Cross covariance
Encyclopedia
In statistics
, the term cross-covariance is sometimes used to refer to the covariance
cov(X, Y) between two random vector
s X and Y, in order to distinguish that concept from the "covariance" of a random vector X, which is understood to be the matrix of covariances
between the scalar components of X.
In signal processing
, the cross-covariance (or sometimes "cross-correlation") is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time
between the signals, is sometimes called the sliding dot product
, and has applications in pattern recognition
and cryptanalysis
.
s whose expected value
and variance
exist, the cross-covariance matrix of X and Y is defined by
where μX and μY are vectors containing the expected values of X and Y. The vectors X and Y need not have the same dimension, and either might be a scalar value. Any element of the cross-covariance matrix is itself a "cross-covariance".
where the sum is over the appropriate values of the integer
j and an asterisk indicates the complex conjugate
. For continuous functions f (x) and g (x) the cross-covariance is defined as
where the integral is over the appropriate values of t.
The cross-covariance is similar in nature to the convolution
of two functions.
by:
so that
if either f or g is an even function. Also:
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the term cross-covariance is sometimes used to refer to the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
cov(X, Y) between two random vector
Coordinate vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
s X and Y, in order to distinguish that concept from the "covariance" of a random vector X, which is understood to be the matrix of covariances
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
between the scalar components of X.
In signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, the cross-covariance (or sometimes "cross-correlation") is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
between the signals, is sometimes called the sliding dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
, and has applications in pattern recognition
Pattern recognition
In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...
and cryptanalysis
Cryptanalysis
Cryptanalysis is the study of methods for obtaining the meaning of encrypted information, without access to the secret information that is normally required to do so. Typically, this involves knowing how the system works and finding a secret key...
.
Statistics
For random vectors, X and Y, each containing random elementRandom element
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line...
s whose expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
exist, the cross-covariance matrix of X and Y is defined by
where μX and μY are vectors containing the expected values of X and Y. The vectors X and Y need not have the same dimension, and either might be a scalar value. Any element of the cross-covariance matrix is itself a "cross-covariance".
Signal processing
For discrete functions fi and gi the cross-covariance is defined aswhere the sum is over the appropriate values of the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
j and an asterisk indicates the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
. For continuous functions f (x) and g (x) the cross-covariance is defined as
where the integral is over the appropriate values of t.
The cross-covariance is similar in nature to the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of two functions.
Properties
The cross-covariance of two signals is related to the convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
by:
so that
if either f or g is an even function. Also:
See also
- ConvolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
- CorrelationCorrelationIn statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
- AutocovarianceAutocovarianceIn statistics, given a real stochastic process X, the autocovariance is the covariance of the variable with itself, i.e. the variance of the variable against a time-shifted version of itself...
External links
- Cross Correlation from Mathworld
- http://scribblethink.org/Work/nvisionInterface/nip.html
- http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
- http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf