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  • library/doc/html/formula.repository

    r590 r608  
    11\form#0:$f(x)$ 
    22\form#1:$x$ 
    3 \form#2:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] 
    4 \form#3:$y_t$ 
    5 \form#4:$ c_t $ 
    6 \form#5:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] 
    7 \form#6:$x=$ 
    8 \form#7:$ x $ 
    9 \form#8:$ f_x()$ 
    10 \form#9:$ [x_1 , x_2 , \ldots \ $ 
    11 \form#10:$ f_x(rv)$ 
    12 \form#11:$x \sim epdf(rv|cond)$ 
    13 \form#12:$ t $ 
    14 \form#13:$ t+1 $ 
    15 \form#14:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ 
    16 \form#15:$t$ 
    17 \form#16:$[y_{t} y_{t-1} ...]$ 
    18 \form#17:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ 
    19 \form#18:$ f(x_t|x_{t-1}) $ 
    20 \form#19:$ f(d_t|x_t) $ 
    21 \form#20:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] 
    22 \form#21:$[\theta r]$ 
    23 \form#22:$\psi=\psi(y_{1:t},u_{1:t})$ 
    24 \form#23:$u_t$ 
    25 \form#24:$e_t$ 
    26 \form#25:\[ e_t \sim \mathcal{N}(0,1). \] 
    27 \form#26:$ y_t $ 
    28 \form#27:$\theta,r$ 
    29 \form#28:$ dt = [y_t psi_t] $ 
    30 \form#29:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] 
    31 \form#30:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] 
    32 \form#31:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] 
    33 \form#32:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] 
    34 \form#33:$\psi$ 
    35 \form#34:$w=[w_1,\ldots,w_n]$ 
    36 \form#35:$\theta_i$ 
    37 \form#36:$\Theta$ 
    38 \form#37:$\Theta = [\theta_1,\ldots,\theta_n,w]$ 
    39 \form#38:$A=Ch' Ch$ 
    40 \form#39:$Ch$ 
    41 \form#40:$f(x) = a$ 
    42 \form#41:$f(x) = Ax+B$ 
    43 \form#42:$f(x,u)$ 
    44 \form#43:$f(x,u) = Ax+Bu$ 
    45 \form#44:$f(x0,u0)$ 
    46 \form#45:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$ 
    47 \form#46:$u$ 
    48 \form#47:$A=\frac{d}{du}f(x,u)|_{x0,u0}$ 
    49 \form#48:\[M = L'DL\] 
    50 \form#49:$L$ 
    51 \form#50:$D$ 
    52 \form#51:$V = V + w v v'$ 
    53 \form#52:$C$ 
    54 \form#53:$V = C*V*C'$ 
    55 \form#54:$V = C'*V*C$ 
    56 \form#55:$V$ 
    57 \form#56:$x= v'*V*v$ 
    58 \form#57:$x= v'*inv(V)*v$ 
    59 \form#58:$U$ 
    60 \form#59:$A'D0 A$ 
    61 \form#60:$L'DL$ 
    62 \form#61:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ 
    63 \form#62:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] 
    64 \form#63:$ f(rvc) = \int f(rv,rvc) d\ rv $ 
    65 \form#64:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] 
    66 \form#65:$f_i(x)$ 
    67 \form#66:$p$ 
    68 \form#67:$p\times$ 
    69 \form#68:$n$ 
    70 \form#69:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] 
    71 \form#70:$\gamma=\sum_i \beta_i$ 
    72 \form#71:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] 
    73 \form#72:$\beta$ 
