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Temporal shaping of time series is the activity of deriving a time series model with a prescribed marginal distribution and some sample path characteristics. Starting with an empirical sample path, one often computes from it an empirical histogram (a step-function density) and empirical autocorrelation function. The corresponding cumulative distribution function is piecewise linear, and so is the inverse distribution function. The so-called inversion method uses the latter to generate the corresponding distribution from a uniform random variable on [0,1), histograms being a special case. This paper shows how to manipulate the inverse histogram and an underlying marginally uniform process, so as to “shape” the model sample paths in an attempt to match the qualitative nature of the empirical sample paths, while maintaining a guaranteed match of the empirical marginal distribution. It proposes a new approach to temporal shaping of time series and identifies a number of operations on a piecewise-linear inverse histogram function, which leave the marginal distribution invariant. For cyclical processes with a prescribed marginal distribution and a prescribed cycle profile, one can also use these transformations to generate sample paths which “conform” to the profile. This approach also improves the ability to approximate the empirical autocorrelation function. [ABSTRACT FROM AUTHOR]

MARM ( Multivariate Autoregressive Modular) processes constitute a versatile class of multidimensional stochastic sequences which can exactly fit arbitrary multi-dimensional empirical histograms and approximately fit the leading empirical autocorrelations and cross-correlations. A companion paper (Part I) presented the general theory of MARM processes. This paper (Part II) proposes practical MARM modeling and forecasting methodologies of considerable generality, suitable for implementation on a computer. The purpose of Part II is twofold: (1) to specialize the general class of MARM processes to a practical subclass, called Empirically-Based MARM ( EB-MARM) processes, suitable for modeling of empirical vector-valued time series, and devise the corresponding fitting and forecasting algorithms; and (2) to illustrate the efficacy of the EB-MARM fitting and forecasting algorithms. Specifically, we shall consider MARM processes with iid step-function innovation densities and distortions based on an empirical multi-dimensional histogram, as well as empirical autocorrelation and cross-correlation functions. Finally, we illustrate the efficacy of these methodologies with an example of a three-dimension time series vector, using a software environment, called MultiArmLab, which supports MARM modeling and forecasting. [ABSTRACT FROM AUTHOR]