Statistics

2010 Submissions

[2] viXra:2010.0257 [pdf] submitted on 2020-10-31 19:46:07

Hidden Markov Model Evaluation from First Principles

Authors: Russell Leidich
Comments: 9 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

Hidden Markov models (HMMs) are a class of generative stochastic process models which seek to explain, in the simplest possible terms subject to inherent structural constraints, a set of equally long sequences (time series) of observations. Given such a set, an HMM can be trivially constructed which will reproduce the set exactly. Such an approach, however, would amount to verfitting the data, yielding a model that fails to generalize to new observations of the same physical system under analysis. It’s therefore important to consider the information cost (entropy) of describing the HMM itself – not just the entropy of reproducing the observations, which would be zero in the foregoing extreme case, but in general would be the negative log of the probability of such reproduction occurring by chance. The sum of these entropies would then be suitable for the purpose of ranking a set of candidate HMMs by their respective likelihoods of having actually generated the observations in the first place. To the author’s knowledge, however, no approach has yet been derived for the purpose of measuring HMM entropy from first principles, which is the subject of this paper, notwithstanding the popular use of the Bayesian information criterion (BIC) for this purpose.
Category: Statistics

[1] viXra:2010.0002 [pdf] submitted on 2020-10-01 10:42:20

Random Walks Are Not So Random, After All

Authors: Arturo Tozzi
Comments: 9 Pages.

Physical and biological phenomena are often portrayed in terms of random walks, white noise, Markov paths, stochastic trajectories with subsequent symmetry breaks. Here we show that this approach from dynamical systems theory is not profitable when random walks occur in phase spaces of dimensions higher than two. The more the dimensions, the more the (seemingly) stochastic paths are constrained, because their trajectories cannot resume to the starting point. This means that high-dimensional tracks, ubiquitous in real world physical/biological phenomena, cannot be operationally treated in terms of closed paths, symplectic manifolds, Betti numbers, Jordan theorem, topological vortexes. This also means that memoryless events disconnected from the past such as Markov chains cannot exist in high dimensions. Once expunged the operational role of random walks in the assessment of experimental phenomena, we take aim to somewhat “redeem” stochasticity. We suggest two methodological accounts alternative to random walks that partially rescue the operational role of white noise and Markov chains. The first option is to assess multidimensional systems in lower dimensions, the second option is to establish a different role for random walks. We diffusely describe the two alternatives and provide heterogeneous examples from boosting chemistry, tunneling nanotubes, backward entropy, chaotic attractors.
Category: Statistics