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Analogue (Continuous-Space) Image Representation
Published in Jiří Jan, Medical Image Processing, Reconstruction and Analysis, 2019
The covariance may be rather transparently interpreted. If both variables are dependent, though stochastically, we may expect that the differences from the respective mean values will be mostly of the same sign (positively related variables) or mostly of opposite signs (inversely related variables). In the first case, most products in the mean (Equation 1.73) will then be positive, leading to a positive value of the average (i.e., of the covariance estimate). With inverse relation, the products will be mostly negative, thus giving a negative covariance estimate value. Strong relations with most products of the same sign provide high absolute covariance values, while products of randomly changing signs, yielding the low absolute value of the covariance estimate, indicate weak dependencies. Independent variables may be expected to have equally frequent differences with identical signs as differences with opposite signs; the mean then approaches zero. Variables, the covariance of which is zero, are called uncorrelated; while it can be shown that independent random variables are always uncorrelated, the opposite is not generally true. Still, it is often necessary to make a hypothesis that variables whose absolute covariance has been found to be small are independent, as the covariance may be the only available information. The hypothesis should be confirmed independently (e.g., by the consistency of some consequential results).
Introduction to Signal Processing
Published in Ralph D. Hippenstiel, Detection Theory, 2017
Correlating two data sequences provides a measure of likeness between the two sequences. If one normalizes the correlation outputs, the values take on a value between minus and plus one. The normalization is a division of the square root by the product of the energy of the two sequences involved. A minus one and plus one correspond to -100 percent and +100 percent, respectively. When the normalized cross-correlation value is +1, the two sequences are identical. Conversely, a value of –1 indicates that the two sequences are identical in a magnitude sense, but differ in-phase by 180°. A correlation of a value of zero indicates that the two sequences are uncorrelated. Assuming that the sequences are of equal length, a typical correlation expression, disregarding the normalization, is given by
Data Tours
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
The fact that two or more signals are unrelated can be expressed in terms of statistical independence. As we know from basic probability and statistics, if random variables (or signals) are statistically independent, then the value of one of the variables would not give us any information about the value of the other variables (or source signals) in the mixture. It is important to note that statistical independence is a much stronger requirement than lack of correlation. Two variables that are statistically independent will also be uncorrelated, but the fact that two variables are uncorrelated does not imply that they are independent. An exception to this is with Gaussian data, because in this case uncorrelated variables are also independent. ICA seeks to separate the data into a set of statistically independent or unrelated component signals or variables, which are then assumed to be from some meaningful sources or factors.
An Optimized Neuro-Fuzzy Network for Software Project Effort Estimation
Published in IETE Journal of Research, 2022
Sudhir Sharma, Shripal Vijayvargiya
Features of a project from a given dataset generate the feature vector, which is subjected to feature selection using correlation coefficient. These selected most significant features are used for the software effort (or cost) estimation process. The correlation coefficient method is applied to determine the similarity measures between two linearly dependent elements in the range of −1–1. For the uncorrelated features, the correlation coefficient is 0. The correlation coefficient is computed using the following formula. Where, represents the standard deviation of , represents the standard deviation of , is the covariance of the rank variables and is the correlation coefficient.
Effect of Microelectrode Recording in Accurate Targeting STN with High Frequency DBS in Parkinson Disease
Published in IETE Journal of Research, 2022
Venkateshwarla Rama Raju, Rukmini Kandadai Mridula, Rupam Borgohain
Given sets of variates denoted , the covariance of and is defined by where and are the means of and . The matrix of the quantities is called the covariance matrix. In the special case , giving the usual variance . Note that statistically independent variables are always uncorrelated, but the converse is not necessarily true. The covariance of two variates and provides a measure of how strongly correlated these variables are, and the derived quantity where and are the standard deviations (SDs), is called statistical correlation of and . The covariance is symmetric since
A conservative chaotic system with coexisting chaotic-like attractors and its application in image encryption
Published in Journal of Control and Decision, 2023
It can be seen from Table 6 and Figure 11 that the correlation between the adjacent pixels of the original image is strong and linear, while the pixel values of the adjacent pixels of the encrypted image are uniformly dense on the whole plane and show random correlation characteristics. The correlation coefficient of an uncorrelated sequence is zero in theory. This paper is an approach to the theoretical values. To sum up, combined with the above analysis, the initial value of the chaotic system is a pseudo-random sequence, which achieves better anti statistical attack performance.