    74 \form#73:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] 
    75 \form#74:$mu=A*rvc+mu_0$ 
    76 \form#75:$\mu$ 
    77 \form#76:$k$ 
    78 \form#77:$\alpha=k$ 
    79 \form#78:$\beta=k/\mu$ 
    80 \form#79:$\mu/\sqrt(k)$ 
    81 \form#80:$ \mu $ 
    82 \form#81:$ k $ 
    83 \form#82:$ \alpha=\mu/k^2+2 $ 
    84 \form#83:$ \beta=\mu(\alpha-1)$ 
    85 \form#84:$ \mu/\sqrt(k)$ 
    86 \form#85:$l$ 
    87 \form#86:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] 
    88 \form#87:$ \log(x)\sim \mathcal{N}(\mu,\sigma^2) $ 
    89 \form#88:\[ x \sim \frac{1}{x\sigma\sqrt{2\pi}}\exp{-\frac{1}{2\sigma^2}(\log(x)-\mu)} \] 
    90 \form#89:$\mathcal{I}$ 
    91 \form#90:$\theta$ 
    92 \form#91:$\alpha$ 
    93 \form#92:$ \Psi $ 
    94 \form#93:$ \nu $ 
    95 \form#94:$ \nu-p-1 $ 
    96 \form#95:$w$ 
    97 \form#96:$x^{(i)}, i=1..n$ 
    98 \form#97:\[ f(x_i|y_i), i=1..n \] 
    99 \form#98:$ \cup [x_i,y_i] $ 
    100 \form#99:\[ f(z_i|y_i,x_i) f(x_i|y_i) f(y_i) i=1..n \] 
    101 \form#100:$ z_i $ 
    102 \form#101:$ y_i={}, z_i={}, \forall i $ 
    103 \form#102:$ f(z_i|x_i,y_i) $ 
    104 \form#103:$ f(D) $ 
    105 \form#104:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \] 
    106 \form#105:$ f(a|b,c) $ 
    107 \form#106:$ f(b) $ 
    108 \form#107:$ f(c) $ 
    109 \form#108:\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} 
    110 \form#109:$ x_t $ 
    111 \form#110:$ A, B, C, D$ 
    112 \form#111:$v_t, w_t$ 
    113 \form#112:$Q, R$ 
    114 \form#113:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray} 
    115 \form#114:$ g(), h() $ 
    116 \form#115:\[ y_t = \theta' \psi_t + \rho e_t \] 
    117 \form#116:$[\theta,\rho]$ 
    118 \form#117:$\psi_t$ 
    119 \form#118:$\mathcal{N}(0,1)$ 
    120 \form#119:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] 
    121 \form#120:\[ \nu_t = \sum_{i=0}^{n} 1 \] 
    122 \form#121:$ \theta_t , r_t $ 
    123 \form#122:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] 
    124 \form#123:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \] 
    125 \form#124:$ \phi $ 
    126 \form#125:$ \phi \in [0,1]$ 
    127 \form#126:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] 
    128 \form#127:$ \phi=0.9 $ 
    129 \form#128:$ V_0 , \nu_0 $ 
    130 \form#129:$ V_t , \nu_t $ 
    131 \form#130:$ \phi<1 $ 
    132 \form#131:$ f(d_{t+1} |d_{t+h-1}, \ldots d_{t}) $ 
    133 \form#132:$ f( x | y) $ 
    134 \form#133:$ y $ 
    135 \form#134:$ \mu=A*\mbox{rvc}+\mu_0 $ 
    136 \form#135:$ \Lambda $ 
    137 \form#136:$ R $ 
    138 \form#137:$ R_e $ 
    139 \form#138:\[ L(y,u) = (y-y_{req})'Q_y (y-y_{req}) + (u-u_{req})' Q_u (u-u_{req}) \] 
    140 \form#139:\[ x_{t+1} = Ax_t + B u_t + R^{1/2} e_t, y_t=Cx_t+Du_t + R^{1/2}w_t, \] 
    141 \form#140:\[ y_t = a y_{t-1} + b u_{t-1}\] 
    142 \form#141:$ x_t = [y_{t-1}, u_{t-1}] $ 
     3\form#2:$ f( x | y) $ 
     4\form#3:$ x $ 
