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 | /*! |
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26 | * \brief Basic elements of linear state-space model |
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27 | |
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28 | Parameter evolution model:\f[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \f] |
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29 | Observation model: \f[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \f] |
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30 | Where $e_t$ and $w_t$ are independent vectors Normal(0,1)-distributed disturbances. |
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31 | */ |
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32 | template<class sq_T> |
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33 | class StateSpace |
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34 | { |
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35 | protected: |
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36 | //! Matrix A |
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37 | mat A; |
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38 | //! Matrix B |
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39 | mat B; |
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40 | //! Matrix C |
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41 | mat C; |
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42 | //! Matrix D |
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43 | mat D; |
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44 | //! Matrix Q in square-root form |
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45 | sq_T Q; |
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46 | //! Matrix R in square-root form |
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47 | sq_T R; |
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48 | public: |
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49 | StateSpace() : A(), B(), C(), D(), Q(), R() {} |
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50 | //!copy constructor |
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51 | StateSpace(const StateSpace<sq_T> &S0) : A(S0.A), B(S0.B), C(S0.C), D(S0.D), Q(S0.Q), R(S0.R) {} |
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52 | //! set all matrix parameters |
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53 | 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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54 | //! validation |
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55 | void validate(); |
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56 | //! not virtual in this case |
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57 | void from_setting (const Setting &set) { |
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58 | UI::get (A, set, "A", UI::compulsory); |
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59 | UI::get (B, set, "B", UI::compulsory); |
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60 | UI::get (C, set, "C", UI::compulsory); |
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61 | UI::get (D, set, "D", UI::compulsory); |
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62 | mat Qtm, Rtm; |
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63 | if(!UI::get(Qtm, set, "Q", UI::optional)){ |
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64 | vec dq; |
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65 | UI::get(dq, set, "dQ", UI::compulsory); |
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66 | Qtm=diag(dq); |
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67 | } |
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68 | if(!UI::get(Rtm, set, "R", UI::optional)){ |
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69 | vec dr; |
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70 | UI::get(dr, set, "dQ", UI::compulsory); |
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71 | Rtm=diag(dr); |
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72 | } |
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73 | R=Rtm; // automatic conversion |
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74 | Q=Qtm; |
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75 | |
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76 | validate(); |
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77 | } |
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78 | //! access function |
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79 | const mat& _A() const {return A;} |
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80 | //! access function |
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81 | const mat& _B()const {return B;} |
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82 | //! access function |
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83 | const mat& _C()const {return C;} |
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84 | //! access function |
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85 | const mat& _D()const {return D;} |
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86 | //! access function |
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87 | const sq_T& _Q()const {return Q;} |
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88 | //! access function |
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89 | const sq_T& _R()const {return R;} |
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90 | }; |
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91 | |
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92 | //! Common abstract base for Kalman filters |
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93 | template<class sq_T> |
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94 | class Kalman: public BM, public StateSpace<sq_T> |
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95 | { |
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96 | protected: |
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97 | //! id of output |
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98 | RV yrv; |
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99 | //! Kalman gain |
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100 | mat _K; |
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101 | //!posterior |
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102 | enorm<sq_T> est; |
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103 | //!marginal on data f(y|y) |
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104 | enorm<sq_T> fy; |
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105 | public: |
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106 | Kalman<sq_T>() : BM(), StateSpace<sq_T>(), yrv(), _K(), est(){} |
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107 | //! Copy constructor |
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108 | Kalman<sq_T>(const Kalman<sq_T> &K0) : BM(K0), StateSpace<sq_T>(K0), yrv(K0.yrv), _K(K0._K), est(K0.est), fy(K0.fy){} |
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109 | //!set statistics of the posterior |
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110 | void set_statistics (const vec &mu0, const mat &P0) {est.set_parameters (mu0, P0); }; |
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111 | //!set statistics of the posterior |
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112 | void set_statistics (const vec &mu0, const sq_T &P0) {est.set_parameters (mu0, P0); }; |
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113 | //! return correctly typed posterior (covariant return) |
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114 | const enorm<sq_T>& posterior() const {return est;} |
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115 | //! load basic elements of Kalman from structure |
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116 | void from_setting (const Setting &set) { |
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117 | StateSpace<sq_T>::from_setting(set); |
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118 | |
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119 | mat P0; vec mu0; |
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120 | UI::get(mu0, set, "mu0", UI::optional); |
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121 | UI::get(P0, set, "P0", UI::optional); |
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122 | set_statistics(mu0,P0); |
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123 | // Initial values |
