Multivariate Cumulants Research Articles

Higher-order functional connectivity analysis of resting-state functional magnetic resonance imaging data using multivariate cumulants.

Blood-level oxygenation-dependent (BOLD) functional magnetic resonance imaging (fMRI) is the most common modality to study functional connectivity in the human brain. Most research to date has focused on connectivity between pairs of brain regions. However, attention has recently turned towards connectivity involving more than two regions, that is, higher-order connectivity. It is not yet clear how higher-order connectivity can best be quantified. The measures that are currently in use cannot distinguish between pairwise (i.e., second-order) and higher-order connectivity. We show that genuine higher-order connectivity can be quantified by using multivariate cumulants. We explore the use of multivariate cumulants for quantifying higher-order connectivity and the performance of block bootstrapping for statistical inference. In particular, we formulate a generative model for fMRI signals exhibiting higher-order connectivity and use it to assess bias, standard errors, and detection probabilities. Application to resting-state fMRI data from the Human Connectome Project demonstrates that spontaneous fMRI signals are organized into higher-order networks that are distinct from second-order resting-state networks. Application to a clinical cohort of patients with multiple sclerosis further demonstrates that cumulants can be used to classify disease groups and explain behavioral variability. Hence, we present a novel framework to reliably estimate genuine higher-order connectivity in fMRI data which can be used for constructing hyperedges, and finally, which can readily be applied to fMRI data from populations with neuropsychiatric disease or cognitive neuroscientific experiments.

KStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno's Formula

kStatistics is a package in `R` that serves as a unified framework for estimating univariate and multivariate cumulants as well as products of univariate and multivariate cumulants of a random sample, using unbiased estimators with minimum variance. The main computational machinery of kStatistics is an algorithm for computing multi-index partitions. The same algorithm underlies the general-purpose multivariate Faà di Bruno's formula, which therefore has been included in the last release of the package. This formula gives the coefficients of formal power series compositions as well as the partial derivatives of multivariable function compositions. One of the most significant applications of this formula is the possibility to generate many well-known polynomial families as special cases. So, in the package, there are special functions for generating very popular polynomial families, such as the Bell polynomials. However, further families can be obtained, for suitable choices of the formal power series involved in the composition or when suitable symbolic strategies are employed. In both cases, we give examples on how to modify the `R` codes of the package to accomplish this task. Future developments are addressed at the end of the paper

Multivariate cumulants in flow analyses: The next generation

We reconcile for the first time the strict mathematical formalism of multivariate cumulants with the use of cumulants in anisotropic flow analyses in high-energy nuclear collisions. This reconciliation yields to the next generation of estimators to be used in flow analyses. We review all fundamental properties of multivariate cumulants and use them as a foundation to establish two simple necessary conditions to determine whether some multivariate random variable is a multivariate cumulant in the basis they are expressed in. We argue that properties of cumulants are preserved only for the stochastic variables on which the cumulant expansion has been performed directly, and if there are no underlying symmetries due to which some terms in the cumulant expansion are identically zero. We illustrate one possibility of new multivariate cumulants of azimuthal angles by defining them event-by-event and by keeping all nonisotropic terms in the cumulant expansion. Furthermore, we introduce new cumulants of flow amplitudes named asymmetric cumulants, which generalize recently introduced symmetric cumulants for the case when flow amplitudes are raised to different powers. Finally, we present the new concept of cumulants of symmetry plane correlations and provide the first realization for the lowest orders. The new estimators can be used directly to constrain the multivariate probability density function of flow fluctuations, since its functional form can be reconstructed only from its true moments or cumulants. The new definition for cumulants of azimuthal angles enables separation of nonflow and flow contributions and offers first insights into how the combinatorial background contributes for small multiplicities to flow measurements with correlation techniques. All the presented results are supported by Monte Carlo studies using state-of-the-art models.

On non-linear dependence of multivariate subordinated Lévy processes

Multivariate subordinated Lévy processes are widely employed in finance for modeling multivariate asset returns. We propose to exploit non-linear dependence among financial assets through multivariate cumulants of these processes, for which we provide a closed form formula by using the multi-index generalized Bell polynomials. Using multivariate cumulants, we perform a sensitivity analysis, to investigate non-linear dependence as a function of the model parameters driving the dependence structure.

Computational multivariate modelling of electrical activity of the porcine uterus during spontaneous and hormone-induced oestrus.

What is the central question of this study? Does oestrous cycle synchronization influence myoelectrical activity of porcine myometrium? What is the main finding and its importance? Exogenous hormones used to synchronize oestrus in pigs altered myoelectrical activity, which was effectively modelled. Higher-order multivariate statistic modelling provided evidence of similar activity in both types of oestrus, but a larger order of EMG signals during induced oestrus. Higher-order statistical analysis of the probabilistic model suggests the beginning of the early follicular phase and the mid-luteal phase to be most important in evaluation of the natural patterns of myoelectrical activity. Higher-order multivariate cumulants are more informative than classical statistics in characterization of myoelectrical activity changes in porcine myometrium. In pig production units, control of the oestrous cycle and synchronization of ovulation have become routine herd management procedures. During the oestrous cycle, in both induced and spontaneous conditions, the ovaries and the uterus undergo hormone-dominated physiological changes, which are consistent with the hypothesis that there is a functional role of uterine contractions in promoting fertilization. We have used electromyography to determine whether the use of exogenous hormones, such as equine chorionic gonadotrophin and human chorionic gonadotrophin, which have the potential to control the timing of ovulation in female pigs, changes the multivariate relationships between parameters of electrical bursts and modulates the patterns of myoelectrical activity. We used the mathematical approach of higher-order multivariate cumulants in complex modelling of the myometrial electrical activity. The experiment was conducted on 12 mature Polish Landrace sows, and uterine activity was recorded during both spontaneous and induced oestrous cycles. The burst parameters were determined using six features in the time domain and, after Fast Fourier transformation, in the frequency domain. Evaluation of myoelectrical activity patterns was conducted based on classical univariate statistical methods and multivariate probabilistic modelling. The classical statistical approach indicated weaker myoelectrical activity after hormonal stimulation, whereas the higher-order multivariate statistical model showed evidence of similar status of activity and a larger order of signals during induced oestrus. Routine oestrous cycle synchronization affects the multivariate probabilistic model of myometrial electrical activity.

An application of higher order multivariate cumulants in modelling of myoelectrical activity of porcine uterus during early pregnancy

The analysis of the uterine contraction have become a general practice in an effort to improve the clinical management of uterine contractions during pregnancy and labour in human beings. The fluctuations in uterine activity may occur without affecting progress of gestation, however the painful and fashion contractions may be the first threat of miscarriage. While pigs were considered as an referential preclinical model, the computational modelling of spontaneous myoelectrical activity of complex systems of porcine myometrium in peri-fertilization period has been proposed. The higher order statistic, multivariate cumulants and Joint Skewness Band Selection method, have been applied to study the dependence structure of electromyographic (EMG) signal with an effective EMG feature. Than the model of recognition of multivariate, myoelectricaly changes according to crucial stages for successful fertilization and early pregnancy maintenance has been estimated. We found that considering together time and frequency features of EMG signal was extremely non-Gaussian distributed and the higher order multivariate statistics such as cumulants, have to be used to determine the pattern of myoelectrical activity in reproductive tract. We confirmed the expectance that the probabilistic model changes on a daily base. We demonstrated the changes in proposed model at the crucial time points of in peri-fertilization period. We speculate the activity of the middle of uterine horn and the power (minimum and maximum) and pauses between myoelectrical burst features are essential for the functional role of uterine contractility in peri-fertilization period.

Bias reduction when data are rounded

Analytical bias reduction methods are developed for univariate rounded data for the first time. Extensions are given to rounding of multivariate data, and to smooth functionals of several distributions. As a by‐product, we give for the first time the relation between rounded and unrounded multivariate cumulants. Estimators obtained by analytical bias reduction are compared with bootstrap and jackknife estimators by simulation.

Finite-Dimensional Distributions of a Square-Root Diffusion

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

Finite-Dimensional Distributions of a Square-Root Diffusion

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

Using third-order cumulants to investigate spatial variation: A case study on the porosity of the Bunter Sandstone

The multivariate cumulants characterize aspects of the spatial variability of a regionalized variable. A centred multivariate Gaussian random variable, for example, has zero third-order cumulants. In this paper it is shown how the third-order cumulants can be used to test the plausibility of the assumption of multivariate normality for the porosity of an important formation, the Bunter Sandstone in the North Sea. The results suggest that the spatial variability of this variable deviates from multivariate normality, and that this assumption may lead to misleading inferences about, for example, the uncertainty attached to kriging predictions.

On the joint distribution of a linear and a quadratic form in skew normal variables

Let z be distributed as multivariate skew normal vector. We derive the joint moment generating function (m.g.f.) of a linear form and a quadratic form in z, and the conditions for their independence. The first two multivariate cumulants of the two forms are derived and applied in special cases. Finally a simulation example is presented.

A Multivariate Faa di Bruno Formula with Applications

A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.

The factorization of a sum of matrices and the multivariate cumulants of a set of quadratic expressions

A factorization of I − B 1 − B 2 − … − B m is given in the form Π n=1 ∞( I − P n) where each P n is a product of B's. Up to terms of degree n there are ∏ r=1 n 1 r ∑ d| μ 1 r m d ! possible factorizations. The P's are called prime words since they have some of the properties of prime numbers. A theorem concerning prime words, resembling the ordinary prime number theorem, is conjectured and partly proved. A simple method is given for writing down, in terms of the traces of the “prime words” and their “powers,” the cumulants of the joint distribution of a set of quadratic expressions in a vector with a multinormal distributions.

The Joint Cumulants of True Values and Errors of Measurement

Replicate fallible measurements on each of a large number of objects are used to estimate the multivariate cumulants of the joint frequency distribution of the true value and the errors of measurement. Simple, explicit formulas are given for all multivariate cumulants up through the fourth order. It is assumed only that each error of measurement has a mean of zero and is uncorrelated with other appropriate chance variables.

THE JOINT CUMULANTS OF TRUE VALUES AND ERRORS OF MEASUREMENT

ABSTRACTReplicate fallible measurements on each of a large number of objects are used to estimate the multivariate cumulants of the joint frequency distribution of the true value and the errors of measurement. Simple, explicit formulas are given for all multivariate cumulants up through the fourth order. It is assumed only that each error of measurement has a mean of zero and is uncorrelated with other appropriate chance variables.

INFERRING THE SHAPE OF THE FREQUENCY DISTRIBUTION OF TRUE SCORES AND OF ERRORS OF MEASUREMENT

ABSTRACTGiven k parallel, fallible measures on each individual in a group, all the multivariate cumulants of k ‐th and lover orders for the joint distribution of true scores and errors of measurement may be expressed as simple linear functions of the multivariate cumulants of the fallible measures. It is assumed only that the number of examinees is large; that over a very large number of different testings, the average error of measurement will be zero for each examinee; and that the errors of measurement for two different examinees, or for two different testings of the same examinee, are independently distributed. The form of the frequency distribution of the errors is unspecified and may be different for different individuals. The results of a numerical application to psychological test data are given.