Dear R list As suggested by Prof Brian Ripley, I have tried to read acf literature. The main problem is I am not the statistician and hence have some problem in understanding the concepts immediately. I came across one literature (http://www.stat.nus.edu.sg/~staxyc/REG32.pdf) on auto-correlation giving the methodology. As per that literature, the auto-correlation is arrived at as per following.
y = c(15.91,9.80,17.16,16.68,15.53,22.66,31.01,8.62,45.82,10.97,45.46,28.69,36.75,37.75, 41.18,42.67,46.05, 43.70,53.08,47.56) t = c(1:20) # defining time variable. Fitting y = a + bt + e, I get the estimates of a and b as a = 9.12 and b = 2.07. So using these estimates I obtain y_fit = c(11.19,13.26,15.33,17.40,19.47,21.54,23.61,25.68,27.75,29.82,31.89,33.96, 36.03,38.10, 40.17,42.24,44.31,46.38,48.45,50.52) # these are fitted values. e_t = (y - y_fit)  # dif between the observed y and fitted value of corresponding y > e_t  [1]  4.72 -3.46  1.83 -0.72 -3.94  1.12  7.40  [8] -17.06 18.07 -18.85 13.57 -5.27  0.72 -0.35 [15]  1.01  0.43  1.74 -2.68  4.63 -2.96 # We define e_t1 = c(-3.46,1.83,-0.72,-3.94,1.12,7.40,-17.06,18.07,-18.85,13.57,-5.27,0.72,-0.35,1.01, 0.43,1.74,-2.68,4.63,-2.96)  # 1 st element of e_t deleted e_t2 = c(4.72,-3.46,1.83,-0.72,-3.94,1.12,7.40,-17.06,18.07,-18.85,13.57,-5.27,0.72,-0.35, 1.01, 0.43,1.74,-2.68,4.63)    # Original series with last element deleted cor(e_t1, e_t2) > cor(e_t1, e_t2) [1] -0.8732316 However, if I use acf(y, 1) Autocorrelations of series âyâ, by lag    0    1 1.000 0.343 I am simply not able to figure out how acf is used? Thanking you in advance. Regards Vincy --- On Wed, 8/24/11, Prof Brian Ripley <rip...@stats.ox.ac.uk> wrote: From: Prof Brian Ripley <rip...@stats.ox.ac.uk> Subject: Re: [R] Autocorrelation using library(tseries) To: "Vincy Pyne" <vincy_p...@yahoo.ca> Cc: r-help@r-project.org Received: Wednesday, August 24, 2011, 9:08 AM Your understanding is wrong. For a start, there is no function acf() in package tseries: it is in stats. And the autocorrelation at lag one is not the correlation omitting the first and last values: it uses the mean and variance estimated from the whole series and divisor n. Have you looked at the reference given on ?acf ? As the help says     (This contains the exact definitions used.) Neither the R help pages nor R-help are intended as tutorials in statistics. On Wed, 24 Aug 2011, Vincy Pyne wrote: > Dear R list > > I am trying to understand the auto-correlation concept. Auto-correlation is > the self-correlation of random variable X with a certain time lag of say t. > > The article > "http://www.mit.tut.fi/MIT-3010/luentokalvot/lk10-11/MDA_lecture16_11.pdf"; > (Page no. 9 and 10) gives the methodology as under. But that is not the definitive reference, and no, it doesn't (and what it does give is not the conventional definition in the time series literature). > Suppose you have a time series observations as say > > X = c(44,41,46,49,49,50,40,44,49,41) > > # For autocorrelation with time lag of 1 we define > > A = c(41,46,49,49,50,40,44,49,41)?? # first element of X not considered > B = c(44,41,46,49,49,50,40,44,49) # Last element of X not considered > >> cor(A,B) > [1] -0.02581234 > > However, if I try the acf command using library tseries I get > > acf(X, 1) > > Autocorrelations of series ???X???, by > lag > > ???????? 0?????????? 1 > ??1.000 -0.019 > > So by usual correlation command (where same random variable X is converted into two series with a time lag of 1), I obtain auto-correlation as -0.02581234 and by acf command I get auto-correlation = -0.019 (for time lag of 1). > > I am not able to figure out where I am going wrong or is it my understanding > of auto-correlation procedure is wrong? > > Will be grateful if someone guides . > > Vincy > > > >    [[alternative HTML version deleted]] > > -- Brian D. Ripley,         rip...@stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford,        Tel: +44 1865 272861 (self) 1 South Parks Road,            +44 1865 272866 (PA) Oxford OX1 3TG, UK        Fax: +44 1865 272595 [[alternative HTML version deleted]]
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