Package 'stats'

Description

R statistical functions

Details

This package contains functions for statistical calculations and random number generation.

For a complete list of functions, use library(help = "stats").

Author(s)

R Core Team and contributors worldwide

Maintainer: R Core Team [email protected]

Description

.checkMFClasses checks if the variables used in a predict method agree in type with those used for fitting.

.MFclass categorizes variables for this purpose.

.getXlevels() extracts factor levels from factor or character variables.

Usage

.checkMFClasses(cl, m, ordNotOK = FALSE)
.MFclass(x)
.getXlevels(Terms, m)

Arguments

Details

For applications involving model.matrix() such as linear models we do not need to differentiate between ordered factors and factors as although these affect the coding, the coding used in the fit is already recorded and imposed during prediction. However, other applications may treat ordered factors differently: rpart does, for example.

Value

.checkMFClasses() checks and either signals an error calling stop() or returns NULL invisibly.

.MFclass() returns a character string, one of "logical", "ordered", "factor", "numeric", "nmatrix.*" (a numeric matrix with a number of columns appended) or "other".

.getXlevels returns a named list of character vectors, possibly empty, or NULL.

Examples

sapply(warpbreaks, .MFclass) # "numeric" plus 2 x "factor"
sapply(iris,       .MFclass) # 4 x "numeric" plus "factor"

mf <- model.frame(Sepal.Width ~ Species,      iris)
mc <- model.frame(Sepal.Width ~ Sepal.Length, iris)

.checkMFClasses("numeric", mc) # nothing else
.checkMFClasses(c("numeric", "factor"), mf)

## simple .getXlevels() cases :
(xl <- .getXlevels(terms(mf), mf)) # a list with one entry " $ Species" with 3 levels:
stopifnot(exprs = {
  identical(xl$Species, levels(iris$Species))
  identical(.getXlevels(terms(mc), mc), xl[0]) # a empty named list, as no factors
  is.null(.getXlevels(terms(x~x), list(x=1)))
})

Description

The function acf computes (and by default plots) estimates of the autocovariance or autocorrelation function. Function pacf is the function used for the partial autocorrelations. Function ccf computes the cross-correlation or cross-covariance of two univariate series.

Usage

acf(x, lag.max = NULL,
    type = c("correlation", "covariance", "partial"),
    plot = TRUE, na.action = na.fail, demean = TRUE, ...)

pacf(x, lag.max, plot, na.action, ...)

## Default S3 method:
pacf(x, lag.max = NULL, plot = TRUE, na.action = na.fail,
    ...)

ccf(x, y, lag.max = NULL, type = c("correlation", "covariance"),
    plot = TRUE, na.action = na.fail, ...)

## S3 method for class 'acf'
x[i, j]

Arguments

Details

For type = "correlation" and "covariance", the estimates are based on the sample covariance. (The lag 0 autocorrelation is fixed at 1 by convention.)

By default, no missing values are allowed. If the na.action function passes through missing values (as na.pass does), the covariances are computed from the complete cases. This means that the estimate computed may well not be a valid autocorrelation sequence, and may contain missing values. Missing values are not allowed when computing the PACF of a multivariate time series.

The partial correlation coefficient is estimated by fitting autoregressive models of successively higher orders up to lag.max.

The generic function plot has a method for objects of class "acf".

The lag is returned and plotted in units of time, and not numbers of observations.

There are print and subsetting methods for objects of class "acf".

Value

An object of class "acf", which is a list with the following elements:

The lag k value returned by ccf(x, y) estimates the correlation between x[t+k] and y[t].

The result is returned invisibly if plot is TRUE.

Author(s)

Original: Paul Gilbert, Martyn Plummer. Extensive modifications and univariate case of pacf by B. D. Ripley.

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer-Verlag.

(This contains the exact definitions used.)

See Also

plot.acf, ARMAacf for the exact autocorrelations of a given ARMA process.

Examples

require(graphics)

## Examples from Venables & Ripley
acf(lh)
acf(lh, type = "covariance")
pacf(lh)

acf(ldeaths)
acf(ldeaths, ci.type = "ma")
acf(ts.union(mdeaths, fdeaths))
ccf(mdeaths, fdeaths, ylab = "cross-correlation")
# (just the cross-correlations)

presidents # contains missing values
acf(presidents, na.action = na.pass)
pacf(presidents, na.action = na.pass)

Description

Compute an AR process exactly fitting an autocorrelation function.

Usage

acf2AR(acf)

Arguments

Value

A matrix, with one row for the computed AR(p) coefficients for 1 <= p <= length(acf).

See Also

ARMAacf, ar.yw which does this from an empirical ACF.

Examples

(Acf <- ARMAacf(c(0.6, 0.3, -0.2)))
acf2AR(Acf)

Description

Compute all the single terms in the scope argument that can be added to or dropped from the model, fit those models and compute a table of the changes in fit.

Usage

add1(object, scope, ...)

## Default S3 method:
add1(object, scope, scale = 0, test = c("none", "Chisq"),
     k = 2, trace = FALSE, ...)

## S3 method for class 'lm'
add1(object, scope, scale = 0, test = c("none", "Chisq", "F"),
     x = NULL, k = 2, ...)

## S3 method for class 'glm'
add1(object, scope, scale = 0,
     test = c("none", "Rao", "LRT", "Chisq", "F"),
     x = NULL, k = 2, ...)

drop1(object, scope, ...)

## Default S3 method:
drop1(object, scope, scale = 0, test = c("none", "Chisq"),
      k = 2, trace = FALSE, ...)

## S3 method for class 'lm'
drop1(object, scope, scale = 0, all.cols = TRUE,
      test = c("none", "Chisq", "F"), k = 2, ...)

## S3 method for class 'glm'
drop1(object, scope, scale = 0,
      test = c("none", "Rao", "LRT", "Chisq", "F"),
      k = 2, ...)

Arguments

Details

For drop1 methods, a missing scope is taken to be all terms in the model. The hierarchy is respected when considering terms to be added or dropped: all main effects contained in a second-order interaction must remain, and so on.

In a scope formula . means ‘what is already there’.

The methods for lm and glm are more efficient in that they do not recompute the model matrix and call the fit methods directly.

The default output table gives AIC, defined as minus twice log likelihood plus $2p$ where $p$ is the rank of the model (the number of effective parameters). This is only defined up to an additive constant (like log-likelihoods). For linear Gaussian models with fixed scale, the constant is chosen to give Mallows' $C_p$, $RSS/scale + 2p - n$. Where $C_p$ is used, the column is labelled as Cp rather than AIC.

The F tests for the "glm" methods are based on analysis of deviance tests, so if the dispersion is estimated it is based on the residual deviance, unlike the F tests of anova.glm.

Value

An object of class "anova" summarizing the differences in fit between the models.

Warning

The model fitting must apply the models to the same dataset. Most methods will attempt to use a subset of the data with no missing values for any of the variables if na.action = na.omit, but this may give biased results. Only use these functions with data containing missing values with great care.

The default methods make calls to the function nobs to check that the number of observations involved in the fitting process remained unchanged.

Note

These are not fully equivalent to the functions in S. There is no keep argument, and the methods used are not quite so computationally efficient.

Their authors' definitions of Mallows' $C_p$ and Akaike's AIC are used, not those of the authors of the models chapter of S.

Author(s)

The design was inspired by the S functions of the same names described in Chambers (1992).

