How is the auto-correlation of vectors defined?











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Suppose $v$ is an $n$-ary vector with entries from the set ${0,1}$ (i.e. a vector of ones and zeros).



A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the correlation operator.



1) What is an auto-correlation sequence of a vector?



2) What is the correlation operator? (I'm assuming it can be applied to two distinct vectors too)





My first guess was that to auto-correlate a vector you try all the possible rotational permutations of the vector and measure the cosine of the angle between each permuted vector with the original. However, Mathematica's CorrelationFunction on ${1,0}$ with $lag=0$ returns 1 and with $lag=1$ returns $-frac{1}{2}$, which shoots down my theory since I would expect orthogonal vectors to have $0$ correlation. So what is Mathematica doing here?










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    favorite












    Suppose $v$ is an $n$-ary vector with entries from the set ${0,1}$ (i.e. a vector of ones and zeros).



    A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the correlation operator.



    1) What is an auto-correlation sequence of a vector?



    2) What is the correlation operator? (I'm assuming it can be applied to two distinct vectors too)





    My first guess was that to auto-correlate a vector you try all the possible rotational permutations of the vector and measure the cosine of the angle between each permuted vector with the original. However, Mathematica's CorrelationFunction on ${1,0}$ with $lag=0$ returns 1 and with $lag=1$ returns $-frac{1}{2}$, which shoots down my theory since I would expect orthogonal vectors to have $0$ correlation. So what is Mathematica doing here?










    share|cite|improve this question
























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Suppose $v$ is an $n$-ary vector with entries from the set ${0,1}$ (i.e. a vector of ones and zeros).



      A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the correlation operator.



      1) What is an auto-correlation sequence of a vector?



      2) What is the correlation operator? (I'm assuming it can be applied to two distinct vectors too)





      My first guess was that to auto-correlate a vector you try all the possible rotational permutations of the vector and measure the cosine of the angle between each permuted vector with the original. However, Mathematica's CorrelationFunction on ${1,0}$ with $lag=0$ returns 1 and with $lag=1$ returns $-frac{1}{2}$, which shoots down my theory since I would expect orthogonal vectors to have $0$ correlation. So what is Mathematica doing here?










      share|cite|improve this question













      Suppose $v$ is an $n$-ary vector with entries from the set ${0,1}$ (i.e. a vector of ones and zeros).



      A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the correlation operator.



      1) What is an auto-correlation sequence of a vector?



      2) What is the correlation operator? (I'm assuming it can be applied to two distinct vectors too)





      My first guess was that to auto-correlate a vector you try all the possible rotational permutations of the vector and measure the cosine of the angle between each permuted vector with the original. However, Mathematica's CorrelationFunction on ${1,0}$ with $lag=0$ returns 1 and with $lag=1$ returns $-frac{1}{2}$, which shoots down my theory since I would expect orthogonal vectors to have $0$ correlation. So what is Mathematica doing here?







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      asked Nov 15 at 16:10









      Craig

      621314




      621314






















          1 Answer
          1






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          up vote
          3
          down vote



          accepted










          The sample correlation of vectors $(X_1, dots, X_n)$ and
          $(Y_1, dots, Y_n)$ is



          $$rho_{(X,Y)} = frac{frac{1}{n-1}sum_{i=1}^n (X_i - bar X)(Y_i - bar Y) }{S_XS_Y},$$
          where $bar X, bar Y$ are the respective sample means and $S_XS_Y$ are the respective sample standard deviations.



          Roughly speaking, the sample autocorrelation of lag $ell$ of a vector $(X_1, dots X_n)$ is the sample correlation of the
          vector $(X_1, dots, X_{n-ell})$ and and the lagged vector
          $(X_ell, X_{ell + 1}, dots, X_n).$



          Various refinements are used in specific applications.
          Perhaps the one you are looking for is of the following
          form:



          $$rho_ell = frac{sum_{i=1}^{n-ell} (X_1 - bar X)(X_ell - bar X) }{(n-1)S_X^2},$$
          Notice that $bar X$ and $S_X^2$ are based on the
          entire sequence. Also, when $ell=0,$ we have $rho_ell = 1.$
          See Wikipedia
          at the last bullet under Estimation.