     5\form#4:$ y $ 
     6\form#5:$ u_t $ 
     7\form#6:$ y_t $ 
     8\form#7:$ d_t=[y_t,u_t, \ldots ]$ 
     9\form#8:\[ f(\theta_t | d_1,\ldots,d_t) = \frac{f(y_t|\theta_t,\cdot) f(\theta_t|d_1,\ldots,d_{t-1})}{f(y_t|d_1,\ldots,d_{t-1})} \] 
     10\form#9:$y_t$ 
     11\form#10:$ c_t $ 
     12\form#11:\[ f(\theta_t | c_t, d_1,\ldots,d_t) \propto f(y_t,\theta_t|c_t,\cdot, d_1,\ldots,d_{t-1}) \] 
     13\form#12:$x=$ 
     14\form#13:$ f_x()$ 
     15\form#14:$ [x_1 , x_2 , \ldots \ $ 
     16\form#15:$ f_x(rv)$ 
     17\form#16:$x \sim epdf(rv|cond)$ 
     18\form#17:$[Up_{t-1},Up_{t-2}, \ldots]$ 
     19\form#18:$ t $ 
     20\form#19:$ t+1 $ 
     21\form#20:$ f(d_{t+1} |d_{t}, \ldots d_{0}) $ 
     22\form#21:$ f(d_{t+1} |d_{t+h-1}, \ldots d_{t}) $ 
     23\form#22:$t$ 
     24\form#23:$[y_{t} y_{t-1} ...]$ 
     25\form#24:$[y_t, u_t, y_{t-1 }, u_{t-1}, \ldots]$ 
     26\form#25:$ f(x_t|x_{t-1}) $ 
     27\form#26:$ f(d_t|x_t) $ 
     28\form#27:\[ L(y,u) = (y-y_{req})'Q_y (y-y_{req}) + (u-u_{req})' Q_u (u-u_{req}) \] 
     29\form#28:\[ y_t = \theta_1 \psi_1 + \theta_2 + \psi_2 +\ldots + \theta_n \psi_n + r e_t \] 
     30\form#29:$[\theta r]$ 
     31\form#30:$\psi=\psi(y_{1:t},u_{1:t})$ 
     32\form#31:$u_t$ 
     33\form#32:$e_t$ 
     34\form#33:\[ e_t \sim \mathcal{N}(0,1). \] 
     35\form#34:$\theta,r$ 
     36\form#35:$ dt = [y_t psi_t] $ 
     37\form#36:\[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \] 
     38\form#37:\[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \] 
     39\form#38:\[ x_{t+1} = Ax_t + B u_t + R^{1/2} e_t, y_t=Cx_t+Du_t + R^{1/2}w_t, \] 
     40\form#39:\[ \left[\begin{array}{cc} R^{0.5}\\ P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\ & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc} R_{y}^{0.5} & KA'\\ & P_{t+1|t}^{0.5}\\ \\\end{array}\right]\] 
     41\form#40:\[ f(y_t|\psi_t, \Theta) = \sum_{i=1}^{n} w_i f(y_t|\psi_t, \theta_i) \] 
     42\form#41:$\psi$ 
     43\form#42:$w=[w_1,\ldots,w_n]$ 
     44\form#43:$\theta_i$ 
     45\form#44:$\Theta$ 
     46\form#45:$\Theta = [\theta_1,\ldots,\theta_n,w]$ 
     47\form#46:$A=Ch' Ch$ 
     48\form#47:$Ch$ 
     49\form#48:$f(x) = a$ 
     50\form#49:$f(x) = Ax+B$ 
     51\form#50:$f(x,u)$ 
     52\form#51:$f(x,u) = Ax+Bu$ 
     53\form#52:$f(x0,u0)$ 
     54\form#53:$A=\frac{d}{dx}f(x,u)|_{x0,u0}$ 
     55\form#54:$u$ 
     56\form#55:$A=\frac{d}{du}f(x,u)|_{x0,u0}$ 
     57\form#56:\[M = L'DL\] 
     58\form#57:$L$ 
     59\form#58:$D$ 
     60\form#59:$V = V + w v v'$ 
     61\form#60:$C$ 
     62\form#61:$V = C*V*C'$ 
     63\form#62:$V = C'*V*C$ 
     64\form#63:$V$ 
     65\form#64:$x= v'*V*v$ 
     66\form#65:$x= v'*inv(V)*v$ 
     67\form#66:$U$ 
     68\form#67:$A'D0 A$ 
     69\form#68:$L'DL$ 
     70\form#69:$A'*diag(D)*A = self.L'*diag(self.D)*self.L$ 
     71\form#70:\[ f(rv|rvc) = \frac{f(rv,rvc)}{f(rvc)} \] 
