Why is cross-correlation not defined in a normalized sense?
When correlation is defined in systems and signals, as well as in the statistical sense, it is defined as a normalized measure with respect to the Cauchy-Schwarz inequality.
$space$
In systems and signals, correlation is defined as
$$rho = frac{ langle x(t), space y(t) rangle}{| x(t)||y(t)|} = frac{1}{sqrt{E_xE_y}}int_{-infty}^{infty}x(t)y^*(t)dt$$
Here, the inner product is normalized by dividing it by the square root of the energies, or the norms, of the two signals in question. The Cauchy-Schwarz inequality for this case is stated as
$$ |langle x(t), space y(t)rangle| leq | x(t)||y(t)|$$
$space$
In statistics, correlation is defined as
$$rho_{xy} = frac{Cov(X,Y)}{sigma_xsigma_y}$$
Here, the covariance is normalized by dividing it by the square root of the variances, or the standard deviations, of the two random variables in question. The Cauchy-Schwarz inequality for this case is stated as
$$|Cov(X,Y)| leq sigma_xsigma_y$$
$space$
So, it seems as if there is a universal trend for defining correlation in a normalized sense. That is until we get to cross-correlation. For some reason cross-correlation is defined as
$$ psi(tau) = langle x(t), space y(t-tau) rangle = int_{-infty}^{infty} x(t)y^*(t-tau)dt $$
Here, the cross-correlation is simply the time-shifted inner product. No normalization has been applied in this definition.
$space$
So, why do we define correlation in a normalized sense across the board, but when it comes to its' extended time delayed version, cross-correlation, we "forget" to normalize it all of a sudden?
correlation
asked Jan 5 at 23:46
GustavGustav
13
|
show 7 more comments
When correlation is defined in systems and signals, as well as in the statistical sense, it is defined as a normalized measure with respect to the Cauchy-Schwarz inequality.
$space$
In systems and signals, correlation is defined as
$$rho = frac{ langle x(t), space y(t) rangle}{| x(t)||y(t)|} = frac{1}{sqrt{E_xE_y}}int_{-infty}^{infty}x(t)y^*(t)dt$$
Here, the inner product is normalized by dividing it by the square root of the energies, or the norms, of the two signals in question. The Cauchy-Schwarz inequality for this case is stated as
$$ |langle x(t), space y(t)rangle| leq | x(t)||y(t)|$$
$space$
In statistics, correlation is defined as
$$rho_{xy} = frac{Cov(X,Y)}{sigma_xsigma_y}$$
Here, the covariance is normalized by dividing it by the square root of the variances, or the standard deviations, of the two random variables in question. The Cauchy-Schwarz inequality for this case is stated as
$$|Cov(X,Y)| leq sigma_xsigma_y$$
$space$
So, it seems as if there is a universal trend for defining correlation in a normalized sense. That is until we get to cross-correlation. For some reason cross-correlation is defined as
$$ psi(tau) = langle x(t), space y(t-tau) rangle = int_{-infty}^{infty} x(t)y^*(t-tau)dt $$
Here, the cross-correlation is simply the time-shifted inner product. No normalization has been applied in this definition.
$space$
So, why do we define correlation in a normalized sense across the board, but when it comes to its' extended time delayed version, cross-correlation, we "forget" to normalize it all of a sudden?
correlation
asked Jan 5 at 23:46
GustavGustav
13
1
It's just words. Don't make too much of it.
– herb steinberg
Jan 5 at 23:51
It's not just words. It's definitions which allow for mental associations between concepts. The more clearly and consistently they are defined the more fluently we can think about them.
– Gustav
Jan 5 at 23:53
I don't know if this helps, but covariance uses subtraction of the means, while cross-correlation does not.
– herb steinberg
Jan 6 at 0:03
@herb steinberg That is in the statistical sense, but in the signal sense no means are subtracted. In the statistical sense this is fine since the variances, which are used to normalize the covariance, also have mean subtraction.
– Gustav
Jan 6 at 0:07
You are never going to get an acceptable answer to this question.
– Somos
Jan 6 at 0:37
|
show 7 more comments
When correlation is defined in systems and signals, as well as in the statistical sense, it is defined as a normalized measure with respect to the Cauchy-Schwarz inequality.
$space$
In systems and signals, correlation is defined as
$$rho = frac{ langle x(t), space y(t) rangle}{| x(t)||y(t)|} = frac{1}{sqrt{E_xE_y}}int_{-infty}^{infty}x(t)y^*(t)dt$$
Here, the inner product is normalized by dividing it by the square root of the energies, or the norms, of the two signals in question. The Cauchy-Schwarz inequality for this case is stated as
$$ |langle x(t), space y(t)rangle| leq | x(t)||y(t)|$$
$space$
In statistics, correlation is defined as
$$rho_{xy} = frac{Cov(X,Y)}{sigma_xsigma_y}$$
Here, the covariance is normalized by dividing it by the square root of the variances, or the standard deviations, of the two random variables in question. The Cauchy-Schwarz inequality for this case is stated as
$$|Cov(X,Y)| leq sigma_xsigma_y$$
$space$
So, it seems as if there is a universal trend for defining correlation in a normalized sense. That is until we get to cross-correlation. For some reason cross-correlation is defined as
$$ psi(tau) = langle x(t), space y(t-tau) rangle = int_{-infty}^{infty} x(t)y^*(t-tau)dt $$
Here, the cross-correlation is simply the time-shifted inner product. No normalization has been applied in this definition.
