root/library/doc/html/formula.repository @ 408

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1\form#0:$f(x)$
2\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 $
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