, const mat Q0, const mat R0 ); |
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204 | |
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205 | //! Here dt = [yt;ut] of appropriate dimensions |
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206 | void bayes ( const vec &dt ); |
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207 | //! set estimates |
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208 | void set_statistics ( vec mu0, mat P0 ) { |
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209 | mu = mu0; |
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210 | P = P0; |
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211 | }; |
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212 | //!dummy! |
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213 | const epdf& posterior() const { |
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214 | return E; |
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215 | }; |
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216 | const enorm<fsqmat>* _e() const { |
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217 | return &E; |
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218 | }; |
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219 | const mat _R() { |
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220 | return P; |
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221 | } |
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222 | }; |
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223 | |
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224 | /*! |
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225 | \brief Extended Kalman Filter |
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226 | |
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227 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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228 | */ |
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229 | template<class sq_T> |
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230 | class EKF : public Kalman<fsqmat> { |
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231 | //! Internal Model f(x,u) |
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232 | diffbifn* pfxu; |
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233 | //! Observation Model h(x,u) |
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234 | diffbifn* phxu; |
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235 | public: |
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236 | //! Default constructor |
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237 | EKF ( RV rvx, RV rvy, RV rvu ); |
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238 | //! copy constructor |
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239 | EKF<sq_T>* _copy_() const { |
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240 | return new EKF<sq_T> ( this ); |
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241 | } |
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242 | //! Set nonlinear functions for mean values and covariance matrices. |
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243 | void set_parameters ( diffbifn* pfxu, diffbifn* phxu, const sq_T Q0, const sq_T R0 ); |
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244 | //! Here dt = [yt;ut] of appropriate dimensions |
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245 | void bayes ( const vec &dt ); |
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246 | }; |
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247 | |
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248 | /*! |
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249 | \brief Extended Kalman Filter in Square root |
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250 | |
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251 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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252 | */ |
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253 | |
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254 | class EKFCh : public KalmanCh { |
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255 | protected: |
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256 | //! Internal Model f(x,u) |
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257 | shared_ptr<diffbifn> pfxu; |
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258 | |
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259 | //! Observation Model h(x,u) |
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260 | shared_ptr<diffbifn> phxu; |
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261 | public: |
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262 | //! copy constructor duplicated - calls different set_parameters |
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263 | BM* _copy_() const { |
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264 | EKFCh* E = new EKFCh; |
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265 | E->set_parameters ( pfxu, phxu, Q, R ); |
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266 | E->set_statistics ( _mu, _P ); |
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267 | return E; |
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268 | } |
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269 | //! Set nonlinear functions for mean values and covariance matrices. |
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270 | void set_parameters ( const shared_ptr<diffbifn> &pfxu, const shared_ptr<diffbifn> &phxu, const chmat Q0, const chmat R0 ); |
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271 | |
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272 | //! Here dt = [yt;ut] of appropriate dimensions |
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273 | void bayes ( const vec &dt ); |
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274 | |
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275 | void from_setting ( const Setting &set ); |
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276 | |
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277 | // TODO dodelat void to_setting( Setting &set ) const; |
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278 | |
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279 | }; |
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280 | |
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281 | UIREGISTER ( EKFCh ); |
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282 | SHAREDPTR ( EKFCh ); |
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283 | |
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284 | |
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285 | /*! |
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286 | \brief Kalman Filter with conditional diagonal matrices R and Q. |
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287 | */ |
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288 | |
