Innovation Method Explained

In statistics, the Innovation Method provides an estimator for the parameters of stochastic differential equations given a time series of (potentially noisy) observations of the state variables. In the framework of continuous-discrete state space models, the innovation estimator is obtained by maximizing the log-likelihood of the corresponding discrete-time innovation process with respect to the parameters. The innovation estimator can be classified as a M-estimator, a quasi-maximum likelihood estimator or a prediction error estimator depending on the inferential considerations that want to be emphasized. The innovation method is a system identification technique for developing mathematical models of dynamical systems from measured data and for the optimal design of experiments.

Background

Stochastic differential equations (SDEs) have become an important mathematical tool for describing the time evolution of several random phenomenon in natural, social and applied sciences. Statistical inference for SDEs is thus of great importance in applications for model building, model selection, model identification and forecasting. To carry out statistical inference for SDEs, measurements of the state variables of these random phenomena are indispensable. Usually, in practice, only a few state variables are measured by physical devices that introduce random measurement errors (observational errors).

Mathematical model for inference

The innovation estimator. for SDEs is defined in the framework of continuous-discrete state space models.[1] These models arise as natural mathematical representation of the temporal evolution of continuous random phenomena and their measurements in a succession of time instants. In the simplest formulation, these continuous-discrete models are expressed in termof a SDE of the form

      dx(t)=f(t,x(t);\theta)dt

m
+\sum
i=1

gi(t,x(t);\theta)dwi(t)       (1)

describing the time evolution of

d

state variables

x

of the phenomenon for all time instant

t\get0

, and an observation equation

     

z
tk

=Cx(tk)+e

tk

      (2)

describing the time series of measurements

z
t0
,...,z
tM-1
of at least one of the variables

x

of the random phenomenon on

M

time instants

t0,...,tM-1

. In the model (1)-(2),

f

and

gi

are differentiable functions,

w=(w1,...,wm)

is an

m

-dimensional standard Wiener process,

\theta\inRp

is a vector of

p

parameters,
\{e
tk

:

e
tk

\sim\Nu(

0,\Pi
tk

)\}k=0,...,M-1

is a sequence of

r

-dimensional i.i.d. Gaussian random vectors independent of

w

,
\Pi
tk
an

r x r

positive definite matrix, and

C

an

r x d

matrix.

Statistical problem to solve

Once the dynamics of a phenomenon is described by a state equation as (1) and the way of measurement the state variables specified by an observation equation as (2), the inference problem to solve is the following:[2] given

M

partial and noisy observations
z
t0
,...,z
tM-1
of the stochastic process

x

on the observation times

t0,...,tM-1

, estimate the unobserved state variable of

x

and the unknown parameters

\theta

in (1) that better fit to the given observations.

Discrete-time innovation process

Let

\{t\}M

be the sequence of

M

observation times

t0,\ldots,tM-1

of the states of (1), and

Z\rho=\{

z
tk

:tk\leq\rho,tk\in\{t\}M\}

the time series of partial and noisy measurements of

x

described by the observation equation (2).

Further, let

xt/\rho=\Epsilon(x(t)|Z\rho)

and

Ut/\rho=E(x(t)x\intercal(t)|Z\rho)-xt/\rho

\intercal
x
t/\rho
be the conditional mean and variance of

x

with

\rho\leqt

, where

E()

denotes the expected value of random vectors.

The random sequence

\{

\nu
tk

\}k=1,\ldots,M-1,

with

     

\nu
tk

=

z
tk

-

Cx
{tk

/{tk-1

}}(\theta), \qquad \qquad (3)

defines the discrete-time innovation process,[3] where

\nu
tk

is proved to be an independent normally distributed random vector with zero mean and variance

     

\Sigma
tk

=

CU
{tk

/tk-1

}(\theta)\ \mathbf^\intercal + \Pi_, \qquad \qquad (4)

for small enough

\Delta=\underset{k}{max}\{tk+1-tk\}

, with

tk,tk+1\in\{t\}M

. In practice,[4] this distribution for the discrete-time innovation is valid when, with a suitable selection of both, the number

M

of observations and the time distance

tk+1-tk

between consecutive observations, the time series of observations
z
t0
,...,z
tM-1
of the SDE contains the main information about the continuous-time process

x

. That is, when the sampling of the continuous-time process

x

has low distortion (aliasing) and when there is a suitable signal-noise ratio.

