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    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:\[M = L'DL\] 
    22 \form#21:$L$ 
    23 \form#22:$D$ 
    24 \form#23:$V = V + w v v'$ 
    25 \form#24:$C$ 
    26 \form#25:$V = C*V*C'$ 
    27 \form#26:$V = C'*V*C$ 
    28 \form#27:$V$ 
    29 \form#28:$x$ 
    30 \form#29:$x= v'*V*v$ 
    31 \form#30:$x= v'*inv(V)*v$ 
    32 \form#31:$U$ 
    33 \form#32:$A'D0 A$ 
    34 \form#33:$L'DL$ 
    35 \form#34:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ 
    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:$[\theta,\rho]$ 
    103 \form#102:$\psi_t$ 
    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] $ 
    118 \form#117:$\omega$ 
    119 \form#118:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] 
    120 \form#119:$ c_t $ 
    121 \form#120:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] 
    122 \form#121:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] 
    123 \form#122:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \] 
    124 \form#123:$ \Psi $ 
    125 \form#124:$ \nu $ 
    126 \form#125:$ \nu-p-1 $ 
    127 \form#126:$[y_{t} y_{t-1} ...]$ 
     1\form#0:$x$ 
     2\form#1:$\omega$ 
     3\form#2:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] 
     4\form#3:$[\theta r]$ 
     5\form#4:$\psi=\psi(y_{1:t},u_{1:t})$ 
     6\form#5:$u_t$ 
     7\form#6:$e_t$ 
     8\form#7:\[ e_t \sim \mathcal{N}(0,1). \] 
     9\form#8:$ y_t $ 
     10\form#9:$\theta,r$ 
     11\form#10:$ dt = [y_t psi_t] $ 
     12\form#11:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] 
     13\form#12:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] 
     14\form#13:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] 
     15\form#14:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] 
     16\form#15:$\psi$ 
     17\form#16:$w=[w_1,\ldots,w_n]$ 
     18\form#17:$\theta_i$ 
     19\form#18:$\Theta$ 
     20\form#19:$\Theta = [\theta_1,\ldots,\theta_n,w]$ 
     21\form#20:$A=Ch' Ch$ 
     22\form#21:$Ch$ 
     23\form#22:\[M = L'DL\] 
     24\form#23:$L$ 
     25\form#24:$D$ 
     26\form#25:$V = V + w v v'$ 
     27\form#26:$C$ 
     28\form#27:$V = C*V*C'$ 
     29\form#28:$V = C'*V*C$ 
     30\form#29:$V$ 
     31\form#30:$x= v'*V*v$ 
     32\form#31:$x= v'*inv(V)*v$ 
     33\form#32:$U$ 
     34\form#33:$A'D0 A$ 
     35\form#34:$L'DL$ 
     36\form#35:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ 
     37\form#36:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] 
     38\form#37:$ f(rvc) = \int f(rv,rvc) d\ rv $ 
     39\form#38:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] 
     40\form#39:$f_i(x)$ 
     41\form#40:$f(x)$ 
     42\form#41:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] 
     43\form#42:$y_t$ 
     44\form#43:$ c_t $ 
     45\form#44:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] 
     46\form#45:$x=$ 
     47\form#46:$ x $ 
     48\form#47:$ f_x()$ 
     49\form#48:$ [x_1 , x_2 , \ldots \ $ 
     50\form#49:$ f_x(rv)$ 
     51\form#50:$x \sim epdf(rv|cond)$ 
     52\form#51:$ t $ 
     53\form#52:$ t+1 $ 
     54\form#53:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ 
     55\form#54:$t$ 
     56\form#55:$[y_{t} y_{t-1} ...]$ 
     57\form#56:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ 
     58\form#57:$ f(x_t|x_{t-1}) $ 
     59\form#58:$ f(d_t|x_t) $ 
     60\form#59:$p$ 
     61\form#60:$p\times$ 
     62\form#61:$n$ 
     63\form#62:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] 
     64\form#63:$\gamma=\sum_i \beta_i$ 
     65\form#64:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] 
     66\form#65:$\beta$ 
     67\form#66:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] 
     68\form#67:$mu=A*rvc+mu_0$ 
     69\form#68:$\mu$ 
     70\form#69:$k$ 
     71\form#70:$\alpha=k$ 
     72\form#71:$\beta=k/\mu$ 
     73\form#72:$\mu/\sqrt(k)$ 
     74\form#73:$ \mu $ 
     75\form#74:$ k $ 
     76\form#75:$ \alpha=\mu/k^2+2 $ 
     77\form#76:$ \beta=\mu(\alpha-1)$ 
     78\form#77:$ \mu/\sqrt(k)$ 
     79\form#78:$l$ 
     80\form#79:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] 
     81\form#80:$\mathcal{I}$ 
     82\form#81:$\alpha$ 
     83\form#82:$ \Psi $ 
     84\form#83:$ \nu $ 
     85\form#84:$ \nu-p-1 $ 
     86\form#85:$w$ 
     87\form#86:$x^{(i)}, i=1..n$ 
     88\form#87:$f(x) = a$ 
     89\form#88:$f(x) = Ax+B$ 
     90\form#89:$f(x,u)$ 
     91\form#90:$f(x,u) = Ax+Bu$ 
     92\form#91:$f(x0,u0)$ 
     93\form#92:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$ 
     94\form#93:$u$ 
     95\form#94:$A=\frac{d}{du}f(x,u)|_{x0,u0}$ 
     96\form#95:$ f(D) $ 
     97\form#96:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \] 
     98\form#97:$ f(a|b,c) $ 
     99\form#98:$ f(b) $ 
     100\form#99:$ f(c) $ 
     101\form#100:\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} 
     102\form#101:$ x_t $ 
     103\form#102:$ A, B, C, D$ 
     104\form#103:$v_t, w_t$ 
     105\form#104:$Q, R$ 
     106\form#105:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray} 
     107\form#106:$ g(), h() $ 
     108\form#107:\[ y_t = \theta' \psi_t + \rho e_t \] 
     109\form#108:$[\theta,\rho]$ 
     110\form#109:$\psi_t$ 
     111\form#110:$\mathcal{N}(0,1)$ 
     112\form#111:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] 
     113\form#112:\[ \nu_t = \sum_{i=0}^{n} 1 \] 
     114\form#113:$ \theta_t , r_t $ 
     115\form#114:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] 
     116\form#115:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \] 
     117\form#116:$ \phi $ 
     118\form#117:$ \phi \in [0,1]$ 
     119\form#118:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] 
     120\form#119:$ \phi=0.9 $ 
     121\form#120:$ V_0 , \nu_0 $ 
     122\form#121:$ V_t , \nu_t $ 
     123\form#122:$ \phi<1 $ 
     124\form#123:$ [d_1, d_2, \ldots d_t] $