A Special Study of the Mixed Weighted Fractional Brownian Motion

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by ηt:=Bt+ξt, where B is a Brownian motion and ξ is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter θ>0 in the mixed weighted fractional Ornstein–Uhlenbeck model of the form X0=0;Xt=θXtdt+dηt. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

Parameter Identification of ARX Models Based on Modified Momentum Gradient Descent Algorithm

The parameter estimation problem of the ARX model is studied in this paper. First, some traditional identification algorithms are briefly introduced, and then a new parameter estimation algorithm—the modified momentum gradient descent algorithm—is developed. Two gradient directions with their corresponding step sizes are derived in each iteration. Compared with the traditional parameter identification algorithms, the modified momentum gradient descent algorithm has a faster convergence rate. A simulation example shows that the proposed algorithm is effective.

Unbiased Least-Squares Modelling

In this paper we analyze the bias in a general linear least-squares parameter estimation problem, when it is caused by deterministic variables that have not been included in the model. We propose a method to substantially reduce this bias, under the hypothesis that some a-priori information on the magnitude of the modelled and unmodelled components of the model is known. We call this method Unbiased Least-Squares (ULS) parameter estimation and present here its essential properties and some numerical results on an applied example.

Assimilation of multiple linearly dependent data vectors

Assimilation of a sequence of linearly dependent data vectors, $\{d_{l}\}^{L}_{l=1}${dl}l=1L such that ${d_{l} = B_{l}d_{L}}^{L-1}_{ l=1}$dl=BldLl=1L−1, is considered for a parameter estimation problem. Such a data sequence can occur, for example, in the context of multilevel data assimilation. Since some information is used several times when linearly dependent data vectors are assimilated, the associated data-error covariances must be modified. I develop a condition that the modified covariances must satisfy in order to sample correctly from the posterior probability density function of the uncertain parameter in the linear-Gaussian case. It is shown that this condition is a generalization of the well-known condition that must be satisfied when assimilating the same data vector multiple times. I also briefly discuss some qualitative and computational issues related to practical use of the developed condition.