$space$
So, why do we define correlation in a normalized sense across the board, but when it comes to its' extended time delayed version, cross-correlation, we "forget" to normalize it all of a sudden?
correlation
asked Jan 5 at 23:46
GustavGustav
13
When correlation is defined in systems and signals, as well as in the statistical sense, it is defined as a normalized measure with respect to the Cauchy-Schwarz inequality.
$space$
In systems and signals, correlation is defined as
$$rho = frac{ langle x(t), space y(t) rangle}{| x(t)||y(t)|} = frac{1}{sqrt{E_xE_y}}int_{-infty}^{infty}x(t)y^*(t)dt$$
Here, the inner product is normalized by dividing it by the square root of the energies, or the norms, of the two signals in question. The Cauchy-Schwarz inequality for this case is stated as
$$ |langle x(t), space y(t)rangle| leq | x(t)||y(t)|$$
$space$
In statistics, correlation is defined as
$$rho_{xy} = frac{Cov(X,Y)}{sigma_xsigma_y}$$
Here, the covariance is normalized by dividing it by the square root of the variances, or the standard deviations, of the two random variables in question. The Cauchy-Schwarz inequality for this case is stated as
$$|Cov(X,Y)| leq sigma_xsigma_y$$
$space$
So, it seems as if there is a universal trend for defining correlation in a normalized sense. That is until we get to cross-correlation. For some reason cross-correlation is defined as
$$ psi(tau) = langle x(t), space y(t-tau) rangle = int_{-infty}^{infty} x(t)y^*(t-tau)dt $$
Here, the cross-correlation is simply the time-shifted inner product. No normalization has been applied in this definition.
$space$
So, why do we define correlation in a normalized sense across the board, but when it comes to its' extended time delayed version, cross-correlation, we "forget" to normalize it all of a sudden?
correlation
correlation
asked Jan 5 at 23:46
GustavGustav
13
asked Jan 5 at 23:46
GustavGustav
13
asked Jan 5 at 23:46
GustavGustav
13
asked Jan 5 at 23:46
GustavGustav
13
asked Jan 5 at 23:46
GustavGustav
13
13
1
It's just words. Don't make too much of it.
– herb steinberg
Jan 5 at 23:51
It's not just words. It's definitions which allow for mental associations between concepts. The more clearly and consistently they are defined the more fluently we can think about them.
– Gustav
Jan 5 at 23:53
I don't know if this helps, but covariance uses subtraction of the means, while cross-correlation does not.
– herb steinberg
Jan 6 at 0:03
@herb steinberg That is in the statistical sense, but in the signal sense no means are subtracted. In the statistical sense this is fine since the variances, which are used to normalize the covariance, also have mean subtraction.
– Gustav
Jan 6 at 0:07
You are never going to get an acceptable answer to this question.
– Somos
Jan 6 at 0:37
|
show 7 more comments
1
It's just words. Don't make too much of it.
– herb steinberg
Jan 5 at 23:51
It's not just words. It's definitions which allow for mental associations between concepts. The more clearly and consistently they are defined the more fluently we can think about them.
– Gustav
Jan 5 at 23:53
I don't know if this helps, but covariance uses subtraction of the means, while cross-correlation does not.
– herb steinberg
Jan 6 at 0:03
@herb steinberg That is in the statistical sense, but in the signal sense no means are subtracted. In the statistical sense this is fine since the variances, which are used to normalize the covariance, also have mean subtraction.
– Gustav
Jan 6 at 0:07
You are never going to get an acceptable answer to this question.
– Somos
Jan 6 at 0:37
1
1
It's just words. Don't make too much of it.
– herb steinberg
Jan 5 at 23:51
It's just words. Don't make too much of it.
– herb steinberg
Jan 5 at 23:51
It's not just words. It's definitions which allow for mental associations between concepts. The more clearly and consistently they are defined the more fluently we can think about them.
– Gustav
Jan 5 at 23:53
It's not just words. It's definitions which allow for mental associations between concepts. The more clearly and consistently they are defined the more fluently we can think about them.
– Gustav
Jan 5 at 23:53
I don't know if this helps, but covariance uses subtraction of the means, while cross-correlation does not.
– herb steinberg
Jan 6 at 0:03
I don't know if this helps, but covariance uses subtraction of the means, while cross-correlation does not.
– herb steinberg
Jan 6 at 0:03
@herb steinberg That is in the statistical sense, but in the signal sense no means are subtracted. In the statistical sense this is fine since the variances, which are used to normalize the covariance, also have mean subtraction.
– Gustav
Jan 6 at 0:07
@herb steinberg That is in the statistical sense, but in the signal sense no means are subtracted. In the statistical sense this is fine since the variances, which are used to normalize the covariance, also have mean subtraction.
– Gustav
Jan 6 at 0:07
You are never going to get an acceptable answer to this question.
– Somos
Jan 6 at 0:37
You are never going to get an acceptable answer to this question.
– Somos
Jan 6 at 0:37
|
show 7 more comments
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1
It's just words. Don't make too much of it.
– herb steinberg
Jan 5 at 23:51
It's not just words. It's definitions which allow for mental associations between concepts. The more clearly and consistently they are defined the more fluently we can think about them.
– Gustav
Jan 5 at 23:53
I don't know if this helps, but covariance uses subtraction of the means, while cross-correlation does not.
– herb steinberg
Jan 6 at 0:03
@herb steinberg That is in the statistical sense, but in the signal sense no means are subtracted. In the statistical sense this is fine since the variances, which are used to normalize the covariance, also have mean subtraction.
– Gustav
Jan 6 at 0:07
You are never going to get an acceptable answer to this question.
– Somos
Jan 6 at 0:37