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124 | UI::get (yrv, set, "yrv", UI::optional); |
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125 | UI::get (rvc, set, "urv", UI::optional); |
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126 | set_yrv(concat(yrv,rvc)); |
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127 | |
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128 | validate(); |
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129 | } |
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130 | //! validate object |
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131 | void validate() { |
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132 | StateSpace<sq_T>::validate(); |
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133 | dimy = this->C.rows(); |
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134 | dimc = this->B.cols(); |
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135 | set_dim(this->A.rows()); |
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136 | |
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137 | bdm_assert(est.dimension(), "Statistics and model parameters mismatch"); |
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138 | } |
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139 | }; |
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140 | /*! |
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141 | * \brief Basic Kalman filter with full matrices |
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142 | */ |
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143 | |
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144 | class KalmanFull : public Kalman<fsqmat> |
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145 | { |
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146 | public: |
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147 | //! For EKFfull; |
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148 | KalmanFull() :Kalman<fsqmat>(){}; |
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149 | //! Here dt = [yt;ut] of appropriate dimensions |
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150 | void bayes (const vec &yt, const vec &cond=empty_vec); |
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151 | BM* _copy_() const { |
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152 | KalmanFull* K = new KalmanFull; |
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153 | K->set_parameters (A, B, C, D, Q, R); |
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154 | K->set_statistics (est._mu(), est._R()); |
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155 | return K; |
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156 | } |
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157 | }; |
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158 | UIREGISTER(KalmanFull); |
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159 | |
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160 | |
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161 | /*! \brief Kalman filter in square root form |
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162 | |
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163 | Trivial example: |
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164 | \include kalman_simple.cpp |
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165 | |
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166 | Complete constructor: |
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167 | */ |
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168 | class KalmanCh : public Kalman<chmat> |
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169 | { |
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170 | protected: |
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171 | //! @{ \name Internal storage - needs initialize() |
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172 | //! pre array (triangular matrix) |
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173 | mat preA; |
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174 | //! post array (triangular matrix) |
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175 | mat postA; |
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176 | //!@} |
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177 | public: |
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178 | //! copy constructor |
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179 | BM* _copy_() const { |
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180 | KalmanCh* K = new KalmanCh; |
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181 | K->set_parameters (A, B, C, D, Q, R); |
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182 | K->set_statistics (est._mu(), est._R()); |
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183 | K->validate(); |
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184 | return K; |
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185 | } |
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186 | //! set parameters for adapt from Kalman |
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187 | 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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188 | //! initialize internal parametetrs |
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189 | void initialize(); |
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190 | |
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191 | /*!\brief Here dt = [yt;ut] of appropriate dimensions |
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192 | |
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193 | The following equality hold::\f[ |
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194 | \left[\begin{array}{cc} |
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195 | R^{0.5}\\ |
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196 | P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ |
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197 | & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} |
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198 | R_{y}^{0.5} & KA'\\ |
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199 | & P_{t+1|t}^{0.5}\\ |
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200 | \\\end{array}\right]\f] |
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201 | |
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202 | Thus this object evaluates only predictors! Not filtering densities. |
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203 | */ |
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204 | void bayes (const vec &yt, const vec &cond=empty_vec); |
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205 | |
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206 | void from_setting(const Setting &set){ |
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207 | Kalman<chmat>::from_setting(set); |
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208 | validate(); |
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209 | } |
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210 | void validate() { |
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211 | Kalman<chmat>::validate(); |
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212 | initialize(); |
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213 | } |
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214 | }; |
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215 | UIREGISTER(KalmanCh); |
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216 | |
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217 | /*! |
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218 | \brief Extended Kalman Filter in full matrices |
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219 | |
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220 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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221 | */ |
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222 | class EKFfull : public KalmanFull |
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223 | { |
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224 | protected: |
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225 | //! Internal Model f(x,u) |
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226 | shared_ptr<diffbifn> pfxu; |
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227 | |
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228 | //! Observation Model h(x,u) |
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229 | shared_ptr<diffbifn> phxu; |