References

Chambers, J. M. (1992) Linear models. Chapter 4 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

step, aov, lm, extractAIC, anova

Examples

require(graphics); require(utils)
## following example(swiss)
lm1 <- lm(Fertility ~ ., data = swiss)
add1(lm1, ~ I(Education^2) + .^2)
drop1(lm1, test = "F")  # So called 'type II' anova

## following example(glm)

drop1(glm.D93, test = "Chisq")
drop1(glm.D93, test = "F")
add1(glm.D93, scope = ~outcome*treatment, test = "Rao") ## Pearson Chi-square

Description

For a given table one can specify which of the classifying factors to expand by one or more levels to hold margins to be calculated. One may for example form sums and means over the first dimension and medians over the second. The resulting table will then have two extra levels for the first dimension and one extra level for the second. The default is to sum over all margins in the table. Other possibilities may give results that depend on the order in which the margins are computed. This is flagged in the printed output from the function.

Usage

addmargins(A, margin = seq_along(dim(A)), FUN = sum, quiet = FALSE)

Arguments

Details

If the functions used to form margins are not commutative, the result depends on the order in which margins are computed. Annotation of margins is done via naming the FUN list.

Value

A table or array with the same number of dimensions as A, but with extra levels of the dimensions mentioned in margin. The number of levels added to each dimension is the length of the entries in FUN. A message with the order of computation of margins is printed.

Author(s)

Bendix Carstensen, Steno Diabetes Center & Department of Biostatistics, University of Copenhagen, https://BendixCarstensen.com, autumn 2003. Margin naming enhanced by Duncan Murdoch.

See Also

table, ftable, margin.table.

Examples

Aye <- sample(c("Yes", "Si", "Oui"), 177, replace = TRUE)
Bee <- sample(c("Hum", "Buzz"), 177, replace = TRUE)
Sea <- sample(c("White", "Black", "Red", "Dead"), 177, replace = TRUE)
(A <- table(Aye, Bee, Sea))
(aA <- addmargins(A))

ftable(A)
ftable(aA)

# Non-commutative functions - note differences between resulting tables:
ftable( addmargins(A, c(3, 1),
                   FUN = list(list(Min = min, Max = max),
                              Sum = sum)))
ftable( addmargins(A, c(1, 3),
                   FUN = list(Sum = sum,
                              list(Min = min, Max = max))))

# Weird function needed to return the N when computing percentages
sqsm <- function(x) sum(x)^2/100
B <- table(Sea, Bee)
round(sweep(addmargins(B, 1, list(list(All = sum, N = sqsm))), 2,
            apply(B, 2, sum)/100, `/`), 1)
round(sweep(addmargins(B, 2, list(list(All = sum, N = sqsm))), 1,
            apply(B, 1, sum)/100, `/`), 1)

# A total over Bee requires formation of the Bee-margin first:
mB <-  addmargins(B, 2, FUN = list(list(Total = sum)))
round(ftable(sweep(addmargins(mB, 1, list(list(All = sum, N = sqsm))), 2,
                   apply(mB, 2, sum)/100, `/`)), 1)

## Zero.Printing table+margins:
set.seed(1)
x <- sample( 1:7, 20, replace = TRUE)
y <- sample( 1:7, 20, replace = TRUE)
tx <- addmargins( table(x, y) )
print(tx, zero.print = ".")

Description

Splits the data into subsets, computes summary statistics for each, and returns the result in a convenient form.

Usage

aggregate(x, ...)

## Default S3 method:
aggregate(x, ...)

## S3 method for class 'data.frame'
aggregate(x, by, FUN, ..., simplify = TRUE, drop = TRUE)

## S3 method for class 'formula'
aggregate(x, data, FUN, ...,
          subset, na.action = na.omit)

## S3 method for class 'ts'
aggregate(x, nfrequency = 1, FUN = sum, ndeltat = 1,
          ts.eps = getOption("ts.eps"), ...)

Arguments

Details

aggregate is a generic function with methods for data frames and time series.

The default method, aggregate.default, uses the time series method if x is a time series, and otherwise coerces x to a data frame and calls the data frame method.

aggregate.data.frame is the data frame method. If x is not a data frame, it is coerced to one, which must have a non-zero number of rows. Then, each of the variables (columns) in x is split into subsets of cases (rows) of identical combinations of the components of by, and FUN is applied to each such subset with further arguments in ... passed to it. The result is reformatted into a data frame containing the variables in by and x. The ones arising from by contain the unique combinations of grouping values used for determining the subsets, and the ones arising from x the corresponding summaries for the subset of the respective variables in x. If simplify is true, summaries are simplified to vectors or matrices if they have a common length of one or greater than one, respectively; otherwise, lists of summary results according to subsets are obtained. Rows with missing values in any of the by variables will be omitted from the result. (Note that versions of R prior to 2.11.0 required FUN to be a scalar function.)

The formula method provides a standard formula interface to aggregate.data.frame. The latter invokes the formula method if by is a formula, in which case aggregate(x, by, FUN) is the same as aggregate(by, x, FUN) for a data frame x.

aggregate.ts is the time series method, and requires FUN to be a scalar function. If x is not a time series, it is coerced to one. Then, the variables in x are split into appropriate blocks of length frequency(x) / nfrequency, and FUN is applied to each such block, with further (named) arguments in ... passed to it. The result returned is a time series with frequency nfrequency holding the aggregated values. Note that this make most sense for a quarterly or yearly result when the original series covers a whole number of quarters or years: in particular aggregating a monthly series to quarters starting in February does not give a conventional quarterly series.

FUN is passed to match.fun, and hence it can be a function or a symbol or character string naming a function.

Value

For the time series method, a time series of class "ts" or class c("mts", "ts").

For the data frame method, a data frame with columns corresponding to the grouping variables in by followed by aggregated columns from x. If the by has names, the non-empty times are used to label the columns in the results, with unnamed grouping variables being named Group.i for by[[i]].

Warning

The first argument of the "formula" method was named formula rather than x prior to R 4.2.0. Portable uses should not name that argument.

Author(s)

Kurt Hornik, with contributions by Arni Magnusson.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

See Also

apply, lapply, tapply.

Examples

## Compute the averages for the variables in 'state.x77', grouped
## according to the region (Northeast, South, North Central, West) that
## each state belongs to.
aggregate(state.x77, list(Region = state.region), mean)

## Compute the averages according to region and the occurrence of more
## than 130 days of frost.
aggregate(state.x77,
          list(Region = state.region,
               Cold = state.x77[,"Frost"] > 130),
          mean)
## (Note that no state in 'South' is THAT cold.)


## example with character variables and NAs
testDF <- data.frame(v1 = c(1,3,5,7,8,3,5,NA,4,5,7,9),
                     v2 = c(11,33,55,77,88,33,55,NA,44,55,77,99) )
by1 <- c("red", "blue", 1, 2, NA, "big", 1, 2, "red", 1, NA, 12)
by2 <- c("wet", "dry", 99, 95, NA, "damp", 95, 99, "red", 99, NA, NA)
aggregate(x = testDF, by = list(by1, by2), FUN = "mean")

# and if you want to treat NAs as a group
fby1 <- factor(by1, exclude = "")
fby2 <- factor(by2, exclude = "")
aggregate(x = testDF, by = list(fby1, fby2), FUN = "mean")


## Formulas, one ~ one, one ~ many, many ~ one, and many ~ many:
aggregate(weight ~ feed, data = chickwts, mean)
aggregate(breaks ~ wool + tension, data = warpbreaks, mean)
aggregate(cbind(Ozone, Temp) ~ Month, data = airquality, mean)
aggregate(cbind(ncases, ncontrols) ~ alcgp + tobgp, data = esoph, sum)

## "complete cases" vs. "available cases"
colSums(is.na(airquality))  # NAs in Ozone but not Temp
## the default is to summarize *complete cases*:
aggregate(cbind(Ozone, Temp) ~ Month, data = airquality, FUN = mean)
## to handle missing values *per variable*:
aggregate(cbind(Ozone, Temp) ~ Month, data = airquality, FUN = mean,
          na.action = na.pass, na.rm = TRUE)

## Dot notation:
aggregate(. ~ Species, data = iris, mean)
aggregate(len ~ ., data = ToothGrowth, mean)

## Often followed by xtabs():
ag <- aggregate(len ~ ., data = ToothGrowth, mean)
xtabs(len ~ ., data = ag)

## Formula interface via 'by' (for pipe operations)
ToothGrowth |> aggregate(len ~ ., FUN = mean)

## Compute the average annual approval ratings for American presidents.
aggregate(presidents, nfrequency = 1, FUN = mean)
## Give the summer less weight.
aggregate(presidents, nfrequency = 1,
          FUN = weighted.mean, w = c(1, 1, 0.5, 1))

Description

Generic function calculating Akaike's ‘An Information Criterion’ for one or several fitted model objects for which a log-likelihood value can be obtained, according to the formula $-2 \mbox{log-likelihood} + k n_{par}$, where $n_{par}$ represents the number of parameters in the fitted model, and $k = 2$ for the usual AIC, or $k = \log(n)$ ($n$ being the number of observations) for the so-called BIC or SBC (Schwarz's Bayesian criterion).

