\form#0:$A=\frac{df}{dx}|_{x0,u0}$
\form#1:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$
\form#2:$A=\frac{d}{du}f(x,u)|_{x0,u0}$
\form#3:$x \sim epdf(rv)$
\form#4:\[ f(x|a,b) = \prod f(x_i|a_i,b_i) \]
\form#5:\[M = L'DL\]
\form#6:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \]
\form#7:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \]
\form#8:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \]
\form#9:$x^{(i)}, i=1..n$
\form#10:$x \sim epdf(rv|cond)$
\form#11:$\alpha=k$
\form#12:$\beta=k/\mu$
\form#13:$\mu/\sqrt(k)$
\form#14:$\mu$
\form#15:$\alpha$
\form#16:$\beta$
\form#17:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\]
\form#18:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\]
\form#19:$A=Ch' Ch$
\form#20:$Ch$
\form#21:$L$
\form#22:$D$
\form#23:$V = V + w v v'$
\form#24:$C$
\form#25:$V = C*V*C'$
\form#26:$V = C'*V*C$
\form#27:$V$
\form#28:$x$
\form#29:$x= v'*V*v$
\form#30:$x= v'*inv(V)*v$
\form#31:$U$
\form#32:$A'D0 A$
\form#33:$L'DL$
\form#34:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$
\form#35:$f(x)$
\form#36:$f(rv|rvc,data)$
\form#37:$x=$
\form#38:$t$
\form#39:$t+1$
\form#40:$mu=A*rvc$
\form#41:$k$
\form#42:$p$
\form#43:$l$
\form#44:$w$
\form#45:$f(x) = a$
\form#46:$f(x) = Ax+B$
\form#47:$f(x,u)$
\form#48:$f(x,u) = Ax+Bu$
\form#49:$f(x0,u0)$
\form#50:$u$
\form#51:$[\theta r]$
\form#52:$\psi=\psi(y_{1:t},u_{1:t})$
\form#53:$u_t$
\form#54:$e_t$
\form#55:$\theta_t,r_t$
\form#56:$\in <0,1>$
\form#57:$\theta,r$
\form#58:$dt = [y_t psi_t] $
\form#59:$epdf(rv)$
\form#60:$\mathcal{I}$
\form#61:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \]
\form#62:\[ e_t \sim \mathcal{N}(0,1). \]
\form#63:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \]
\form#64:$f_i(x)$
\form#65:$\omega$
\form#66:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \]
\form#67:$\psi$
\form#68:$w=[w_1,\ldots,w_n]$
\form#69:$\theta_i$
\form#70:$\Theta$
\form#71:$\Theta = [\theta_1,\ldots,\theta_n,w]$
\form#72:$p\times$
\form#73:$n$
\form#74:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^(\beta_i-1) \]
\form#75:$\gamma=\sum_i beta_i$
\form#76:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \]
\form#77:$\gamma=\sum_i \beta_i$
\form#78:$mu=A*rvc+mu_0$
\form#79:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \]
\form#80:$ f(rvc) = \int f(rv,rvc) d\ rv $
\form#81:\[ f(\theta|D) =\frac{f(D|\theta)f(\theta)}{f(D)}\]
\form#82:$ \theta $
\form#83:$ D $
\form#84:$ f(D|\theta) $
\form#85:$ f(\theta) $
\form#86:$ f(D) $
\form#87:$\alpha=\mu/k+2$
\form#88:$\beta=\mu(\alpha-1)$
\form#89:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]
\form#90:$ f(a|b,c) $
\form#91:$ f(b) $
\form#92:$ f(c) $
\form#93:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \]
\form#94:$y_t$
\form#95:$[\theta,\rho]$
\form#96:$\phi_t$
\form#97:$\mathcal{N}(0,1)$
\form#98:$\phi$
\form#99:\[ y_t = \theta' \phi_t + \rho e_t \]
\form#100:$[u_t, y_{t-1 }, u_{t-1}, \ldots]$
\form#101:\[ y_t = \theta' \psi_t + \rho e_t \]
\form#102:$\psi_t$
\form#103:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$
\form#104:$ f(x_t|x_{t-1}) $
\form#105:$ f(d_t|x_t) $
\form#106:$ x $
\form#107:$ f_x()$
\form#108:$ [x_1 , x_2 , \ldots \ $
\form#109:$ f_x(rv)$
\form#110:$ t $
\form#111:$ t+1 $
\form#112:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $
\form#113:$ \mu $
\form#114:$ k $
\form#115:$ \alpha=\mu/k^2+2 $
\form#116:$ \beta=\mu(\alpha-1)$
\form#117:$ \mu/\sqrt(k)$
\form#118:$ y_t $
\form#119:$ dt = [y_t psi_t] $
\form#120:$ [d_1, d_2, \ldots d_t] $
\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}
\form#122:$ x_t $
\form#123:$ A, B, C, D$
\form#124:$v_t, w_t$
\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 $
\form#126:$Q, R$
\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}
\form#128:$ g(), h() $
\form#129:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \]
\form#130:\[ \nu_t = \sum_{i=0}^{n} 1 \]
\form#131:$ \theta_t , r_t $
\form#132:\[ V_t = V_{t-1} + \phi \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \]
\form#133:\[ \nu_t = \nu_{t-1} + \phi + (1-\phi) \nu_0 \]
\form#134:$ \phi $
\form#135:$ \phi \in [0,1]$
\form#136:\[ \mathrm{win_length} = \frac{1}{1-\phi}\]
\form#137:$ \phi=0.9 $
\form#138:$ V_0 , \nu_0 $
\form#139:$ V_t , \nu_t $
\form#140:$ \phi<1 $