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230 | |
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231 | public: |
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232 | //! Default constructor |
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233 | EKFfull (); |
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234 | |
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235 | //! Set nonlinear functions for mean values and covariance matrices. |
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236 | void set_parameters (const shared_ptr<diffbifn> &pfxu, const shared_ptr<diffbifn> &phxu, const mat Q0, const mat R0); |
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237 | |
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238 | //! Here dt = [yt;ut] of appropriate dimensions |
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239 | void bayes (const vec &yt, const vec &cond=empty_vec); |
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240 | //! set estimates |
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241 | void set_statistics (const vec &mu0, const mat &P0) { |
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242 | est.set_parameters (mu0, P0); |
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243 | }; |
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244 | //! access function |
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245 | const mat _R() { |
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246 | return est._R().to_mat(); |
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247 | } |
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248 | void from_setting (const Setting &set) { |
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249 | BM::from_setting(set); |
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250 | shared_ptr<diffbifn> IM = UI::build<diffbifn> ( set, "IM", UI::compulsory ); |
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251 | shared_ptr<diffbifn> OM = UI::build<diffbifn> ( set, "OM", UI::compulsory ); |
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252 | |
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253 | //statistics |
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254 | int dim = IM->dimension(); |
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255 | vec mu0; |
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256 | if ( !UI::get ( mu0, set, "mu0" ) ) |
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257 | mu0 = zeros ( dim ); |
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258 | |
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259 | mat P0; |
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260 | vec dP0; |
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261 | if ( UI::get ( dP0, set, "dP0" ) ) |
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262 | P0 = diag ( dP0 ); |
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263 | else if ( !UI::get ( P0, set, "P0" ) ) |
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264 | P0 = eye ( dim ); |
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265 | |
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266 | set_statistics ( mu0, P0 ); |
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267 | |
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268 | //parameters |
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269 | vec dQ, dR; |
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270 | UI::get ( dQ, set, "dQ", UI::compulsory ); |
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271 | UI::get ( dR, set, "dR", UI::compulsory ); |
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272 | set_parameters ( IM, OM, diag ( dQ ), diag ( dR ) ); |
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273 | |
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274 | string options; |
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275 | if ( UI::get ( options, set, "options" ) ) |
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276 | set_options ( options ); |
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277 | // pfxu = UI::build<diffbifn>(set, "IM", UI::compulsory); |
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278 | // phxu = UI::build<diffbifn>(set, "OM", UI::compulsory); |
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279 | // |
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280 | // mat R0; |
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281 | // UI::get(R0, set, "R",UI::compulsory); |
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282 | // mat Q0; |
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283 | // UI::get(Q0, set, "Q",UI::compulsory); |
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284 | // |
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285 | // |
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286 | // mat P0; vec mu0; |
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287 | // UI::get(mu0, set, "mu0", UI::optional); |
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288 | // UI::get(P0, set, "P0", UI::optional); |
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289 | // set_statistics(mu0,P0); |
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290 | // // Initial values |
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291 | // UI::get (yrv, set, "yrv", UI::optional); |
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292 | // UI::get (urv, set, "urv", UI::optional); |
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293 | // set_drv(concat(yrv,urv)); |
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294 | // |
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295 | // // setup StateSpace |
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296 | // pfxu->dfdu_cond(mu0, zeros(pfxu->_dimu()), A,true); |
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297 | // phxu->dfdu_cond(mu0, zeros(pfxu->_dimu()), C,true); |
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298 | // |
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299 | validate(); |
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300 | } |
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301 | void validate() { |
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302 | // check stats and IM and OM |
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303 | } |
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304 | }; |
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305 | UIREGISTER(EKFfull); |
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306 | |
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307 | |
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308 | /*! |
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309 | \brief Extended Kalman Filter in Square root |
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310 | |
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311 | An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean. |
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312 | */ |
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313 | |
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314 | class EKFCh : public KalmanCh |
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315 | { |
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316 | protected: |
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317 | //! Internal Model f(x,u) |
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318 | shared_ptr<diffbifn> pfxu; |
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319 | |
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320 | //! Observation Model h(x,u) |
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321 | shared_ptr<diffbifn> phxu; |
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322 | public: |
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323 | //! copy constructor duplicated - calls different set_parameters |
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324 | BM* _copy_() const { |
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325 | EKFCh* E = new EKFCh; |
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326 | E->set_parameters (pfxu, phxu, Q, R); |
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327 | E->set_statistics (est._mu(), est._R()); |
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328 | return E; |
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329 | } |
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330 | //! Set nonlinear functions for mean values and covariance matrices. |