          As I recall, this is used in the R function acf:



          set.seed(1115)
          x = round(rnorm(10,200,15))-20*(1:10); x
          [1] 176 169 127 99 96 92 45 70 10 12
          acf(x)
          acf(x, plot=F)

          Autocorrelations of series ‘x’, by lag

          0 1 2 3 4 5 6 7 8 9
          1.000 0.625 0.299 0.138 -0.060 -0.177 -0.304 -0.363 -0.434 -0.223


          enter image description here






          share|cite|improve this answer























          • Thanks! Very clear and concise.
            – Craig
            Nov 16 at 14:43











          Your Answer





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          1 Answer
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          1 Answer
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          active

          oldest

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          up vote
          3
          down vote



          accepted










          The sample correlation of vectors $(X_1, dots, X_n)$ and
          $(Y_1, dots, Y_n)$ is



          $$rho_{(X,Y)} = frac{frac{1}{n-1}sum_{i=1}^n (X_i - bar X)(Y_i - bar Y) }{S_XS_Y},$$
          where $bar X, bar Y$ are the respective sample means and $S_XS_Y$ are the respective sample standard deviations.



          Roughly speaking, the sample autocorrelation of lag $ell$ of a vector $(X_1, dots X_n)$ is the sample correlation of the
          vector $(X_1, dots, X_{n-ell})$ and and the lagged vector
          $(X_ell, X_{ell + 1}, dots, X_n).$



          Various refinements are used in specific applications.
          Perhaps the one you are looking for is of the following
          form:



          $$rho_ell = frac{sum_{i=1}^{n-ell} (X_1 - bar X)(X_ell - bar X) }{(n-1)S_X^2},$$
          Notice that $bar X$ and $S_X^2$ are based on the
          entire sequence. Also, when $ell=0,$ we have $rho_ell = 1.$
          See Wikipedia
          at the last bullet under Estimation.



          As I recall, this is used in the R function acf:



          set.seed(1115)
          x = round(rnorm(10,200,15))-20*(1:10); x
          [1] 176 169 127 99 96 92 45 70 10 12
          acf(x)
          acf(x, plot=F)

          Autocorrelations of series ‘x’, by lag

          0 1 2 3 4 5 6 7 8 9
          1.000 0.625 0.299 0.138 -0.060 -0.177 -0.304 -0.363 -0.434 -0.223


          enter image description here






          share|cite|improve this answer























          • Thanks! Very clear and concise.
            – Craig
            Nov 16 at 14:43















          up vote
          3
          down vote



          accepted










          The sample correlation of vectors $(X_1, dots, X_n)$ and
          $(Y_1, dots, Y_n)$ is



          $$rho_{(X,Y)} = frac{frac{1}{n-1}sum_{i=1}^n (X_i - bar X)(Y_i - bar Y) }{S_XS_Y},$$
          where $bar X, bar Y$ are the respective sample means and $S_XS_Y$ are the respective sample standard deviations.



          Roughly speaking, the sample autocorrelation of lag $ell$ of a vector $(X_1, dots X_n)$ is the sample correlation of the
          vector $(X_1, dots, X_{n-ell})$ and and the lagged vector
          $(X_ell, X_{ell + 1}, dots, X_n).$



          Various refinements are used in specific applications.
          Perhaps the one you are looking for is of the following
          form:



          $$rho_ell = frac{sum_{i=1}^{n-ell} (X_1 - bar X)(X_ell - bar X) }{(n-1)S_X^2},$$
          Notice that $bar X$ and $S_X^2$ are based on the
          entire sequence. Also, when $ell=0,$ we have $rho_ell = 1.$
          See Wikipedia
          at the last bullet under Estimation.



          As I recall, this is used in the R function acf:



          set.seed(1115)
          x = round(rnorm(10,200,15))-20*(1:10); x
          [1] 176 169 127 99 96 92 45 70 10 12
          acf(x)
          acf(x, plot=F)

          Autocorrelations of series ‘x’, by lag

          0 1 2 3 4 5 6 7 8 9
          1.000 0.625 0.299 0.138 -0.060 -0.177 -0.304 -0.363 -0.434 -0.223


          enter image description here






          share|cite|improve this answer























          • Thanks! Very clear and concise.
            – Craig
            Nov 16 at 14:43













          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          The sample correlation of vectors $(X_1, dots, X_n)$ and
          $(Y_1, dots, Y_n)$ is



          $$rho_{(X,Y)} = frac{frac{1}{n-1}sum_{i=1}^n (X_i - bar X)(Y_i - bar Y) }{S_XS_Y},$$
          where $bar X, bar Y$ are the respective sample means and $S_XS_Y$ are the respective sample standard deviations.



          Roughly speaking, the sample autocorrelation of lag $ell$ of a vector $(X_1, dots X_n)$ is the sample correlation of the
          vector $(X_1, dots, X_{n-ell})$ and and the lagged vector
          $(X_ell, X_{ell + 1}, dots, X_n).$



          Various refinements are used in specific applications.
          Perhaps the one you are looking for is of the following
          form:



          $$rho_ell = frac{sum_{i=1}^{n-ell} (X_1 - bar X)(X_ell - bar X) }{(n-1)S_X^2},$$
          Notice that $bar X$ and $S_X^2$ are based on the
          entire sequence. Also, when $ell=0,$ we have $rho_ell = 1.$
          See Wikipedia
          at the last bullet under Estimation.



          As I recall, this is used in the R function acf:



          set.seed(1115)
          x = round(rnorm(10,200,15))-20*(1:10); x
          [1] 176 169 127 99 96 92 45 70 10 12
          acf(x)
          acf(x, plot=F)

          Autocorrelations of series ‘x’, by lag

          0 1 2 3 4 5 6 7 8 9
          1.000 0.625 0.299 0.138 -0.060 -0.177 -0.304 -0.363 -0.434 -0.223


          enter image description here






          share|cite|improve this answer














          The sample correlation of vectors $(X_1, dots, X_n)$ and
          $(Y_1, dots, Y_n)$ is



          $$rho_{(X,Y)} = frac{frac{1}{n-1}sum_{i=1}^n (X_i - bar X)(Y_i - bar Y) }{S_XS_Y},$$
          where $bar X, bar Y$ are the respective sample means and $S_XS_Y$ are the respective sample standard deviations.



          Roughly speaking, the sample autocorrelation of lag $ell$ of a vector $(X_1, dots X_n)$ is the sample correlation of the
          vector $(X_1, dots, X_{n-ell})$ and and the lagged vector
          $(X_ell, X_{ell + 1}, dots, X_n).$



          Various refinements are used in specific applications.
          Perhaps the one you are looking for is of the following
          form:



          $$rho_ell = frac{sum_{i=1}^{n-ell} (X_1 - bar X)(X_ell - bar X) }{(n-1)S_X^2},$$
          Notice that $bar X$ and $S_X^2$ are based on the
          entire sequence. Also, when $ell=0,$ we have $rho_ell = 1.$
          See Wikipedia
          at the last bullet under Estimation.



          As I recall, this is used in the R function acf:



          set.seed(1115)
          x = round(rnorm(10,200,15))-20*(1:10); x
          [1] 176 169 127 99 96 92 45 70 10 12
          acf(x)
          acf(x, plot=F)

          Autocorrelations of series ‘x’, by lag

          0 1 2 3 4 5 6 7 8 9
          1.000 0.625 0.299 0.138 -0.060 -0.177 -0.304 -0.363 -0.434 -0.223


          enter image description here







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 16 at 6:14

























          answered Nov 16 at 5:51









          BruceET

          34.8k71440




          34.8k71440












          • Thanks! Very clear and concise.
            – Craig
            Nov 16 at 14:43


















          • Thanks! Very clear and concise.
            – Craig
            Nov 16 at 14:43
















          Thanks! Very clear and concise.
          – Craig
          Nov 16 at 14:43




          Thanks! Very clear and concise.
          – Craig
          Nov 16 at 14:43


















           

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