Indicator function and complex coding for mixed fractional factorial designs

In a general fractional factorial design, the $n$-levels of a factor are coded by the $n$-th roots of the unity. This device allows a full generalization to mixed-level designs of the theory of the polynomial indicator function which has already been introduced for two level designs by Fontana and the Authors (2000). the properties of orthogonal arrays and regular fractions are discussed

Two polynomial representations of experimental design

In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the literature: Groebner bases and indicator functions. We briefly describe them both, how they are used in the analysis and planning of a design and how to switch between them. Examples include fractions of full factorial designs and designs for mixture experiments.Comment: 13 page

Toric Statistical Models: Ising and Markov

This is a review of current research in Markov chains as toric statistical models. Its content is mixture of background information, results from the relevant recent literature, new results, and work in progress.Comment: 26 pages. Submitted Oct 10, 201

Cubature rules and expected value of some complex functions

The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for nite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined

Aberration in qualitative multilevel designs

Generalized Word Length Pattern (GWLP) is an important and widely-used tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction ${\mathcal F}$ using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. Moreover, using mean aberrations for symmetric $s^m$ designs with $s$ prime, we derive a new formula for computing the GWLP of ${\mathcal F}$. It is computationally easy, does not use complex numbers and also provides a clear way to interpret the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used as a further tool to discriminate between fractions.Comment: 16 pages, 1 figur

Differential effects of open- and closed-loop intracortical microstimulation on firing patterns of neurons in distant cortical areas

Intracortical microstimulation can be used successfully to modulate neuronal activity. Activity dependent stimulation (ADS), in which action potentials recorded extracellularly from a single neuron are used to trigger stimulation at another cortical location (closed-loop), is an effective treatment for behavioral recovery after brain lesion, but the related neurophysiological changes are still not clear. Here we investigated the ability of ADS and random stimulation (RS) to alter firing patterns of distant cortical locations. We recorded 591 neuronal units from 23 Long-Evan healthy anesthetized rats. Stimulation was delivered to either forelimb or barrel field somatosensory cortex, using either RS or ADS triggered from spikes recorded in the rostral forelimb area (RFA). Both RS and ADS stimulation protocols rapidly altered spike firing within RFA compared with no stimulation. We observed increase in firing rates and change of spike patterns. ADS was more effective than RS in increasing evoked spikes during the stimulation periods, by producing a reliable, progressive increase in stimulus-related activity over time and an increased coupling of the trigger channel with the network. These results are critical for understanding the efficacy of closed-loop electrical microstimulation protocols in altering activity patterns in interconnected brain networks, thus modulating cortical state and functional connectivity