Usage

AIC(object, ..., k = 2)

BIC(object, ...)

Arguments

Details

When comparing models fitted by maximum likelihood to the same data, the smaller the AIC or BIC, the better the fit.

The theory of AIC requires that the log-likelihood has been maximized: whereas AIC can be computed for models not fitted by maximum likelihood, their AIC values should not be compared.

Examples of models not ‘fitted to the same data’ are where the response is transformed (accelerated-life models are fitted to log-times) and where contingency tables have been used to summarize data.

These are generic functions (with S4 generics defined in package stats4): however methods should be defined for the log-likelihood function logLik rather than these functions: the action of their default methods is to call logLik on all the supplied objects and assemble the results. Note that in several common cases logLik does not return the value at the MLE: see its help page.

The log-likelihood and hence the AIC/BIC is only defined up to an additive constant. Different constants have conventionally been used for different purposes and so extractAIC and AIC may give different values (and do for models of class "lm": see the help for extractAIC). Particular care is needed when comparing fits of different classes (with, for example, a comparison of a Poisson and gamma GLM being meaningless since one has a discrete response, the other continuous).

BIC is defined as AIC(object, ..., k = log(nobs(object))). This needs the number of observations to be known: the default method looks first for a "nobs" attribute on the return value from the logLik method, then tries the nobs generic, and if neither succeed returns BIC as NA.

Value

If just one object is provided, a numeric value with the corresponding AIC (or BIC, or ..., depending on k).

If multiple objects are provided, a data.frame with rows corresponding to the objects and columns representing the number of parameters in the model (df) and the AIC or BIC.

Author(s)

Originally by José Pinheiro and Douglas Bates, more recent revisions by R-core.

References

Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.

See Also

extractAIC, logLik, nobs.

Examples

lm1 <- lm(Fertility ~ . , data = swiss)
AIC(lm1)
stopifnot(all.equal(AIC(lm1),
                    AIC(logLik(lm1))))
BIC(lm1)

lm2 <- update(lm1, . ~ . -Examination)
AIC(lm1, lm2)
BIC(lm1, lm2)

Description

Find aliases (linearly dependent terms) in a linear model specified by a formula.

Usage

alias(object, ...)

## S3 method for class 'formula'
alias(object, data, ...)

## S3 method for class 'lm'
alias(object, complete = TRUE, partial = FALSE,
      partial.pattern = FALSE, ...)

Arguments

Details

Although the main method is for class "lm", alias is most useful for experimental designs and so is used with fits from aov. Complete aliasing refers to effects in linear models that cannot be estimated independently of the terms which occur earlier in the model and so have their coefficients omitted from the fit. Partial aliasing refers to effects that can be estimated less precisely because of correlations induced by the design.

Some parts of the "lm" method require recommended package MASS to be installed.

Value

A list (of class "listof") containing components

Note

The aliasing pattern may depend on the contrasts in use: Helmert contrasts are probably most useful.

The defaults are different from those in S.

Author(s)

The design was inspired by the S function of the same name described in Chambers et al. (1992).

References

Chambers, J. M., Freeny, A and Heiberger, R. M. (1992) Analysis of variance; designed experiments. Chapter 5 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

Examples

op <- options(contrasts = c("contr.helmert", "contr.poly"))
npk.aov <- aov(yield ~ block + N*P*K, npk)
alias(npk.aov)
options(op)  # reset

Description

Compute analysis of variance (or deviance) tables for one or more fitted model objects.

Usage

anova(object, ...)

Arguments

Value

This (generic) function returns an object of class anova. These objects represent analysis-of-variance and analysis-of-deviance tables. When given a single argument it produces a table which tests whether the model terms are significant.

When given a sequence of objects, anova tests the models against one another in the order specified.

The print method for anova objects prints tables in a ‘pretty’ form.

Warning

The comparison between two or more models will only be valid if they are fitted to the same dataset. This may be a problem if there are missing values and R's default of na.action = na.omit is used.

References

Chambers, J. M. and Hastie, T. J. (1992) Statistical Models in S, Wadsworth & Brooks/Cole.

See Also

coefficients, effects, fitted.values, residuals, summary, drop1, add1.

Description

Compute an analysis of deviance table for one or more generalized linear model fits.

Usage

## S3 method for class 'glm'
anova(object, ..., dispersion = NULL, test = NULL)

Arguments

Details

Specifying a single object gives a sequential analysis of deviance table for that fit. That is, the reductions in the residual deviance as each term of the formula is added in turn are given in as the rows of a table, plus the residual deviances themselves.

If more than one object is specified, the table has a row for the residual degrees of freedom and deviance for each model. For all but the first model, the change in degrees of freedom and deviance is also given. (This only makes statistical sense if the models are nested.) It is conventional to list the models from smallest to largest, but this is up to the user.

The table will optionally contain test statistics (and P values) comparing the reduction in deviance for the row to the residuals. For models with known dispersion (e.g., binomial and Poisson fits) the chi-squared test is most appropriate, and for those with dispersion estimated by moments (e.g., gaussian, quasibinomial and quasipoisson fits) the F test is most appropriate. If anova.glm can determine which of these cases applies then by default it will use one of the above tests. If the dispersion argument is supplied, the dispersion is considered known and the chi-squared test will be used. Argument test=FALSE suppresses the test statistics and P values. Mallows' $C_p$ statistic is the residual deviance plus twice the estimate of $\sigma^2$ times the residual degrees of freedom, which is closely related to AIC (and a multiple of it if the dispersion is known). You can also choose "LRT" and "Rao" for likelihood ratio tests and Rao's efficient score test. The former is synonymous with "Chisq" (although both have an asymptotic chi-square distribution).

The dispersion estimate will be taken from the largest model, using the value returned by summary.glm. As this will in most cases use a Chi-squared-based estimate, the F tests are not based on the residual deviance in the analysis of deviance table shown.

Value

An object of class "anova" inheriting from class "data.frame".

Warning

The comparison between two or more models will only be valid if they are fitted to the same dataset. This may be a problem if there are missing values and R's default of na.action = na.omit is used, and anova will detect this with an error.

References

Hastie, T. J. and Pregibon, D. (1992) Generalized linear models. Chapter 6 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

glm, anova.

drop1 for so-called ‘type II’ ANOVA where each term is dropped one at a time respecting their hierarchy.

Examples

## --- Continuing the Example from  '?glm':

anova(glm.D93, test = FALSE)
anova(glm.D93, test = "Cp")
anova(glm.D93, test = "Chisq")
glm.D93a <-
   update(glm.D93, ~treatment*outcome) # equivalent to Pearson Chi-square
anova(glm.D93, glm.D93a, test = "Rao")

Description

Compute an analysis of variance table for one or more linear model fits.

Usage

## S3 method for class 'lm'
anova(object, ...)

## S3 method for class 'lmlist'
anova(object, ..., scale = 0, test = "F")

Arguments

Details

Specifying a single object gives a sequential analysis of variance table for that fit. That is, the reductions in the residual sum of squares as each term of the formula is added in turn are given in as the rows of a table, plus the residual sum of squares.

The table will contain F statistics (and P values) comparing the mean square for the row to the residual mean square.

If more than one object is specified, the table has a row for the residual degrees of freedom and sum of squares for each model. For all but the first model, the change in degrees of freedom and sum of squares is also given. (This only make statistical sense if the models are nested.) It is conventional to list the models from smallest to largest, but this is up to the user.

Optionally the table can include test statistics. Normally the F statistic is most appropriate, which compares the mean square for a row to the residual sum of squares for the largest model considered. If scale is specified chi-squared tests can be used. Mallows' $C_p$ statistic is the residual sum of squares plus twice the estimate of $\sigma^2$ times the residual degrees of freedom.

Value

An object of class "anova" inheriting from class "data.frame".

Warning

The comparison between two or more models will only be valid if they are fitted to the same dataset. This may be a problem if there are missing values and R's default of na.action = na.omit is used, and anova.lmlist will detect this with an error.

References

Chambers, J. M. (1992) Linear models. Chapter 4 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

The model fitting function lm, anova.

drop1 for so-called ‘type II’ ANOVA where each term is dropped one at a time respecting their hierarchy.

Examples

## sequential table
fit <- lm(sr ~ ., data = LifeCycleSavings)
anova(fit)

## same effect via separate models
fit0 <- lm(sr ~ 1, data = LifeCycleSavings)
fit1 <- update(fit0, . ~ . + pop15)
fit2 <- update(fit1, . ~ . + pop75)
fit3 <- update(fit2, . ~ . + dpi)
fit4 <- update(fit3, . ~ . + ddpi)
anova(fit0, fit1, fit2, fit3, fit4, test = "F")

anova(fit4, fit2, fit0, test = "F") # unconventional order

Description

Compute a (generalized) analysis of variance table for one or more multivariate linear models.

Usage

## S3 method for class 'mlm'
anova(object, ...,
      test = c("Pillai", "Wilks", "Hotelling-Lawley", "Roy",
               "Spherical"),
      Sigma = diag(nrow = p), T = Thin.row(Proj(M) - Proj(X)),
      M = diag(nrow = p), X = ~0,
      idata = data.frame(index = seq_len(p)), tol = 1e-7)

Arguments

Details

The anova.mlm method uses either a multivariate test statistic for the summary table, or a test based on sphericity assumptions (i.e. that the covariance is proportional to a given matrix).

For the multivariate test, Wilks' statistic is most popular in the literature, but the default Pillai–Bartlett statistic is recommended by Hand and Taylor (1987). See summary.manova for further details.

For the "Spherical" test, proportionality is usually with the identity matrix but a different matrix can be specified using Sigma. Corrections for asphericity known as the Greenhouse–Geisser, respectively Huynh–Feldt, epsilons are given and adjusted $F$ tests are performed.

It is common to transform the observations prior to testing. This typically involves transformation to intra-block differences, but more complicated within-block designs can be encountered, making more elaborate transformations necessary. A transformation matrix T can be given directly or specified as the difference between two projections onto the spaces spanned by M and X, which in turn can be given as matrices or as model formulas with respect to idata (the tests will be invariant to parametrization of the quotient space M/X).

As with anova.lm, all test statistics use the SSD matrix from the largest model considered as the (generalized) denominator.

Contrary to other anova methods, the intercept is not excluded from the display in the single-model case. When contrast transformations are involved, it often makes good sense to test for a zero intercept.

Value

An object of class "anova" inheriting from class "data.frame"

Note

The Huynh–Feldt epsilon differs from that calculated by SAS (as of v. 8.2) except when the DF is equal to the number of observations minus one. This is believed to be a bug in SAS, not in R.

References

Hand, D. J. and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures. Chapman and Hall.

See Also

summary.manova

Examples

require(graphics)
utils::example(SSD) # Brings in the mlmfit and reacttime objects

mlmfit0 <- update(mlmfit, ~0)

### Traditional tests of intrasubj. contrasts
## Using MANOVA techniques on contrasts:
anova(mlmfit, mlmfit0, X = ~1)

## Assuming sphericity
anova(mlmfit, mlmfit0, X = ~1, test = "Spherical")


### tests using intra-subject 3x2 design
idata <- data.frame(deg = gl(3, 1, 6, labels = c(0, 4, 8)),
                    noise = gl(2, 3, 6, labels = c("A", "P")))

anova(mlmfit, mlmfit0, X = ~ deg + noise,
      idata = idata, test = "Spherical")
anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ noise,
      idata = idata, test = "Spherical" )
anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ deg,
      idata = idata, test = "Spherical" )

f <- factor(rep(1:2, 5)) # bogus, just for illustration
mlmfit2 <- update(mlmfit, ~f)
anova(mlmfit2, mlmfit, mlmfit0, X = ~1, test = "Spherical")
anova(mlmfit2, X = ~1, test = "Spherical")
# one-model form, eqiv. to previous

### There seems to be a strong interaction in these data
plot(colMeans(reacttime))

Description

Performs the Ansari-Bradley two-sample test for a difference in scale parameters.

Usage

ansari.test(x, ...)

## Default S3 method:
ansari.test(x, y,
            alternative = c("two.sided", "less", "greater"),
            exact = NULL, conf.int = FALSE, conf.level = 0.95,
            ...)

## S3 method for class 'formula'
ansari.test(formula, data, subset, na.action, ...)

Arguments

Details

Suppose that x and y are independent samples from distributions with densities $f((t-m)/s)/s$ and $f(t-m)$, respectively, where $m$ is an unknown nuisance parameter and $s$, the ratio of scales, is the parameter of interest. The Ansari-Bradley test is used for testing the null that $s$ equals 1, the two-sided alternative being that $s \ne 1$ (the distributions differ only in variance), and the one-sided alternatives being $s > 1$ (the distribution underlying x has a larger variance, "greater") or $s < 1$ ("less").

By default (if exact is not specified), an exact p-value is computed if both samples contain less than 50 finite values and there are no ties. Otherwise, a normal approximation is used.

Optionally, a nonparametric confidence interval and an estimator for $s$ are computed. If exact p-values are available, an exact confidence interval is obtained by the algorithm described in Bauer (1972), and the Hodges-Lehmann estimator is employed. Otherwise, the returned confidence interval and point estimate are based on normal approximations.

Note that mid-ranks are used in the case of ties rather than average scores as employed in Hollander & Wolfe (1973). See, e.g., Hajek, Sidak and Sen (1999), pages 131ff, for more information.

Value

A list with class "htest" containing the following components:

Note

To compare results of the Ansari-Bradley test to those of the F test to compare two variances (under the assumption of normality), observe that $s$ is the ratio of scales and hence $s^2$ is the ratio of variances (provided they exist), whereas for the F test the ratio of variances itself is the parameter of interest. In particular, confidence intervals are for $s$ in the Ansari-Bradley test but for $s^2$ in the F test.

References

David F. Bauer (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association, 67, 687–690. doi:10.1080/01621459.1972.10481279.

Jaroslav Hajek, Zbynek Sidak and Pranab K. Sen (1999). Theory of Rank Tests. San Diego, London: Academic Press.

Myles Hollander and Douglas A. Wolfe (1973). Nonparametric Statistical Methods. New York: John Wiley & Sons. Pages 83–92.

See Also

fligner.test for a rank-based (nonparametric) $k$-sample test for homogeneity of variances; mood.test for another rank-based two-sample test for a difference in scale parameters; var.test and bartlett.test for parametric tests for the homogeneity in variance.

ansari_test in package coin for exact and approximate conditional p-values for the Ansari-Bradley test, as well as different methods for handling ties.

Examples

## Hollander & Wolfe (1973, p. 86f):
## Serum iron determination using Hyland control sera
ramsay <- c(111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
            101, 96, 97, 102, 107, 113, 116, 113, 110, 98)
jung.parekh <- c(107, 108, 106, 98, 105, 103, 110, 105, 104,
            100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99)
ansari.test(ramsay, jung.parekh)

ansari.test(rnorm(10), rnorm(10, 0, 2), conf.int = TRUE)

## try more points - failed in 2.4.1
ansari.test(rnorm(100), rnorm(100, 0, 2), conf.int = TRUE)

Description

Fit an analysis of variance model by a call to lm (for each stratum if an Error(.) is used).

Usage

aov(formula, data = NULL, projections = FALSE, qr = TRUE,
    contrasts = NULL, ...)

Arguments

Details

This provides a wrapper to lm for fitting linear models to balanced or unbalanced experimental designs.

The main difference from lm is in the way print, summary and so on handle the fit: this is expressed in the traditional language of the analysis of variance rather than that of linear models.

If the formula contains a single Error term, this is used to specify error strata, and appropriate models are fitted within each error stratum.

The formula can specify multiple responses.

Weights can be specified by a weights argument, but should not be used with an Error term, and are incompletely supported (e.g., not by model.tables).

Value

An object of class c("aov", "lm") or for multiple responses of class c("maov", "aov", "mlm", "lm") or for multiple error strata of class c("aovlist", "listof"). There are print and summary methods available for these.

Note

aov is designed for balanced designs, and the results can be hard to interpret without balance: beware that missing values in the response(s) will likely lose the balance. If there are two or more error strata, the methods used are statistically inefficient without balance, and it may be better to use lme in package nlme.

Balance can be checked with the replications function.

The default ‘contrasts’ in R are not orthogonal contrasts, and aov and its helper functions will work better with such contrasts: see the examples for how to select these.

Author(s)

The design was inspired by the S function of the same name described in Chambers et al. (1992).

References

Chambers, J. M., Freeny, A and Heiberger, R. M. (1992) Analysis of variance; designed experiments. Chapter 5 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

lm, summary.aov, replications, alias, proj, model.tables, TukeyHSD

Examples

## From Venables and Ripley (2002) p.165.

## Set orthogonal contrasts.
op <- options(contrasts = c("contr.helmert", "contr.poly"))
( npk.aov <- aov(yield ~ block + N*P*K, npk) )
summary(npk.aov)
coefficients(npk.aov)

## to show the effects of re-ordering terms contrast the two fits
aov(yield ~ block + N * P + K, npk)
aov(terms(yield ~ block + N * P + K, keep.order = TRUE), npk)


## as a test, not particularly sensible statistically
npk.aovE <- aov(yield ~  N*P*K + Error(block), npk)
npk.aovE
summary(npk.aovE)
options(op)  # reset to previous

Description

Return a list of points which linearly interpolate given data points, or a function performing the linear (or constant) interpolation.

Usage

approx   (x, y = NULL, xout, method = "linear", n = 50,
          yleft, yright, rule = 1, f = 0, ties = mean, na.rm = TRUE)

approxfun(x, y = NULL,       method = "linear",
          yleft, yright, rule = 1, f = 0, ties = mean, na.rm = TRUE)

Arguments

Details

The inputs can contain missing values which are deleted (if na.rm is true, i.e., by default), so at least two complete (x, y) pairs are required (for method = "linear", one otherwise). If there are duplicated (tied) x values and ties contains a function it is applied to the y values for each distinct x value to produce (x,y) pairs with unique x. Useful functions in this context include mean, min, and max.

If ties = "ordered" the x values are assumed to be already ordered (and unique) and ties are not checked but kept if present. This is the fastest option for large length(x).

If ties is a list of length two, ties[[2]] must be a function to be applied to ties, see above, but if ties[[1]] is identical to "ordered", the x values are assumed to be sorted and are only checked for ties. Consequently, ties = list("ordered", mean) will be slightly more efficient than the default ties = mean in such a case.

The first y value will be used for interpolation to the left and the last one for interpolation to the right.

Value

approx returns a list with components x and y, containing n coordinates which interpolate the given data points according to the method (and rule) desired.

The function approxfun returns a function performing (linear or constant) interpolation of the given data points. For a given set of x values, this function will return the corresponding interpolated values. It uses data stored in its environment when it was created, the details of which are subject to change.

Warning

The value returned by approxfun contains references to the code in the current version of R: it is not intended to be saved and loaded into a different R session. This is safer for R >= 3.0.0.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

See Also

spline and splinefun for spline interpolation.

Examples

require(graphics)

x <- 1:10
y <- rnorm(10)
par(mfrow = c(2,1))
plot(x, y, main = "approx(.) and approxfun(.)")
points(approx(x, y), col = 2, pch = "*")
points(approx(x, y, method = "constant"), col = 4, pch = "*")

f <- approxfun(x, y)
curve(f(x), 0, 11, col = "green2")
points(x, y)
is.function(fc <- approxfun(x, y, method = "const")) # TRUE
curve(fc(x), 0, 10, col = "darkblue", add = TRUE)
## different extrapolation on left and right side :
plot(approxfun(x, y, rule = 2:1), 0, 11,
     col = "tomato", add = TRUE, lty = 3, lwd = 2)

### Treatment of 'NA's -- are kept if  na.rm=FALSE :

xn <- 1:4
yn <- c(1,NA,3:4)
xout <- (1:9)/2
## Default behavior (na.rm = TRUE): NA's omitted; extrapolation gives NA
data.frame(approx(xn,yn, xout))
data.frame(approx(xn,yn, xout, rule = 2))# -> *constant* extrapolation
## New (2019-2020)  na.rm = FALSE: NA's are "kept"
data.frame(approx(xn,yn, xout, na.rm=FALSE, rule = 2))
data.frame(approx(xn,yn, xout, na.rm=FALSE, rule = 2, method="constant"))

## NA's in x[] are not allowed:
stopifnot(inherits( try( approx(yn,yn, na.rm=FALSE) ), "try-error"))

## Give a nice overview of all possibilities  rule * method * na.rm :
##             -----------------------------  ====   ======   =====
## extrapolations "N":= NA;   "C":= Constant :
rules <- list(N=1, C=2, NC=1:2, CN=2:1)
methods <- c("constant","linear")
ry <- sapply(rules, function(R) {
       sapply(methods, function(M)
        sapply(setNames(,c(TRUE,FALSE)), function(na.)
                 approx(xn, yn, xout=xout, method=M, rule=R, na.rm=na.)$y),
        simplify="array")
   }, simplify="array")
names(dimnames(ry)) <- c("x = ", "na.rm", "method", "rule")
dimnames(ry)[[1]] <- format(xout)
ftable(aperm(ry, 4:1)) # --> (4 * 2 * 2) x length(xout)  =  16 x 9 matrix


## Show treatment of 'ties' :

x <- c(2,2:4,4,4,5,5,7,7,7)
y <- c(1:6, 5:4, 3:1)
(amy <- approx(x, y, xout = x)$y) # warning, can be avoided by specifying 'ties=':
op <- options(warn=2) # warnings would be error
stopifnot(identical(amy, approx(x, y, xout = x, ties=mean)$y))
(ay <- approx(x, y, xout = x, ties = "ordered")$y)
stopifnot(amy == c(1.5,1.5, 3, 5,5,5, 4.5,4.5, 2,2,2),
          ay  == c(2, 2,    3, 6,6,6, 4, 4,    1,1,1))
approx(x, y, xout = x, ties = min)$y
approx(x, y, xout = x, ties = max)$y
options(op) # revert 'warn'ing level

Description

Fit an autoregressive time series model to the data, by default selecting the complexity by AIC.

