Multi-resolution Inference of Stochastic Models from Partially - - PowerPoint PPT Presentation
Multi-resolution Inference of Stochastic Models from Partially Observed Data Samuel Kou Department of Statistics Harvard University Joint with Ben Olding Stochastic Differential Equations n Models based on stochastic differential equations
n Models based on stochastic differential equations (SDE)
are widely used in science and engineering.
n General form
Yt : the process, θ : the underlying parameters
n Models based on stochastic differential equations (SDE)
are widely used in science and engineering.
n General form
Yt : the process, θ : the underlying parameters
n Example 1. In chemistry and biology.
A reversible enzymatic reaction A↔B typically modeled as
U(y) y: reaction coordinate
t t t
θ : the energy barrier heights etc.
EA EB
t t t t
n Models based on stochastic differential equations (SDE)
are widely used in science and engineering.
n General form
Yt : the process, θ : the underlying parameters
n Example 2. In finance and economics.
Feller process (a.k.a. CIR process) has been used to model interest rates
t t t t
dB Y dt Y dY σ µ γ + − = ) (
n Models based on stochastic differential equations (SDE)
are widely used in science and engineering.
n General form
Yt : the process, θ : the underlying parameters
n Example 2. In finance and economics.
Feller process (a.k.a. CIR process) has been used to model interest rates
n Given a stochastic model, infer the parameter values
from data
n Major complication: the continuous-time model is only
Example: (i) Biology or chemistry experiments can track movement of molecules only at discrete camera frames (ii) Finance or economics, interest rates, price index, etc.
n Data (Y1,t1), (Y2,t2),…(Yn,tn) from n Likelihood
f(y|x,t,θ): transition density
n In most cases, f does not permit analytical form; solving
a PDE numerically is not feasible either
n Idea: approximate an SDE
by a difference equation
n Obtain approx likelihood from the difference eqn n Works well only if ∆t is small
Generated from Ornstein-Uhlenbeck process
Generated from Ornstein-Uhlenbeck process
Euler Approx.
Generated from Ornstein-Uhlenbeck process
Euler Approx. Exact
n
If ∆t is not “sufficiently small”
¤
Choose a ∆t small enough so that the Euler approximation is appropriate.
¤
Treat the unobserved values of Yt as missing data.
n
n
Data augmentation:
n
Use Monte Carlo to perform the augmentation
yobs yobs yobs ymis ymis ymis ymis ymis ymis
∝
mis mis
dy y y P y P ) ( ) | , ( ) | ( θ π θ θ ) ( ) | , ( ) | , ( θ π θ θ
mis
mis
y y P y y P ∝
Exact k=31
Idea appeared simultaneously in stats & econ literature in late 1990s: Elerian, Chib, Shephard (2001); Eraker (2001); Jones (1998)
n The smaller the ∆t , the more accurate the approximation n However, the smaller the ∆t, the more missing data we
need to augment: dimensionality goes way up!
n The missing data are dependent as well!
very slow convergence
at small ∆t
n The smaller the ∆t , the more accurate the approximation n However, the smaller the ∆t, the more missing data we
need to augment: dimensionality goes way up!
n The missing data are dependent as well! n The dilemma:
¤ Low resolution (big ∆t) runs quickly, but result
inaccurate
¤ High resolution (small ∆t) good approximation, but
painfully slow
n Utilize the strength of different resolutions, while avoid their
weakness
n Simultaneously work on multiple resolutions
“rough” approximations quickly locate the important regions “fine” approximations get jump start, and then accurately explore the space
n Consider multiple resolutions (i.e., approximation levels)
n Start from the lowest level with a MC (such as Gibbs sampler);
record the results
n Move on to the 2nd level
¤ In each MC update, with prob p do Gibbs ¤ With prob 1- p, draw y from previous lower level chain
augment y to (y, y') by “upsampling” accept (y, y') with probability
n Move on to the 3rd level ……
} ) ( ) ( ) , ( ) ( ) ( ) , ( , 1 min{
) ( ) 1 ( ) ( ) 1 (
y y T y L y y L y y T y L y y L r
k
k
k k
′ → ′ ′ → ′ =
+ +
Likelihood at level k
y1 y2 ym
y1 y2 ym
with prob p Gibbs with prob 1-p accept with
} ) ( ) ( ) , ( ) ( ) ( ) , ( , 1 min{
) ( ) 1 ( ) ( ) 1 (
y y T y L y y L y y T y L y y L r
k
k
k k
′ → ′ ′ → ′ =
+ +
multi-resolution vanilla Gibbs
n Observation: A by-product of the Multiresolution
sampler is that we obtain multiple approximations to the same distribution
n Question: Can we combine them together for inference,
instead of using only the finest resolution?
n Idea: Look for trend from successive approximations
and leap forward
) ( lim h A A
h→
=
n Richardson (1927) n If
is what we want With resolution h But for resolution h/2
) ( ) (
1 2 1
2 1 k k k k k
h O h a A h a h a h a A h A + + = + + + + =
( 2 ) 2 (
1
k k
h O h a A h A + + =
) ( 1 2 ) ( ) 2 ( 2 ) ( ~
1
k k k
h O A h A h A h A + = − − ≡
is an order of magnitude better!
n We have multiple posterior distributions from the multi-
resolution sampler
n Extrapolate the entire distribution by quantiles k = 3 k = 7
n n Model for interest rate, bond rate, exchange rate n No analytical solution for the transition density n The data
, ) (
t t t t
dB Y dt Y dY
ψ
σ µ γ + − =
) , , , ( θ ψ σ µ γ =
n Posterior distribution
K = inf K = inf
vanilla Gibbs multiresolution
3+7 extrap.
n Posterior distribution n Autocorrelation plots
K = inf 3+7 extrap. K = inf
Faster and more accurate!
n In Bayesian analysis, use MC samples to approximate
posterior quantities of interest (eg., mean, median, etc.)
n Use quantiles from MC sample to construct interval
estimate
) | (
Y E θ θ →
) | ( ) ( ˆ
) ( ) (
Y Q Q θ θ
α α
→
n Compare ratio of Mean Square Error given same
n 3-month Eurodollar deposit rate n Use GCIR model
5 10 15 20 25 1/8/1971 1/8/1973 1/8/1975 1/8/1977 1/8/1979 1/8/1981 1/8/1983 1/8/1985 1/8/1987 1/8/1989 1/8/1991 1/8/1993 1/8/1995 1/8/1997 1/8/1999 1/8/2001 1/8/2003 1/8/2005 1/8/2007
n Posterior mean, median and interval
, ) (
t t t t
dB Y dt Y dY
ψ
σ µ γ + − =
) , , , ( θ ψ σ µ γ =
n McCann et al. (1999): Data of a particle in a bistable trap
n Again compare ratio of Mean Square Error given
n We introduce the multi-resolution framework n Efficient Monte Carlo with the multi-resolution sampler n Accurate inference with the multi-resolution extrapolation n Extendible to higher dimensions n Extendible to state space (HMM) models
n SØrensen (2004) – Survey paper n Elerian et al. (2001) – Original Gibbs paper n Roberts & Stramer (2001) – SDE transformations n Kloeden & Platen (1992) – Book on Numerical Solution of
SDEs
n Ben Olding n Jun Liu n Xiaoli Meng n Wing Wong n NSF, NIH
n Assuming:
¤ The diffusion & volatility functions (•) and σ2(•) have
linear growth
¤ (•) and σ2(•) are twice continuously differentiable with
bounded derivatives
¤ σ2(•) is bounded from below
n Then for any integrable function g(θ):
n Taking g(θ) to be an indicator function, then if a posterior
cdf F has non-zero derivative at all points, its quantiles can be expanded as: