Distance covariance for stochastic processes

In this paper, using a Brownian motion we define a new type of covariance for random vectors with a finite second moment. The advantage of Brownian covariance is in the fact that it is equal to zero if and only if the random vectors are independent. We can apply Brownian covariance to random vectors, generally speaking, of different sizes if it is invariant with respect to rotations of axes where the data are given. If, in the definition of Brownian covariance, we replace Brownian motion by an identical function, then we obtain a modulus of the classical Pearson covariance coefficient, and therefore, Brownian covariance generalizes the classical. Brownian covariance is applied to the proof of necessary and sufficient conditions for the validity of the central limit theorem for random sequences which are stationary in a narrow sense.

Distance correlation is a novel class of multivariate dependence measure, taking positive values between 0 and 1, and applicable to random vectors of not necessarily equal arbitrary dimensions. It offers several advantages over the well-known Pearson correlation coefficient, the most important being that distance correlation equals zero if-and-only if- the random vectors are independent. There are two different estimators of the distance correlation available in the literature. The first estimator, proposed by Székely et al. (Ann Stat 35:2769–279 2007), is based on an asymptotically unbiased estimator of the distance covariance, which is a V-statistic. The second builds on an unbiased estimator of the distance covariance proposed in Székely and Rizzo (Stat 42:2382–2412 2014), shown to be a U-statistic by Huo and Székely (Technometrics 58:435–447 2016). This study evaluates their efficiency (mean squared error) and compares computational times for both methods under different dependence structures. Under conditions of independence or near-independence, the V-estimates are biased, while the U-estimator frequently cannot be computed due to negative values. To address this challenge, a convex linear combination of the former estimators is proposed and studied, yielding good results regardless of the level of dependence. Additionally, a medical database is studied and discussed.

A fast algorithm for computing distance correlation

Among the various statistical methods for identifying gene–gene interactions in qualitative genome-wide association studies (GWAS), gene-based methods have recently grown in popularity because they confer advantages in both statistical power and biological interpretability. However, most of these methods make strong assumptions about the form of the relationship between traits and single-nucleotide polymorphisms, which result in limited statistical power. In this paper, we propose a gene-based method based on the distance correlation coefficient called gene-based gene-gene interaction via distance correlation coefficient (GBDcor). The distance correlation (dCor) is a measurement of the dependency between two random vectors with arbitrary, and not necessarily equal, dimensions. We used the difference in dCor in case and control datasets as an indicator of gene–gene interaction, which was based on the assumption that the joint distribution of two genes in case subjects and in control subjects should not be significantly different if the two genes do not interact. We designed a permutation-based statistical test to evaluate the difference between dCor in cases and controls for a pair of genes, and we provided the p-value for the statistic to represent the significance of the interaction between the two genes. In experiments with both simulated and real-world data, our method outperformed previous approaches in detecting interactions accurately.

On the uniqueness of distance covariance

This paper proposes a multivariate distance nonlinear causality test (MDNC) using the partial distance correlation in a time series framework. Partial distance correlation as an extension of the Brownian distance correlation calculates the distance correlation between random vectors X and Y controlling for a random vector Z. Our test can detect nonlinear lagged relationships between time series, and when integrated with machine learning methods it can improve the forecasting power. We apply our method as a feature selection procedure and combine it with the support vector machine and random forests algorithms to study the forecast of the main energy financial time series (oil, coal, and natural gas futures). It shows substantial improvement in forecasting the fuel energy time series in comparison to the classical Granger causality method in time series.

A record among a sequence of iid random variables $$X_1,X_2,\dots $$ on the real line is defined as a member $$X_n$$ such that $$X_n>\max (X_1,\cdots ,X_{n-1})$$ . Trying to generalize this concept to random vectors, or even stochastic processes with continuous sample paths, we introduce two different concepts: A simple record is a stochastic process (or a random vector) $${\varvec{X}}_n$$ that is larger than $${\varvec{X}}_1,\cdots ,{\varvec{X}}_{n-1}$$ in at least one component, whereas a complete record has to be larger than its predecessors in all components. In particular, the probability that a stochastic process $${\varvec{X}}_n$$ is a record as n tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the conditional distribution of $${\varvec{X}}_n$$ given that $${\varvec{X}}_n$$ is a record is derived.