Usage

ar(x, aic = TRUE, order.max = NULL,
   method = c("yule-walker", "burg", "ols", "mle", "yw"),
   na.action, series, ...)

ar.burg(x, ...)
## Default S3 method:
ar.burg(x, aic = TRUE, order.max = NULL,
        na.action = na.fail, demean = TRUE, series,
        var.method = 1, ...)
## S3 method for class 'mts'
ar.burg(x, aic = TRUE, order.max = NULL,
        na.action = na.fail, demean = TRUE, series,
        var.method = 1, ...)

ar.yw(x, ...)
## Default S3 method:
ar.yw(x, aic = TRUE, order.max = NULL,
      na.action = na.fail, demean = TRUE, series, ...)
## S3 method for class 'mts'
ar.yw(x, aic = TRUE, order.max = NULL,
      na.action = na.fail, demean = TRUE, series,
      var.method = 1, ...)

ar.mle(x, aic = TRUE, order.max = NULL, na.action = na.fail,
       demean = TRUE, series, ...)

## S3 method for class 'ar'
predict(object, newdata, n.ahead = 1, se.fit = TRUE, ...)

Arguments

Details

For definiteness, note that the AR coefficients have the sign in

$x_t - \mu = a_1(x_{t-1} - \mu) + \cdots + a_p(x_{t-p} - \mu) + e_t$

ar is just a wrapper for the functions ar.yw, ar.burg, ar.ols and ar.mle.

Order selection is done by AIC if aic is true. This is problematic, as of the methods here only ar.mle performs true maximum likelihood estimation. The AIC is computed as if the variance estimate were the MLE, omitting the determinant term from the likelihood. Note that this is not the same as the Gaussian likelihood evaluated at the estimated parameter values. In ar.yw the variance matrix of the innovations is computed from the fitted coefficients and the autocovariance of x.

ar.burg allows two methods to estimate the innovations variance and hence AIC. Method 1 is to use the update given by the Levinson-Durbin recursion (Brockwell and Davis, 1991, (8.2.6) on page 242), and follows S-PLUS. Method 2 is the mean of the sum of squares of the forward and backward prediction errors (as in Brockwell and Davis, 1996, page 145). Percival and Walden (1998) discuss both. In the multivariate case the estimated coefficients will depend (slightly) on the variance estimation method.

Remember that ar includes by default a constant in the model, by removing the overall mean of x before fitting the AR model, or (ar.mle) estimating a constant to subtract.

Value

For ar and its methods a list of class "ar" with the following elements:

For predict.ar, a time series of predictions, or if se.fit = TRUE, a list with components pred, the predictions, and se, the estimated standard errors. Both components are time series.

Note

Only the univariate case of ar.mle is implemented.

Fitting by method="mle" to long series can be very slow.

If x contains missing values, see NA, also consider using arima(), possibly with method = "ML".

Author(s)

Martyn Plummer. Univariate case of ar.yw, ar.mle and C code for univariate case of ar.burg by B. D. Ripley.

References

Brockwell, P. J. and Davis, R. A. (1991). Time Series and Forecasting Methods, second edition. Springer, New York. Section 11.4.

Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York. Sections 5.1 and 7.6.

Percival, D. P. and Walden, A. T. (1998). Spectral Analysis for Physical Applications. Cambridge University Press.

Whittle, P. (1963). On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix. Biometrika, 40, 129–134. doi:10.2307/2333753.

See Also

ar.ols, arima for ARMA models; acf2AR, for AR construction from the ACF.

arima.sim for simulation of AR processes.

Examples

ar(lh)
ar(lh, method = "burg")
ar(lh, method = "ols")
ar(lh, FALSE, 4) # fit ar(4)

(sunspot.ar <- ar(sunspot.year))
predict(sunspot.ar, n.ahead = 25)
## try the other methods too

ar(ts.union(BJsales, BJsales.lead))
## Burg is quite different here, as is OLS (see ar.ols)
ar(ts.union(BJsales, BJsales.lead), method = "burg")

Description

Fit an autoregressive time series model to the data by ordinary least squares, by default selecting the complexity by AIC.

Usage

ar.ols(x, aic = TRUE, order.max = NULL, na.action = na.fail,
       demean = TRUE, intercept = demean, series, ...)

Arguments

Details

ar.ols fits the general AR model to a possibly non-stationary and/or multivariate system of series x. The resulting unconstrained least squares estimates are consistent, even if some of the series are non-stationary and/or co-integrated. For definiteness, note that the AR coefficients have the sign in

$x_t - \mu = a_0 + a_1(x_{t-1} - \mu) + \cdots + a_p(x_{t-p} - \mu) + e_t$

where $a_0$ is zero unless intercept is true, and $\mu$ is the sample mean if demean is true, zero otherwise.

Order selection is done by AIC if aic is true. This is problematic, as ar.ols does not perform true maximum likelihood estimation. The AIC is computed as if the variance estimate (computed from the variance matrix of the residuals) were the MLE, omitting the determinant term from the likelihood. Note that this is not the same as the Gaussian likelihood evaluated at the estimated parameter values.

Some care is needed if intercept is true and demean is false. Only use this is the series are roughly centred on zero. Otherwise the computations may be inaccurate or fail entirely.

Value

A list of class "ar" with the following elements:

Author(s)

Adrian Trapletti, Brian Ripley.

References

Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis. Springer Verlag, NY, pp. 368–370.

See Also

ar

Examples

ar(lh, method = "burg")
ar.ols(lh)
ar.ols(lh, FALSE, 4) # fit ar(4)

ar.ols(ts.union(BJsales, BJsales.lead))

x <- diff(log(EuStockMarkets))
ar.ols(x, order.max = 6, demean = FALSE, intercept = TRUE)

Description

Fit an ARIMA model to a univariate time series.

Usage

arima(x, order = c(0L, 0L, 0L),
      seasonal = list(order = c(0L, 0L, 0L), period = NA),
      xreg = NULL, include.mean = TRUE,
      transform.pars = TRUE,
      fixed = NULL, init = NULL,
      method = c("CSS-ML", "ML", "CSS"), n.cond,
      SSinit = c("Gardner1980", "Rossignol2011"),
      optim.method = "BFGS",
      optim.control = list(), kappa = 1e6)

Arguments

Details

Different definitions of ARMA models have different signs for the AR and/or MA coefficients. The definition used here has

$X_t= a_1 X_{t-1}+\cdots+ a_p X_{t-p} + e_t + b_1 e_{t-1}+\cdots+b_q e_{t-q}$

and so the MA coefficients differ in sign from those used in documentation written for S-PLUS. Further, if include.mean is true (the default for an ARMA model), this formula applies to $X - m$ rather than $X$. For ARIMA models with differencing, the differenced series follows a zero-mean ARMA model. If an xreg term is included, a linear regression (with a constant term if include.mean is true and there is no differencing) is fitted with an ARMA model for the error term.

The variance matrix of the estimates is found from the Hessian of the log-likelihood, and so may only be a rough guide.

Optimization is done by optim. It will work best if the columns in xreg are roughly scaled to zero mean and unit variance, but does attempt to estimate suitable scalings.

Value

A list of class "Arima" with components:

Fitting methods

The exact likelihood is computed via a state-space representation of the ARIMA process, and the innovations and their variance found by a Kalman filter. The initialization of the differenced ARMA process uses stationarity and is based on Gardner et al. (1980). For a differenced process the non-stationary components are given a diffuse prior (controlled by kappa). Observations which are still controlled by the diffuse prior (determined by having a Kalman gain of at least 1e4) are excluded from the likelihood calculations. (This gives comparable results to arima0 in the absence of missing values, when the observations excluded are precisely those dropped by the differencing.)

Missing values are allowed, and are handled exactly in method "ML".

If transform.pars is true, the optimization is done using an alternative parametrization which is a variation on that suggested by Jones (1980) and ensures that the model is stationary. For an AR(p) model the parametrization is via the inverse tanh of the partial autocorrelations: the same procedure is applied (separately) to the AR and seasonal AR terms. The MA terms are not constrained to be invertible during optimization, but they will be converted to invertible form after optimization if transform.pars is true.

Conditional sum-of-squares is provided mainly for expositional purposes. This computes the sum of squares of the fitted innovations from observation n.cond on, (where n.cond is at least the maximum lag of an AR term), treating all earlier innovations to be zero. Argument n.cond can be used to allow comparability between different fits. The ‘part log-likelihood’ is the first term, half the log of the estimated mean square. Missing values are allowed, but will cause many of the innovations to be missing.

When regressors are specified, they are orthogonalized prior to fitting unless any of the coefficients is fixed. It can be helpful to roughly scale the regressors to zero mean and unit variance.

Note

arima is very similar to arima0 for ARMA models or for differenced models without missing values, but handles differenced models with missing values exactly. It is somewhat slower than arima0, particularly for seasonally differenced models.

References

Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York. Sections 3.3 and 8.3.

Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford University Press.

Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980). Algorithm AS 154: An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics, 29, 311–322. doi:10.2307/2346910.

Harvey, A. C. (1993). Time Series Models. 2nd Edition. Harvester Wheatsheaf. Sections 3.3 and 4.4.

Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22, 389–395. doi:10.2307/1268324.

Ripley, B. D. (2002). “Time series in R 1.5.0”. R News, 2(2), 2–7. https://www.r-project.org/doc/Rnews/Rnews_2002-2.pdf

See Also

predict.Arima, arima.sim for simulating from an ARIMA model, tsdiag, arima0, ar

Examples

arima(lh, order = c(1,0,0))
arima(lh, order = c(3,0,0))
arima(lh, order = c(1,0,1))

arima(lh, order = c(3,0,0), method = "CSS")

arima(USAccDeaths, order = c(0,1,1), seasonal = list(order = c(0,1,1)))
arima(USAccDeaths, order = c(0,1,1), seasonal = list(order = c(0,1,1)),
      method = "CSS") # drops first 13 observations.
# for a model with as few years as this, we want full ML

arima(LakeHuron, order = c(2,0,0), xreg = time(LakeHuron) - 1920)

## presidents contains NAs
## graphs in example(acf) suggest order 1 or 3
require(graphics)
(fit1 <- arima(presidents, c(1, 0, 0)))
nobs(fit1)
tsdiag(fit1)
(fit3 <- arima(presidents, c(3, 0, 0)))  # smaller AIC
tsdiag(fit3)
BIC(fit1, fit3)
## compare a whole set of models; BIC() would choose the smallest
AIC(fit1, arima(presidents, c(2,0,0)),
          arima(presidents, c(2,0,1)), # <- chosen (barely) by AIC
    fit3, arima(presidents, c(3,0,1)))

## An example of using the  'fixed'  argument:
## Note that the period of the seasonal component is taken to be
## frequency(presidents), i.e. 4.
(fitSfx <- arima(presidents, order=c(2,0,1), seasonal=c(1,0,0),
                 fixed=c(NA, NA, 0.5, -0.1, 50), transform.pars=FALSE))
## The partly-fixed & smaller model seems better (as we "knew too much"):
AIC(fitSfx, arima(presidents, order=c(2,0,1), seasonal=c(1,0,0)))

## An example of ARIMA forecasting:
predict(fit3, 3)

Description

Simulate from an ARIMA model.

Usage

arima.sim(model, n, rand.gen = rnorm, innov = rand.gen(n, ...),
          n.start = NA, start.innov = rand.gen(n.start, ...),
          ...)

Arguments

Details

See arima for the precise definition of an ARIMA model.

The ARMA model is checked for stationarity.

ARIMA models are specified via the order component of model, in the same way as for arima. Other aspects of the order component are ignored, but inconsistent specifications of the MA and AR orders are detected. The un-differencing assumes previous values of zero, and to remind the user of this, those values are returned.

Random inputs for the ‘burn-in’ period are generated by calling rand.gen.

Value

A time-series object of class "ts".

See Also

arima

Examples

require(graphics)

arima.sim(n = 63, list(ar = c(0.8897, -0.4858), ma = c(-0.2279, 0.2488)),
          sd = sqrt(0.1796))
# mildly long-tailed
arima.sim(n = 63, list(ar = c(0.8897, -0.4858), ma = c(-0.2279, 0.2488)),
          rand.gen = function(n, ...) sqrt(0.1796) * rt(n, df = 5))

# An ARIMA simulation
ts.sim <- arima.sim(list(order = c(1,1,0), ar = 0.7), n = 200)
ts.plot(ts.sim)

Description

Fit an ARIMA model to a univariate time series, and forecast from the fitted model.

Usage

arima0(x, order = c(0, 0, 0),
       seasonal = list(order = c(0, 0, 0), period = NA),
       xreg = NULL, include.mean = TRUE, delta = 0.01,
       transform.pars = TRUE, fixed = NULL, init = NULL,
       method = c("ML", "CSS"), n.cond, optim.control = list())

## S3 method for class 'arima0'
predict(object, n.ahead = 1, newxreg, se.fit = TRUE, ...)

Arguments

Details

Different definitions of ARMA models have different signs for the AR and/or MA coefficients. The definition here has

$X_t = a_1X_{t-1} + \cdots + a_pX_{t-p} + e_t + b_1e_{t-1} + \dots + b_qe_{t-q}$

and so the MA coefficients differ in sign from those given by S-PLUS. Further, if include.mean is true, this formula applies to $X-m$ rather than $X$. For ARIMA models with differencing, the differenced series follows a zero-mean ARMA model.

The variance matrix of the estimates is found from the Hessian of the log-likelihood, and so may only be a rough guide, especially for fits close to the boundary of invertibility.

Optimization is done by optim. It will work best if the columns in xreg are roughly scaled to zero mean and unit variance, but does attempt to estimate suitable scalings.

Finite-history prediction is used. This is only statistically efficient if the MA part of the fit is invertible, so predict.arima0 will give a warning for non-invertible MA models.

Value

For arima0, a list of class "arima0" with components:

For predict.arima0, a time series of predictions, or if se.fit = TRUE, a list with components pred, the predictions, and se, the estimated standard errors. Both components are time series.

Fitting methods

The exact likelihood is computed via a state-space representation of the ARMA process, and the innovations and their variance found by a Kalman filter based on Gardner et al. (1980). This has the option to switch to ‘fast recursions’ (assume an effectively infinite past) if the innovations variance is close enough to its asymptotic bound. The argument delta sets the tolerance: at its default value the approximation is normally negligible and the speed-up considerable. Exact computations can be ensured by setting delta to a negative value.

If transform.pars is true, the optimization is done using an alternative parametrization which is a variation on that suggested by Jones (1980) and ensures that the model is stationary. For an AR(p) model the parametrization is via the inverse tanh of the partial autocorrelations: the same procedure is applied (separately) to the AR and seasonal AR terms. The MA terms are also constrained to be invertible during optimization by the same transformation if transform.pars is true. Note that the MLE for MA terms does sometimes occur for MA polynomials with unit roots: such models can be fitted by using transform.pars = FALSE and specifying a good set of initial values (often obtainable from a fit with transform.pars = TRUE).