Innovation estimator

The innovation estimator for the parameters of the SDE (1) is the one that maximizes the likelihood function of the discrete-time innovation process

\{\nu
tk

\}k=1,\ldots,M-1

with respect to the parameters. More precisely, given

M

measurements
Z
tM-1
of the state space model (1)-(2) with

\theta=\theta0

on

\{t\}M,

the innovation estimator for the parameters

\theta0

of (1) is defined by

      \hat{\theta}M=\operatorname\arg\{\underset{\theta}{min}UM(\theta,

Z
tM-1

)\},       (5)

where

      UM(\theta,Z

tM-1

)=

M-1
(M-1)ln(2\pi)+\sum
k=1
ln(\det(\Sigma
tk

))+

-1
\nu
tk
\nu
tk

,

being

\nu
t{k
}the discrete-time innovation (3) and
\Sigma
tk
the innovation variance (4) of the model (1)-(2) at

tk

, for all

k=1,...,M-1.

In the above expression for

UM(\theta,

Z
tM-1

),

the conditional mean
x
tk/tk-1

(\theta)

and variance
U
tk/tk-1

(\theta)

are computed by the continuous-discrete filtering algorithm for the evolution of the moments (Section 6.4 in), for all

k=1,...,M-1.

Differences with the maximum likelihood estimator

The maximum likelihood estimator of the parameters

\theta

in the model (1)-(2) involves the evaluation of the - usually unknown - transition density function

p\theta(tk+1-tk,x(tk),x(tk+1))

between the states

x(tk)

and

x(tk+1)

of the diffusion process

x

for all the observation times

tk

and

tk+1

.[5] Instead of this, the innovation estimator (5) is obtained by maximizing the likelihood of the discrete-time innovation process

\{

\nu
tk

\}k=1,...,M-1,

taking into account that

\nu

t1

,...,\nu

tM-1
are Gaussian and independent random vectors. Remarkably, whereas the transition density function

p\theta(tk+1-tk,x(tk),x(tk+1))

changes when the SDE for

x

does, the transition density function

ak{p}\theta(tk+1-tk,

\nu
tk
,\nu
tk+1

)

for the innovation process remains Gaussian independently of the SDEs for

x

. Only in the case that the diffusion

x

is described by a linear SDE with additive noise, the density function

p\theta(tk+1-tk,x(tk),x(tk+1))

is Gaussian and equal to

ak{p}\theta(tk+1-tk,

\nu
tk

,

\nu
tk+1

),

and so the maximum likelihood and the innovation estimator coincide.[6] Otherwise, the innovation estimator is an approximation to the maximum likelihood estimator and, in this sense, the innovation estimator is a Quasi-Maximum Likelihood estimator. In addition, the innovation method is a particular instance of the Prediction Error method according to the definition given in.[7] Therefore, the asymptotic results obtained in for that general class of estimators are valid for the innovation estimators.[8] [9] Intuitively, by following the typical control engineering viewpoint, it is expected that the innovation process - viewed as a measure of the prediction errors of the fitted model - be approximately a white noise process when the models fit the data,[10] which can be used as a practical tool for designing of models and for optimal experimental design.

Properties

The innovation estimator (5) has a number of important attributes:

-UM,h(\widehat{\theta}M,

Z
tM-1

)

of the innovation estimator (5) can be used to compute the Akaike or Bayesian information criterion.

100(1-\alpha)\%

confidence limits

\widehat{\theta}M\pmtriangleup

for the innovation estimator

\widehat{\theta}M

is estimated with

        triangleup=t1-\alpha,\sqrt{

diag(Var(\widehat{\theta
M))}{M-p}},

where

t1-\alpha,

is the t-student distribution with

100(1-\alpha)\%

significance level, and

M-p-1

degrees of freedom . Here,

Var(\widehat{\theta}M)=(I

-1
(\widehat{\theta}
M))
denotes the variance of the innovation estimator

\widehat{\theta}M

, where

        I(\widehat{\theta}M

M-1
)=\sum
k=1

Ik(\widehat{\theta}M)

is the Fisher Information matrix the innovation estimator

\widehat{\theta}M

of

\theta0

and

      \lbrackIk(\widehat{\theta}M)]m,n=

\partial\mu\intercal
\partial\thetam

\Sigma-1

\partial\mu+
\partial\thetan
1
2

trace(\Sigma-1

\partial\Sigma
\partial\thetam

\Sigma-1

\partial\Sigma
\partial\thetan

)

is the entry

(m,n)

of the matrix

Ik(\widehat{\theta}M)

with

\mu=

Cx
tk/tk-1

(\widehat{\theta}M)

and

\Sigma

=\Sigma
tk

(\widehat{\theta}M)

