The problem of variation

Chapter 7 maps out a broad framework for considering the problem of variation in evolution. Under the neo-Darwinian view that variation merely plays the role of supplying random infinitesimal raw materials, with no dispositional influence on the course of evolution, a substantive theory of form and its variation is not required to specify a complete theory of evolution. This view has been breaking down from the moment it was proposed, and is now seriously challenged by results from evo-devo, comparative genomics, molecular evolution, and quantitative genetics. For instance, the multivariate generalization of quantitative genetics indicates that selection cannot possibly act as an independent governing force. Replacing a theory of variation as fuel with a theory of variation as a dispositional factor will require, at minimum, an understanding of tendencies of variation (source laws), and an understanding of how those tendencies affect evolution (consequence laws).

Robust Characterization of Multidimensional Scaling Relations between Size Measures for Business Firms

Although the sizes of business firms have been a subject of intensive research, the definition of a “size” of a firm remains unclear. In this study, we empirically characterize in detail the scaling relations between size measures of business firms, analyzing them based on allometric scaling. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994–2015), we examined up to the trivariate relations between corporate size measures: annual sales, capital stock, total assets, and numbers of employees and trading partners. The data were examined using a multivariate generalization of a previously proposed method for analyzing bivariate scalings. We found that relations between measures other than the capital stock are marked by allometric scaling relations. Power–law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that appeared to correlate with national economic conditions. We established theoretical relations between the exponents. We expect these results to allow direct estimation of the effects of using alternative size measures of business firms in regression analyses, to facilitate the modeling of firms, and to enhance the current theoretical understanding of complex systems.

Minimal factorizations of a cycle: a multivariate generating function

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

A multi-parameter generalization of the symmetric algorithm

The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

Sifting Common Information from Many Variables

Measuring the relationship between any pair of variables is a rich and active area of research that is central to scientific practice. In contrast, characterizing the common information among any group of variables is typically a theoretical exercise with few practical methods for high-dimensional data. A promising solution would be a multivariate generalization of the famous Wyner common information, but this approach relies on solving an apparently intractable optimization problem. We leverage the recently introduced information sieve decomposition to formulate an incremental version of the common information problem that admits a simple fixed point solution, fast convergence, and complexity that is linear in the number of variables. This scalable approach allows us to demonstrate the usefulness of common information in high-dimensional learning problems.The sieve outperforms standard methods on dimensionality reduction tasks, solves a blind source separation problem that cannot be solved with ICA, and accurately recovers structure in brain imaging data.

The Spatial Sign Covariance Matrix and Its Application for Robust Correlation Estimation

We summarize properties of the spatial sign covariance matrix and especially consider the relationship between its eigenvalues and those of the shape matrix of an elliptical distribution. The explicit relationship known in the bivariate case was used to construct the spatial sign correlation coefficient, which is a non-parametric and robust estimator for the correlation coefficient within the elliptical model. We consider a multivariate generalization, which we call the multivariate spatial sign correlation matrix. A small simulation study indicates that the new estimator is very efficient under various elliptical distributions if the dimension is large. We furthermore derive its influence function under certain conditions which indicates that the multivariate spatial sign correlation becomes more sensitive to outliers as the dimension increases.

MULTIVARIATE TREND–CYCLE EXTRACTION WITH THE HODRICK–PRESCOTT FILTER

The Hodrick–Prescott filter represents one of the most popular methods for trend–cycle extraction in macroeconomic time series. In this paper we provide a multivariate generalization of the Hodrick–Prescott filter, based on the seemingly unrelated time series approach. We first derive closed-form expressions linking the signal–noise matrix ratio to the parameters of the VARMA representation of the model. We then show that the parameters can be estimated using a recently introduced method, called “Moment Estimation Through Aggregation (META).” This method replaces traditional multivariate likelihood estimation with a procedure that requires estimating univariate processes only. This makes the estimation simpler, faster, and better behaved numerically. We prove that our estimation method is consistent and asymptotically normal distributed for the proposed framework. Finally, we present an empirical application focusing on the industrial production of several European countries.

Non-Traditional Correlation Analysis: Explanatory Power and Opportunities for Knowledge Discovery in Democracy Studies

The possibility of successful applications of the modified correlation coefficient is demonstrated. The latter was proposed by Lukashin nearly twenty five years ago and has been unused since then. A multivariate generalization of this coefficient is proposed. The modified correlation coefficients provide an efficient tool to develop a new multivariate classification method, i.e. a technique for grouping of objects that occurs together with their ranking. As an example of application of the new method, the data of Freedom House is used. NCA (Non-traditional Correlation Analysis), along with similar unconventional methods as FCA (Formal Concept Analysis) and QCA (Qualitative Comparative Analysis) allow to gain additional knowledge from existing databases and numerous ratings which are produced by different agencies. The latters often lack time and opportunities to deeply analyze them, even to go beyond a simple “averaging”. NCA may give additional opportunities for social researchers to understand social phenomena in its complexity, for in-depth analysis and interpretation of structure of data, to build “hierarchical typologies”, and broadly, for data mining and additional knowledge discovery.

Transformation Formulas for Bilinear Sums of Basic Hypergeometric Series

A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author’s previous results on a transformation formula for Milne’s multivariate generalization of basic hypergeometric series of type A with diòerent dimensions and it can be considered as a generalization of theWhipple–Sears transformation formula for terminating balanced µh series. As an application of the master formula, the one-variable cases of some transformation formulas for bilinear sums of basic hypergeometric series are given as examples. The bilinear transformation formulas seemto be new in the literature, even in the one-variable case.