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289 | class KFcondQR : public Kalman<ldmat> { |
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290 | //protected: |
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291 | public: |
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292 | void condition ( const vec &QR ) { |
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293 | it_assert_debug ( QR.length() == ( dimx + dimy ), "KFcondRQ: conditioning by incompatible vector" ); |
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294 | |
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295 | Q.setD ( QR ( 0, dimx - 1 ) ); |
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296 | R.setD ( QR ( dimx, -1 ) ); |
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297 | }; |
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298 | }; |
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299 | |
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300 | /*! |
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301 | \brief Kalman Filter with conditional diagonal matrices R and Q. |
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302 | */ |
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303 | |
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304 | class KFcondR : public Kalman<ldmat> { |
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305 | //protected: |
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306 | public: |
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307 | //!Default constructor |
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308 | KFcondR ( ) : Kalman<ldmat> ( ) {}; |
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309 | |
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310 | void condition ( const vec &R0 ) { |
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311 | it_assert_debug ( R0.length() == ( dimy ), "KFcondR: conditioning by incompatible vector" ); |
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312 | |
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313 | R.setD ( R0 ); |
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314 | }; |
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315 | |
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316 | }; |
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317 | |
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318 | //////// INstance |
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319 | |
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320 | template<class sq_T> |
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321 | Kalman<sq_T>::Kalman ( const Kalman<sq_T> &K0 ) : BM ( ), rvy ( K0.rvy ), rvu ( K0.rvu ), |
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322 | dimx ( K0.dimx ), dimy ( K0.dimy ), dimu ( K0.dimu ), |
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323 | A ( K0.A ), B ( K0.B ), C ( K0.C ), D ( K0.D ), |
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324 | Q ( K0.Q ), R ( K0.R ), |
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325 | est ( K0.est ), fy ( K0.fy ), _yp ( fy._mu() ), _Ry ( fy._R() ), _mu ( est._mu() ), _P ( est._R() ) { |
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326 | |
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327 | // copy values in pointers |
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328 | // _mu = K0._mu; |
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329 | // _P = K0._P; |
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330 | // _yp = K0._yp; |
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331 | // _Ry = K0._Ry; |
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332 | |
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333 | } |
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334 | |
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335 | template<class sq_T> |
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336 | Kalman<sq_T>::Kalman ( ) : BM (), est ( ), fy (), _yp ( fy._mu() ), _Ry ( fy._R() ), _mu ( est._mu() ), _P ( est._R() ) { |
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337 | }; |
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338 | |
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339 | template<class sq_T> |
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340 | void Kalman<sq_T>::set_parameters ( const mat &A0, const mat &B0, const mat &C0, const mat &D0, const sq_T &Q0, const sq_T &R0 ) { |
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341 | dimx = A0.rows(); |
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342 | dimy = C0.rows(); |
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343 | dimu = B0.cols(); |
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344 | |
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345 | it_assert_debug ( A0.cols() == dimx, "Kalman: A is not square" ); |
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346 | it_assert_debug ( B0.rows() == dimx, "Kalman: B is not compatible" ); |
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347 | it_assert_debug ( C0.cols() == dimx, "Kalman: C is not square" ); |
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348 | it_assert_debug ( ( D0.rows() == dimy ) || ( D0.cols() == dimu ), "Kalman: D is not compatible" ); |
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349 | it_assert_debug ( ( R0.cols() == dimy ) || ( R0.rows() == dimy ), "Kalman: R is not compatible" ); |
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350 | it_assert_debug ( ( Q0.cols() == dimx ) || ( Q0.rows() == dimx ), "Kalman: Q is not compatible" ); |
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351 | |
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352 | A = A0; |
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353 | B = B0; |
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354 | C = C0; |
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355 | D = D0; |
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356 | R = R0; |
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357 | Q = Q0; |
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358 | } |
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359 | |
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360 | template<class sq_T> |
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361 | void Kalman<sq_T>::bayes ( const vec &dt ) { |
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362 | it_assert_debug ( dt.length() == ( dimy + dimu ), "KalmanFull::bayes wrong size of dt" ); |
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363 | |
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364 | sq_T iRy ( dimy ); |
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365 | vec u = dt.get ( dimy, dimy + dimu - 1 ); |
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366 | vec y = dt.get ( 0, dimy - 1 ); |
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367 | //Time update |
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368 | _mu = A * _mu + B * u; |
