归一化互相关（转）

归一化互相关（转自https://anomaly.io/understand-auto-cross-correlation-normalized-shift/index.html）

Normalized Cross-Correlation

There are three problems with cross-correlation:

It is difficult to understand the scoring value.
Both metrics must have the same amplitude. If Graph B has the same shape as Graph A but values two times smaller, the correlation will not be detected.
corr(a, a/2) = 19.5
Due to the formula, a zero value will not be taken into account, since 00=0 and 0200=0.
To solve these problems we use normalized cross-correlation:

$$norm\_corr(x,y)=\frac{{\sum }_{n=0}^{n-1}x[n]\ast y[n]}{\sqrt{{\sum }_{n=0}^{n-1}x[n{]}^2\ast {\sum }_{n=0}^{n-1}y[n{]}^2}}$$

Using this formula let’s compute the normalized cross-correlation of AB and AC.

$$\begin{gathered} norm\_corr(a,b)=\frac{1\ast 2+2\ast 3+-2\ast -2+4\ast 3+2\ast 2+3\ast 4+1\ast 1+0\ast -1}{\sqrt{(1+4+4+16+4+9+1+0)\ast (4+9+4+9+4+16+1+1)}} \\ =\frac{41}{\sqrt{(39)\ast (48)}} \\ =0.947 \end{gathered}$$

$$\begin{gathered} norm\_corr(a,c)=\frac{1\ast -2+2\ast 0+-2\ast 4+4\ast 0+2\ast 1+3\ast 1+1\ast 0+0\ast -2}{\sqrt{(1+4+4+16+4+9+1+0)\ast (4+0+16+0+1+1+0+4)}} \\ =\frac{-5}{\sqrt{(39)\ast (26)}} \\ =-0.157 \end{gathered}$$

Graphs A and B correlate, with a high value of 0.947.
Graphs A and C don’t correlate, showing a low value of -0.157.

Normalized cross-correlation scoring is easy to understand:
– The higher the value, the higher the correlation is.
– The maximum value is 1 when two signals are exactly the same:
norm_corr(a,a)=1
– The minimum value is -1 when two signals are exactly opposite:
norm_corr(a, -a) = -1
Normalized cross-correlation can detect the correlation of two signals with different amplitudes: norma_corr(a, a/2) = 1.
Notice we have perfect correlation between signal A and the same signal with half the amplitude!