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331 | void set_parameters (const shared_ptr<diffbifn> &pfxu, const shared_ptr<diffbifn> &phxu, const chmat Q0, const chmat R0); |
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332 | |
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333 | //! Here dt = [yt;ut] of appropriate dimensions |
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334 | void bayes (const vec &yt, const vec &cond=empty_vec); |
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335 | |
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336 | void from_setting (const Setting &set); |
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337 | |
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338 | void validate(){}; |
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339 | // TODO dodelat void to_setting( Setting &set ) const; |
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340 | |
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341 | }; |
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342 | |
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343 | UIREGISTER (EKFCh); |
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344 | SHAREDPTR (EKFCh); |
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345 | |
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346 | |
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347 | //////// INstance |
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348 | |
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349 | /*! \brief (Switching) Multiple Model |
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350 | The model runs several models in parallel and evaluates thier weights (fittness). |
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351 | |
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352 | The statistics of the resulting density are merged using (geometric?) combination. |
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353 | |
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354 | The next step is performed with the new statistics for all models. |
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355 | */ |
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356 | class MultiModel: public BM |
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357 | { |
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358 | protected: |
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359 | //! List of models between which we switch |
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360 | Array<EKFCh*> Models; |
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361 | //! vector of model weights |
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362 | vec w; |
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363 | //! cache of model lls |
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364 | vec _lls; |
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365 | //! type of switching policy [1=maximum,2=...] |
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366 | int policy; |
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367 | //! internal statistics |
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368 | enorm<chmat> est; |
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369 | public: |
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370 | //! set internal parameters |
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371 | void set_parameters (Array<EKFCh*> A, int pol0 = 1) { |
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372 | Models = A;//TODO: test if evalll is set |
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373 | w.set_length (A.length()); |
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374 | _lls.set_length (A.length()); |
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375 | policy = pol0; |
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376 | |
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377 | est.set_rv (RV ("MM", A (0)->posterior().dimension(), 0)); |
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378 | est.set_parameters (A (0)->posterior().mean(), A (0)->posterior()._R()); |
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379 | } |
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380 | void bayes (const vec &yt, const vec &cond=empty_vec) { |
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381 | int n = Models.length(); |
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382 | int i; |
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383 | for (i = 0; i < n; i++) { |
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384 | Models (i)->bayes (yt); |
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385 | _lls (i) = Models (i)->_ll(); |
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386 | } |
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387 | double mlls = max (_lls); |
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388 | w = exp (_lls - mlls); |
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389 | w /= sum (w); //normalization |
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390 | //set statistics |
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391 | switch (policy) { |
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392 | case 1: { |
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393 | int mi = max_index (w); |
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394 | const enorm<chmat> &st = Models (mi)->posterior() ; |
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395 | est.set_parameters (st.mean(), st._R()); |
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396 | } |
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397 | break; |
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398 | default: |
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399 | bdm_error ("unknown policy"); |
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400 | } |
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401 | // copy result to all models |
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402 | for (i = 0; i < n; i++) { |
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403 | Models (i)->set_statistics (est.mean(), est._R()); |
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404 | } |
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405 | } |
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406 | //! return correctly typed posterior (covariant return) |
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407 | const enorm<chmat>& posterior() const { |
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408 | return est; |
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409 | } |
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410 | |
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411 | void from_setting (const Setting &set); |
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412 | |
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413 | // TODO dodelat void to_setting( Setting &set ) const; |
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414 | |
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415 | }; |
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416 | |
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417 | UIREGISTER (MultiModel); |
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418 | SHAREDPTR (MultiModel); |
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419 | |
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420 | //! conversion of outer ARX model (mlnorm) to state space model |
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421 | /*! |
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422 | The model is constructed as: |
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423 | \f[ x_{t+1} = Ax_t + B u_t + R^{1/2} e_t, y_t=Cx_t+Du_t + R^{1/2}w_t, \f] |
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424 | For example, for: |
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425 | Using Frobenius form, see []. |
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426 | |
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427 | For easier use in the future, indeces theta_in_A and theta_in_C are set. TODO - explain |
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428 | */ |
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429 | //template<class sq_T> |
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430 | class StateCanonical: public StateSpace<fsqmat>{ |
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431 | protected: |
