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3 Autocorrelation
Autocorrelation refers to the correlation of a time series with its own past and future values
...
Positive
autocorrelation might be considered a specific form of “persistence”, a tendency for a system to
remain in the same state from one observation to the next
...
Geophysical time series are
frequently autocorrelated because of inertia or carryover processes in the physical system
...
Or the slow drainage of groundwater reserves might impart
correlation to successive annual flows of a river
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Autocorrelation complicates the
application of statistical tests by reducing the number of independent observations
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g
...
Autocorrelation can be exploited
for predictions: an autocorrelated time series is predictable, probabilistically, because future
values depend on current and past values
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3
...
1)
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Positive autocorrelation might show up in a time series plot as
unusually long runs, or stretches, of several consecutive observations above or below the mean
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Because the
“departures” for computing autocorrelation are relative the mean, a horizontal line plotted at the
sample mean is useful in evaluating autocorrelation with the time series plot
...
Statistical tests based on the observed number of runs above and
below the mean are available (e
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, Draper and Smith 1981), though none are covered in this
course
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If nothing else, this inspection might show that the persistence is much more
prevalent in some parts of the series than in others
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2 Lagged scatterplot
The simplest graphical summary of autocorrelation in a time series is the lagged scatterplot,
which is a scatterplot of the time series against itself offset in time by one to several time steps
(Figure 3
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Let the time series of length N be x i , i  1,
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The lagged scatterplot for lag k
is a scatterplot of the last N  k observations against the first N  k observations
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A random scattering of points in the lagged scatterplot indicates a lack of autocorrelation
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Alignment from lower left to upper right in the lagged scatterplot
indicates positive autocorrelation
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Figure 3
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Time series plot illustrating signatures of persistence
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Figure 3
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Lagged scatterplots of tree-ring series MEAF
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1 with itself offset by 1, 2, 3 and 4 years
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05)
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At an offset of 3
years, the juxtaposition of high-growth 1999 with low-growth 2002 exerts high influence
(point in red rectangle)
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An assumption of linear dependence is not necessary
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Such nonlinear dependence might not be effectively summarized by other
methods (e
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, the autocorrelation function [acf], which is described later)
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The scatter plot in Figure 3
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Influence of outliers would not be detectable from the acf alone
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A straight line can be fit to the points in a lagged scatterplot to facilitate
evaluation linearity and strength of relationship of current with past values
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g
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Correlation coefficient and 95% significance level
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It
is helpful to compare the computed correlation coefficient with critical level of correlation
required to reject the null hypothesis that the sample comes from a population with zero
correlation at the indicated lag
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It follows that the approximate threshold, or critical, level of
correlation for 95% significance (  0
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9 5  0  2 / N , where N is the sample size
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3)
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3 Autocorrelation function
(correlogram)

Figure 3
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Critical level of correlation
coefficient (95 percent significance) as a function
of sample size
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20 for a sample size of 100 to r=0
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 x

r 



 xi  x

Notes_3, GEOS 585A, Spring 2013



i

2

 x



1/ 2

An important guide to the persistence in
a time series is given by the series of
quantities called the sample autocorrelation
coefficients, which measure the correlation
between observations at different times
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An analogy can be drawn between the
autocorrelation coefficient and the productmoment correlation coefficient
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The correlation coefficient between x and
y is given by
  yi

 y




yi  y






2




1/ 2

(1)

3

where the summations are over the N observations
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Instead of two different time series, the correlation is computed between one time series and the
same series lagged by one or more time units
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The first-order autocorrelation coefficient is the simple correlation coefficient of the
first N  1 observations, x t , t  1, 2 ,
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, N
...
As the correlation coefficient given by (2) measures correlation between
successive observations, it is called the autocorrelation coefficient or serial correlation
coefficient
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First, the difference between the sub-period means x (1 ) and x ( 2 ) can be ignored
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Accordingly, r1 can be approximated by
N 1

 x

  xt 1

 x

r1 

 x



t

 x



t 1

(3)

N
t

 x

2

t 1
N

where

x 



xt

is the overall mean
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The plot of the
autocorrelation function as a function of lag is also called the correlogram
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The correlation coefficients for the lagged
scatterplots at lags k  1, 2 ,
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Notes_3, GEOS 585A, Spring 2013

4

Link between acf and autocovariance function (acvf)
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By analogy the autocovariance of a time series is defined as
the average product of departures at times t and t+k
ck 

1
N

N k

 x

t

 x

  xt  k

 x



(5)

t 1

where c k is the autocovariance coefficient at lag k
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By combining equations (4) and (5), the autocorrelation at lag k can be written in terms
of the autocovariance:
(6)
rk  c k c 0
Alternative equation for autocovariance function
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The acvf is sometimes
computed with the alternative equation
1