     72\form#71:$ f(rvc) = \int f(rv,rvc) d\ rv $ 
     73\form#72:\[ f(x) = \sum_{i=1}^{n} w_{i} f_i(x), \quad \sum_{i=1}^n w_i = 1. \] 
     74\form#73:$f_i(x)$ 
     75\form#74:$p$ 
     76\form#75:$p\times$ 
     77\form#76:$n$ 
     78\form#77:\[ f(x|\beta) = \frac{\Gamma[\gamma]}{\prod_{i=1}^{n}\Gamma(\beta_i)} \prod_{i=1}^{n}x_i^{\beta_i-1} \] 
     79\form#78:$\gamma=\sum_i \beta_i$ 
     80\form#79:\[ f(x|\alpha,\beta) = \prod f(x_i|\alpha_i,\beta_i) \] 
     81\form#80:$\beta$ 
     82\form#81:\[ x\sim iG(a,b) => 1/x\sim G(a,1/b) \] 
     83\form#82:$ \mu=A*\mbox{rvc}+\mu_0 $ 
     84\form#83:$\mu$ 
     85\form#84:$k$ 
     86\form#85:$\alpha=k$ 
     87\form#86:$\beta=k/\mu$ 
     88\form#87:$\mu/\sqrt(k)$ 
     89\form#88:$ \mu $ 
     90\form#89:$ k $ 
     91\form#90:$ \alpha=\mu/k^2+2 $ 
     92\form#91:$ \beta=\mu(\alpha-1)$ 
     93\form#92:$ \mu/\sqrt(k)$ 
     94\form#93:$l$ 
     95\form#94:\[ \mu = \mu_{t-1} ^{l} p^{1-l}\] 
     96\form#95:$ \log(x)\sim \mathcal{N}(\mu,\sigma^2) $ 
     97\form#96:\[ x \sim \frac{1}{x\sigma\sqrt{2\pi}}\exp{-\frac{1}{2\sigma^2}(\log(x)-\mu)} \] 
     98\form#97:$\mathcal{I}$ 
     99\form#98:$\theta$ 
     100\form#99:$\alpha$ 
     101\form#100:$ \Lambda $ 
     102\form#101:$ R $ 
     103\form#102:$ R_e $ 
     104\form#103:$ \Psi $ 
     105\form#104:$ \nu $ 
     106\form#105:$ \nu-p-1 $ 
     107\form#106:$w$ 
     108\form#107:$x^{(i)}, i=1..n$ 
     109\form#108:\[ f(x_i|y_i), i=1..n \] 
     110\form#109:$ \cup [x_i,y_i] $ 
     111\form#110:\[ f(z_i|y_i,x_i) f(x_i|y_i) f(y_i) i=1..n \] 
     112\form#111:$ z_i $ 
     113\form#112:$ y_i={}, z_i={}, \forall i $ 
     114\form#113:$ f(z_i|x_i,y_i) $ 
     115\form#114:$ f(D) $ 
     116\form#115:\[ f(a,b,c) = f(a|b,c) f(b) f(c) \] 
     117\form#116:$ f(a|b,c) $ 
     118\form#117:$ f(b) $ 
     119\form#118:$ f(c) $ 
     120\form#119:\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} 
     121\form#120:$ x_t $ 
     122\form#121:$ A, B, C, D$ 
     123\form#122:$v_t, w_t$ 
     124\form#123:$Q, R$ 
     125\form#124:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray} 
     126\form#125:$ g(), h() $ 
     127\form#126:\[ y_t = \theta' \psi_t + \rho e_t \] 
     128\form#127:$[\theta,\rho]$ 
     129\form#128:$\psi_t$ 
     130\form#129:$\mathcal{N}(0,1)$ 
     131\form#130:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] 
     132\form#131:\[ \nu_t = \sum_{i=0}^{n} 1 \] 
     133\form#132:$ \theta_t , r_t $ 
     134\form#133:\[ V_t = \phi V_{t-1} + \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] 
     135\form#134:\[ \nu_t = \phi \nu_{t-1} + 1 + (1-\phi) \nu_0 \] 
     136\form#135:$ \phi $ 
     137\form#136:$ \phi \in [0,1]$ 
     138\form#137:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] 
     139\form#138:$ \phi=0.9 $ 
     140\form#139:$ V_0 , \nu_0 $ 
     141\form#140:$ V_t , \nu_t $ 
     142\form#141:$ \phi<1 $