In this work we derive a representation for the variation of the difference of singular components in the Skorokhod problem. in which besides the other terms the integrals from some local times are presented. In the special cases when these local times turn out to be identically equal to zero, we come to the bound of the above-mentioned variation which, in turn, is applied to the investigation of the existence and uniqueness of a strong solution of the stochastic differential system in a half-space and to the estimation of the distance in variation between the local times of two reflected (at zero) stochastic processes with the same diffusion and different drift coefficients

<!-- *** Custom HTML *** --> A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$. This is equivalent (see \cite{MOU}) with the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that $${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta, %% \eqno{(\ast)}$$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define generalized convolution of measures defined by the formula $${\cal L}(Q_1) \oplus_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$$ if the equation $(\ast)$ holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\Theta)$. We study here basic properties of this convolution, basic properties of $\oplus_{\mu}$-infinitely divisible distributions, $\oplus_{\mu}$-stable distributions and give a series of examples.

In this chapter we introduce some basic ideas of time series analysis and stochastic processes. Of particular importance are the concepts Of stationarity and the autocovariance and sample autocovariance functions. Some standard techniques are described for the estimation and removal of trend and seasonality (of known period) from an observed time series. These are illustrated with reference to the data sets in Section 1.1. The calculations in all the examples can be carried out using the programs supplied on the enclosed diskette. The data sets are contained in files with names ending in.DAT. For example, the Australian red wine sales are filed as WINE.DAT. Most of the topics covered in this chapter will be developed more fully in later sections of the book. The reader who is not already familiar with random variables and random vectors should first read Appendix A where a concise account of the required background is given.KeywordsAutocorrelation FunctionTime Series ModelQuadratic TrendStationary Time SeriesSeasonal ComponentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Patient anatomical variation during the radiotherapy course can be modeled using a stochastic process. In this process, spatial position of each subvolume in patient organs of interest is defined as a random vector described using a probability distribution function (pdf). Two main parameters, the mean and the standard deviation, of the pdf have been historically used to characterize patient anatomical variation during the radiation treatment. It has been demonstrated that treatment dose distribution in an organ of interest can be evaluated approximately using these two parameters alone, without the full knowledge of organ motion distribution. The approximation is, however, dependent on the scale of the standard deviation as well as the number of treatment delivery fractions. It is straightforward to estimate these two parameters if patient anatomical variation process is stationary. In this case, the two parameters are constants or time‐invariance during the treatment course. However, the estimation will be relatively difficult if patient anatomical variation process is non‐stationary.Patient anatomical variation in radiotherapy can be systematically managed using image feedback adaptive treatment technique. The fundamental difference between adaptive treatment technique and other image‐guided techniques is the use of patient‐specific treatment information. Adaptive technique intends to use all patient‐specific dose information ‐ including what has been delivered in the previous treatments, what can be delivered at the present treatment, and what would be delivered in future treatments ‐ in the design of treatment plan. Therefore, treatment plan designed in adaptive radiotherapy is called 4D adaptive plan, which is in principle a treatment control law to manage treatment process. Mathematically, treatment control law is a spatial mapping from the parameter space of patient variation to the parameter space of treatment delivery control, which can be determined including the patient variation in the planning optimization or inverse planning. The objectives in adaptive planning optimization are constructed based on a selection of control strategies that could be either the online or the offline with one control action, multiple actions or continue actions. Selection of control strategy and number of control actions is, of course, dependent on the nature of patient variation process as well as the estimation uncertainties, and has to be determined considering also the clinical practice.The lecture will provide an overview of the models and characteristics of patient anatomical variation process during the radiotherapy, the 4D dose summation methodology and the strategies of adaptive treatment process.Educational Objectives:1. Understand the characteristics and dynamic model for patient anatomical variation during the course of radiotherapy.2. Understand the model and methodology of 4D dose summation.3. Understand the options of control strategy for image guided adaptive radiation treatment.

Probabilistic study of Induced Ordered Linear Fusion Operators for time series forecasting

Chapter 7 Implementing Representations of Uncertainty

In this chapter, we introduce some important inequalities about stochastic processes and Banach space valued random elements. Because the image space of the random elements is more complicated than those for random variables or vectors, the proofs are generally more involved or need specific approaches. Therefore, proofs of some inequalities will be omitted and some references are given.

The Sharpe ratios and PNL curves of minimum variance, market and eigenportfolios for stocks in the Dow Jones Industrial Average (DJIA) index are calculated. We employed in this study a) the exponential function to approximate the measured cross correlations of the end of day (EOD) returns for US equities in DJIA, and b) their empirical correlation and covariance matrices to design the three portfolio types, and compare their market performance from May 4, 1999 to November 1, 2018. It is shown that the performances of portfolios derived by using exponential model based and empirical correlation and covariance matrices are consistent. We also displayed the PNL curve of DIA for performance comparison. It is observed from these PNLs that the first eigenportfolio significantly outperforms the other portfolios and DIA.