, for

1\leqm,n\leqp

.
\{\nu
tk
:\nu
tk
=z
tk
-Cx
tk/tk-1

(\widehat{\theta}M)\}k=1,\ldots

measures the goodness of fit of the model to the data.

h

, nonlinear observation equations of the form

     

z
tk

=h(tk,x(tk

))+e
tk

,       (6)

can be transformed to the simpler one (2), and the innovation estimator (5) can be applied.

Approximate Innovation estimators

In practice, close form expressions for computing

x
tk/tk-1

(\theta)

and
U
tk/tk-1

(\theta)

in (5) are only available for a few models (1)-(2). Therefore, approximate filtering algorithms as the following are used in applications.

Given

M

measurements
Z
tM-1
and the initial filter estimates
y
t0/t0
=x
t0/t0
,
V
t0/t0
=U
t0/t0
, the approximate Linear Minimum Variance (LMV) filter for the model (1)-(2) is iteratively defined at each observation time

tk\in\{t\}M

by the prediction estimates[12]

     

y
tk+1/tk

=E(y(tk+1

)|Z
tk

)

and

V
tk+1/tk

=E(y(tk+1)y \intercal(tk+1

)|Z
tk
)-y
tk+1/tk
\intercal
y
tk+1/tk

,    (7)

with initial conditions

y
tk/tk

and
V
tk/tk

, and the filter estimates

     

y
tk+1/tk+1
=y
tk+1/tk
+K
tk+1
(z
tk+1
-Cy
tk+1/tk

)

and

V
tk+1/tk+1
=V
tk+1/tk
-K
tk+1
CV
tk+1/tk

   (8)

with filter gain

     

K
tk+1
=V
tk+1/tk

C\intercal

CV
tk+1/tk

(C\intercal

+\Pi
tk+1

)-1

for all

tk,tk+1\in\{t\}M

, where

y

is an approximation to the solution

x

of (1) on the observation times

\{t\}M

.

Given

M

measurements
Z
tM-1

of the state space model (1)-(2) with

\theta=\theta0

on

\{t\}M

, the approximate innovation estimator for the parameters

\theta0

of (1) is defined by

      \widehat{\vartheta

}_=\arg \, \qquad \qquad (9)

where

      \widetilde{U}M

(\theta,Z
tM-1

)=(M-1)ln

M-1
(2\pi)+\sum\limits
k=1

ln(\det(\widetilde{\Sigma

}_))+\widetilde_^(\widetilde_)^\widetilde_,

being

      \widetilde{\nu

}_=\mathbf_-\mathbf_(\theta) \qquad and

   \widetilde{\Sigma

}_=\mathbf_(\theta)\mathbf^+\mathbf_

approximations to the discrete-time innovation (3) and innovation variance (4), respectively, resulting from the filtering algorithm (7)-(8).

For models with complete observations free of noise (i.e., with

C=I

and
\Pi
tk

=0

in (2)), the approximate innovation estimator (9) reduces to the known Quasi-Maximum Likelihood estimators for SDEs.

Main conventional-type estimators

Conventional-type innovation estimators are those (9) derived from conventional-type continuous-discrete ordiscrete-discrete approximate filtering algorithms. With approximate continuous-discrete filters there are the innovation estimators based on Local Linearization (LL) filters,[13] on the extended Kalman filter,[14] [15] and on the second order filters. Approximate innovation estimators based on discrete-discrete filters result from the discretization of the SDE (1) by means of a numerical scheme.[16] [17] Typically, the effectiveness of these innovation estimators is directly related to the stability of the involved filtering algorithms.

A shared drawback of these conventional-type filters is that, once the observations are given, the error between the approximate and the exact innovation process is fixed and completely settled by the time distance between observations. This might set a large bias of the approximate innovation estimators in some applications, bias that cannot be corrected by increasing the number of observations. However, the conventional-type innovation estimators are useful in many practical situations for which only medium or low accuracy for the parameter estimation is required.

