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. 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
d x ( t ) = f ( t , x ( t ) ; θ ) d t + ∑ i = 1 m g i ( t , x ( t ) ; θ ) d w i ( t ) ( 1 ) {\displaystyle \qquad \qquad d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t);\theta )dt+\sum _{i=1}^{m}\mathbf {g} _{i}\ (t,\mathbf {x} (t);\theta )\ d\mathbf {w} ^{i}(t)\qquad \qquad (1)}
describing the time evolution of d {\displaystyle d} state variables x {\displaystyle \mathbf {x} } of the phenomenon for all time instant t ≥ t 0 {\displaystyle t\geq t_{0}} , and an observation equation
z t k = C x ( t k ) + e t k ( 2 ) {\displaystyle \qquad \qquad \mathbf {z} _{t_{k}}=\mathbf {Cx} (t_{k})+\mathbf {e} _{t_{k}}\qquad \qquad (2)}
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![Innovation method: Fig. 2 Histograms and confidence limits for the innovation estimators
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{\displaystyle ({\widehat {\alpha }}_{h,M},{\widehat {\sigma }}_{h,M})}
and
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{\displaystyle ({\widehat {\alpha }}_{\cdot ,M},{\widehat {\sigma }}_{\cdot ,M})}
of
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{\displaystyle (\alpha ,\sigma )}
computed with the deterministic order-1 LL filter on uniform
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and adaptive
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time discretizations, respectively, from
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noisy realizations of the Van der Pol model (13)-(15) with sampling period
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on the time interval
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and
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. Observe the bias reduction of the estimated parameter as
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decreases.](https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/WikiInnFigure.jpg/500px-WikiInnFigure.jpg?utm_source=en.wikipedia.org&utm_campaign=parser&utm_content=thumbnail)