Missing values are allowed, but any missing values will force delta to be ignored and full recursions used. Note that missing values will be propagated by differencing, so the procedure used in this function is not fully efficient in that case.

Conditional sum-of-squares is provided mainly for expositional purposes. This computes the sum of squares of the fitted innovations from observation n.cond on, (where n.cond is at least the maximum lag of an AR term), treating all earlier innovations to be zero. Argument n.cond can be used to allow comparability between different fits. The ‘part log-likelihood’ is the first term, half the log of the estimated mean square. Missing values are allowed, but will cause many of the innovations to be missing.

When regressors are specified, they are orthogonalized prior to fitting unless any of the coefficients is fixed. It can be helpful to roughly scale the regressors to zero mean and unit variance.

Note

This is a preliminary version, and will be replaced by arima.

The standard errors of prediction exclude the uncertainty in the estimation of the ARMA model and the regression coefficients.

The results are likely to be different from S-PLUS's arima.mle, which computes a conditional likelihood and does not include a mean in the model. Further, the convention used by arima.mle reverses the signs of the MA coefficients.

References

Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York. Sections 3.3 and 8.3.

Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980). Algorithm AS 154: An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics, 29, 311–322. doi:10.2307/2346910.

Harvey, A. C. (1993). Time Series Models. 2nd Edition. Harvester Wheatsheaf. Sections 3.3 and 4.4.

Harvey, A. C. and McKenzie, C. R. (1982). Algorithm AS 182: An algorithm for finite sample prediction from ARIMA processes. Applied Statistics, 31, 180–187. doi:10.2307/2347987.

Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22, 389–395. doi:10.2307/1268324.

See Also

arima, ar, tsdiag

Examples

## Not run: arima0(lh, order = c(1,0,0))
arima0(lh, order = c(3,0,0))
arima0(lh, order = c(1,0,1))
predict(arima0(lh, order = c(3,0,0)), n.ahead = 12)

arima0(lh, order = c(3,0,0), method = "CSS")

# for a model with as few years as this, we want full ML
(fit <- arima0(USAccDeaths, order = c(0,1,1),
               seasonal = list(order=c(0,1,1)), delta = -1))
predict(fit, n.ahead = 6)

arima0(LakeHuron, order = c(2,0,0), xreg = time(LakeHuron)-1920)
## Not run: 
## presidents contains NAs
## graphs in example(acf) suggest order 1 or 3
(fit1 <- arima0(presidents, c(1, 0, 0), delta = -1))  # avoid warning
tsdiag(fit1)
(fit3 <- arima0(presidents, c(3, 0, 0), delta = -1))  # smaller AIC
tsdiag(fit3)
## End(Not run)

Description

Compute the theoretical autocorrelation function or partial autocorrelation function for an ARMA process.

Usage

ARMAacf(ar = numeric(), ma = numeric(), lag.max = r, pacf = FALSE)

Arguments

Details

The methods used follow Brockwell & Davis (1991, section 3.3). Their equations (3.3.8) are solved for the autocovariances at lags $0, \dots, \max(p, q+1)$, and the remaining autocorrelations are given by a recursive filter.

Value

A vector of (partial) autocorrelations, named by the lags.

References

Brockwell, P. J. and Davis, R. A. (1991) Time Series: Theory and Methods, Second Edition. Springer.

See Also

arima, ARMAtoMA, acf2AR for inverting part of ARMAacf; further filter.

Examples

ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10)

## Example from Brockwell & Davis (1991, pp.92-4)
## answer: 2^(-n) * (32/3 + 8 * n) /(32/3)
n <- 1:10
a.n <- 2^(-n) * (32/3 + 8 * n) /(32/3)
(A.n <- ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10))
stopifnot(all.equal(unname(A.n), c(1, a.n)))

ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10, pacf = TRUE)
zapsmall(ARMAacf(c(1.0, -0.25), lag.max = 10, pacf = TRUE))

## Cov-Matrix of length-7 sub-sample of AR(1) example:
toeplitz(ARMAacf(0.8, lag.max = 7))

Description

Convert ARMA process to infinite MA process.

Usage

ARMAtoMA(ar = numeric(), ma = numeric(), lag.max)

Arguments

Value

A vector of coefficients.

References

Brockwell, P. J. and Davis, R. A. (1991) Time Series: Theory and Methods, Second Edition. Springer.

See Also

arima, ARMAacf.

Examples

ARMAtoMA(c(1.0, -0.25), 1.0, 10)
## Example from Brockwell & Davis (1991, p.92)
## answer (1 + 3*n)*2^(-n)
n <- 1:10; (1 + 3*n)*2^(-n)

Description

Converts objects from other hierarchical clustering functions to class "hclust".

Usage

as.hclust(x, ...)

Arguments

Details

Currently there is only support for converting objects of class "twins" as produced by the functions diana and agnes from the package cluster. The default method throws an error unless passed an "hclust" object.

Value

An object of class "hclust".

See Also

hclust, and from package cluster, diana and agnes

Examples

x <- matrix(rnorm(30), ncol = 3)
hc <- hclust(dist(x), method = "complete")

if(require("cluster", quietly = TRUE)) {# is a recommended package
  ag <- agnes(x, method = "complete")
  hcag <- as.hclust(ag)
  ## The dendrograms order slightly differently:
  op <- par(mfrow = c(1,2))
  plot(hc) ;  mtext("hclust", side = 1)
  plot(hcag); mtext("agnes",  side = 1)
  detach("package:cluster")
}

Description

Names, calls, expressions (first element), numeric values, and character strings are converted to one-sided formulae associated with the global environment. If the input is a formula, it must be one-sided, in which case it is returned unaltered.

Usage

asOneSidedFormula(object)

Arguments

Value

a one-sided formula representing object

Author(s)

José Pinheiro and Douglas Bates

See Also

formula

Examples

(form <- asOneSidedFormula("age"))
stopifnot(exprs = {
    identical(form, asOneSidedFormula(form))
    identical(form, asOneSidedFormula(as.name("age")))
    identical(form, asOneSidedFormula(expression(age)))
})
asOneSidedFormula(quote(log(age)))
asOneSidedFormula(1)

Description

Subsets of x[] are averaged, where each subset consist of those observations with the same factor levels.

Usage

ave(x, ..., FUN = mean)

Arguments

Value

A numeric vector, say y of length length(x). If ... is g1, g2, e.g., y[i] is equal to FUN(x[j], for all j with g1[j] == g1[i] and g2[j] == g2[i]).

See Also

mean, median.

Examples

require(graphics)

ave(1:3)  # no grouping -> grand mean

attach(warpbreaks)
ave(breaks, wool)
ave(breaks, tension)
ave(breaks, tension, FUN = function(x) mean(x, trim = 0.1))
plot(breaks, main =
     "ave( Warpbreaks )  for   wool  x  tension  combinations")
lines(ave(breaks, wool, tension              ), type = "s", col = "blue")
lines(ave(breaks, wool, tension, FUN = median), type = "s", col = "green")
legend(40, 70, c("mean", "median"), lty = 1,
      col = c("blue","green"), bg = "gray90")
detach()

Description

Bandwidth selectors for Gaussian kernels in density.

Usage

bw.nrd0(x)

bw.nrd(x)

bw.ucv(x, nb = 1000, lower = 0.1 * hmax, upper = hmax,
       tol = 0.1 * lower)

bw.bcv(x, nb = 1000, lower = 0.1 * hmax, upper = hmax,
       tol = 0.1 * lower)

bw.SJ(x, nb = 1000, lower = 0.1 * hmax, upper = hmax,
      method = c("ste", "dpi"), tol = 0.1 * lower)

Arguments