Order-β innovation estimators

Let us consider the finer time discretization

\left(\tau\right)h>0=\{\taun:\taun+1-\taun\leqhforn=0,1,\ldots,N\}

of the time interval

[t0,tM-1]

satisfying the condition

\left(\tau\right)h\supset\{t\}M

. Further, let

yn

be the approximate value of

x(\taun)

obtained from a discretization of the equation (1) for all

\left(\tau\right)h

, and

      y=\{y(t),t\in\lbrackt0,tM-1]:y(\taun)=yn,

for all

\taun\in\left(\tau\right)h\}       (10)

a continuous-time approximation to

x

.

A order-

\beta

LMV filter. is an approximate LMV filter for which

y

is an order-

\beta

weak approximation to

x

satisfying (10) and the weak convergence condition

      \underset{tk\leqt\leqtk+1

}\left\vert E\left(g(\mathbf (t))|Z_\right) -E\left(g(\mathbf(t))|Z_\right) \right\vert\leq L_h^

for all

tk,tk+1\in\{t\}M

and any

2(\beta+1)

times continuously differentiable functions

g:RdR

for which

g

and all its partial derivatives up to order

2(\beta+1)

have polynomial growth, being

Lk

a positive constant. This order-

\beta

LMV filter converges with rate

\beta

to the exact LMV filter as

h

goes to zero, where

h

is the maximum stepsize of the timediscretization

(\tau)h\supset\{t\}M

on which the approximation

y

to

x

is defined.

A order-

\beta

innovation estimator is an approximate innovation estimator (9) for which the approximations to the discrete-time innovation (3) and innovation variance (4), respectively, resulting from an order-

\beta

LMV filter.

Approximations

y

of any kind converging to

x

in a weak sense (as, e.g., those in [18]) can be used to design an order-

\beta

LMV filter and, consequently, an order-

\beta

innovation estimator. These order-

\beta

innovation estimators are intended for the recurrent practical situation in which a diffusion process should be identified from a reduced number of observations distant in time or when high accuracy for the estimated parameters is required.

Properties

An order-

\beta

innovation estimator

\widehat{\theta

}_(h) has a number of important properties:
Z
tM-1
of

M

observations,

\widehat{\theta

}_(h) converges to the exact innovation estimator

\widehat{\theta

}_ as the maximum stepsize

h

of the time discretization

\left(\tau\right)h\supset\{t\}M

goes to zero.

M

observations, the expected value of

\widehat{\theta

}_(h) converges to the expected value of the exact innovation estimator

\widehat{\theta

}_ as

h

goes to zero.

\widehat{\theta

}_(h) is asymptotically normal distributed and its bias decreases when

h

goes to zero.

\beta

LMV filter to the exact LMV filter, for the convergence and asymptotic properties of

\widehat{\theta

}_(h) there are no constraints on the time distance

tk+1-tk

between two consecutive observations
z
tk
and
z
tk+1
, nor on the time discretization

(\tau)h\supset\{t\}M.

\widehat{\theta

}_ by its approximation

\widehat{\theta

}_(h). These approximations converge to the corresponding exact one when the maximum stepsize

h

of the time discretization

\left(\tau\right)h\supset\{t\}M

goes to zero.

\{\widetilde{\nu

}_:\widetilde_=\mathbf_-\mathbf_(\widehat_(h))\}_ measures the goodness of fit of the model to the data, which is also used as a practical tool for designing of models and for optimal experimental design.

h

, nonlinear observation equations of the form (6) can be transformed to the simpler one (2), and the order-

\beta

innovation estimator can be applied.

Figure 1 presents the histograms of the differences

(\widehat{\alpha}M

D
-\widehat{\alpha}
h,M

,\widehat{\sigma}M

D
-\widehat{\sigma}
h,M

)

and

(\widehat{\alpha}M-\widehat{\alpha}h,M,\widehat{\sigma}M-\widehat{\sigma}h,M)

between the exact innovation estimator

(\widehat{\alpha}M,\widehat{\sigma}M)

with the conventional
D
(\widehat{\alpha}
h,M
D
,\widehat{\sigma}
h,M

)

and order-

1

(\widehat{\alpha}h,M,\widehat{\sigma}h,M)

innovation estimators for the parameters

\alpha=-0.1

and

\sigma=0.1

of the equation

   dx=txdt+\sigma\sqrt{t}xdw(11)

obtained from 100 time series

z
t0
,..,z
tM-1
of

M

noisy observations

  

z
tk

=x(tk

)+e
tk

,fork=0,1,..,M-1,(12)

of

x

on the observation times

\{t\}M=10=\{tk=0.5+k\Delta:k=0,\ldotsM-1

,

\Delta=1\}

, with

x(0.5)=1

and

\Pik=0.0001

. The classical and the order-

1

Local Linearization filters of the innovationestimators
D
(\widehat{\alpha}
h,M
D
,\widehat{\sigma}
h,M

