PROPOSED DETECTION SYSTEMPROPOSED DETECTION SYSTEM
5.4 Motion detection using Correlation Network Motion detection using Correlation Network
5.4 Motion detection using Correlation Network
A correlation neural network (CNN) which accounts for velocity sensitive A correlation neural network (CNN) which accounts for velocity sensitive res
responponses ses of of neuneuronrons s is is susuitaitable ble for for anaanalog log circcircuit uit impimplemelementantatiotion n of of motmotion ion--detection systems and has been successfully implemented on CMOS. The CNN detection systems and has been successfully implemented on CMOS. The CNN utilizes local motion detectors to correlate signals sampled at one location in the utilizes local motion detectors to correlate signals sampled at one location in the image with those sampled after a delay at adjacent locations; however, an image with those sampled after a delay at adjacent locations; however, an edge-detection process is required in practical motion edge-detection systems with the CNNs.
detection process is required in practical motion detection systems with the CNNs.
The term correlation can also mean the cross-correlation
The term correlation can also mean the cross-correlation of twoof two functions or functions or ele
electroctron n corcorrelarelatiotion n in in molmolecuecular lar sysystestemsms.. In In proprobabbabiliility ty thetheory ory and and stastatististictics,s, correlation, also called correlation coefficient, indicates the strength and direction of correlation, also called correlation coefficient, indicates the strength and direction of a
a linlinear ear relrelatioationshnship ip betbetween ween two two ranrandom dom varvariabiablesles. . In In gengeneraeral l stastatististictical al ususageage,, correlation or co-relation refers to the departure of two variables from independence, correlation or co-relation refers to the departure of two variables from independence, although correlation does not imply causation. In this broad sense there are several although correlation does not imply causation. In this broad sense there are several coefficients, measuring the degree of correlation, adapted to the nature of data. A coefficients, measuring the degree of correlation, adapted to the nature of data. A numb
number er of different coefficientof different coefficients s are used are used for different situatiofor different situations. The ns. The best known isbest known is the Pearson product-moment correlation coefficient, which is obtained by dividing the Pearson product-moment correlation coefficient, which is obtained by dividing the covariance of the two variables by the product of their standard deviations.
the covariance of the two variables by the product of their standard deviations.
5.4.1 Mathematical properties 5.4.1 Mathematical properties
The correlation ρ
The correlation ρX, YX, Y between two random variablesbetween two random variables X X andand Y Y with expected values μwith expected values μ X X
and μ
and μY Y and standard deviations σand standard deviations σ X X and σand σY Y is defined as:is defined as:
Where E is the expected value of the variable and cov means covariance. Since μ Where E is the expected value of the variable and cov means covariance. Since μXX == E(X), σ
E(X), σXX22= E(X= E(X22) − E) − E22(X) and likewise for (X) and likewise for Y Y , we may also write, we may also write
The correlation is defined only if both of the standard deviations are finite and both The correlation is defined only if both of the standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the of them are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.
correlation cannot exceed 1 in absolute value.
The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a de
decrecreasasining g lilinenear ar relrelatatioionsnshihip, p, anand d sosome me vavalulue e in in bebetwtweeeen n in in alall l ototheher r cascaseses,, indicating the degree of linear dependence between the variables. The closer the indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
coefficient is to either −1 or 1, the stronger the correlation between the variables.
If the variables are independent then the correlation is 0, but the converse is not true If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two because the correlation coefficient detects only linear dependencies between two
va
variariablbleses. . HeHere re is is an an exexamamplple: e: SuSupppposose e ththe e ranrandodom m vavariariablble e X X is is ununifoiformrmlyly distributed on the interval from −1 t
distributed on the interval from −1 to 1, ando 1, and Y Y = X= X22. Then. Then Y Y is completely determinedis completely determined b
by y X, X, so so ththat at X X anand d Y Y arare e dedepependndenent, t, bubut t ththeieir r cocorrrrelelatatioion n is is zezeroro; ; ththey ey araree un
uncocorrerrelalatedted. . HoHowewevever, r, in in ththe e spspececial ial cascase e whwhen en X X anand d Y Y are are jojoinintltly y nonormrmalal,, independence is equivalent to uncorrelated ness. A correlation between two variables independence is equivalent to uncorrelated ness. A correlation between two variables is diluted in the presence of measurement error around estimates of one or both is diluted in the presence of measurement error around estimates of one or both variables, in which case disattenuation provides a more accurate coefficient.
variables, in which case disattenuation provides a more accurate coefficient.
