Show
Ignore:
Timestamp:
02/17/09 11:15:56 (16 years ago)
Author:
smidl
Message:

doc

Files:
1 modified

Legend:

Unmodified
Added
Removed

  • doc/html/formula.repository

    r271 r275  
    1 \form#0:$A=\frac{df}{dx}|_{x0,u0}$ 
    2 \form#1:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$ 
    3 \form#2:$A=\frac{d}{du}f(x,u)|_{x0,u0}$ 
    4 \form#3:$x \sim epdf(rv)$ 
    5 \form#4:\[ f(x|a,b) = \prod f(x_i|a_i,b_i) \] 
    6 \form#5:\[M = L'DL\] 
    7 \form#6:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] 
    8 \form#7:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] 
    9 \form#8:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] 
    10 \form#9:$x^{(i)}, i=1..n$ 
    11 \form#10:$x \sim epdf(rv|cond)$ 
    12 \form#11:$\alpha=k$ 
    13 \form#12:$\beta=k/\mu$ 
    14 \form#13:$\mu/\sqrt(k)$ 
    15 \form#14:$\mu$ 
    16 \form#15:$\alpha$ 
    17 \form#16:$\beta$ 
    18 \form#17:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] 
    19 \form#18:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] 
    20 \form#19:$A=Ch' Ch$ 
    21 \form#20:$Ch$ 
     1\form#0:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] 
     2\form#1:$[\theta r]$ 
     3\form#2:$\psi=\psi(y_{1:t},u_{1:t})$ 
     4\form#3:$u_t$ 
     5\form#4:$e_t$ 
     6\form#5:\[ e_t \sim \mathcal{N}(0,1). \] 
     7\form#6:$ y_t $ 
     8\form#7:$\theta,r$ 
     9\form#8:$ dt = [y_t psi_t] $ 
     10\form#9:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] 
     11\form#10:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] 
     12\form#11:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] 
     13\form#12:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] 
     14\form#13:$\psi$ 
     15\form#14:$w=[w_1,\ldots,w_n]$ 
     16\form#15:$\theta_i$ 
     17\form#16:$\Theta$ 
     18\form#17:$\Theta = [\theta_1,\ldots,\theta_n,w]$ 
     19\form#18:$A=Ch' Ch$ 
     20\form#19:$Ch$ 
     21\form#20:\[M = L'DL\] 
    2222\form#21:$L$ 
    2323\form#22:$D$ 
    … …  
    3434\form#33:$L'DL$ 
    3535\form#34:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ 
    36 \form#35:$f(x)$ 
    37 \form#36:$f(rv|rvc,data)$ 
    38 \form#37:$x=$ 
    39 \form#38:$t$ 
    40 \form#39:$t+1$ 
    41 \form#40:$mu=A*rvc$ 
    42 \form#41:$k$ 
    43 \form#42:$p$ 
    44 \form#43:$l$ 
    45 \form#44:$w$ 
    46 \form#45:$f(x) = a$ 
    47 \form#46:$f(x) = Ax+B$ 
    48 \form#47:$f(x,u)$ 
    49 \form#48:$f(x,u) = Ax+Bu$ 
    50 \form#49:$f(x0,u0)$ 
    51 \form#50:$u$ 
    52 \form#51:$[\theta r]$ 
    53 \form#52:$\psi=\psi(y_{1:t},u_{1:t})$ 
    54 \form#53:$u_t$ 
    55 \form#54:$e_t$ 
    56 \form#55:$\theta_t,r_t$ 
    57 \form#56:$\in <0,1>$ 
    58 \form#57:$\theta,r$ 
    59 \form#58:$dt = [y_t psi_t] $ 
    60 \form#59:$epdf(rv)$ 
    61 \form#60:$\mathcal{I}$ 
    62 \form#61:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] 
    63 \form#62:\[ e_t \sim \mathcal{N}(0,1). \] 
    64 \form#63:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] 
    65 \form#64:$f_i(x)$ 
    66 \form#65:$\omega$ 
    67 \form#66:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] 
    68 \form#67:$\psi$ 
    69 \form#68:$w=[w_1,\ldots,w_n]$ 
    70 \form#69:$\theta_i$ 
    71 \form#70:$\Theta$ 
    72 \form#71:$\Theta = [\theta_1,\ldots,\theta_n,w]$ 
    73 \form#72:$p\times$ 
