Online Covariance
by Joshua Burkholder
online_covariance.pdf
online_covariance.docx

Given the following set of two-dimensional inputs:

{ ( x 1 , y 1 ),( x 2 , y 2 ),,( x n1 , y n1 ),( x n , y n ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada qadaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyEamaa BaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaacYcadaqadaqaai aadIhadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyEamaaBaaaleaa caaIYaaabeaaaOGaayjkaiaawMcaaiaacYcacqWIMaYscaGGSaWaae WaaeaacaWG4bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaa cYcacaWG5bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaOGaay jkaiaawMcaaiaacYcadaqadaqaaiaadIhadaWgaaWcbaGaamOBaaqa baGccaGGSaGaamyEamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawM caaaGaay5Eaiaaw2haaaaa@5894@

Let n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ be the number of two-dimensional inputs, X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@36D4@  represent the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@  dimension, Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@36D5@  represent the y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@  dimension, Co v n ( X,Y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaaaaa@3DCB@  be the biased sample covariance of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@  and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@  dimensions for the first n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ two-dimensional inputs, Co v n1 ( X,Y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakmaabmaa baGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaaaaa@3F73@ be the biased sample covariance of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ dimensions for the first n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaaigdaaaa@3892@ two-dimensional inputs, x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGUbaabeaaaaa@3813@ be the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ value of the n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ -th two-dimensional input, x ¯ n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara WaaSbaaSqaaiaad6gaaeqaaaaa@382B@ be the sample mean of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ values for the first n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ two-dimensional inputs, y n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGUbaabeaaaaa@3814@ be the y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ value of the n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ -th two-dimensional input, and y ¯ n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaaa@39D4@ be the sample mean of the y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ values for the first n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaaigdaaaa@3892@ two-dimensional inputs.  Then, the recurrence equation for the biased sample covariance (a.k.a. online covariance) is:

Co v n ( X,Y )=Co v n1 ( X,Y ) Co v n1 ( X,Y )( x n x ¯ n )( y n y ¯ n1 ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpcaWGdbGaam4BaiaadAhada WgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGa aiilaiaadMfaaiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadoeaca WGVbGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqa daqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgkHiTmaabm aabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqe amaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaGaam yEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaaBaaa leaacaWGUbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaeaaca WGUbaaaaaa@6404@

Note: The recurrence equation above also applies when computing the online covariance matrix:

Σ n [ j,k ]= Σ n1 [ j,k ] Σ n1 [ j,k ]( x n [ j ] x[ j ] ¯ n )( x n [ k ] x[ k ] ¯ n1 ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacceGae83Odm 1aaSbaaSqaaiaad6gaaeqaaOWaamWaaeaacaWGQbGaaiilaiaadUga aiaawUfacaGLDbaacqGH9aqpcqWFJoWudaWgaaWcbaGaamOBaiabgk HiTiaaigdaaeqaaOWaamWaaeaacaWGQbGaaiilaiaadUgaaiaawUfa caGLDbaacqGHsisldaWcaaqaaiab=n6atnaaBaaaleaacaWGUbGaey OeI0IaaGymaaqabaGcdaWadaqaaiaadQgacaGGSaGaam4AaaGaay5w aiaaw2faaiabgkHiTmaabmaabaGaaCiEamaaBaaaleaacaWGUbaabe aakmaadmaabaGaamOAaaGaay5waiaaw2faaiabgkHiTmaanaaabaGa aCiEamaadmaabaGaamOAaaGaay5waiaaw2faaaaadaWgaaWcbaGaam OBaaqabaaakiaawIcacaGLPaaadaqadaqaaiaahIhadaWgaaWcbaGa amOBaaqabaGcdaWadaqaaiaadUgaaiaawUfacaGLDbaacqGHsislda qdaaqaaiaahIhadaWadaqaaiaadUgaaiaawUfacaGLDbaaaaWaaSba aSqaaiaad6gacqGHsislcaaIXaaabeaaaOGaayjkaiaawMcaaaqaai aad6gaaaaaaa@6D91@ .

However, we will restrict ourselves to the online covariance computation of two-dimensional input in this post and explore the online covariance matrix computation of m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E9@ -dimensional input in a later post.

 

Proof:

The definition of the biased sample covariance of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@  and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ dimensions for the first n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ two-dimensional inputs is defined as:

Co v n ( X,Y )= i=1 n ( x i x ¯ n )( y i y ¯ n ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaaqahabaWaae WaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmiEayaa raWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaaca WG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmyEayaaraWaaSba aSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0 JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaaiaad6gaaaaaaa@536E@ .

If we expand this definition, we have:

Co v n ( X,Y )= i=1 n1 ( x i x ¯ n )( y i y ¯ n ) +( x n x ¯ n )( y n y ¯ n ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaaqahabaWaae WaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmiEayaa raWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaaca WG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmyEayaaraWaaSba aSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0 JaaGymaaqaaiaad6gacqGHsislcaaIXaaaniabggHiLdGccqGHRaWk daqadaqaaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislceWG4b GbaebadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaadaqadaqa aiaadMhadaWgaaWcbaGaamOBaaqabaGccqGHsislceWG5bGbaebada WgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaaaeaacaWGUbaaaaaa @63AE@ .

Since the recurrence equations for the sample mean of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ values are:

x ¯ n = x ¯ n1 x ¯ n1 x n n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara WaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JabmiEayaaraWaaSbaaSqa aiaad6gacqGHsislcaaIXaaabeaakiabgkHiTmaalaaabaGabmiEay aaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaa dIhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaaa@460A@  and y ¯ n = y ¯ n1 y ¯ n1 y n n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JabmyEayaaraWaaSbaaSqa aiaad6gacqGHsislcaaIXaaabeaakiabgkHiTmaalaaabaGabmyEay aaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaa dMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaaa@460E@ ,

then we have:

Co v n ( X,Y )= i=1 n1 ( x i ( x ¯ n1 x ¯ n1 x n n ) )( y i ( y ¯ n1 y ¯ n1 y n n ) ) +( x n x ¯ n )( y n y ¯ n ) n Co v n ( X,Y )= i=1 n1 ( x i x ¯ n1 + x ¯ n1 x n n )( y i y ¯ n1 + y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n ) n Co v n ( X,Y )= i=1 n1 ( x i y i x i y ¯ n1 + x i ( y ¯ n1 y n n ) x ¯ n1 y i + x ¯ n1 y ¯ n1 x ¯ n1 ( y ¯ n1 y n n ) +( x ¯ n1 x n n ) y i ( x ¯ n1 x n n ) y ¯ n1 +( x ¯ n1 x n n )( y ¯ n1 y n n ) ) +( x n x ¯ n )( y n y ¯ n ) n Co v n ( X,Y )= ( i=1 n1 ( x i y i x i y ¯ n1 x ¯ n1 y i + x ¯ n1 y ¯ n1 ) + i=1 n1 ( x i ( y ¯ n1 y n n ) x ¯ n1 ( y ¯ n1 y n n ) ) + i=1 n1 ( ( x ¯ n1 x n n ) y i ( x ¯ n1 x n n ) y ¯ n1 +( x ¯ n1 x n n )( y ¯ n1 y n n ) ) +( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( i=1 n1 ( x i x ¯ n1 )( y i y ¯ n1 ) +( y ¯ n1 y n n ) i=1 n1 ( x i x ¯ n1 ) +( x ¯ n1 x n n ) i=1 n1 ( y i y ¯ n1 +( y ¯ n1 y n n ) ) +( x n x ¯ n )( y n y ¯ n ) ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGdb Gaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfa caGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maalaaabaWaaabCae aadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsisldaqa daqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqaba GccqGHsisldaWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOe I0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaa GcbaGaamOBaaaaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaqadaqa aiaadMhadaWgaaWcbaGaamyAaaqabaGccqGHsisldaqadaqaaiqadM hagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsisl daWcaaqaaiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaa qabaGccqGHsislcaWG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOB aaaaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqaaiaadMgacqGH9a qpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabgUca RmaabmaabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadI hagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaa baGaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeam aaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaaqaaiaad6gaaaaa baGaam4qaiaad+gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaae aacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqa amaaqahabaWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey OeI0IabmiEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaa kiabgUcaRmaalaaabaGabmiEayaaraWaaSbaaSqaaiaad6gacqGHsi slcaaIXaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaamOBaaqabaaa keaacaWGUbaaaaGaayjkaiaawMcaamaabmaabaGaamyEamaaBaaale aacaWGPbaabeaakiabgkHiTiqadMhagaqeamaaBaaaleaacaWGUbGa eyOeI0IaaGymaaqabaGccqGHRaWkdaWcaaqaaiqadMhagaqeamaaBa aaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG5bWaaSba aSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaaaSqaai aadMgacqGH9aqpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0Gaeyye IuoakiabgUcaRmaabmaabaGaamiEamaaBaaaleaacaWGUbaabeaaki abgkHiTiqadIhagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaa wMcaamaabmaabaGaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTi qadMhagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaaqa aiaad6gaaaaabaGaam4qaiaad+gacaWG2bWaaSbaaSqaaiaad6gaae qaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH 9aqpdaWcaaqaamaaqahabaWaaeWaaqaabeqaaiaadIhadaWgaaWcba GaamyAaaqabaGccaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0Ia amiEamaaBaaaleaacaWGPbaabeaakiqadMhagaqeamaaBaaaleaaca WGUbGaeyOeI0IaaGymaaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaa dMgaaeqaaOWaaeWaaeaadaWcaaqaaiqadMhagaqeamaaBaaaleaaca WGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaa d6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaacqGHsislceWG4b GbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaamyEamaa BaaaleaacaWGPbaabeaakiabgUcaRiqadIhagaqeamaaBaaaleaaca WGUbGaeyOeI0IaaGymaaqabaGcceWG5bGbaebadaWgaaWcbaGaamOB aiabgkHiTiaaigdaaeqaaOGaeyOeI0IabmiEayaaraWaaSbaaSqaai aad6gacqGHsislcaaIXaaabeaakmaabmaabaWaaSaaaeaaceWG5bGb aebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0Iaam yEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaacaGLOaGaayzk aaaabaGaey4kaSYaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBaaale aacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqa aiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyOeI0YaaeWaaeaadaWcaaqaaiqadIha gaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislca WG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGL PaaaceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaO Gaey4kaSYaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaacaWG UbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6 gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaadaqadaqaamaalaaa baGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaki abgkHiTiaadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGa ayjkaiaawMcaaaaacaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gacqGHsislcaaIXaaaniabggHiLdGccqGHRaWkdaqa daqaaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislceWG4bGbae badaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaadaqadaqaaiaa dMhadaWgaaWcbaGaamOBaaqabaGccqGHsislceWG5bGbaebadaWgaa WcbaGaamOBaaqabaaakiaawIcacaGLPaaaaeaacaWGUbaaaaqaaiaa doeacaWGVbGaamODamaaBaaaleaacaWGUbaabeaakmaabmaabaGaam iwaiaacYcacaWGzbaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaqa daabaeqabaWaaabCaeaadaqadaqaaiaadIhadaWgaaWcbaGaamyAaa qabaGccaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaamiEamaa BaaaleaacaWGPbaabeaakiqadMhagaqeamaaBaaaleaacaWGUbGaey OeI0IaaGymaaqabaGccqGHsislceWG4bGbaebadaWgaaWcbaGaamOB aiabgkHiTiaaigdaaeqaaOGaamyEamaaBaaaleaacaWGPbaabeaaki abgUcaRiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqa baGcceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaa GccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6ga cqGHsislcaaIXaaaniabggHiLdGccqGHRaWkdaaeWbqaamaabmaaba GaamiEamaaBaaaleaacaWGPbaabeaakmaabmaabaWaaSaaaeaaceWG 5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0 IaamyEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaacaGLOaGa ayzkaaGaeyOeI0IabmiEayaaraWaaSbaaSqaaiaad6gacqGHsislca aIXaaabeaakmaabmaabaWaaSaaaeaaceWG5bGbaebadaWgaaWcbaGa amOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaaca WGUbaabeaaaOqaaiaad6gaaaaacaGLOaGaayzkaaaacaGLOaGaayzk aaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6gacqGHsislcaaIXa aaniabggHiLdaakeaacqGHRaWkdaaeWbqaamaabmaabaWaaeWaaeaa daWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaa qabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOB aaaaaiaawIcacaGLPaaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaey OeI0YaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGa eyOeI0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaae qaaaGcbaGaamOBaaaaaiaawIcacaGLPaaaceWG5bGbaebadaWgaaWc baGaamOBaiabgkHiTiaaigdaaeqaaOGaey4kaSYaaeWaaeaadaWcaa qaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGc cqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaai aawIcacaGLPaaadaqadaqaamaalaaabaGabmyEayaaraWaaSbaaSqa aiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcba GaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaaaGaayjkaiaa wMcaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbGaeyOeI0IaaG ymaaqdcqGHris5aOGaey4kaSYaaeWaaeaacaWG4bWaaSbaaSqaaiaa d6gaaeqaaOGaeyOeI0IabmiEayaaraWaaSbaaSqaaiaad6gaaeqaaa GccaGLOaGaayzkaaWaaeWaaeaacaWG5bWaaSbaaSqaaiaad6gaaeqa aOGaeyOeI0IabmyEayaaraWaaSbaaSqaaiaad6gaaeqaaaGccaGLOa GaayzkaaaaaiaawIcacaGLPaaaaeaacaWGUbaaaaqaaiaadoeacaWG VbGaamODamaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiwaiaacY cacaWGzbaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaqadaabaeqa baWaaabCaeaadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccq GHsislceWG4bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqa aaGccaGLOaGaayzkaaWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMgaae qaaOGaeyOeI0IabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaI XaaabeaaaOGaayjkaiaawMcaaaWcbaGaamyAaiabg2da9iaaigdaae aacaWGUbGaeyOeI0IaaGymaaqdcqGHris5aOGaey4kaSYaaeWaaeaa daWcaaqaaiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaa qabaGccqGHsislcaWG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOB aaaaaiaawIcacaGLPaaadaaeWbqaamaabmaabaGaamiEamaaBaaale aacaWGPbaabeaakiabgkHiTiqadIhagaqeamaaBaaaleaacaWGUbGa eyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaSqaaiaadMgacqGH9a qpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoaaOqaaiab gUcaRmaabmaabaWaaSaaaeaaceWG4bGbaebadaWgaaWcbaGaamOBai abgkHiTiaaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaa beaaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaabCaeaadaqadaqaai aadMhadaWgaaWcbaGaamyAaaqabaGccqGHsislceWG5bGbaebadaWg aaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaey4kaSYaaeWaaeaada WcaaqaaiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqa baGccqGHsislcaWG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaa aaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqp caaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabgUcaRm aabmaabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIha gaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaaba GaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaa BaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaaaacaGLOaGaayzkaa aabaGaamOBaaaaaaaa@50C8@

Since the biased sample covariance of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ dimensions for the first n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaaigdaaaa@3892@ two-dimensional inputs is defined as:

Co v n1 ( X,Y )= i=1 n1 ( x i x ¯ n1 )( y i y ¯ n1 ) n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakmaabmaa baGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaGaeyypa0ZaaSaaae aadaaeWbqaamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaakiab gkHiTiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqaba aakiaawIcacaGLPaaadaqadaqaaiaadMhadaWgaaWcbaGaamyAaaqa baGccqGHsislceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaig daaeqaaaGccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqa aiaad6gacqGHsislcaaIXaaaniabggHiLdaakeaacaWGUbGaeyOeI0 IaaGymaaaaaaa@5BB6@ ,

then we also have:

i=1 n1 ( x i x ¯ n1 )( y i y ¯ n1 ) =( n1 )Co v n1 ( X,Y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada qadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsislceWG4bGb aebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaGccaGLOaGaay zkaaWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0Ia bmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaOGaay jkaiaawMcaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbGaeyOe I0IaaGymaaqdcqGHris5aOGaeyypa0ZaaeWaaeaacaWGUbGaeyOeI0 IaaGymaaGaayjkaiaawMcaaiaadoeacaWGVbGaamODamaaBaaaleaa caWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaadIfacaGGSaGaam ywaaGaayjkaiaawMcaaaaa@5D2F@ .

With this, we have:

Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( y ¯ n1 y n n ) i=1 n1 ( x i x ¯ n1 ) +( x ¯ n1 x n n ) i=1 n1 ( y i y ¯ n1 +( y ¯ n1 y n n ) ) +( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( y ¯ n1 y n n )( i=1 n1 ( x i ) + i=1 n1 ( x ¯ n1 ) ) +( x ¯ n1 x n n )( i=1 n1 ( y i ) + i=1 n1 ( y ¯ n1 ) + i=1 n1 ( y ¯ n1 y n n ) )+( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( y ¯ n1 y n n )( i=1 n1 ( x i ) x ¯ n1 i=1 n1 ( 1 ) ) +( x ¯ n1 x n n )( i=1 n1 ( y i ) y ¯ n1 i=1 n1 ( 1 ) +( y ¯ n1 y n n ) i=1 n1 ( 1 ) )+( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( y ¯ n1 y n n )( i=1 n1 ( x i ) x ¯ n1 ( n1 ) ) +( x ¯ n1 x n n )( i=1 n1 ( y i ) y ¯ n1 ( n1 )+( y ¯ n1 y n n )( n1 ) )+( x n x ¯ n )( y n y ¯ n ) ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGdb Gaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfa caGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaq aabeqaamaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaa caWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaae qaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH RaWkdaqadaqaamaalaaabaGabmyEayaaraWaaSbaaSqaaiaad6gacq GHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqa baaakeaacaWGUbaaaaGaayjkaiaawMcaamaaqahabaWaaeWaaeaaca WG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmiEayaaraWaaSba aSqaaiaad6gacqGHsislcaaIXaaabeaaaOGaayjkaiaawMcaaaWcba GaamyAaiabg2da9iaaigdaaeaacaWGUbGaeyOeI0IaaGymaaqdcqGH ris5aaGcbaGaey4kaSYaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBa aaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG4bWaaSba aSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaadaaeWb qaamaabmaabaGaamyEamaaBaaaleaacaWGPbaabeaakiabgkHiTiqa dMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHRa WkdaqadaqaamaalaaabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGH sislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqaba aakeaacaWGUbaaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaWcbaGa amyAaiabg2da9iaaigdaaeaacaWGUbGaeyOeI0IaaGymaaqdcqGHri s5aOGaey4kaSYaaeWaaeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGa eyOeI0IabmiEayaaraWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaay zkaaWaaeWaaeaacaWG5bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Ia bmyEayaaraWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaai aawIcacaGLPaaaaeaacaWGUbaaaaqaaiaadoeacaWGVbGaamODamaa BaaaleaacaWGUbaabeaakmaabmaabaGaamiwaiaacYcacaWGzbaaca GLOaGaayzkaaGaeyypa0ZaaSaaaeaadaqadaabaeqabaWaaeWaaeaa caWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaadoeacaWGVbGaam ODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaa dIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgUcaRmaabmaabaWaaS aaaeaaceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqa aOGaeyOeI0IaamyEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaa aacaGLOaGaayzkaaWaaeWaaeaadaaeWbqaamaabmaabaGaamiEamaa BaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaWcbaGaamyAaiabg2 da9iaaigdaaeaacaWGUbGaeyOeI0IaaGymaaqdcqGHris5aOGaey4k aSYaaabCaeaadaqadaqaaiabgkHiTiqadIhagaqeamaaBaaaleaaca WGUbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaSqaaiaadMga cqGH9aqpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoaaO GaayjkaiaawMcaaaqaaiabgUcaRmaabmaabaWaaSaaaeaaceWG4bGb aebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0Iaam iEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaacaGLOaGaayzk aaWaaeWaaeaadaaeWbqaamaabmaabaGaamyEamaaBaaaleaacaWGPb aabeaaaOGaayjkaiaawMcaaaWcbaGaamyAaiabg2da9iaaigdaaeaa caWGUbGaeyOeI0IaaGymaaqdcqGHris5aOGaey4kaSYaaabCaeaada qadaqaaiabgkHiTiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0Ia aGymaaqabaaakiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXa aabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabgUcaRmaaqaha baWaaeWaaeaadaWcaaqaaiqadMhagaqeamaaBaaaleaacaWGUbGaey OeI0IaaGymaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaad6gaaeqa aaGcbaGaamOBaaaaaiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpca aIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoaaOGaayjkaiaa wMcaaiabgUcaRmaabmaabaGaamiEamaaBaaaleaacaWGUbaabeaaki abgkHiTiqadIhagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaa wMcaamaabmaabaGaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTi qadMhagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaaaa caGLOaGaayzkaaaabaGaamOBaaaaaeaacaWGdbGaam4BaiaadAhada WgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfacaGGSaGaamywaaGa ayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaqaabeqaamaabmaaba GaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGdbGaam4Baiaa dAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaaca WGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGHRaWkdaqadaqaamaa laaabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabe aakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaa aaGaayjkaiaawMcaamaabmaabaWaaabCaeaadaqadaqaaiaadIhada WgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaSqaaiaadMgacqGH 9aqpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabgk HiTiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGc daaeWbqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaWcbaGaamyAai abg2da9iaaigdaaeaacaWGUbGaeyOeI0IaaGymaaqdcqGHris5aaGc caGLOaGaayzkaaaabaGaey4kaSYaaeWaaeaadaWcaaqaaiqadIhaga qeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG 4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPa aadaqadaqaamaaqahabaWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqaai aad6gacqGHsislcaaIXaaaniabggHiLdGccqGHsislceWG5bGbaeba daWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaabCaeaadaqada qaaiaaigdaaiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaa baGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabgUcaRmaabmaaba WaaSaaaeaaceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigda aeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6 gaaaaacaGLOaGaayzkaaWaaabCaeaadaqadaqaaiaaigdaaiaawIca caGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaiabgkHiTi aaigdaa0GaeyyeIuoaaOGaayjkaiaawMcaaiabgUcaRmaabmaabaGa amiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqeamaaBa aaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaGaamyEamaa BaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaaBaaaleaaca WGUbaabeaaaOGaayjkaiaawMcaaaaacaGLOaGaayzkaaaabaGaamOB aaaaaeaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcda qadaqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maa laaabaWaaeWaaqaabeqaamaabmaabaGaamOBaiabgkHiTiaaigdaai aawIcacaGLPaaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawI cacaGLPaaacqGHRaWkdaqadaqaamaalaaabaGabmyEayaaraWaaSba aSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaa WcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaamaabmaa baWaaabCaeaadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaki aawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaiab gkHiTiaaigdaa0GaeyyeIuoakiabgkHiTiqadIhagaqeamaaBaaale aacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaad6gacqGHsisl caaIXaaacaGLOaGaayzkaaaacaGLOaGaayzkaaaabaGaey4kaSYaae WaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0Ia aGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcba GaamOBaaaaaiaawIcacaGLPaaadaqadaqaamaaqahabaWaaeWaaeaa caWG5bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaleaaca WGPbGaeyypa0JaaGymaaqaaiaad6gacqGHsislcaaIXaaaniabggHi LdGccqGHsislceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaig daaeqaaOWaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMca aiabgUcaRmaabmaabaWaaSaaaeaaceWG5bGbaebadaWgaaWcbaGaam OBaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWG UbaabeaaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaeWaaeaacaWGUb GaeyOeI0IaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiabgUca RmaabmaabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadI hagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaa baGaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeam aaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaaaacaGLOaGaayzk aaaabaGaamOBaaaaaaaa@20E3@

Since the sample mean for the first n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaaigdaaaa@3892@ x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ values are defined as:

x ¯ n1 = i=1 n1 x i n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara WaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabg2da9maalaaa baWaaabCaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoaaOqa aiaad6gacqGHsislcaaIXaaaaaaa@4730@ and y ¯ n1 = i=1 n1 y i n1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabg2da9maalaaa baWaaabCaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoaaOqa aiaad6gacqGHsislcaaIXaaaaaaa@4732@ ,

then we also have:

i=1 n1 x i = x ¯ n1 ( n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WG4bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaa baGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabg2da9iqadIhaga qeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaa d6gacqGHsislcaaIXaaacaGLOaGaayzkaaaaaa@48A9@ and i=1 n1 y i = y ¯ n1 ( n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WG5bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaa baGaamOBaiabgkHiTiaaigdaa0GaeyyeIuoakiabg2da9iqadMhaga qeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaa d6gacqGHsislcaaIXaaacaGLOaGaayzkaaaaaa@48AB@ .

With that, we have:

Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( y ¯ n1 y n n )( x ¯ n1 ( n1 ) x ¯ n1 ( n1 ) ) +( x ¯ n1 x n n )( y ¯ n1 ( n1 ) y ¯ n1 ( n1 )+( y ¯ n1 y n n )( n1 ) )+( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( y ¯ n1 y n n )( x ¯ n1 ( n1 ) x ¯ n1 ( n1 ) ) +( x ¯ n1 x n n )( y ¯ n1 ( n1 ) y ¯ n1 ( n1 ) +( y ¯ n1 y n n )( n1 ) )+( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( x ¯ n1 x n n )( y ¯ n1 y n n )( n1 )+( x n x ¯ n )( y n y ¯ n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) + x n y n x n y ¯ n x ¯ n y n + x ¯ n y ¯ n ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGdb Gaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfa caGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaq aabeqaamaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaa caWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaae qaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH RaWkdaqadaqaamaalaaabaGabmyEayaaraWaaSbaaSqaaiaad6gacq GHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqa baaakeaacaWGUbaaaaGaayjkaiaawMcaamaabmaabaGabmiEayaara WaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakmaabmaabaGaamOB aiabgkHiTiaaigdaaiaawIcacaGLPaaacqGHsislceWG4bGbaebada WgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGUbGa eyOeI0IaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiabgU caRmaabmaabaWaaSaaaeaaceWG4bGbaebadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaabe aaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaeWaaeaaceWG5bGbaeba daWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGUb GaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgkHiTiqadMhagaqeamaa BaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaad6gacq GHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaadaWcaaqa aiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccq GHsislcaWG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaa wIcacaGLPaaadaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaay zkaaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacaWG4bWaaSbaaSqa aiaad6gaaeqaaOGaeyOeI0IabmiEayaaraWaaSbaaSqaaiaad6gaae qaaaGccaGLOaGaayzkaaWaaeWaaeaacaWG5bWaaSbaaSqaaiaad6ga aeqaaOGaeyOeI0IabmyEayaaraWaaSbaaSqaaiaad6gaaeqaaaGcca GLOaGaayzkaaaaaiaawIcacaGLPaaaaeaacaWGUbaaaaqaaiaadoea caWGVbGaamODamaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiwai aacYcacaWGzbaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaqadaab aeqabaWaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaai aadoeacaWGVbGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqa baGcdaqadaqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgU caRmaabmaabaWaaSaaaeaaceWG5bGbaebadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGUbaabe aaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaeWaaeaadaajcaqaaiqa dIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqada qaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaaaamaaKiaabaGa eyOeI0IabmiEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabe aakmaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaaa caGLOaGaayzkaaaabaGaey4kaSYaaeWaaeaadaWcaaqaaiqadIhaga qeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG 4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPa aadaqadaqaamaaKiaabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGH sislcaaIXaaabeaakmaabmaabaGaamOBaiabgkHiTiaaigdaaiaawI cacaGLPaaaaaWaaqIaaeaacqGHsislceWG5bGbaebadaWgaaWcbaGa amOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGUbGaeyOeI0IaaG ymaaGaayjkaiaawMcaaaaacqGHRaWkdaqadaqaamaalaaabaGabmyE ayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTi aadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaa wMcaamaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaaai aawIcacaGLPaaacqGHRaWkdaqadaqaaiaadIhadaWgaaWcbaGaamOB aaqabaGccqGHsislceWG4bGbaebadaWgaaWcbaGaamOBaaqabaaaki aawIcacaGLPaaadaqadaqaaiaadMhadaWgaaWcbaGaamOBaaqabaGc cqGHsislceWG5bGbaebadaWgaaWcbaGaamOBaaqabaaakiaawIcaca GLPaaaaaGaayjkaiaawMcaaaqaaiaad6gaaaaabaGaam4qaiaad+ga caWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiilai aadMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaabmaabaWaaeWa aeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaadoeacaWGVb GaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqa aiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgUcaRmaabmaaba WaaSaaaeaaceWG4bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigda aeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6 gaaaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiqadMhagaqeamaa BaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG5bWaaS baaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaadaqa daqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSYaae WaaeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IabmiEayaa raWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaaca WG5bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IabmyEayaaraWaaSba aSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaaba GaamOBaaaaaeaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaaqa baGcdaqadaqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabg2 da9maalaaabaWaaeWaaqaabeqaamaabmaabaGaamOBaiabgkHiTiaa igdaaiaawIcacaGLPaaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaam OBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGaaiilaiaadMfa aiaawIcacaGLPaaacqGHRaWkdaqadaqaaiaad6gacqGHsislcaaIXa aacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBaaa leaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaS qaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaadaqadaqa amaalaaabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXa aabeaakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWG UbaaaaGaayjkaiaawMcaaaqaaiabgUcaRiaadIhadaWgaaWcbaGaam OBaaqabaGccaWG5bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamiE amaaBaaaleaacaWGUbaabeaakiqadMhagaqeamaaBaaaleaacaWGUb aabeaakiabgkHiTiqadIhagaqeamaaBaaaleaacaWGUbaabeaakiaa dMhadaWgaaWcbaGaamOBaaqabaGccqGHRaWkceWG4bGbaebadaWgaa WcbaGaamOBaaqabaGcceWG5bGbaebadaWgaaWcbaGaamOBaaqabaaa aOGaayjkaiaawMcaaaqaaiaad6gaaaaaaaa@9CD9@

Since the recurrence equation for the sample mean of the y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ values is:

y ¯ n = y ¯ n1 y ¯ n1 y n n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JabmyEayaaraWaaSbaaSqa aiaad6gacqGHsislcaaIXaaabeaakiabgkHiTmaalaaabaGabmyEay aaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaa dMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaaa@460E@ ,

then we have:

Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) + x n y n x n ( y ¯ n1 y ¯ n1 y n n ) x ¯ n y n + x ¯ n ( y ¯ n1 y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) + x n y n x n y ¯ n1 + x n ( y ¯ n1 y n n ) x ¯ n y n + x ¯ n y ¯ n1 x ¯ n ( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) + x n y n x n y ¯ n1 x ¯ n y n + x ¯ n y ¯ n1 +( x n x ¯ n )( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n1 )+( x n x ¯ n )( y ¯ n1 y n n ) ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGdb Gaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfa caGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaq aabeqaamaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaa caWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaae qaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH RaWkdaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaWaae WaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0Ia aGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcba GaamOBaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGabmyEayaa raWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadM hadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMca aaqaaiabgUcaRiaadIhadaWgaaWcbaGaamOBaaqabaGccaWG5bWaaS baaSqaaiaad6gaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaa beaakmaabmaabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislca aIXaaabeaakiabgkHiTmaalaaabaGabmyEayaaraWaaSbaaSqaaiaa d6gacqGHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaam OBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaaiabgkHiTiqadIha gaqeamaaBaaaleaacaWGUbaabeaakiaadMhadaWgaaWcbaGaamOBaa qabaGccqGHRaWkceWG4bGbaebadaWgaaWcbaGaamOBaaqabaGcdaqa daqaaiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqaba GccqGHsisldaWcaaqaaiqadMhagaqeamaaBaaaleaacaWGUbGaeyOe I0IaaGymaaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaad6gaaeqaaa GcbaGaamOBaaaaaiaawIcacaGLPaaaaaGaayjkaiaawMcaaaqaaiaa d6gaaaaabaGaam4qaiaad+gacaWG2bWaaSbaaSqaaiaad6gaaeqaaO WaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH9aqp daWcaaqaamaabmaaeaqabeaadaqadaqaaiaad6gacqGHsislcaaIXa aacaGLOaGaayzkaaGaam4qaiaad+gacaWG2bWaaSbaaSqaaiaad6ga cqGHsislcaaIXaaabeaakmaabmaabaGaamiwaiaacYcacaWGzbaaca GLOaGaayzkaaGaey4kaSYaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGa ayjkaiaawMcaamaabmaabaWaaSaaaeaaceWG4bGbaebadaWgaaWcba GaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaa caWGUbaabeaaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaeWaaeaada WcaaqaaiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqa baGccqGHsislcaWG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaa aaaiaawIcacaGLPaaaaeaacqGHRaWkcaWG4bWaaSbaaSqaaiaad6ga aeqaaOGaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTiaadIhada WgaaWcbaGaamOBaaqabaGcceWG5bGbaebadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOGaey4kaSIaamiEamaaBaaaleaacaWGUbaabe aakmaabmaabaWaaSaaaeaaceWG5bGbaebadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGUbaabe aaaOqaaiaad6gaaaaacaGLOaGaayzkaaGaeyOeI0IabmiEayaaraWa aSbaaSqaaiaad6gaaeqaaOGaamyEamaaBaaaleaacaWGUbaabeaaki abgUcaRiqadIhagaqeamaaBaaaleaacaWGUbaabeaakiqadMhagaqe amaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislceWG4b GbaebadaWgaaWcbaGaamOBaaqabaGcdaqadaqaamaalaaabaGabmyE ayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTi aadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaa wMcaaaaacaGLOaGaayzkaaaabaGaamOBaaaaaeaacaWGdbGaam4Bai aadAhadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfacaGGSaGa amywaaGaayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaqaabeqaam aabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGdbGa am4BaiaadAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaae WaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGHRaWkdaqa daqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaada WcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqa baGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaa aaaiaawIcacaGLPaaadaqadaqaamaalaaabaGabmyEayaaraWaaSba aSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaa WcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaaaqaaiab gUcaRiaadIhadaWgaaWcbaGaamOBaaqabaGccaWG5bWaaSbaaSqaai aad6gaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaabeaakiqa dMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsi slceWG4bGbaebadaWgaaWcbaGaamOBaaqabaGccaWG5bWaaSbaaSqa aiaad6gaaeqaaOGaey4kaSIabmiEayaaraWaaSbaaSqaaiaad6gaae qaaOGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaa kiabgUcaRmaabmaabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgk HiTiqadIhagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMca amaabmaabaWaaSaaaeaaceWG5bGbaebadaWgaaWcbaGaamOBaiabgk HiTiaaigdaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGUbaabeaa aOqaaiaad6gaaaaacaGLOaGaayzkaaaaaiaawIcacaGLPaaaaeaaca WGUbaaaaqaaiaadoeacaWGVbGaamODamaaBaaaleaacaWGUbaabeaa kmaabmaabaGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaGaeyypa0 ZaaSaaaeaadaqadaabaeqabaWaaeWaaeaacaWGUbGaeyOeI0IaaGym aaGaayjkaiaawMcaaiaadoeacaWGVbGaamODamaaBaaaleaacaWGUb GaeyOeI0IaaGymaaqabaGcdaqadaqaaiaadIfacaGGSaGaamywaaGa ayjkaiaawMcaaiabgUcaRmaabmaabaGaamOBaiabgkHiTiaaigdaai aawIcacaGLPaaadaqadaqaamaalaaabaGabmiEayaaraWaaSbaaSqa aiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadIhadaWgaaWcba GaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaamaabmaabaWa aSaaaeaaceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaae qaaOGaeyOeI0IaamyEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6ga aaaacaGLOaGaayzkaaaabaGaey4kaSYaaeWaaeaacaWG4bWaaSbaaS qaaiaad6gaaeqaaOGaeyOeI0IabmiEayaaraWaaSbaaSqaaiaad6ga aeqaaaGccaGLOaGaayzkaaWaaeWaaeaacaWG5bWaaSbaaSqaaiaad6 gaaeqaaOGaeyOeI0IabmyEayaaraWaaSbaaSqaaiaad6gacqGHsisl caaIXaaabeaaaOGaayjkaiaawMcaaiabgUcaRmaabmaabaGaamiEam aaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqeamaaBaaaleaa caWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaWaaSaaaeaaceWG5b GbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0Ia amyEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaacaGLOaGaay zkaaaaaiaawIcacaGLPaaaaeaacaWGUbaaaaaaaa@A4F4@

Since the recurrence equation for the sample mean of the x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ values is:

x ¯ n = x ¯ n1 x ¯ n1 x n n x ¯ n = n x ¯ n1 n + x ¯ n1 + x n n x ¯ n = ( n1 ) x ¯ n1 + x n n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaceWG4b GbaebadaWgaaWcbaGaamOBaaqabaGccqGH9aqpceWG4bGbaebadaWg aaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0YaaSaaaeaace WG4bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOe I0IaamiEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaaaabaGabm iEayaaraWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaacaWG UbGabmiEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaO qaaiaad6gaaaGaey4kaSYaaSaaaeaacqGHsislceWG4bGbaebadaWg aaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamiEamaaBa aaleaacaWGUbaabeaaaOqaaiaad6gaaaaabaGabmiEayaaraWaaSba aSqaaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaad6gacq GHsislcaaIXaaacaGLOaGaayzkaaGabmiEayaaraWaaSbaaSqaaiaa d6gacqGHsislcaaIXaaabeaakiabgUcaRiaadIhadaWgaaWcbaGaam OBaaqabaaakeaacaWGUbaaaiaacYcaaaaa@68FB@

then we have:

Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n1 )+( x n ( ( n1 ) x ¯ n1 + x n n ) )( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n1 )+( n x n n + ( n1 ) x ¯ n1 x n n )( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n1 )+( ( n1 ) x ¯ n1 +( n1 ) x n n )( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y )+( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n1 )( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( ( n1 )Co v n1 ( X,Y ) +( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) +( x n x ¯ n )( y n y ¯ n1 ) ( n1 )( x ¯ n1 x n n )( y ¯ n1 y n n ) ) n Co v n ( X,Y )= ( n1 )Co v n1 ( X,Y )+( x n x ¯ n )( y n y ¯ n1 ) n Co v n ( X,Y )= nCo v n1 ( X,Y )Co v n1 ( X,Y )+( x n x ¯ n )( y n y ¯ n1 ) n Co v n ( X,Y )= nCo v n1 ( X,Y ) n + ( Co v n1 ( X,Y )( x n x ¯ n )( y n y ¯ n1 ) ) n Co v n ( X,Y )=Co v n1 ( X,Y ) Co v n1 ( X,Y )( x n x ¯ n )( y n y ¯ n1 ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGdb Gaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfa caGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaq aabeqaamaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaa caWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaae qaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGH RaWkdaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaWaae WaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0Ia aGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcba GaamOBaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGabmyEayaa raWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadM hadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMca aaqaaiabgUcaRmaabmaabaGaamiEamaaBaaaleaacaWGUbaabeaaki abgkHiTiqadIhagaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaa wMcaamaabmaabaGaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTi qadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaakiaa wIcacaGLPaaacqGHRaWkdaqadaqaaiaadIhadaWgaaWcbaGaamOBaa qabaGccqGHsisldaqadaqaamaalaaabaWaaeWaaeaacaWGUbGaeyOe I0IaaGymaaGaayjkaiaawMcaaiqadIhagaqeamaaBaaaleaacaWGUb GaeyOeI0IaaGymaaqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiaad6ga aeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaaaiaawIcacaGLPaaada qadaqaamaalaaabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsisl caaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqabaaake aacaWGUbaaaaGaayjkaiaawMcaaaaacaGLOaGaayzkaaaabaGaamOB aaaaaeaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaaqabaGcda qadaqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabg2da9maa laaabaWaaeWaaqaabeqaamaabmaabaGaamOBaiabgkHiTiaaigdaai aawIcacaGLPaaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawI cacaGLPaaacqGHRaWkdaqadaqaaiaad6gacqGHsislcaaIXaaacaGL OaGaayzkaaWaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaaca WGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaa d6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaadaqadaqaamaala aabaGabmyEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaa kiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaa GaayjkaiaawMcaaaqaaiabgUcaRmaabmaabaGaamiEamaaBaaaleaa caWGUbaabeaakiabgkHiTiqadIhagaqeamaaBaaaleaacaWGUbaabe aaaOGaayjkaiaawMcaamaabmaabaGaamyEamaaBaaaleaacaWGUbaa beaakiabgkHiTiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaG ymaaqabaaakiaawIcacaGLPaaacqGHRaWkdaqadaqaamaalaaabaGa amOBaiaadIhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaiabgU caRmaalaaabaGaeyOeI0YaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGa ayjkaiaawMcaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaG ymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGa amOBaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGabmyEayaara WaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadMha daWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaaa aacaGLOaGaayzkaaaabaGaamOBaaaaaeaacaWGdbGaam4BaiaadAha daWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadIfacaGGSaGaamywaa GaayjkaiaawMcaaiabg2da9maalaaabaWaaeWaaqaabeqaamaabmaa baGaamOBaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGdbGaam4Bai aadAhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaa caWGybGaaiilaiaadMfaaiaawIcacaGLPaaacqGHRaWkdaqadaqaai aad6gacqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqa aiqadIhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccq GHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaa wIcacaGLPaaadaqadaqaamaalaaabaGabmyEayaaraWaaSbaaSqaai aad6gacqGHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGa amOBaaqabaaakeaacaWGUbaaaaGaayjkaiaawMcaaaqaaiabgUcaRm aabmaabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIha gaqeamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaaba GaamyEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaa BaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaacq GHRaWkdaqadaqaamaalaaabaGaeyOeI0YaaeWaaeaacaWGUbGaeyOe I0IaaGymaaGaayjkaiaawMcaaiqadIhagaqeamaaBaaaleaacaWGUb GaeyOeI0IaaGymaaqabaGccqGHRaWkdaqadaqaaiaad6gacqGHsisl caaIXaaacaGLOaGaayzkaaGaamiEamaaBaaaleaacaWGUbaabeaaaO qaaiaad6gaaaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiqadMha gaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislca WG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGL PaaaaaGaayjkaiaawMcaaaqaaiaad6gaaaaabaGaam4qaiaad+gaca WG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiilaiaa dMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaabmaaeaqabeaada qadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaam4qaiaa d+gacaWG2bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakmaabm aabaGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaGaey4kaSYaaeWa aeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaabaWaaS aaaeaaceWG4bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqa aOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaabeaaaOqaaiaad6gaaa aacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiqadMhagaqeamaaBaaa leaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislcaWG5bWaaSbaaS qaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGLPaaaaeaacqGH RaWkdaqadaqaaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGHsislce WG4bGbaebadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaadaqa daqaaiaadMhadaWgaaWcbaGaamOBaaqabaGccqGHsislceWG5bGbae badaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaGccaGLOaGaayzk aaGaeyOeI0YaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawM caamaabmaabaWaaSaaaeaaceWG4bGbaebadaWgaaWcbaGaamOBaiab gkHiTiaaigdaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGUbaabe aaaOqaaiaad6gaaaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiqa dMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsi slcaWG5bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIca caGLPaaaaaGaayjkaiaawMcaaaqaaiaad6gaaaaabaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaabmaaeaqabe aadaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaam4q aiaad+gacaWG2bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakm aabmaabaGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaWaaqIaaeaa cqGHRaWkdaqadaqaaiaad6gacqGHsislcaaIXaaacaGLOaGaayzkaa WaaeWaaeaadaWcaaqaaiqadIhagaqeamaaBaaaleaacaWGUbGaeyOe I0IaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaad6gaaeqaaa GcbaGaamOBaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGabmyE ayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgkHiTi aadMhadaWgaaWcbaGaamOBaaqabaaakeaacaWGUbaaaaGaayjkaiaa wMcaaaaaaeaacqGHRaWkdaqadaqaaiaadIhadaWgaaWcbaGaamOBaa qabaGccqGHsislceWG4bGbaebadaWgaaWcbaGaamOBaaqabaaakiaa wIcacaGLPaaadaqadaqaaiaadMhadaWgaaWcbaGaamOBaaqabaGccq GHsislceWG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqa aaGccaGLOaGaayzkaaWaaqIaaeaacqGHsisldaqadaqaaiaad6gacq GHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiqadIha gaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHsislca WG4bWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOBaaaaaiaawIcacaGL PaaadaqadaqaamaalaaabaGabmyEayaaraWaaSbaaSqaaiaad6gacq GHsislcaaIXaaabeaakiabgkHiTiaadMhadaWgaaWcbaGaamOBaaqa baaakeaacaWGUbaaaaGaayjkaiaawMcaaaaaaaGaayjkaiaawMcaaa qaaiaad6gaaaaabaGaam4qaiaad+gacaWG2bWaaSbaaSqaaiaad6ga aeqaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGLPaaacq GH9aqpdaWcaaqaamaabmaabaGaamOBaiabgkHiTiaaigdaaiaawIca caGLPaaacaWGdbGaam4BaiaadAhadaWgaaWcbaGaamOBaiabgkHiTi aaigdaaeqaaOWaaeWaaeaacaWGybGaaiilaiaadMfaaiaawIcacaGL PaaacqGHRaWkdaqadaqaaiaadIhadaWgaaWcbaGaamOBaaqabaGccq GHsislceWG4bGbaebadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGL PaaadaqadaqaaiaadMhadaWgaaWcbaGaamOBaaqabaGccqGHsislce WG5bGbaebadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaGccaGL OaGaayzkaaaabaGaamOBaaaaaeaacaWGdbGaam4BaiaadAhadaWgaa WcbaGaamOBaaqabaGcdaqadaqaaiaadIfacaGGSaGaamywaaGaayjk aiaawMcaaiabg2da9maalaaabaGaamOBaiaadoeacaWGVbGaamODam aaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaadIfa caGGSaGaamywaaGaayjkaiaawMcaaiabgkHiTiaadoeacaWGVbGaam ODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaa dIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaam iEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqeamaaBaaa leaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaGaamyEamaaBa aaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaaBaaaleaacaWG UbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaeaacaWGUbaaaa qaaiaadoeacaWGVbGaamODamaaBaaaleaacaWGUbaabeaakmaabmaa baGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaGaeyypa0ZaaSaaae aacaWGUbGaam4qaiaad+gacaWG2bWaaSbaaSqaaiaad6gacqGHsisl caaIXaaabeaakmaabmaabaGaamiwaiaacYcacaWGzbaacaGLOaGaay zkaaaabaGaamOBaaaacqGHRaWkdaWcaaqaaiabgkHiTmaabmaabaGa am4qaiaad+gacaWG2bWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabe aakmaabmaabaGaamiwaiaacYcacaWGzbaacaGLOaGaayzkaaGaeyOe I0YaaeWaaeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Iabm iEayaaraWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaeWa aeaacaWG5bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IabmyEayaara WaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaOGaayjkaiaawMca aaGaayjkaiaawMcaaaqaaiaad6gaaaaabaGaam4qaiaad+gacaWG2b WaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiilaiaadMfa aiaawIcacaGLPaaacqGH9aqpcaWGdbGaam4BaiaadAhadaWgaaWcba GaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGaaiilaiaa dMfaaiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadoeacaWGVbGaam ODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaa dIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgkHiTmaabmaabaGaam iEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqeamaaBaaa leaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaGaamyEamaaBa aaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaaBaaaleaacaWG UbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaeaacaWGUbaaaa aaaa@BC53@