)

and

(\widehat{\alpha}h,M,\widehat{\sigma}h,M)

are defined as in, respectively, on the uniform time discretizations

\left(\tau\right)h=\Delta\equiv\{t\}M

and

\left(\tau \right)h=\Delta/2,\Delta/8,\Delta/32=\{\taun:\taun=0.5+nh

, with

n=0,1,\ldots,(M-1)/h\}

. The number of stochastic simulations of the order-

1

Local Linearization filter is estimated via an adaptive sampling algorithm with moderate tolerance. The Figure 1 illustrates the convergence of the order-

1

innovation estimator

(\widehat{\alpha}h,M,\widehat{\sigma}h,M)

to the exact innovation estimators

(\widehat{\alpha}M,\widehat{\sigma}M)

as

h

decreases, which substantially improves the estimation provided by the conventional innovation estimator
D
(\widehat{\alpha}
\Delta,M
D
,\widehat{\sigma}
\Delta,M

)

.

Deterministic approximations

The order-

\beta

innovation estimators overcome the drawback of the conventional-type innovation estimators concerning the impossibility of reducing bias. However, the viable bias reduction of an order-

\beta

innovation estimators might eventually require that the associated order-

\beta

LMV filter performs a large number of stochastic simulations. In situations where only low or medium precision approximate estimators are needed, an alternative deterministic filter algorithm - called deterministic order-

\beta

LMV filter - can be obtained by tracking the first two conditional moments

\mu

and

Λ

of the order-

\beta

weak approximation

y

at all the time instants

\taun\in\left(\tau\right)h

in between two consecutive observation times

tk

and

tk+1

. That is, the value of the predictions
y
tk+1/tk
and
P
tk+1/tk
in the filtering algorithm are computed from the recursive formulas

     

y
\taun+1/tk

=\mu(\taun

,y
\taun/tk

;hn)

and

P
\taun+1/tk

(\taun

,P
\tau n/tk

;hn),

with

\taun,\taun+1\in(\tau)h\cap\lbracktk,tk+1],

and with

hn=\taun+1-\taun

. The approximate innovation estimators

\widehat{\theta

}_ defined with these deterministic order-

\beta

LMV filters not longer converge to the exact innovation estimator, but allow a significant bias reduction in the estimated parameters for a given finite sample with a lower computational cost.

Figure 2 presents the histograms and the confidence limits of the approximate innovation estimators

(\widehat{\alpha}h,M,\widehat{\sigma}h,M)

and

(\widehat{\alpha},\widehat{\sigma})

for the parameters

\alpha=1

and

\sigma=1

of the Van der Pol oscillator with random frequency

   dx1=x2dt(13)

   dx2

2
=(-(x
1

-1)x2-\alphax1)dt+\sigmax1dw(14)

obtained from 100 time series

z
t0
,..,z
tM-1
of

M

partial and noisy observations

  

z
tk

=x1(tk

)+e
tk

,fork=0,1,..,M-1,(15)

of

x

on the observation times

\{t\}M=30=\{tk=k\Delta:k=0,\ldotsM-1

,

\Delta=1\}

, with

(x1(0),x1(0))=(1,1)

and

\Pik=0.001

. The deterministic order-

1

Local Linearization filter of the innovation estimators

(\widehat{\alpha}h,,M,\widehat{\sigma}h,M)

and

(\widehat{\alpha},\widehat{\sigma})

is defined, for each estimator, on uniform time discretizations

\left(\tau\right)h=\{\taun:\taun=nh

, with

n=0,1,\ldots,(M-1)/h\}

and on an adaptive time-stepping discretization

\left(\tau\right)

with moderate relative and absolute tolerances, respectively. Observe the bias reduction of the estimated parameter as

h

decreases.