    74 \form#73:$n$ 
    75 \form#74:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^(\beta_i-1) \] 
    76 \form#75:$\gamma=\sum_i beta_i$ 
    77 \form#76:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] 
    78 \form#77:$\gamma=\sum_i \beta_i$ 
    79 \form#78:$mu=A*rvc+mu_0$ 
    80 \form#79:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] 
    81 \form#80:$ f(rvc) = \int f(rv,rvc) d\ rv $ 
    82 \form#81:\[ f(\theta|D) =\frac{f(D|\theta)f(\theta)}{f(D)}\] 
    83 \form#82:$ \theta $ 
    84 \form#83:$ D $ 
    85 \form#84:$ f(D|\theta) $ 
    86 \form#85:$ f(\theta) $ 
    87 \form#86:$ f(D) $ 
    88 \form#87:$\alpha=\mu/k+2$ 
    89 \form#88:$\beta=\mu(\alpha-1)$ 
    90 \form#89:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \] 
    91 \form#90:$ f(a|b,c) $ 
    92 \form#91:$ f(b) $ 
    93 \form#92:$ f(c) $ 
    94 \form#93:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] 
    95 \form#94:$y_t$ 
    96 \form#95:$[\theta,\rho]$ 
    97 \form#96:$\phi_t$ 
    98 \form#97:$\mathcal{N}(0,1)$ 
    99 \form#98:$\phi$ 
    100 \form#99:\[ y_t = \theta' \phi_t + \rho e_t \] 
    101 \form#100:$[u_t, y_{t-1 }, u_{t-1}, \ldots]$ 
    102 \form#101:\[ y_t = \theta' \psi_t + \rho e_t \] 
     36\form#35:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] 
     37\form#36:$ f(rvc) = \int f(rv,rvc) d\ rv $ 
     38\form#37:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] 
     39\form#38:$f_i(x)$ 
     40\form#39:$f(x)$ 
     41\form#40:$f(rv|rvc,data)$ 
     42\form#41:$x=$ 
     43\form#42:$ x $ 
     44\form#43:$ f_x()$ 
     45\form#44:$ [x_1 , x_2 , \ldots \ $ 
     46\form#45:$ f_x(rv)$ 
     47\form#46:$x \sim epdf(rv|cond)$ 
     48\form#47:$ t $ 
     49\form#48:$ t+1 $ 
     50\form#49:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ 
     51\form#50:$t$ 
     52\form#51:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ 
     53\form#52:$ f(x_t|x_{t-1}) $ 
     54\form#53:$ f(d_t|x_t) $ 
     55\form#54:$p$ 
     56\form#55:$p\times$ 
     57\form#56:$n$ 
     58\form#57:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] 
     59\form#58:$\gamma=\sum_i \beta_i$ 
     60\form#59:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] 
     61\form#60:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] 
     62\form#61:$mu=A*rvc+mu_0$ 
     63\form#62:$\mu$ 
     64\form#63:$k$ 
     65\form#64:$\alpha=k$ 
     66\form#65:$\beta=k/\mu$ 
     67\form#66:$\mu/\sqrt(k)$ 
     68\form#67:$ \mu $ 
     69\form#68:$ k $ 
     70\form#69:$ \alpha=\mu/k^2+2 $ 
     71\form#70:$ \beta=\mu(\alpha-1)$ 
     72\form#71:$ \mu/\sqrt(k)$ 
     73\form#72:$l$ 
     74\form#73:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] 
     75\form#74:$\mathcal{I}$ 
     76\form#75:$\alpha$ 
     77\form#76:$\beta$ 
     78\form#77:$w$ 
     79\form#78:$x^{(i)}, i=1..n$ 
     80\form#79:$f(x) = a$ 
     81\form#80:$f(x) = Ax+B$ 
     82\form#81:$f(x,u)$ 
     83\form#82:$f(x,u) = Ax+Bu$ 
     84\form#83:$f(x0,u0)$ 
     85\form#84:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$ 
     86\form#85:$u$ 
     87\form#86:$A=\frac{d}{du}f(x,u)|_{x0,u0}$ 
     88\form#87:$ f(D) $ 
     89\form#88:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \] 
     90\form#89:$ f(a|b,c) $ 
     91\form#90:$ f(b) $ 
     92\form#91:$ f(c) $ 
     93\form#92:\begin{eqnarray} x_t &= &A x_{t-1} + B u_{t} + v_t,\\ y_t &= &C x_{t} + D u_{t} + w_t, \end{eqnarray} 
     94\form#93:$ x_t $ 
     95\form#94:$ A, B, C, D$ 
     96\form#95:$v_t, w_t$ 
     97\form#96:$Q, R$ 
     98\form#97:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray} 
     99\form#98:$ g(), h() $ 
     100\form#99:\[ y_t = \theta' \psi_t + \rho e_t \] 
     101\form#100:$y_t$ 
     102\form#101:$[\theta,\rho]$ 
    103103\form#102:$\psi_t$ 
    104 \form#103:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ 
    105 \form#104:$ f(x_t|x_{t-1}) $ 
    106 \form#105:$ f(d_t|x_t) $ 
    107 \form#106:$ x $ 
    108 \form#107:$ f_x()$ 
    109 \form#108:$ [x_1 , x_2 , \ldots \ $ 
    110 \form#109:$ f_x(rv)$ 
    111 \form#110:$ t $ 
    112 \form#111:$ t+1 $ 
    113 \form#112:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ 
    114 \form#113:$ \mu $ 
    115 \form#114:$ k $ 
    116 \form#115:$ \alpha=\mu/k^2+2 $ 
    117 \form#116:$ \beta=\mu(\alpha-1)$ 
    118 \form#117:$ \mu/\sqrt(k)$ 
    119 \form#118:$ y_t $ 
    120 \form#119:$ dt = [y_t psi_t] $ 
    121 \form#120:$ [d_1, d_2, \ldots d_t] $ 
    122 \form#121:\begin{eqnarray} x_t &= &A x_{t-1} + B u_{t} + v_t,\\ y_t &= &C x_{t} + D u_{t} + w_t, \end{eqnarray} 
    123 \form#122:$ x_t $ 
    124 \form#123:$ A, B, C, D$ 
    125 \form#124:$v_t, w_t$ 
    126 \form#125:$Q, R\$, respectively. Both prior and posterior densities on the state are Gaussian, i.e. of the class enorm. There is a range of classes that implements this functionality, namely: - KalmanFull which implements the estimation algorithm on full matrices, - KalmanCh which implements the estimation algorithm using choleski decompositions and QR algorithm. \section ekf Extended Kalman Filtering Extended Kalman filtering arise by linearization of non-linear state space model: \f{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \f} where $ 
    127 \form#126:$Q, R$ 
    128 \form#127:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray} 
    129 \form#128:$ g(), h() $ 
    130 \form#129:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] 
    131 \form#130:\[ \nu_t = \sum_{i=0}^{n} 1 \] 
    132 \form#131:$ \theta_t , r_t $ 
    133 \form#132:\[ V_t = V_{t-1} + \phi \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] 
    134 \form#133:\[ \nu_t = \nu_{t-1} + \phi + (1-\phi) \nu_0 \] 
    135 \form#134:$ \phi $ 
    136 \form#135:$ \phi \in [0,1]$ 
    137 \form#136:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] 
    138 \form#137:$ \phi=0.9 $ 
    139 \form#138:$ V_0 , \nu_0 $ 
    140 \form#139:$ V_t , \nu_t $ 
    141 \form#140:$ \phi<1 $ 
     104\form#103:$\mathcal{N}(0,1)$ 
     105\form#104:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] 
     106\form#105:\[ \nu_t = \sum_{i=0}^{n} 1 \] 
     107\form#106:$ \theta_t , r_t $ 
     108\form#107:\[ V_t = V_{t-1} + \phi \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] 
     109\form#108:\[ \nu_t = \nu_{t-1} + \phi + (1-\phi) \nu_0 \] 
     110\form#109:$ \phi $ 
     111\form#110:$ \phi \in [0,1]$ 
     112\form#111:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] 
     113\form#112:$ \phi=0.9 $ 
     114\form#113:$ V_0 , \nu_0 $ 
     115\form#114:$ V_t , \nu_t $ 
     116\form#115:$ \phi<1 $ 
     117\form#116:$ [d_1, d_2, \ldots d_t] $