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432 | //! remember connection from theta ->A |
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433 | datalink_part th2A; |
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434 | //! remember connection from theta ->C |
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435 | datalink_part th2C; |
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436 | //! remember connection from theta ->D |
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437 | datalink_part th2D; |
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438 | //!cached first row of A |
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439 | vec A1row; |
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440 | //!cached first row of C |
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441 | vec C1row; |
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442 | //!cached first row of D |
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443 | vec D1row; |
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444 | |
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445 | public: |
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446 | //! set up this object to match given mlnorm |
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447 | void connect_mlnorm(const mlnorm<fsqmat> &ml){ |
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448 | //get ids of yrv |
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449 | const RV &yrv = ml._rv(); |
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450 | //need to determine u_t - it is all in _rvc that is not in ml._rv() |
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451 | RV rgr0 = ml._rvc().remove_time(); |
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452 | RV urv = rgr0.subt(yrv); |
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453 | |
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454 | //We can do only 1d now... :( |
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455 | bdm_assert(yrv._dsize()==1, "Only for SISO so far..." ); |
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456 | |
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457 | // create names for |
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458 | RV xrv; //empty |
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459 | RV Crv; //empty |
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460 | int td=ml._rvc().mint(); |
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461 | // assuming strictly proper function!!! |
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462 | for (int t=-1;t>=td;t--){ |
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463 | xrv.add(yrv.copy_t(t)); |
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464 | Crv.add(urv.copy_t(t)); |
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465 | } |
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466 | |
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467 | // get mapp |
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468 | th2A.set_connection(xrv, ml._rvc()); |
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469 | th2C.set_connection(Crv, ml._rvc()); |
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470 | th2D.set_connection(urv, ml._rvc()); |
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471 | |
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472 | //set matrix sizes |
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473 | this->A=zeros(xrv._dsize(),xrv._dsize()); |
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474 | for (int j=1; j<xrv._dsize(); j++){A(j,j-1)=1.0;} // off diagonal |
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475 | this->B=zeros(xrv._dsize(),1); |
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476 | this->B(0) = 1.0; |
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477 | this->C=zeros(1,xrv._dsize()); |
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478 | this->D=zeros(1,urv._dsize()); |
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479 | this->Q = zeros(xrv._dsize(),xrv._dsize()); |
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480 | // R is set by update |
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481 | |
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482 | //set cache |
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483 | this->A1row = zeros(xrv._dsize()); |
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484 | this->C1row = zeros(xrv._dsize()); |
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485 | this->D1row = zeros(urv._dsize()); |
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486 | |
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487 | update_from(ml); |
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488 | validate(); |
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489 | }; |
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490 | //! fast function to update parameters from ml - not checked for compatibility!! |
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491 | void update_from(const mlnorm<fsqmat> &ml){ |
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492 | |
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493 | vec theta = ml._A().get_row(0); // this |
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494 | |
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495 | th2A.filldown(theta,A1row); |
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496 | th2C.filldown(theta,C1row); |
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497 | th2D.filldown(theta,D1row); |
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498 | |
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499 | R = ml._R(); |
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500 | |
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501 | A.set_row(0,A1row); |
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502 | C.set_row(0,C1row+D1row*A1row); |
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503 | D.set_row(0,D1row); |
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504 | |
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505 | } |
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506 | }; |
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507 | |
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508 | /////////// INSTANTIATION |
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509 | |
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510 | template<class sq_T> |
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511 | void StateSpace<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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512 | { |
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513 | |
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514 | A = A0; |
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515 | B = B0; |
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516 | C = C0; |
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517 | D = D0; |
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518 | R = R0; |
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519 | Q = Q0; |
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520 | validate(); |
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521 | } |
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522 | |
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523 | template<class sq_T> |
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524 | void StateSpace<sq_T>::validate(){ |
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525 | bdm_assert (A.cols() == A.rows(), "KalmanFull: A is not square"); |
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526 | bdm_assert (B.rows() == A.rows(), "KalmanFull: B is not compatible"); |
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527 | bdm_assert (C.cols() == A.rows(), "KalmanFull: C is not compatible"); |
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528 | bdm_assert ( (D.rows() == C.rows()) && (D.cols() == B.cols()), "KalmanFull: D is not compatible"); |
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529 | bdm_assert ( (Q.cols() == A.rows()) && (Q.rows() == A.rows()), "KalmanFull: Q is not compatible"); |
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530 | bdm_assert ( (R.cols() == C.rows()) && (R.rows() == C.rows()), "KalmanFull: R is not compatible"); |
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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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