N k

(7)
  xt  x   xt  k  x 
 N  k  t 1
The acvf by (7) has a lower bias than the acvf by (5), but is conjectured to have a higher mean
square error (Jenkins and Watts 1968, chapter 5)
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4 Testing for randomness with the correlogram
The first question that can be answered with the correlogram is whether the series is
random or not
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It can be shown that if x1
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The 95%
confidence limits for the correlogram can therefore be plotted at
further approximated to 0  2

N

1 / N  2

N

, and are often


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2 0
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Factors that must be considered in judging whether a sample autocorrelation outside the
confidence limits indicates an autocorrelated process or population are (1) how many lags are
being examined, (2) the magnitude of rk , and (3) at what lag k the large coefficient occurs
...
And a large rk at a low lag (e
...
, k  1 ) is more likely to represent persistence in most
physical systems than an isolated large rk at some higher lag
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5 Large-lag standard error
While the confidence bands described above are horizontal lines above and below zero on the
correlogram, the confidence bands you see in the assignment script may appear to be narrowest at
Notes_3, GEOS 585A, Spring 2013

5

lag 1 and to widen slightly at higher lags
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8)
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This interdependence makes it difficult to assess just at how
many lags the correlogram is significant
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The variance of rk , with the adjustment, is given by
1 
2 
 1  2  ri 
N 
i 1

K

V a r( rk )

(10)

where K  k
...
8)
...
For example, the variance of the lag-3 autocorrelation coefficient,
V a r  r3  , is greater than 1 / N by an amount that depends on the autocorrelation coefficients at
lags 1 and 2
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Assessment of the significance of lag-k
autocorrelation by the large-lag standard error essentially assumes that the theoretical
autocorrelation has “died out” by lag k, but does not assume that the lower-lag theoretical
autocorrelations are zero (Box and Jenkins 1976, p
...
Thus the null hypothesis is NOT that
the series is random, as lower-lag autocorrelations in the generating process may be non-zero
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4)
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Thus, the autocorrelation at lag 5, say, is
judged significant under the null hypothesis that the series is random, but is not judged significant
if the theoretical autocorrelation function is considered to not have died out until lag 5
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4
...
Dotted line is
simple approximate confidence interval at ± 2 / N , where N is
the sample size
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Notes_3, GEOS 585A, Spring 2013

6

3
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The first-order
autocorrelation coefficient, r1 , can be tested against the null hypothesis that the corresponding
population value  1  0
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g
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For the one-tailed hypothesis, the
alternative hypothesis is usually that the true first-order autocorrelation is greater than zero:
(11)
H1 :   0
For the two-tailed test, the alternative hypothesis is that the true first-order autocorrelation is
different from zero, with no specification of whether it is positive or negative:
(12)
H1 :   0
Which alternative hypothesis to use depends on the problem
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g
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Otherwise, the two-sided test is best
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9 5 

 1  1
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More generally, following Salas et al
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6 4 5

N  k 1

N  k

o n e s id e d

(14)
rk ( 9 5 % ) 

 1  1
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Equation (13) comes from substitution of k=1 into
equation (14)
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7 Effective Sample Size
If a time series of length N is autocorrelated, the number of independent observations is fewer
than N
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The reduction in number of
independent observations has implications for hypothesis testing
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The way of circumventing the
problem of autocorrelation is to adjust the sample size for autocorrelation
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Below is an equation for computing so-called “effective” sample size, or sample size adjusted for
autocorrelation
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The equation was derived based on the assumption that the autocorrelation in the
series represents first-order autocorrelation (dependence on lag-1 only)
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Computation of the effective sample

Notes_3, GEOS 585A, Spring 2013

7

size requires only the sample size and first-order sample autocorrelation coefficient
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The ratio  1 

r1 is

the first-order

 1  r1  is a scaling factor multiplied by the original
sample size to compute the effective sample size
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50 has an adjusted sample size of
N '  100

(1  0
...
5

r1 



3 3 ye a rs

1
...
5 
The adjustment to effective sample size becomes less important the lower the autocorrelation,
but a first-order autocorrelation coefficient as small as r1=0
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5)
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L
...
Statistics, v
...

1, p
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Anderson, O
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182 pp
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5
...
For a given first-order
autocorrelation, the scaling factor is multiplied by the
original time series
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E
...
, and Jenkins, G
...
,
1976,
Time
series
analysis:
forecasting and control: San
Francisco, Holden Day, p
...


Chatfield, C
...

Dawdy, D
...
, and Matalas, N
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, 1964, Statistical and probability analysis of hydrologic data,
part III: Analysis of variance, covariance and time series, in Ven Te Chow, ed
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8
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90
...
M
...
G
...

Salas, J
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, Delleur, J
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, Yevjevich, V
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, and Lane, W
...
, 1980, Applied modeling of
hydrologic time series: Littleton, Colorado, Water Resources Publications, 484 pp
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79: Climatic Change, WMO-No,
195
...
100, Geneva, 80 pp

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