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Extrema and Variational Calculus
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
Another useful statistical quantity is the covariance. The covariance is a measure of the relationship between two variables. It is defined as Cov(x,y)=1N∑i=1N(xj−μx)(yj−μy).
Causality Analysis of Multivariate Neural Data
Published in Hualou Liang, Joseph D. Bronzino, Donald R. Peterson, Biosignal Processing, 2012
The value of covariance Rxy(s) or correlation Cxy(s) tells how much the two signals X and Y change together. Correlation may be treated as a normalized version of covariance: its value, unlike covariance, is limited to the [–1, 1] range. Big absolute values of these functions indicate that changes in signal X and changes in signal Y appear in common. Negative values of covariance or correlation indicate that the changes in Y coincide with changes in X, but are in opposite direction. The covariance (correlation) defined above is in fact the covariance (correlation) function. The argument s of these functions, called a time lag, tells us about common changes (of the signals) which are delayed by the value of the time lag (positive or negative). If R is calculated for two different signals it is called cross-covariance function; if X = Y we call R autocovariance function.
Basics of probability and statistics
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
In Eq. (2.45), Cov(X,Y) represents the covariance of X and Y. Covariance is a measure of the association or dependence between two random variables, X and Y. The covariance between two random variables depends on extend of the data values. Covariance can be either positive or negative.
Covariance Logarithmic Aggregation Operators in Decision-Making Processes
Published in Cybernetics and Systems, 2023
Miriam Edith Pérez-Romero, Víctor G. Alfaro-García, José M. Merigó, Martha Beatriz Flores-Romero
During the last few decades, there have been some interesting advances in decision-making modeling. Aggregation operators have proven useful in decision-making situations in a variety of fields, such as engineering, economics, education, biology, computer science and statistics (Torra and Narukawa 2007; Beliakov, Pradera, and Calvo 2007); particularly in this last area, they have been introduced for the study of variance, covariance and the Pearson coefficient (Yager 1996; Yager 2006; Merigó 2012; Merigó, Guillén, and Sarabia 2015), in addition to the fact that the aggregation operators include arithmetic mean, weighted average and the estimated value as special cases (Kacprzyk, Yager, and Merigó 2019). Specifically, covariance is a fundamental concept in statistics that is used to measure data dispersion; it is viewed as a method of averaging the individual dispersions and the interaction between two sets of variables (Merigó, Guillén, and Sarabia 2015). It is a method for determining how two random variables are related and it reflects the degree of joint variation of the variables with respect to their means (Blanco-Mesa, León-Castro, and Merigó 2020).
Semi-Supervised Clustering Ensemble Based on Cluster Consensus Selection
Published in Cybernetics and Systems, 2022
Yanxi Liu, Ali Hussein Demin Al-Khafaji
Covariance is a way to show the relationship between two variables (Li, Rezaeipanah, and El Din 2022). In general, the positive covariance value indicates a direct relationship between two variables. This means that as one variable increases, another variable also increases. Also, the negative covariance indicates an inverse relationship between two variables. This means that as one variable increases another decrease. Since the covariance depends on the data measurement unit, it is not possible to compare the covariance of two variables without considering their measurement unit. The correlation coefficient is a unitless index and is used to solve this problem. Like covariance, correlation coefficient can also measure the dependence between two variables. The correlation coefficient of two variables is in the range of −1 and +1, where values farther than 0 indicate that the two variables are in the same direction or opposite.
Multivariate Fragility Functions for Seismic Landslide Hazard Assessment
Published in Journal of Earthquake Engineering, 2021
Yaser Jafarian, Ali Lashgari, Mohsen Miraiei
where is mean vector and a vector of linear functions of X, and is the covariance matrix which depends on X and composed of the corresponding submatrix of variance-covariance (). Covariance refers to the measure of how two random variables will change together and it is used to calculate the correlation between the input variables. These formulations show that if and X are not independent, then , and the conditional distribution of given X, , is multivariate normal with and . and can be determined by the following equations: