Compute autocorrelations and related quantities
autocorrelations.RdGeneric functions for computation of autocorrelations, autocovariances and related quantities. The idea is to free the user from the need to look for specific functions that compute the desired property for their object.
Usage
autocovariances(x, maxlag, ...)
autocorrelations(x, maxlag, lag_0, ...)
partialAutocorrelations(x, maxlag, lag_0 = TRUE, ...)
partialAutocovariances(x, maxlag, ...)
partialVariances(x, ...)Arguments
- x
an object for which the requested property makes sense.
- maxlag
the maximal lag to include in the result.
- lag_0
if TRUE include lag zero.
- ...
further arguments for methods.
Details
autocorrelations is a generic function for computation of
autocorrelations. It deduces the appropriate type of autocorrelation
from the class of the object. For example, for models it computes
theoretical autocorrelations, while for time series it computes sample
autocorrelations.
The other functions described are similar for other second order
properties of x.
These functions return objects from suitable classes, all inheriting
from "Lagged". The latter means that indexing starts from zero,
so the value for lag zero is accessed by r[0]). Subscripting
always returns the underlying data unclassed (i.e. ordinary vectors or
arrays). In particular, "[]" extracts the underlying data.
Functions computing autocorrelations and partial autocorrelations have
argument lag_0 --- if it is set to FALSE, the value for
lag zero is dropped from the result and the returned object is an
ordinary vector or array, as appropriate.
See the individual methods for the format of the result and further details.
There are plot methods for sample autocorrelations and sample partial
autocorrelations with overlaid significance limits under null
hypotheses for independence or weak white noise, see
plot-methods and the examples there. More details can be
found in the vignettes, see section ‘See also’ below.
See also
plot-methods for plotting with significance limits
computed under strong white noise and weak white noise hypotheses;
autocorrelations-methods,
partialAutocorrelations-methods
for details on individual methods;
vignette("white_noise_tests", package = "sarima") and vignette("garch_tests_example", package = "sarima") for
extensive worked examples.
Examples
set.seed(1234)
v1 <- rnorm(100)
autocorrelations(v1)
#> An object of class "SampleAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 1.00000000 0.12957682 0.04402061 0.06106303 0.05826444 0.11109644
#> Lag_6 Lag_7 Lag_8 Lag_9 Lag_10 Lag_11
#> 0.13135298 0.26913701 -0.08548724 0.12722915 -0.01484862 -0.01438639
#> Lag_12 Lag_13 Lag_14 Lag_15 Lag_16 Lag_17
#> 0.09282460 0.11136127 0.04431534 -0.14147363 -0.02991080 -0.02545693
#> Lag_18 Lag_19 Lag_20
#> 0.11824060 -0.01218368 -0.10847365
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
v1.acf <- autocorrelations(v1, maxlag = 10)
v1.acf[1:10] # drop lag zero value (and the class)
#> [1] 0.12957682 0.04402061 0.06106303 0.05826444 0.11109644 0.13135298
#> [7] 0.26913701 -0.08548724 0.12722915 -0.01484862
autocorrelations(v1, maxlag = 10, lag_0 = FALSE) # same
#> [1] 0.12957682 0.04402061 0.06106303 0.05826444 0.11109644 0.13135298
#> [7] 0.26913701 -0.08548724 0.12722915 -0.01484862
partialAutocorrelations(v1)
#> An object of class "SamplePartialAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 1.00000000 0.12957682 0.02769547 0.05285558 0.04377385 0.09714201
#> Lag_6 Lag_7 Lag_8 Lag_9 Lag_10 Lag_11
#> 0.10383731 0.24233653 -0.16783708 0.14648145 -0.10189429 -0.02638830
#> Lag_12 Lag_13 Lag_14 Lag_15 Lag_16 Lag_17
#> 0.04045013 0.07193251 -0.04576705 -0.10297684 -0.09458730 0.04627240
#> Lag_18 Lag_19 Lag_20
#> 0.09892282 -0.08337618 -0.11082180
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocorrelations(v1, maxlag = 10)
#> An object of class "SamplePartialAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 1.00000000 0.12957682 0.02769547 0.05285558 0.04377385 0.09714201
#> Lag_6 Lag_7 Lag_8 Lag_9 Lag_10
#> 0.10383731 0.24233653 -0.16783708 0.14648145 -0.10189429
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
## compute 2nd order properties from raw data
autocovariances(v1)
#> An object of class "SampleAutocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 0.99874171 0.12941377 0.04396522 0.06098619 0.05819112 0.11095664
#> Lag_6 Lag_7 Lag_8 Lag_9 Lag_10 Lag_11
#> 0.13118770 0.26879836 -0.08537967 0.12706906 -0.01482994 -0.01436829
#> Lag_12 Lag_13 Lag_14 Lag_15 Lag_16 Lag_17
#> 0.09270780 0.11122115 0.04425957 -0.14129562 -0.02987317 -0.02542490
#> Lag_18 Lag_19 Lag_20
#> 0.11809182 -0.01216835 -0.10833715
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
autocovariances(v1, maxlag = 10)
#> An object of class "SampleAutocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 0.99874171 0.12941377 0.04396522 0.06098619 0.05819112 0.11095664
#> Lag_6 Lag_7 Lag_8 Lag_9 Lag_10
#> 0.13118770 0.26879836 -0.08537967 0.12706906 -0.01482994
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocovariances(v1, maxlag = 6)
#> An object of class "SamplePartialAutocovariances"
#> 0 1 2 3 4 5 6
#> 0.99874171 0.12724089 0.02717533 0.05171803 0.04274969 0.09397397 0.09936784
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocovariances(v1)
#> An object of class "SamplePartialAutocovariances"
#> 0 1 2 3 4 5
#> 0.99874171 0.12724089 0.02717533 0.05171803 0.04274969 0.09397397
#> 6 7 8 9 10 11
#> 0.09936784 0.21828652 -0.14692191 0.12547617 -0.08637655 -0.02235398
#> 12 13 14 15 16 17
#> 0.03420994 0.06052078 -0.03842568 -0.08554178 -0.07786971 0.03801254
#> 18 19 20
#> 0.08046935 -0.06735137 -0.08842251
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
partialVariances(v1, maxlag = 6)
#> An object of class "SamplePartialVariances"
#> 0 1 2 3 4 5 6
#> 0.9987417 0.9819727 0.9812195 0.9784782 0.9766033 0.9673875 0.9569570
#> Slot n:
#> [1] 100
#> Slot varnames: <not set>
#> Slot objectname: x
pv1 <- partialVariances(v1)
## compute 2nd order properties from raw data
autocovariances(AirPassengers, maxlag = 6)
#> An object of class "SampleAutocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 14291.973 13549.467 12513.692 11529.066 10756.502 10201.181 9743.318
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
autocorrelations(AirPassengers, maxlag = 6)
#> An object of class "SampleAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 1.0000000 0.9480473 0.8755748 0.8066812 0.7526254 0.7137700 0.6817336
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocorrelations(AirPassengers, maxlag = 6)
#> An object of class "SamplePartialAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 1.000000000 0.948047341 -0.229421874 0.038147781 0.093785438 0.073606698
#> Lag_6
#> 0.007727603
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocovariances(AirPassengers, maxlag = 6)
#> An object of class "SamplePartialAutocovariances"
#> 0 1 2 3 4 5
#> 14291.97333 1371.29064 -314.37784 52.19801 127.19886 99.29006
#> 6
#> 10.42335
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialVariances(AirPassengers, maxlag = 6)
#> An object of class "SamplePartialVariances"
#> 0 1 2 3 4 5 6
#> 14291.973 1446.437 1370.305 1368.310 1356.275 1348.927 1348.846
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
acv <- autocovariances(AirPassengers, maxlag = 6)
autocovariances(acv) # no-op
#> An object of class "SampleAutocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 14291.973 13549.467 12513.692 11529.066 10756.502 10201.181 9743.318
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
autocovariances(acv, maxlag = 4) # trim the available lags
#> An object of class "SampleAutocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4
#> 14291.97 13549.47 12513.69 11529.07 10756.50
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
## compute 2nd order properties from sample autocovariances
acr <- autocorrelations(acv)
acr
#> An object of class "SampleAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 1.0000000 0.9480473 0.8755748 0.8066812 0.7526254 0.7137700 0.6817336
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocorrelations(acv)
#> An object of class "SamplePartialAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 1.000000000 0.948047341 -0.229421874 0.038147781 0.093785438 0.073606698
#> Lag_6
#> 0.007727603
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocovariances(acv)
#> An object of class "SamplePartialAutocovariances"
#> 0 1 2 3 4 5
#> 14291.97333 1371.29064 -314.37784 52.19801 127.19886 99.29006
#> 6
#> 10.42335
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialVariances(acv)
#> An object of class "SamplePartialVariances"
#> 0 1 2 3 4 5 6
#> 14291.973 1446.437 1370.305 1368.310 1356.275 1348.927 1348.846
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
## compute 2nd order properties from sample autocorrelations
acr
#> An object of class "SampleAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 1.0000000 0.9480473 0.8755748 0.8066812 0.7526254 0.7137700 0.6817336
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialAutocorrelations(acr)
#> An object of class "SamplePartialAutocorrelations"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5
#> 1.000000000 0.948047341 -0.229421874 0.038147781 0.093785438 0.073606698
#> Lag_6
#> 0.007727603
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
## These cannot be computed, since the variance is needed but unknown:
## autocovariances(acr)
## partialAutocovariances(acr)
## partialVariances(acr)
## to treat autocorrelations as autocovariances,
## convert them to autocovariances explicitly:
as(acr, "Autocovariances")
#> An object of class "Autocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 1.0000000 0.9480473 0.8755748 0.8066812 0.7526254 0.7137700 0.6817336
as(acr, "SampleAutocovariances")
#> An object of class "SampleAutocovariances"
#> Lag_0 Lag_1 Lag_2 Lag_3 Lag_4 Lag_5 Lag_6
#> 1.0000000 0.9480473 0.8755748 0.8066812 0.7526254 0.7137700 0.6817336
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x
partialVariances(as(acr, "Autocovariances"))
#> An object of class "PartialVariances"
#> 0 1 2 3 4 5 6
#> 1.00000000 0.10120624 0.09587931 0.09573978 0.09489768 0.09438353 0.09437790
partialVariances(as(acr, "SampleAutocovariances"))
#> An object of class "SamplePartialVariances"
#> 0 1 2 3 4 5 6
#> 1.00000000 0.10120624 0.09587931 0.09573978 0.09489768 0.09438353 0.09437790
#> Slot n:
#> [1] 144
#> Slot varnames: <not set>
#> Slot objectname: x