Advanced Probability and Statistics

Chapter 3 Some Study Projects on Applied Signal Processing Remarks About Related Contributions of Scientists

 [1] Linear models: Time series models like AR, MA, ARMA, casting these models in the form

$$X\left(n\right)=H\left(n\right)\theta +V\left(n\right)$$

where X(n), H(n) are data vectors and data matrices. V(n) is noise. H(n) ∈ ℝ^{n × p}, X(n), V(n) ∈ ℝⁿ. If R_υ = Coυ(V(n)) and V(n) are iid zero mean Gaussian, then the MLE of θ based on data collected upto time n is given by

$$\widehat{\theta }\left(n\right)={({\displaystyle \sum _{k=1}^{n}H{\left(k\right)}^T{R}_{\upsilon }H\left(k\right)})}^{-1}({\displaystyle \sum _{k=1}^{n}H{\left(n\right)}^T{R}_{\upsilon }X\left(k\right)})$$

Since X(n), H(n) are data matrices, they are also random and we wish to determine the mean and covariance of θ̂(n) in terms of the statistics of these data matrices.

 [2] Innovations process and ...