Hypothetical Matrices Research Articles

Morphological integration and cranial modularity in six genera of echimyid rodents (Rodentia: Echimyidae)

Abstract Rodents of the family Echimyidae present a wide variety of life histories and ecomorphological adaptations. This study evaluated morphological integration patterns, modularity, and evolutionary flexibility in six Echimyid genera representing ecomorphological extremes within the family. The relationships between traits were evaluated by comparing estimated covariance and correlations matrices of populations. The presence of modules was investigated by comparing the patterns of integration between traits and using hypothetical matrices based on shared development/function and masticatory stress. The results point to a common covariance and correlation pattern among the six echimyid genera, suggesting a conserved pattern of covariation (associations among traits) throughout the evolution of this group. The overall magnitude of integration, however, varied greatly. We also found a high degree of modularity in all six echimyid genera. Finally, we observed a clear association between flexibility, i.e., the ability of a species to respond to the direction of selection, with the overall magnitude of integration and degree of modularization. The results of this study provide hypotheses concerning the underlying effects of the association among traits, which may have facilitated or constrained the evolution of morphological variation in the diverse family Echimyidae.

A New Two-Dimensional Rank Correlation Coefficient

Spearman’s rank correlation coefficient might be the best known nonparametric measure of association currently in use. It assesses the linear relationships between the ranks of monotonically related variables even if the relationship between the variables is not linear. This study presents a new method for calculating two-dimensional (horizontal and vertical) rank correlation coefficients between matrices composed of variables which are not necessarily in linear association. The matrices contain the ranks of measurements instead of raw values. The averages of all rows are used for calculating the horizontal rank correlation and the averages of all columns are used for calculating the vertical rank correlation instead of considering the averages of the whole matrices. This approach enables a separate determination of the degree of horizontal and vertical relationships between the compared data matrices by using the horizontal and vertical variance and covariance values that constitute the base of the two-dimensional correlation method. The presented method is first applied on 5 simple hypothetical matrices and then on the monthly total precipitation records of 6 stations in southwest Turkey. The results have shown that the presented rank correlation approach successfully assesses the two-dimensional associations between the hypothetical matrices and time series data like precipitation, provides a measure for exactly determining the monotonic relationships not determined by the conventional Pearson’s and Spearman’s correlation approaches, it is much more robust to outliers and normality is not a prerequisite. The software developed for calculating two-dimensional rank correlation coefficients is freely provided together with this paper.

A novel two‐dimensional correlation coefficient for assessing associations in time series data

ABSTRACTThe widely used Pearson's correlation coefficient calculated for assessing the linear relationship between two variables might produce misleading results especially in the comparison of periodic variables. A single correlation coefficient provides a measure of the overall dependence structure and generally might not be sufficient for assessing local differences between the variables (e.g. associations between each individual year might vary in hydrologic series). The reason for this deficiency is the consideration of the averages of the whole series while ignoring the variations of the local averages (e.g. annual averages or long year averages of months) throughout the observations. This study presents a two‐dimensional horizontal (row wise) and vertical (column wise) correlation calculation approach where the compared series are considered as two‐dimensional matrices in which each row represents a sub‐period (e.g. one calendar year of the precipitation data) of the investigated time series data. The method applies a normalization procedure by considering the averages of all rows (namely local averages) for calculating the horizontal correlation and the averages of all columns for calculating the vertical correlation instead of considering the averages of the whole matrices. This enables a separate determination of the degree of relationships between the rows and columns of the compared data matrices by using the horizontal and vertical variance and covariance values that constitute the base of the two‐dimensional correlation. The method is applied on 14 different linearly varying hypothetical matrices, 6 matrices for testing the influence of seasonal and inter‐annual variations and the monthly total precipitation records of 6 stations in southwest Turkey. The results have shown that the developed correlation approach assesses the two‐dimensional behaviour of time series data like precipitation and provides a measure which enables separate assessment of the contributions from the seasonal cycle vs. inter‐annual variability in the association between two time series.

Hypothetical Matrix for Institutional Modeling of the Basis for Economic Development in the Countries of Southeast Europe

In this paper we propose three hypothetical matrices for institutional modeling of the basis for economic development. They apply selectively chosen elements of Neo-Institutional Theory (NIT) and convention theory, which are in line with affirmation of institutional pluralism and criticism of institutional monism.

Biodistance analysis of the Moche sacrificial victims from Huaca de la Luna plaza 3C: Matrix method test of their origins

The purpose of this study is to test two competing models regarding the origins of Early Intermediate Period (AD 200-750) sacrificial victims from the Huacas de Moche site using the matrix correlation method. The first model posits the sacrificial victims represent local elites who lost competitions in ritual battles with one another, while the other model suggests the victims were nonlocal warriors captured during warfare with nearby polities. We estimate biodistances for sacrificial victims from Huaca de la Luna Plaza 3C (AD 300-550) with eight previously reported samples from the north coast of Peru using both the mean measure of divergence (MMD) and Mahalanobis' distance (d2). Hypothetical matrices are developed based upon the assumptions of each of the two competing models regarding the origins of Moche sacrificial victims. When the MMD matrix is compared to the two hypothetical matrices using a partial-Mantel test (Smouse et al.: Syst Zool 35 (1986) 627-632), the ritual combat model (i.e. local origins) has a low and nonsignificant correlation (r = 0.134, P = 0.163), while the nonlocal origins model is highly correlated and significant (r = 0.688, P = 0.001). Comparisons of the d2 results and the two hypothetical matrices also produced low and nonsignificant correlation for the ritual combat model (r = 0.210, P = 0.212), while producing a higher and statistically significant result with the nonlocal origins model (r = 0.676, P = 0.002). We suggest that the Moche sacrificial victims represent nonlocal warriors captured in territorial combat with nearby competing polities.

Limitations of relative apparent synapomorphy analysis (RASA) for measuring phylogenetic signal.

In this paper we use hypothetical and empirical data matrices to evaluate the ability of relative apparent synapomorphy analysis (RASA) to measure phylogenetic signal, select outgroups, and identify terminals subject to long-branch attraction. In all cases, except for equal character-state frequencies, RASA indicated extraordinarily high levels of phylogenetic information for hypothetical data matrices that are uninformative regarding relationships among the terminals. Yet, regardless of the number of characters or character-state frequencies, RASA failed to detect phylogenetic signal for hypothetical matrices with strong phylogenetic signal. In our empirical example, RASA indicated increasing phylogenetic signal for matrices for which the strict consensus of the most parsimonious trees is increasingly poorly resolved, clades are increasingly poorly supported, and for which many relationships are in conflict with more widely sampled analyses. RASA is an ineffective approach to identify outgroup terminal(s) with the most plesiomorphic character states for the ingroup. Our hypothetical example demonstrated that RASA preferred outgroup terminals with increasing numbers of convergent character states with ingroup terminals, and rejected the outgroup terminal with all plesiomorphic character states. Our empirical example demonstrated that RASA, in all three cases examined, selected an ingroup terminal, rather than an outgroup terminal, as the best outgroup. In no case was one of the two outgroup terminals even close to being considered the optimal outgroup by RASA. RASA is an ineffective means of identifying problematic long-branch terminals. In our hypothetical example, RASA indicated a terminal as being a problematic long-branch terminal in spite of the terminal being on a zero-length branch and having no possibility of undergoing long-branch attraction with another terminal. RASA also failed to identify actual problematic long-branch terminals that did undergo long-branch attraction, but only after following Lyons-Weiler and Hoelzer's (1997) three-step process to identify and remove terminals subject to long-branch attraction. We conclude that RASA should not be used for any of these purposes.