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369 | //P = A*P*A.transpose() + Q; in sq_T |
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370 | _P.mult_sym ( A ); |
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371 | _P += Q; |
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372 | |
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373 | //Data update |
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374 | //_Ry = C*P*C.transpose() + R; in sq_T |
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375 | _P.mult_sym ( C, _Ry ); |
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376 | _Ry += R; |
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377 | |
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378 | mat Pfull = _P.to_mat(); |
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379 | |
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380 | _Ry.inv ( iRy ); // result is in _iRy; |
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381 | _K = Pfull * C.transpose() * ( iRy.to_mat() ); |
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382 | |
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383 | sq_T pom ( ( int ) Pfull.rows() ); |
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384 | iRy.mult_sym_t ( C*Pfull, pom ); |
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385 | ( _P ) -= pom; // P = P -PC'iRy*CP; |
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386 | ( _yp ) = C * _mu + D * u; //y prediction |
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387 | ( _mu ) += _K * ( y - _yp ); |
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388 | |
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389 | |
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390 | if ( evalll == true ) { //likelihood of observation y |
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391 | ll = fy.evallog ( y ); |
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392 | } |
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393 | |
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394 | //cout << "y: " << y-(*_yp) <<" R: " << _Ry->to_mat() << " iR: " << _iRy->to_mat() << " ll: " << ll <<endl; |
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395 | |
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396 | }; |
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397 | |
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398 | /*! \brief (Switching) Multiple Model |
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399 | The model runs several models in parallel and evaluates thier weights (fittness). |
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400 | |
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401 | The statistics of the resulting density are merged using (geometric?) combination. |
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402 | |
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403 | The next step is performed with the new statistics for all models. |
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404 | */ |
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405 | class MultiModel: public BM { |
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406 | protected: |
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407 | //! List of models between which we switch |
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408 | Array<EKFCh*> Models; |
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409 | //! vector of model weights |
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410 | vec w; |
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411 | //! cache of model lls |
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412 | vec _lls; |
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413 | //! type of switching policy [1=maximum,2=...] |
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414 | int policy; |
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415 | //! internal statistics |
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416 | enorm<chmat> est; |
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417 | public: |
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418 | void set_parameters ( Array<EKFCh*> A, int pol0 = 1 ) { |
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419 | Models = A;//TODO: test if evalll is set |
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420 | w.set_length ( A.length() ); |
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421 | _lls.set_length ( A.length() ); |
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422 | policy = pol0; |
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423 | |
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424 | est.set_rv ( RV ( "MM", A ( 0 )->posterior().dimension(), 0 ) ); |
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425 | est.set_parameters ( A ( 0 )->_e()->mean(), A ( 0 )->_e()->_R() ); |
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426 | } |
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427 | void bayes ( const vec &dt ) { |
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428 | int n = Models.length(); |
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429 | int i; |
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430 | for ( i = 0; i < n; i++ ) { |
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431 | Models ( i )->bayes ( dt ); |
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432 | _lls ( i ) = Models ( i )->_ll(); |
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433 | } |
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434 | double mlls = max ( _lls ); |
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435 | w = exp ( _lls - mlls ); |
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436 | w /= sum ( w ); //normalization |
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437 | //set statistics |
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438 | switch ( policy ) { |
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439 | case 1: { |
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440 | int mi = max_index ( w ); |
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441 | const enorm<chmat>* st = ( Models ( mi )->_e() ); |
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442 | est.set_parameters ( st->mean(), st->_R() ); |
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443 | } |
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444 | break; |
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445 | default: |
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446 | it_error ( "unknown policy" ); |
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447 | } |
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448 | // copy result to all models |
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449 | for ( i = 0; i < n; i++ ) { |
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450 | Models ( i )->set_statistics ( est.mean(), est._R() ); |
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451 | } |
