# Data-driven Techniques For Estimation And Stochastic Reduction Of Multiscale Systems

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Parametric estimation of stochastic processes is one of the most widely used techniques for obtaining effective models, given a discrete dataset. Often, the experimental or observational datasets are not explicitly generated by the underlying stochastic model and are, thus, expected to agree with the model only in approximate statistical sense, also referred as \textit{Indirect Observability}. Therefore, there may be inherent difference between the data and the model, leading to the inconsistency of the parametric estimation.

The thesis is presented in three parts. In part I (chapter \ref{chapter 2}) of the thesis, the goal is to develop an efficient and accurate parametric estimation procedure for the reduced model (SDE for slow variables alone) when the given data is the time series of the slow variables in the multi-scale high dimensional Lorenz-96 model. First estimator considered for the reduced model is approximate Maximum Likelihood estimator which is highly dependent on the subsampling time-step of the given data. There is no feasible solution to compute optimal subsampling time-step for consistent approximate Maximum Likelihood estimator. Next, we look at the moment estimator which is weakly dependent on subsampling time-step of the given data but is valid only if the mean of the slow variables in the full model is relatively large. Both estimators give acceptable values for the parameters in reduced model. Given the situation of non-linear multiscale model to be reduced to stochastic model of slow variables alone, moment estimator is preferred if mean of the slow variables is relatively large and the reduced model is not very sensitive to the change in parameters.

The second part is Chapter \ref{chapter 3} in which we consider the multiscale model having energy conserving fast subsystem and the stochastic terms been added to the equation of slow variables alone. In such situations, the energy of the fast variables changes with time due to the coupling between the slow and fast dynamics and hence, considered as an additional hidden slow variable. We develop a stochastic mode reduction technique to derive an efficient stochastic model for the original slow variables in full model and additional slow variable given by energy of the fast subsystem.

In the third part (chapter \ref{chapter 4}), similar parametric estimation procedure is studied under \textit{indirect observability} as done for Lorenz-96 model in chapter \ref{chapter 2} but here we consider multiscale fast-oscillating potential model. We get linear reduced model which simplifies certain analytical computations and we specify explicit conditions for which the estimators of the reduced model are consistent under indirect observability. Another important aspect discussed in the chapter is estimation of an effective model from a dataset generated with a fixed but unknown value of the scale separation parameter