Cross-validation estimate of E [J(h)]
0 12
nJ
XX
RbCV (J) = @Yi j(xi)bj;( i)A
i=1 j=1
20 Stochastic Processes
Stochastic Process
X |
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: t |
T |
T = f0; 1; : : : g = Z |
discrete |
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f |
t |
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2 g |
([0; |
1 |
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continuous |
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Notations Xt, X(t)
State space X
Index set T
20.1Markov Chains
Markov chain
P [Xn = x j X0; : : : ; Xn 1] = P [Xn = x j Xn 1] 8n 2 T; x 2 X
Transition probabilities
pij P [Xn+1 = j j Xn = i]
pij(n) P [Xm+n = j j Xm = i] n-step
Transition matrix P (n-step: Pn)
(i; j) element is pij
pij > 0
P
i pij = 1
Chapman-Kolmogorov
X
pij(m + n) = pij(m)pkj(n)
k
Pm+n = PmPn
Pn = P P = Pn
Marginal probability
n = ( n(1); : : : ; n(N)) where i(i) = P [Xn = i]
0 , initial distribution
n = 0Pn
20.2Poisson Processes
Poisson process
fXt : t 2 [0; 1)g = number of events up to and including time t
X0 = 0
Independent increments:
8t0 < < tn : Xt1 Xt0 ?? ?? Xtn Xtn 1
Intensity function (t)
{P [Xt+h Xt = 1] = (t)h + o(h)
{P [Xt+h Xt = 2] = o(h)
Xs+t Xs Po (m(s + t) m(s)) where m(t) = R0t (s) ds
Waiting times
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1 |
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Wt Gamma t; |
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St = Wt+1 Wt |
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St |
Exp |
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1 |
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St |
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Wt 1 |
Wt |
t |
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Homogeneous Poisson process |
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(t) =) Xt Po ( t) |
> 0 |
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Wt := time at which Xt occurs
Interarrival times
22