5.4.2 Geometric Interpretation of correlation 5.4.2 Geometric Interpretation of correlation
The correlation coefficient can also be viewed as the cosine of the angle The correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables.
between the two vectors of samples drawn from the two random variables.
This method only works with centered data, i.e., data
This method only works with centered data, i.e., data which have been shiftedwhich have been shifted by the sample mean so as to have an average of zero. Some practitioners prefer an by the sample mean so as to have an average of zero. Some practitioners prefer an
uncen
uncentered tered (non-(non-PearsoPearson-comn-compliantpliant) ) correlacorrelation tion coefficcoefficient. ient. See See the the exampexample le belowbelow for a comparison.
for a comparison.
As an example, suppose five countries are found to have gross national As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18%
countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18%
poverty. Then let
poverty. Then let xx andand yy be ordered 5-element vectors containing the above data:be ordered 5-element vectors containing the above data: xx
= (1, 2, 3, 5, 8) and
= (1, 2, 3, 5, 8) and yy = (0.11, 0.12, 0.13, 0.15, 0.18). By the usual procedure for = (0.11, 0.12, 0.13, 0.15, 0.18). By the usual procedure for finding the angle between two vectors, the uncentered correlation coefficient is:
finding the angle between two vectors, the uncentered correlation coefficient is:
Note that the above data were deliberately chosen to be perfectly correlated:
Note that the above data were deliberately chosen to be perfectly correlated: y y = 0.10= 0.10 +
+ 0.0.0101 x x. . ThThe e PePearsarson on cocorrrrelaelatiotion n cocoefefficficieient nt mumust st ththereerefofore re be be exexacactltly y onone.e.
Centering the data (shifting
Centering the data (shifting xx by E(by E(xx) = 3.8 and) = 3.8 and yy by E(by E(yy) = 0.138) yields) = 0.138) yields xx = (-2.8,= (-2.8, -1.8, -0.8, 1.2, 4.2) and
-1.8, -0.8, 1.2, 4.2) and yy = (-0.028, -0.018, -0.008, 0.012, 0.042), from = (-0.028, -0.018, -0.008, 0.012, 0.042), from whichwhich
as expected.
as expected.
5.4.3 Interpretation of the size of a correlation 5.4.3 Interpretation of the size of a correlation
Several authors have offered guidelines for the interpretation of a correlation Several authors have offered guidelines for the interpretation of a correlation coeffi
coefficient. Cohen cient. Cohen (198(1988), 8), for for exampexample, le, has has suggsuggested the ested the followfollowing ing interpinterpretatioretationsns for correlations in psychological research, in the table in the bottom.
for correlations in psychological research, in the table in the bottom.
As Cohen himself has observed, however, all such criteria are in some ways As Cohen himself has observed, however, all such criteria are in some ways arbitrary and should not be observed too strictly. This is because the interpretation of arbitrary and should not be observed too strictly. This is because the interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.9 a correlation coefficient depends on the context and purposes. A correlation of 0.9 may be very low if one is verifying a physical law using high-quality instruments, may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a greater but may be regarded as very high in the social sciences where there may be a greater
contribution from complicating factors contribution from complicating factors
C
Coorrrreellaattiioonn NNeeggaattiivvee PPoossiittiivvee S
Smmaallll −−00..229 9 tto o −−00..1100 00..110 0 tto o 00..2299 M
Meeddiiuumm −−00..449 9 tto o −−00..3300 00..330 0 tto o 00..4499 L
Laarrggee −−11..000 0 tto o −−00..5500 00..550 0 tto o 11..0000 Table 1
Table 1
Fig 15 An unit network of
Fig 15 An unit network of two-dimensional CCN.two-dimensional CCN.
Fig 16 Flow chart for correlation Fig 16 Flow chart for correlation