Parallelization algorithms for modeling ARM processes (Q5932232)

Let \(U_0\) have uniform distribution on \([0,1)\) and let \(\{V_n\}\) be iid random variables independent of \(U_0\). Denote the fractional part of \(x\) by \(\langle x \rangle\). A background TES\({}^+\) process \(U_n^+\) is defined by \(U_0^+=U_0\), \(U_n^+ = \langle U_{n-1}^+ + V_n \rangle\) for \(n>0\). A foreground TES\({}^+\) process \(X_n^+\) is given by \(X_n^+ = D(U_n^+)\), where \(D\) is a measurable transformation from \([0,1)\) to the reals, called a distortion. Given an empirical time series \(Y_0,\dots,Y_N\), the authors deal with \(D_{Y,\xi}(x)= \hat H_Y^{-1}(S_{\xi}(x))\), \(x\in[0,1)\), where \(S_{\xi}(y)=y/\xi\) for \(0\leq y\leq \xi\) and \(S_{\xi}(y)=(1-y)/(1-\xi)\) for \(\xi\leq y <1\) is a stitching transformation and \(\hat H_Y^{-1}\) is the inverse of the empirical distribution function of \(Y_0,\dots,Y_N\). Then \(X_n^+=\hat H_Y^{-1}(S_{\xi}(U_n^+))\) has empirical distribution \(\hat H_y\) regardless of innovation density \(f_V\) and parameter \(\xi\in[0,1]\). TES (Transform Expand Sample) processes are a subclass of ARM (AutoRegressive Modular) processes which can capture empirical distributions and autocovariance functions simultaneously.NEWLINENEWLINENEWLINEThe modelling procedure has two steps: a global search for locating minima of a nonlinear function over a large parametric space and a local optimization of models found in the first step. Two space-partitioning methods, called interleaving and segmentation, are investigated in this paper and their performances are compared. The methods are based on parallelization of the global search using parallel processors.