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.Jazwinski A.H., Stochastic Processes and Filtering Theory, Academic Press, New York, 1970. 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 term of a SDE of the form
describing the time evolution of state variables of the phenomenon for all time instant , and an observation equation
describing the time series of measurements of at least one of the variables of the random phenomenon on time instants . In the model (1)-(2), and are differentiable functions, is an -dimensional standard Wiener process, is a vector of parameters, is a sequence of -dimensional i.i.d. Gaussian random vectors independent of , an positive definite matrix, and an 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: given partial and noisy observations of the stochastic process on the observation times , estimate the unobserved state variable of and the unknown parameters in (1) that better fit to the given observations.
Discrete-time innovation process Let be the sequence of observation times of the states of (1), and the time series of partial and noisy measurements of described by the observation equation (2).
Further, let and be the conditional mean and variance of with , where denotes the expected value of random vectors.
The random sequence with
defines the discrete-time innovation process,Kailath T., Lectures on Wiener and Kalman Filtering. New York: Springer-Verlag, 1981. where is proved to be an independent normally distributed random vector with zero mean and variance
for small enough , with . In practice, this distribution for the discrete-time innovation is valid when, with a suitable selection of both, the number of observations and the time distance between consecutive observations, the time series of observations of the SDE contains the main information about the continuous-time process . That is, when the sampling of the continuous-time process 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 with respect to the parameters. More precisely, given measurements of the state space model (1)-(2) with on the innovation estimator for the parameters of (1) is defined by
where
being the discrete-time innovation (3) and the innovation variance (4) of the model (1)-(2) at , for all In the above expression for the conditional mean and variance are computed by the continuous-discrete filtering algorithm for the evolution of the moments (Section 6.4 in), for all
Differences with the maximum likelihood estimator The maximum likelihood estimator of the parameters in the model (1)-(2) involves the evaluation of the - usually unknown - transition density function between the states and of the diffusion process for all the observation times and . Instead of this, the innovation estimator (5) is obtained by maximizing the likelihood of the discrete-time innovation process taking into account that are Gaussian and independent random vectors. Remarkably, whereas the transition density function changes when the SDE for does, the transition density function for the innovation process remains Gaussian independently of the SDEs for . Only in the case that the diffusion is described by a linear SDE with additive noise, the density function is Gaussian and equal to and so the maximum likelihood and the innovation estimator coincide. 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.Ljung L., System Identification, Theory for the User (2nd edn). Englewood Cliffs: Prentice Hall, 1999. Therefore, the asymptotic results obtained in for that general class of estimators are valid for the innovation estimators.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. 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, 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:
Under conventional regularity conditions, the innovation estimator (5) is consistent and asymptoticaly normal distributed. For selecting models, the maximum log-likelihood of the innovation estimator (5) can be used to compute the Akaike or Bayesian information criterion. The confidence limits for the innovation estimator is estimated with
where is the t-student distribution with significance level, and degrees of freedom . Here, denotes the variance of the innovation estimator , where
is the Fisher Information matrix the innovation estimator of and
is the entry of the matrix with and , for .
The distribution of the fitting-innovation process measures the goodness of fit of the model to the data.
For smooth enough function , nonlinear observation equations of the form
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 and in (5) are only available for a few models (1)-(2). Therefore, approximate filtering algorithms as the following are used in applications.
Given measurements and the initial filter estimates , , the approximate Linear Minimum Variance (LMV) filter for the model (1)-(2) is iteratively defined at each observation time by the prediction estimates
and
with initial conditions and , and the filter estimates
and
with filter gain
for all , where is an approximation to the solution of (1) on the observation times .
Given measurements of the state space model (1)-(2) with on , the approximate innovation estimator for the parameters of (1) is defined by
where
being
and
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 and 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 or discrete-discrete approximate filtering algorithms. With approximate continuous-discrete filters there are the innovation estimators based on Local Linearization (LL) filters, on the extended Kalman filter, 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. 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 of the time interval satisfying the condition . Further, let be the approximate value of obtained from a discretization of the equation (1) for all , and
for all
a continuous-time approximation to .
A order- LMV filter. is an approximate LMV filter for which is an order- weak approximation to satisfying (10) and the weak convergence condition
for all and any times continuously differentiable functions for which and all its partial derivatives up to order have polynomial growth, being a positive constant. This order- LMV filter converges with rate to the exact LMV filter as goes to zero, where is the maximum stepsize of the time discretization on which the approximation to is defined.
A order- 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- LMV filter.
Approximations of any kind converging to in a weak sense (as, e.g., those in Kloeden P.E., Platen E., Numerical Solution of Stochastic Differential Equations, 3rd edn. Berlin: Springer, 1999.) can be used to design an order- LMV filter and, consequently, an order- innovation estimator. These order- 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- innovation estimator has a number of important properties:
For each given data of observations, converges to the exact innovation estimator as the maximum stepsize of the time discretization goes to zero. For finite samples of observations, the expected value of converges to the expected value of the exact innovation estimator as goes to zero. For an increasing number of observations, is asymptotically normal distributed and its bias decreases when goes to zero.