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452 | } |
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453 | //all posterior densities are equal => return the first one |
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454 | const enorm<chmat>* _e() const { |
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455 | return &est; |
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456 | } |
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457 | //all posterior densities are equal => return the first one |
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458 | const enorm<chmat>& posterior() const { |
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459 | return est; |
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460 | } |
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461 | |
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462 | void from_setting ( const Setting &set ); |
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463 | |
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464 | // TODO dodelat void to_setting( Setting &set ) const; |
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465 | |
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466 | }; |
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467 | |
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468 | UIREGISTER ( MultiModel ); |
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469 | SHAREDPTR ( MultiModel ); |
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470 | |
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471 | |
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472 | |
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473 | //TODO why not const pointer?? |
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474 | |
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475 | template<class sq_T> |
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476 | EKF<sq_T>::EKF ( RV rvx0, RV rvy0, RV rvu0 ) : Kalman<sq_T> ( rvx0, rvy0, rvu0 ) {} |
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477 | |
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478 | template<class sq_T> |
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479 | void EKF<sq_T>::set_parameters ( diffbifn* pfxu0, diffbifn* phxu0, const sq_T Q0, const sq_T R0 ) { |
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480 | pfxu = pfxu0; |
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481 | phxu = phxu0; |
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482 | |
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483 | //initialize matrices A C, later, these will be only updated! |
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484 | pfxu->dfdx_cond ( _mu, zeros ( dimu ), A, true ); |
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485 | // pfxu->dfdu_cond ( *_mu,zeros ( dimu ),B,true ); |
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486 | B.clear(); |
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487 | phxu->dfdx_cond ( _mu, zeros ( dimu ), C, true ); |
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488 | // phxu->dfdu_cond ( *_mu,zeros ( dimu ),D,true ); |
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489 | D.clear(); |
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490 | |
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491 | R = R0; |
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492 | Q = Q0; |
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493 | } |
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494 | |
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495 | template<class sq_T> |
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496 | void EKF<sq_T>::bayes ( const vec &dt ) { |
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497 | it_assert_debug ( dt.length() == ( dimy + dimu ), "KalmanFull::bayes wrong size of dt" ); |
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498 | |
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499 | sq_T iRy ( dimy, dimy ); |
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500 | vec u = dt.get ( dimy, dimy + dimu - 1 ); |
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501 | vec y = dt.get ( 0, dimy - 1 ); |
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502 | //Time update |
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503 | _mu = pfxu->eval ( _mu, u ); |
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504 | pfxu->dfdx_cond ( _mu, u, A, false ); //update A by a derivative of fx |
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505 | |
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506 | //P = A*P*A.transpose() + Q; in sq_T |
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507 | _P.mult_sym ( A ); |
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508 | _P += Q; |
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509 | |
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510 | //Data update |
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511 | phxu->dfdx_cond ( _mu, u, C, false ); //update C by a derivative hx |
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512 | //_Ry = C*P*C.transpose() + R; in sq_T |
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513 | _P.mult_sym ( C, _Ry ); |
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514 | ( _Ry ) += R; |
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515 | |
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516 | mat Pfull = _P.to_mat(); |
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517 | |
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518 | _Ry.inv ( iRy ); // result is in _iRy; |
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519 | _K = Pfull * C.transpose() * ( iRy.to_mat() ); |
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520 | |
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521 | sq_T pom ( ( int ) Pfull.rows() ); |
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522 | iRy.mult_sym_t ( C*Pfull, pom ); |
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523 | ( _P ) -= pom; // P = P -PC'iRy*CP; |
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524 | _yp = phxu->eval ( _mu, u ); //y prediction |
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525 | ( _mu ) += _K * ( y - _yp ); |
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526 | |
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527 | if ( evalll == true ) { |
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528 | ll += fy.evallog ( y ); |
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529 | } |
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530 | }; |
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531 | |
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532 | |
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533 | } |
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534 | #endif // KF_H |
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535 | |
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