Software

A Matlab implementation of various approximate innovation estimators is provided by the SdeEstimation toolbox.[19] This toolbox has Local Linearization filters, including deterministic and stochastic options with fixed step sizes and sample numbers. It also offers adaptive time stepping and sampling algorithms, along with local and global optimization algorithms for innovation estimation. For models with complete observations free of noise, various approximations to the Quasi-Maximum Likelihood estimator are implemented in R.[20]

Notes and References

  1. Jazwinski A.H., Stochastic Processes and Filtering Theory, Academic Press, New York, 1970.
  2. Nielsen . Jan Nygaard . Vestergaard . Martin . 2000 . Estimation in continuous-time stochastic volatility models using nonlinear filters . International Journal of Theoretical and Applied Finance . 03 . 2 . 279–308 . 10.1142/S0219024900000139 . 0219-0249.
  3. Kailath T., Lectures on Wiener and Kalman Filtering. New York: Springer-Verlag, 1981.
  4. Jimenez . J. C. . Yoshimoto . A. . Miwakeichi . F. . 2021-08-24 . State and parameter estimation of stochastic physical systems from uncertain and indirect measurements . The European Physical Journal Plus . en . 136. 8. 136, 869 . 10.1140/epjp/s13360-021-01859-1 . 2021EPJP..136..869J . 238846267 . 2190-5444.
  5. Schweppe . F. . 1965 . Evaluation of likelihood functions for Gaussian signals . IEEE Transactions on Information Theory . 11 . 1 . 61–70 . 10.1109/TIT.1965.1053737 . 1557-9654.
  6. Jimenez . J. C. . Ozaki . T. . 2006 . An Approximate Innovation Method For The Estimation Of Diffusion Processes From Discrete Data . Journal of Time Series Analysis . en . 27 . 1 . 77–97 . 10.1111/j.1467-9892.2005.00454.x . 18072651 . 0143-9782.
  7. Ljung L., System Identification, Theory for the User (2nd edn). Englewood Cliffs: Prentice Hall, 1999.
  8. Lennart . Ljung . Caines . Peter E. . 1980 . Asymptotic normality of prediction error estimators for approximate system models . Stochastics . en . 3 . 1–4 . 29–46 . 10.1080/17442507908833135 . 43397253 . 0090-9491.
  9. Nolsoe K., Nielsen, J.N., Madsen H. (2000) "Prediction-based estimating function for diffusion processes with measurement noise", Technical Reports 2000, No. 10, Informatics and Mathematical Modelling, Technical University of Denmark.
  10. Ozaki . T. . Jimenez . J. C. . Haggan-Ozaki . V. . 2000 . The Role of the Likelihood Function in the Estimation of Chaos Models . Journal of Time Series Analysis . en . 21 . 4 . 363–387 . 10.1111/1467-9892.00189 . 122681657 . 0143-9782.
  11. Jimenez . J.C. . 2020 . Bias reduction in the estimation of diffusion processes from discrete observations . 2023-07-06 . IMA Journal of Mathematical Control and Information . 37 . 4 . 1468–1505 . 10.1093/imamci/dnaa021.
  12. Jimenez . J.C. . 2019 . Approximate linear minimum variance filters for continuous-discrete state space models: convergence and practical adaptive algorithms . 2023-07-06 . IMA Journal of Mathematical Control and Information . 36 . 2 . 341–378 . 10.1093/imamci/dnx047. free .
  13. Shoji . Isao . 1998 . A comparative study of maximum likelihood estimators for nonlinear dynamical system models . International Journal of Control . en . 71 . 3 . 391–404 . 10.1080/002071798221731 . 0020-7179.
  14. Nielsen . Jan Nygaard . Madsen . Henrik . 2001-01-01 . Applying the EKF to stochastic differential equations with level effects . Automatica . en . 37 . 1 . 107–112 . 10.1016/S0005-1098(00)00128-X . 0005-1098.
  15. Singer . Hermann . 2002 . Parameter Estimation of Nonlinear Stochastic Differential Equations: Simulated Maximum Likelihood versus Extended Kalman Filter and Itô-Taylor Expansion . Journal of Computational and Graphical Statistics . en . 11 . 4 . 972–995 . 10.1198/106186002808 . 120719418 . 1061-8600.
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  17. Book: Peng . H. . Ozaki . T. . Jimenez . J.C. . Proceedings of the 41st IEEE Conference on Decision and Control, 2002 . Modeling and control for foreign exchange based on a continuous time stochastic microstructure model . 2002 . https://ieeexplore.ieee.org/document/1185071 . 4 . 4440–4445 vol.4 . 10.1109/CDC.2002.1185071. 0-7803-7516-5 . 8239063 .
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