Therefore, the recurrence equation for the biased sample covariance (a.k.a. online covariance) is:

Co v n ( X,Y )=Co v n1 ( X,Y ) Co v n1 ( X,Y )( x n x ¯ n )( y n y ¯ n1 ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpcaWGdbGaam4BaiaadAhada WgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGa aiilaiaadMfaaiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadoeaca WGVbGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqa daqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgkHiTmaabm aabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqe amaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaabmaabaGaam yEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadMhagaqeamaaBaaa leaacaWGUbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPaaaaeaaca WGUbaaaaaa@6404@

 

Note: We can manipulate this recurrence equation such as that we also have:

Co v n ( X,Y )=Co v n1 ( X,Y ) Co v n1 ( X,Y )( x n x ¯ n1 )( y n y ¯ n ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpcaWGdbGaam4BaiaadAhada WgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGa aiilaiaadMfaaiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadoeaca WGVbGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqa daqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgkHiTmaabm aabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiqadIhagaqe amaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaakiaawIcacaGLPa aadaqadaqaaiaadMhadaWgaaWcbaGaamOBaaqabaGccqGHsislceWG 5bGbaebadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaaaeaaca WGUbaaaaaa@6404@ ,

Co v n ( X,Y )=Co v n1 ( X,Y ) Co v n1 ( X,Y )( n1 n )( x n x ¯ n1 )( y n y ¯ n1 ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpcaWGdbGaam4BaiaadAhada WgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWGybGa aiilaiaadMfaaiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadoeaca WGVbGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqa daqaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgkHiTmaabm aabaWaaSaaaeaacaWGUbGaeyOeI0IaaGymaaqaaiaad6gaaaaacaGL OaGaayzkaaWaaeWaaeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaey OeI0IabmiEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaa aOGaayjkaiaawMcaamaabmaabaGaamyEamaaBaaaleaacaWGUbaabe aakiabgkHiTiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGym aaqabaaakiaawIcacaGLPaaaaeaacaWGUbaaaaaa@6AD3@ ,

and

Co v n ( X,Y )= ( n1 )( Co v n1 ( X,Y )+ ( x n x ¯ n1 )( y n y ¯ n1 ) n ) n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaad+ gacaWG2bWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWGybGaaiil aiaadMfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaabmaabaGaam OBaiabgkHiTiaaigdaaiaawIcacaGLPaaadaqadaqaaiaadoeacaWG VbGaamODamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaqada qaaiaadIfacaGGSaGaamywaaGaayjkaiaawMcaaiabgUcaRmaalaaa baWaaeWaaeaacaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Iabm iEayaaraWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaOGaayjk aiaawMcaamaabmaabaGaamyEamaaBaaaleaacaWGUbaabeaakiabgk HiTiqadMhagaqeamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaa kiaawIcacaGLPaaaaeaacaWGUbaaaaGaayjkaiaawMcaaaqaaiaad6 gaaaaaaa@61E8@ .

Reference:
http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance


Example of C++ code that computes the online covariance:
// Filename: main.cpp
#include <iostream>
#include <iomanip>

int main () {
    
    double x;
    double y;
    double n = 0;
    double mean_x = 0;  // mean of the x values
    double mean_y = 0;  // mean of the y values
    double cov = 0;     // covariance of the x and y values
    double prev_mean_x; // previous mean of the x values
    double prev_mean_y; // previous mean of the y values
    double prev_cov;    // previous covariance of the x and y values
    
    if ( std::cin >> x && std::cin >> y ) {
        ++n;
        mean_x = x;
        mean_y = y;
        cov = 0;
        while ( std::cin >> x && std::cin >> y ) {
            prev_mean_x = mean_x;
            prev_mean_y = mean_y;
            prev_cov = cov;
            ++n;
            mean_x = prev_mean_x - ( prev_mean_x - x ) / n;
            mean_y = prev_mean_y - ( prev_mean_y - y ) / n;
            cov = prev_cov - ( prev_cov - ( x - mean_x ) * ( y - prev_mean_y ) ) / n;
        }
    }
    
    std::cout << "n:      " << n << '\n';
    std::cout << "mean_x: " << std::setprecision( 17 ) << mean_x << '\n';
    std::cout << "mean_y: " << std::setprecision( 17 ) << mean_y << '\n';
    std::cout << "cov:    " << std::setprecision( 17 ) << cov    << '\n';
    
}

Example of data.txt:
-281.189       612.083
974.663        -24.0965
25.8526        401.539
.              .
.              .
.              .

Command Line:
g++ -o main.exe main.cpp -std=c++11 -march=native -O3 -Wall -Wextra -Werror -static
./main.exe < data.txt

 

Note: Mathematica's Covariance[] function computes the unbiased sample covariance matrix, not the biased sample covariance matrix; therefore, the biased sample covariance matrix is computed in Mathematica as:

( ( Length[ list ] - 1 ) / Length[ list ] ) * Covariance[ list ]