Version:	9.0.0
Title:	Forecasting Functions for Time Series and Linear Models
Description:	Methods and tools for displaying and analysing univariate time series forecasts including exponential smoothing via state space models and automatic ARIMA modelling.
Depends:	R (≥ 4.1.0)
Imports:	colorspace, fracdiff, generics (≥ 0.1.2), ggplot2 (≥ 3.4.0), graphics, lmtest, magrittr, nnet, parallel, Rcpp (≥ 0.11.0), stats, timeDate, tseries, urca, withr, zoo
Suggests:	forecTheta, knitr, methods, rmarkdown, rticles, scales, seasonal, testthat (≥ 3.3.0), uroot
LinkingTo:	Rcpp (≥ 0.11.0), RcppArmadillo (≥ 0.2.35)
LazyData:	yes
ByteCompile:	TRUE
BugReports:	https://github.com/robjhyndman/forecast/issues
License:	GPL-3
URL:	https://pkg.robjhyndman.com/forecast/, https://github.com/robjhyndman/forecast
VignetteBuilder:	knitr
RoxygenNote:	7.3.3
Encoding:	UTF-8
Config/testthat/edition:	3
NeedsCompilation:	yes
Packaged:	2026-01-10 05:29:15 UTC; hyndman
Author:	Rob Hyndman [aut, cre, cph], George Athanasopoulos [aut], Christoph Bergmeir [aut], Gabriel Caceres [aut], Leanne Chhay [aut], Kirill Kuroptev [aut], Maximilian Mücke [aut], Mitchell O'Hara-Wild [aut], Fotios Petropoulos [aut], Slava Razbash [aut], Earo Wang [aut], Farah Yasmeen [aut], Federico Garza [ctb], Daniele Girolimetto [ctb], Ross Ihaka [ctb, cph], R Core Team [ctb, cph], Daniel Reid [ctb], David Shaub [ctb], Yuan Tang [ctb], Xiaoqian Wang [ctb], Zhenyu Zhou [ctb]
Maintainer:	Rob Hyndman <Rob.Hyndman@monash.edu>
Repository:	CRAN
Date/Publication:	2026-01-11 08:40:02 UTC

forecast: Forecasting Functions for Time Series and Linear Models

Description

Methods and tools for displaying and analysing univariate time series forecasts including exponential smoothing via state space models and automatic ARIMA modelling.

Author(s)

Maintainer: Rob Hyndman Rob.Hyndman@monash.edu (ORCID) [copyright holder]

Authors:

• George Athanasopoulos (ORCID)

• Christoph Bergmeir (ORCID)

• Gabriel Caceres (ORCID)

• Leanne Chhay

• Kirill Kuroptev

• Maximilian Mücke (ORCID)

• Mitchell O'Hara-Wild (ORCID)

• Fotios Petropoulos (ORCID)

• Slava Razbash

• Earo Wang (ORCID)

• Farah Yasmeen (ORCID)

Other contributors:

• Federico Garza [contributor]

• Daniele Girolimetto [contributor]

• Ross Ihaka [contributor, copyright holder]

• R Core Team [contributor, copyright holder]

• Daniel Reid [contributor]

• David Shaub [contributor]

• Yuan Tang (ORCID) [contributor]

• Xiaoqian Wang [contributor]

• Zhenyu Zhou [contributor]

See Also

Useful links:

• https://pkg.robjhyndman.com/forecast/

• https://github.com/robjhyndman/forecast

• Report bugs at https://github.com/robjhyndman/forecast/issues

(Partial) Autocorrelation and Cross-Correlation Function Estimation

Description

The function Acf computes (and by default plots) an estimate of the autocorrelation function of a (possibly multivariate) time series. Function Pacf computes (and by default plots) an estimate of the partial autocorrelation function of a (possibly multivariate) time series. 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.contiguous,
  demean = TRUE,
  ...
)

Pacf(
  x,
  lag.max = NULL,
  plot = TRUE,
  na.action = na.contiguous,
  demean = TRUE,
  ...
)

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

taperedacf(
  x,
  lag.max = NULL,
  type = c("correlation", "partial"),
  plot = TRUE,
  calc.ci = TRUE,
  level = 95,
  nsim = 100,
  ...
)

taperedpacf(x, ...)

Arguments

Details

The functions improve the stats::acf(), stats::pacf() and stats::ccf() functions. The main differences are that Acf does not plot a spike at lag 0 when type = "correlation" (which is redundant) and the horizontal axes show lags in time units rather than seasonal units.

The tapered versions implement the ACF and PACF estimates and plots described in Hyndman (2015), based on the banded and tapered estimates of autocovariance proposed by McMurry and Politis (2010).

Value

The Acf, Pacf and Ccf functions return objects of class "acf" as described in stats::acf() from the stats package. The taperedacf and taperedpacf functions return objects of class "mpacf".

Author(s)

Rob J Hyndman

References

Hyndman, R.J. (2015). Discussion of “High-dimensional autocovariance matrices and optimal linear prediction”. Electronic Journal of Statistics, 9, 792-796.

McMurry, T. L., & Politis, D. N. (2010). Banded and tapered estimates for autocovariance matrices and the linear process bootstrap. Journal of Time Series Analysis, 31(6), 471-482.

See Also

stats::acf(), stats::pacf(), stats::ccf(), tsdisplay()

Examples

Acf(wineind)
Pacf(wineind)
## Not run: 
taperedacf(wineind, nsim = 50)
taperedpacf(wineind, nsim = 50)

## End(Not run)

Fit ARIMA model to univariate time series

Description

Largely a wrapper for the stats::arima() function in the stats package. The main difference is that this function allows a drift term. It is also possible to take an ARIMA model from a previous call to Arima and re-apply it to the data y.

Usage

Arima(
  y,
  order = c(0, 0, 0),
  seasonal = c(0, 0, 0),
  xreg = NULL,
  include.mean = TRUE,
  include.drift = FALSE,
  include.constant = NULL,
  lambda = model$lambda,
  biasadj = attr(lambda, "biasadj"),
  method = c("CSS-ML", "ML", "CSS"),
  model = NULL,
  x = y,
  ...
)

Arguments

Details

The fitted model is a regression with ARIMA(p,d,q) errors

y_t = c + \beta' x_t + z_t

where x_t is a vector of regressors at time t and z_t is an ARMA(p,d,q) error process. If there are no regressors, and d=0, then c is an estimate of the mean of y_t. For more information, see Hyndman & Athanasopoulos (2018). For details of the estimation algorithm, see the stats::arima() function in the stats package.

Value

See the stats::arima() function in the stats package. The additional objects returned are:

Author(s)

Rob J Hyndman

References

Hyndman, R.J. and Athanasopoulos, G. (2018) "Forecasting: principles and practice", 2nd ed., OTexts, Melbourne, Australia. https://OTexts.com/fpp2/.

See Also

auto.arima(), forecast.Arima().

Examples

library(ggplot2)
WWWusage |>
  Arima(order = c(3, 1, 0)) |>
  forecast(h = 20) |>
  autoplot()

# Fit model to first few years of AirPassengers data
air.model <- Arima(
  window(AirPassengers, end = 1956 + 11 / 12),
  order = c(0, 1, 1),
  seasonal = list(order = c(0, 1, 1), period = 12),
  lambda = 0
)
plot(forecast(air.model, h = 48))
lines(AirPassengers)

# Apply fitted model to later data
air.model2 <- Arima(window(AirPassengers, start = 1957), model = air.model)

# Forecast accuracy measures on the log scale.
# in-sample one-step forecasts.
accuracy(air.model)
# out-of-sample one-step forecasts.
accuracy(air.model2)
# out-of-sample multi-step forecasts
accuracy(
  forecast(air.model, h = 48, lambda = NULL),
  log(window(AirPassengers, start = 1957))
)

Box Cox Transformation

Description

BoxCox() returns a transformation of the input variable using a Box-Cox transformation. InvBoxCox() reverses the transformation.

Usage

BoxCox(x, lambda)

InvBoxCox(x, lambda, biasadj = FALSE, fvar = NULL)

Arguments

Details

The Box-Cox transformation (as given by Bickel & Doksum 1981) is given by

f_\lambda(x) =(sign(x)|x|^\lambda - 1)/\lambda

if \lambda\ne0. For \lambda=0,

f_0(x)=\log(x)

.

Value

a numeric vector of the same length as x.

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

References

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. JRSS B 26 211–246. Bickel, P. J. and Doksum K. A. (1981) An Analysis of Transformations Revisited. JASA 76 296-311.

See Also

BoxCox.lambda()

Examples

lambda <- BoxCox.lambda(lynx)
lynx.fit <- ar(BoxCox(lynx, lambda))
plot(forecast(lynx.fit, h = 20, lambda = lambda))

Automatic selection of Box Cox transformation parameter

Description

If method = "guerrero", Guerrero's (1993) method is used, where lambda minimizes the coefficient of variation for subseries of x.

Usage

BoxCox.lambda(x, method = c("guerrero", "loglik"), lower = -1, upper = 2)

Arguments

Details

If method = "loglik", the value of lambda is chosen to maximize the profile log likelihood of a linear model fitted to x. For non-seasonal data, a linear time trend is fitted while for seasonal data, a linear time trend with seasonal dummy variables is used.

Value

a number indicating the Box-Cox transformation parameter.

Author(s)

Leanne Chhay and Rob J Hyndman

References

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. JRSS B 26 211–246.

Guerrero, V.M. (1993) Time-series analysis supported by power transformations. Journal of Forecasting, 12, 37–48.

See Also

BoxCox()

Examples

lambda <- BoxCox.lambda(AirPassengers, lower = 0)
air.fit <- Arima(
  AirPassengers,
  order = c(0, 1, 1),
  seasonal = list(order = c(0, 1, 1), period = 12),
  lambda = lambda
)
plot(forecast(air.fit))

Cross-validation statistic

Description

Computes the leave-one-out cross-validation statistic (the mean of PRESS – prediction residual sum of squares), AIC, corrected AIC, BIC and adjusted R^2 values for a linear model.

Usage

CV(obj)

Arguments

Value

Numerical vector containing CV, AIC, AICc, BIC and AdjR2 values.

Author(s)

Rob J Hyndman

See Also

stats::AIC()

Examples

y <- ts(rnorm(120, 0, 3) + 20 * sin(2 * pi * (1:120) / 12), frequency = 12)
fit1 <- tslm(y ~ trend + season)
fit2 <- tslm(y ~ season)
CV(fit1)
CV(fit2)

k-fold Cross-Validation applied to an autoregressive model

Description

CVar computes the errors obtained by applying an autoregressive modelling function to subsets of the time series y using k-fold cross-validation as described in Bergmeir, Hyndman and Koo (2015). It also applies a Ljung-Box test to the residuals. If this test is significant (see returned pvalue), there is serial correlation in the residuals and the model can be considered to be underfitting the data. In this case, the cross-validated errors can underestimate the generalization error and should not be used.

Usage

CVar(
  y,
  k = 10,
  FUN = nnetar,
  cvtrace = FALSE,
  blocked = FALSE,
  LBlags = 24,
  ...
)

Arguments

Value

A list containing information about the model and accuracy for each fold, plus other summary information computed across folds.

Author(s)

Gabriel Caceres and Rob J Hyndman

References

Bergmeir, C., Hyndman, R.J., Koo, B. (2018) A note on the validity of cross-validation for evaluating time series prediction. Computational Statistics & Data Analysis, 120, 70-83. https://robjhyndman.com/publications/cv-time-series/.

See Also

CV(), tsCV().

Examples

modelcv <- CVar(lynx, k = 5, lambda = 0.15)
print(modelcv)
print(modelcv$fold1)

library(ggplot2)
autoplot(lynx, series = "Data") +
  autolayer(modelcv$testfit, series = "Fits") +
  autolayer(modelcv$residuals, series = "Residuals")
ggAcf(modelcv$residuals)

Forecast plot

Description

Generates forecasts from forecast.ts and adds them to the plot. Forecasts can be modified via sending forecast specific arguments above.

Usage

StatForecast

GeomForecast

geom_forecast(
  mapping = NULL,
  data = NULL,
  stat = "forecast",
  position = "identity",
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  PI = TRUE,
  showgap = TRUE,
  series = NULL,
  ...
)

Arguments

Format

An object of class StatForecast (inherits from Stat, ggproto, gg) of length 3.

An object of class GeomForecast (inherits from Geom, ggproto, gg) of length 7.

Details

Multivariate forecasting is supported by having each time series on a different group.

You can also pass geom_forecast a forecast object to add it to the plot.

The aesthetics required for the forecasting to work includes forecast observations on the y axis, and the time of the observations on the x axis. Refer to the examples below. To automatically set up aesthetics, use autoplot.

Value

A layer for a ggplot graph.

Author(s)

Mitchell O'Hara-Wild

See Also

generics::forecast(), ggplot2::ggproto()

Examples

## Not run: 
library(ggplot2)
autoplot(USAccDeaths) + geom_forecast()

lungDeaths <- cbind(mdeaths, fdeaths)
autoplot(lungDeaths) + geom_forecast()

# Using fortify.ts
p <- ggplot(aes(x = x, y = y), data = USAccDeaths)
p <- p + geom_line()
p + geom_forecast()

# Without fortify.ts
data <- data.frame(USAccDeaths = as.numeric(USAccDeaths),
                   time = as.numeric(time(USAccDeaths)))
p <- ggplot(aes(x = time, y = USAccDeaths), data = data)
p <- p + geom_line()
p + geom_forecast()

p + geom_forecast(h = 60)
p <- ggplot(aes(x = time, y = USAccDeaths), data = data)
p + geom_forecast(level = c(70, 98))
p + geom_forecast(level = c(70, 98), colour = "lightblue")

#Add forecasts to multivariate series with colour groups
lungDeaths <- cbind(mdeaths, fdeaths)
autoplot(lungDeaths) + geom_forecast(forecast(mdeaths), series = "mdeaths")

## End(Not run)

Accuracy measures for a forecast model

Description

Returns range of summary measures of the forecast accuracy. If x is provided, the function measures test set forecast accuracy based on x - f. If x is not provided, the function only produces training set accuracy measures of the forecasts based on f["x"] - fitted(f). All measures are defined and discussed in Hyndman and Koehler (2006).

Usage

## S3 method for class 'forecast'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

## S3 method for class 'mforecast'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

## S3 method for class 'fc_model'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

## S3 method for class 'Arima'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

## S3 method for class 'lm'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

## S3 method for class 'ts'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

## S3 method for class 'numeric'
accuracy(object, x, test = NULL, d = NULL, D = NULL, ...)

Arguments

Details

The measures calculated are:

• ME: Mean Error

• RMSE: Root Mean Squared Error

• MAE: Mean Absolute Error

• MPE: Mean Percentage Error

• MAPE: Mean Absolute Percentage Error

• MASE: Mean Absolute Scaled Error

• ACF1: Autocorrelation of errors at lag 1.

By default, the MASE calculation is scaled using MAE of training set naive forecasts for non-seasonal time series, training set seasonal naive forecasts for seasonal time series and training set mean forecasts for non-time series data. If f is a numerical vector rather than a forecast object, the MASE will not be returned as the training data will not be available.

See Hyndman and Koehler (2006) and Hyndman and Athanasopoulos (2014, Section 2.5) for further details.

Value

Matrix giving forecast accuracy measures.

Author(s)

Rob J Hyndman

References

Hyndman, R.J. and Koehler, A.B. (2006) "Another look at measures of forecast accuracy". International Journal of Forecasting, 22(4), 679-688.

Hyndman, R.J. and Athanasopoulos, G. (2018) "Forecasting: principles and practice", 2nd ed., OTexts, Melbourne, Australia. Section 3.4 "Evaluating forecast accuracy". https://otexts.com/fpp2/accuracy.html.

Examples

fit1 <- rwf(EuStockMarkets[1:200, 1], h = 100)
fit2 <- meanf(EuStockMarkets[1:200, 1], h = 100)
accuracy(fit1)
accuracy(fit2)
accuracy(fit1, EuStockMarkets[201:300, 1])
accuracy(fit2, EuStockMarkets[201:300, 1])
plot(fit1)
lines(EuStockMarkets[1:300, 1])

Fit a fractionally differenced ARFIMA model

Description

An ARFIMA(p,d,q) model is selected and estimated automatically using the Hyndman-Khandakar (2008) algorithm to select p and q and the Haslett and Raftery (1989) algorithm to estimate the parameters including d.

Usage

arfima(
  y,
  drange = c(0, 0.5),
  estim = c("mle", "ls"),
  model = NULL,
  lambda = NULL,
  biasadj = FALSE,
  xreg = NULL,
  x = y,
  ...
)

Arguments

Details

This function combines fracdiff::fracdiff() and auto.arima() to automatically select and estimate an ARFIMA model. The fractional differencing parameter is chosen first assuming an ARFIMA(2,d,0) model. Then the data are fractionally differenced using the estimated d and an ARMA model is selected for the resulting time series using auto.arima(). Finally, the full ARFIMA(p,d,q) model is re-estimated using fracdiff::fracdiff(). If estim = "mle", the ARMA coefficients are refined using stats::arima().

Value

A list object of S3 class fracdiff, which is described in the fracdiff::fracdiff() documentation. A few additional objects are added to the list including x (the original time series), and the residuals and fitted values.

Author(s)

Rob J Hyndman and Farah Yasmeen

References

J. Haslett and A. E. Raftery (1989) Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource (with discussion); Applied Statistics 38, 1-50.

Hyndman, R.J. and Khandakar, Y. (2008) "Automatic time series forecasting: The forecast package for R", Journal of Statistical Software, 26(3).

See Also

fracdiff::fracdiff(), auto.arima(), forecast.fracdiff().

Examples

library(fracdiff)
x <- fracdiff.sim(100, ma = -0.4, d = 0.3)$series
fit <- arfima(x)
tsdisplay(residuals(fit))

Errors from a regression model with ARIMA errors

Description

Returns time series of the regression residuals from a fitted ARIMA model.

Usage

arima.errors(object)

Arguments

Details

This is a deprecated function which is identical to residuals.Arima(object, type="regression") Regression residuals are equal to the original data minus the effect of any regression variables. If there are no regression variables, the errors will be identical to the original series (possibly adjusted to have zero mean).

Value

A ts object

Author(s)

Rob J Hyndman

See Also

residuals.Arima().

Return the order of an ARIMA or ARFIMA model

Description

Returns the order of a univariate ARIMA or ARFIMA model.

Usage

arimaorder(object)

Arguments

Value

A numerical vector giving the values p, d and q of the ARIMA or ARFIMA model. For a seasonal ARIMA model, the returned vector contains the values p, d, q, P, D, Q and m, where m is the period of seasonality.

Author(s)

Rob J Hyndman

See Also

stats::ar(), auto.arima, Arima(), stats::arima(), arfima().

Examples

WWWusage |> auto.arima() |> arimaorder()

Fit best ARIMA model to univariate time series

Description

Returns best ARIMA model according to either AIC, AICc or BIC value. The function conducts a search over possible model within the order constraints provided.

Usage

auto.arima(
  y,
  d = NA,
  D = NA,
  max.p = 5,
  max.q = 5,
  max.P = 2,
  max.Q = 2,
  max.order = 5,
  max.d = 2,
  max.D = 1,
  start.p = 2,
  start.q = 2,
  start.P = 1,
  start.Q = 1,
  stationary = FALSE,
  seasonal = TRUE,
  ic = c("aicc", "aic", "bic"),
  stepwise = TRUE,
  nmodels = 94,
  trace = FALSE,
  approximation = (length(x) > 150 || frequency(x) > 12),
  method = NULL,
  truncate = NULL,
  xreg = NULL,
  test = c("kpss", "adf", "pp"),
  test.args = list(),
  seasonal.test = c("seas", "ocsb", "hegy", "ch"),
  seasonal.test.args = list(),
  allowdrift = TRUE,
  allowmean = TRUE,
  lambda = NULL,
  biasadj = FALSE,
  parallel = FALSE,
  num.cores = 2,
  x = y,
  ...
)

Arguments

Details

The default arguments are designed for rapid estimation of models for many time series. If you are analysing just one time series, and can afford to take some more time, it is recommended that you set stepwise = FALSE and approximation = FALSE.

Non-stepwise selection can be slow, especially for seasonal data. The stepwise algorithm outlined in Hyndman & Khandakar (2008) is used except that the default method for selecting seasonal differences is now based on an estimate of seasonal strength (Wang, Smith & Hyndman, 2006) rather than the Canova-Hansen test. There are also some other minor variations to the algorithm described in Hyndman and Khandakar (2008).

Value

Same as for Arima()

Author(s)

Rob J Hyndman

References

Hyndman, RJ and Khandakar, Y (2008) "Automatic time series forecasting: The forecast package for R", Journal of Statistical Software, 26(3).

Wang, X, Smith, KA, Hyndman, RJ (2006) "Characteristic-based clustering for time series data", Data Mining and Knowledge Discovery, 13(3), 335-364.

See Also

Arima()

Examples

fit <- auto.arima(WWWusage)
plot(forecast(fit, h = 20))

Create a ggplot layer appropriate to a particular data type

Description

autolayer() uses ggplot2 to draw a particular layer for an object of a particular class in a single command. This defines the S3 generic that other classes and packages can extend.

Usage

autolayer(object, ...)

Arguments

Value

a ggplot layer

See Also

Other plotting automation topics: automatic_plotting, autoplot(), fortify()

Automatically create a ggplot for time series objects

Description

autoplot takes an object of type ts or mts and creates a ggplot object suitable for usage with stat_forecast.

Usage

## S3 method for class 'mts'
autolayer(object, colour = TRUE, series = NULL, ...)

## S3 method for class 'msts'
autolayer(object, series = NULL, ...)

## S3 method for class 'ts'
autolayer(object, colour = TRUE, series = NULL, ...)

## S3 method for class 'ts'
autoplot(
  object,
  series = NULL,
  xlab = "Time",
  ylab = deparse1(substitute(object)),
  main = NULL,
  ...
)

## S3 method for class 'mts'
autoplot(
  object,
  colour = TRUE,
  facets = FALSE,
  xlab = "Time",
  ylab = deparse1(substitute(object)),
  main = NULL,
  ...
)

## S3 method for class 'msts'
autoplot(object, ...)

## S3 method for class 'ts'
fortify(model, data, ...)

Arguments

Details

fortify.ts takes a ts object and converts it into a data frame (for usage with ggplot2).

Value

None. Function produces a ggplot graph.

Author(s)

Mitchell O'Hara-Wild

See Also

stats::plot.ts(), ggplot2::fortify()

Examples

library(ggplot2)
autoplot(USAccDeaths)

lungDeaths <- cbind(mdeaths, fdeaths)
autoplot(lungDeaths)
autoplot(lungDeaths, facets = TRUE)

ggplot (Partial) Autocorrelation and Cross-Correlation Function Estimation and Plotting

Description

Produces a ggplot object of their equivalent Acf, Pacf, Ccf, taperedacf and taperedpacf functions.

Usage

## S3 method for class 'acf'
autoplot(object, ci = 0.95, ...)

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

ggPacf(
  x,
  lag.max = NULL,
  plot = TRUE,
  na.action = na.contiguous,
  demean = TRUE,
  ...
)

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

## S3 method for class 'mpacf'
autoplot(object, ...)

ggtaperedacf(
  x,
  lag.max = NULL,
  type = c("correlation", "partial"),
  plot = TRUE,
  calc.ci = TRUE,
  level = 95,
  nsim = 100,
  ...
)

ggtaperedpacf(x, ...)

Arguments

Details

If autoplot is given an acf or mpacf object, then an appropriate ggplot object will be created.

ggtaperedpacf

Value

A ggplot object.

Author(s)

Mitchell O'Hara-Wild

See Also

stats::plot.acf() Acf(), [stats::acf(), taperedacf()

Examples

library(ggplot2)
ggAcf(wineind)
wineind |> Acf(plot = FALSE) |> autoplot()
## Not run: 
wineind |> taperedacf(plot = FALSE) |> autoplot()
ggtaperedacf(wineind)
ggtaperedpacf(wineind)

## End(Not run)
ggCcf(mdeaths, fdeaths)

Plot time series decomposition components using ggplot

Description

Produces a ggplot object of seasonally decomposed time series for objects of class stl (created with stats::stl(), class seas (created with seasonal::seas()), or class decomposed.ts (created with stats::decompose()).

Usage

## S3 method for class 'decomposed.ts'
autoplot(object, labels = NULL, range.bars = NULL, ...)

## S3 method for class 'stl'
autoplot(object, labels = NULL, range.bars = TRUE, ...)

## S3 method for class 'StructTS'
autoplot(object, labels = NULL, range.bars = TRUE, ...)

## S3 method for class 'seas'
autoplot(object, labels = NULL, range.bars = NULL, ...)

## S3 method for class 'mstl'
autoplot(object, ...)

Arguments

Value

Returns an object of class ggplot.

Author(s)

Mitchell O'Hara-Wild

See Also

seasonal::seas(), stats::stl(), stats::decompose(), stats::StructTS(), stats::plot.stl().

Examples

library(ggplot2)
co2 |>
  decompose() |>
  autoplot()
nottem |>
  stl(s.window = "periodic") |>
  autoplot()
## Not run: 
library(seasonal)
seas(USAccDeaths) |> autoplot()

## End(Not run)

Multivariate forecast plot

Description

Plots historical data with multivariate forecasts and prediction intervals.

Usage

## S3 method for class 'mforecast'
autoplot(object, PI = TRUE, facets = TRUE, colour = FALSE, ...)

## S3 method for class 'mforecast'
autolayer(object, series = NULL, PI = TRUE, ...)

## S3 method for class 'mforecast'
plot(x, main = paste("Forecasts from", unique(x$method)), xlab = "time", ...)

Arguments

Details

autoplot will produce an equivalent plot as a ggplot object.

Author(s)

Mitchell O'Hara-Wild

References

Hyndman and Athanasopoulos (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. https://otexts.com/fpp2/

See Also

plot.forecast(), stats::plot.ts()

Examples

library(ggplot2)

lungDeaths <- cbind(mdeaths, fdeaths)
fit <- tslm(lungDeaths ~ trend + season)
fcast <- forecast(fit, h = 10)
plot(fcast)
autoplot(fcast)

carPower <- as.matrix(mtcars[, c("qsec", "hp")])
carmpg <- mtcars[, "mpg"]
fit <- lm(carPower ~ carmpg)
fcast <- forecast(fit, newdata = data.frame(carmpg = 30))
plot(fcast, xlab = "Year")
autoplot(fcast, xlab = rep("Year", 2))

Forecasting using a bagged model

Description

The bagged model forecasting method.

Usage

baggedModel(y, bootstrapped_series = bld.mbb.bootstrap(y, 100), fn = ets, ...)

baggedETS(y, bootstrapped_series = bld.mbb.bootstrap(y, 100), ...)

Arguments

Details

This function implements the bagged model forecasting method described in Bergmeir et al. By default, the ets() function is applied to all bootstrapped series. Base models other than ets() can be given by the parameter fn. Using the default parameters, the function bld.mbb.bootstrap() is used to calculate the bootstrapped series with the Box-Cox and Loess-based decomposition (BLD) bootstrap. The function forecast.baggedModel() can then be used to calculate forecasts.

baggedETS is a wrapper for baggedModel, setting fn to "ets". This function is included for backwards compatibility only, and may be deprecated in the future.

Value

Returns an object of class baggedModel.

The function print is used to obtain and print a summary of the results.

Author(s)

Christoph Bergmeir, Fotios Petropoulos

References

Bergmeir, C., R. J. Hyndman, and J. M. Benitez (2016). Bagging Exponential Smoothing Methods using STL Decomposition and Box-Cox Transformation. International Journal of Forecasting 32, 303-312.

Examples

fit <- baggedModel(WWWusage)
fcast <- forecast(fit)
plot(fcast)

BATS model (Exponential smoothing state space model with Box-Cox transformation, ARMA errors, Trend and Seasonal components)

Description

Fits a BATS model applied to y, as described in De Livera, Hyndman & Snyder (2011). Parallel processing is used by default to speed up the computations.

Usage

bats(
  y,
  use.box.cox = NULL,
  use.trend = NULL,
  use.damped.trend = NULL,
  seasonal.periods = NULL,
  use.arma.errors = TRUE,
  use.parallel = length(y) > 1000,
  num.cores = 2,
  bc.lower = 0,
  bc.upper = 1,
  biasadj = FALSE,
  model = NULL,
  ...
)

Arguments

Value

An object of class bats. The generic accessor functions fitted.values and residuals extract useful features of the value returned by bats and associated functions. The fitted model is designated BATS(omega, p,q, phi, m1,...mJ) where omega is the Box-Cox parameter and phi is the damping parameter; the error is modelled as an ARMA(p,q) process and m1,...,mJ list the seasonal periods used in the model.

Author(s)

Slava Razbash and Rob J Hyndman

References

De Livera, A.M., Hyndman, R.J., & Snyder, R. D. (2011), Forecasting time series with complex seasonal patterns using exponential smoothing, Journal of the American Statistical Association, 106(496), 1513-1527.

Examples

## Not run: 
fit <- bats(USAccDeaths)
plot(forecast(fit))

taylor.fit <- bats(taylor)
plot(forecast(taylor.fit))

## End(Not run)

Number of trading days in each season

Description

Returns number of trading days in each month or quarter of the observed time period in a major financial center.

Usage

bizdays(x, FinCenter = c("New York", "London", "NERC", "Toronto", "Zurich"))

Arguments

Details

Useful for trading days length adjustments. More on how to define "business days", please refer to timeDate::isBizday().

Value

Time series

Author(s)

Earo Wang

See Also

monthdays()

Examples

x <- ts(rnorm(30), start = c(2013, 2), frequency = 12)
bizdays(x, FinCenter = "New York")

Box-Cox and Loess-based decomposition bootstrap.

Description

Generates bootstrapped versions of a time series using the Box-Cox and Loess-based decomposition bootstrap.

Usage

bld.mbb.bootstrap(x, num, block_size = NULL)

Arguments

Details

The procedure is described in Bergmeir et al. Box-Cox decomposition is applied, together with STL or Loess (for non-seasonal time series), and the remainder is bootstrapped using a moving block bootstrap.

Value

A list with bootstrapped versions of the series. The first series in the list is the original series.

Author(s)

Christoph Bergmeir, Fotios Petropoulos

References

Bergmeir, C., R. J. Hyndman, and J. M. Benitez (2016). Bagging Exponential Smoothing Methods using STL Decomposition and Box-Cox Transformation. International Journal of Forecasting 32, 303-312.

See Also

baggedETS().

Examples

bootstrapped_series <- bld.mbb.bootstrap(WWWusage, 100)

Check that residuals from a time series model look like white noise

Description

If plot = TRUE, produces a time plot of the residuals, the corresponding ACF, and a histogram. If test is not FALSE, the output from either a Ljung-Box test or Breusch-Godfrey test is printed.

Usage

checkresiduals(object, lag, test, plot = TRUE, ...)

Arguments

Value

None

Author(s)

Rob J Hyndman

See Also

ggtsdisplay(), stats::Box.test(), [lmtest::bgtest()

Examples

fit <- ets(WWWusage)
checkresiduals(fit)

Croston forecast model

Description

Based on Croston's (1972) method for intermittent demand forecasting, also described in Shenstone and Hyndman (2005). Croston's method involves using simple exponential smoothing (SES) on the non-zero elements of the time series and a separate application of SES to the times between non-zero elements of the time series. Returns a model object that can be used to generate forecasts using Croston's method for intermittent demand time series. It isn't a true statistical model in that it doesn't describe a data generating process that would lead to the forecasts produced using Croston's method.

Usage

croston_model(y, alpha = 0.1, type = c("croston", "sba", "sbj"))

Arguments

Details

Note that prediction intervals are not computed as Croston's method has no underlying stochastic model.

There are two variant methods available which apply multiplicative correction factors to the forecasts that result from the original Croston's method. For the Syntetos-Boylan approximation (type = "sba"), this factor is 1 - \alpha / 2, and for the Shale-Boylan-Johnston method (type = "sbj"), this factor is 1 - \alpha / (2 - \alpha), where \alpha is the smoothing parameter for the interval SES application.

Value

An object of class croston_model

Author(s)

Rob J Hyndman

References

Croston, J. (1972) "Forecasting and stock control for intermittent demands", Operational Research Quarterly, 23(3), 289-303.

Shale, E.A., Boylan, J.E., & Johnston, F.R. (2006). Forecasting for intermittent demand: the estimation of an unbiased average. Journal of the Operational Research Society, 57(5), 588-592.

Shenstone, L., and Hyndman, R.J. (2005) "Stochastic models underlying Croston's method for intermittent demand forecasting". Journal of Forecasting, 24, 389-402.

Syntetos A.A., Boylan J.E. (2001). On the bias of intermittent demand estimates. International Journal of Production Economics, 71, 457–466.

Examples

y <- rpois(20, lambda = 0.3)
fit <- croston_model(y)
forecast(fit) |> autoplot()

Diebold-Mariano test for predictive accuracy

Description

The Diebold-Mariano test compares the forecast accuracy of two forecast methods.

Usage

dm.test(
  e1,
  e2,
  alternative = c("two.sided", "less", "greater"),
  h = 1,
  power = 2,
  varestimator = c("acf", "bartlett")
)

Arguments

Details

This function implements the modified test proposed by Harvey, Leybourne and Newbold (1997). The null hypothesis is that the two methods have the same forecast accuracy. For alternative = "less", the alternative hypothesis is that method 2 is less accurate than method 1. For alternative = "greater", the alternative hypothesis is that method 2 is more accurate than method 1. For alternative = "two.sided", the alternative hypothesis is that method 1 and method 2 have different levels of accuracy. The long-run variance estimator can either the auto-correlation estimator varestimator = "acf", or the estimator based on Bartlett weights varestimator = "bartlett" which ensures a positive estimate. Both long-run variance estimators are proposed in Diebold and Mariano (1995).

Value

A list with class htest containing the following components:

Author(s)

George Athanasopoulos and Kirill Kuroptev

References

Diebold, F.X. and Mariano, R.S. (1995) Comparing predictive accuracy. Journal of Business and Economic Statistics, 13, 253-263.

Harvey, D., Leybourne, S., & Newbold, P. (1997). Testing the equality of prediction mean squared errors. International Journal of forecasting, 13(2), 281-291.

Examples

# Test on in-sample one-step forecasts
f1 <- ets(WWWusage)
f2 <- auto.arima(WWWusage)
accuracy(f1)
accuracy(f2)
dm.test(residuals(f1), residuals(f2), h = 1)

# Test on out-of-sample one-step forecasts
f1 <- ets(WWWusage[1:80])
f2 <- auto.arima(WWWusage[1:80])
f1.out <- ets(WWWusage[81:100], model = f1)
f2.out <- Arima(WWWusage[81:100], model = f2)
accuracy(f1.out)
accuracy(f2.out)
dm.test(residuals(f1.out), residuals(f2.out), h = 1)

Double-Seasonal Holt-Winters Forecasting

Description

Returns forecasts using Taylor's (2003) Double-Seasonal Holt-Winters method.

Usage

dshw(
  y,
  period1 = NULL,
  period2 = NULL,
  h = 2 * max(period1, period2),
  alpha = NULL,
  beta = NULL,
  gamma = NULL,
  omega = NULL,
  phi = NULL,
  lambda = NULL,
  biasadj = FALSE,
  armethod = TRUE,
  model = NULL
)

Arguments

Details

Taylor's (2003) double-seasonal Holt-Winters method uses additive trend and multiplicative seasonality, where there are two seasonal components which are multiplied together. For example, with a series of half-hourly data, one would set period1 = 48 for the daily period and period2 = 336 for the weekly period. The smoothing parameter notation used here is different from that in Taylor (2003); instead it matches that used in Hyndman et al (2008) and that used for the ets() function.

Value

An object of class forecast.

Author(s)

Rob J Hyndman

References

Taylor, J.W. (2003) Short-term electricity demand forecasting using double seasonal exponential smoothing. Journal of the Operational Research Society, 54, 799-805.

Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. https://robjhyndman.com/expsmooth/.

See Also

stats::HoltWinters(), ets().

Examples

## Not run: 
fcast <- dshw(taylor)
plot(fcast)

t <- seq(0, 5, by = 1 / 20)
x <- exp(sin(2 * pi * t) + cos(2 * pi * t * 4) + rnorm(length(t), 0, 0.1))
fit <- dshw(x, 20, 5)
plot(fit)

## End(Not run)

Easter holidays in each season

Description

Returns a vector of 0's and 1's or fractional results if Easter spans March and April in the observed time period. Easter is defined as the days from Good Friday to Easter Sunday inclusively, plus optionally Easter Monday if easter.mon = TRUE.

Usage

easter(x, easter.mon = FALSE)

Arguments

Details

Useful for adjusting calendar effects.

Value

Time series

Author(s)

Earo Wang

Examples

easter(wineind, easter.mon = TRUE)

Exponential smoothing state space model

Description

Returns ets model applied to y.

Usage

ets(
  y,
  model = "ZZZ",
  damped = NULL,
  alpha = NULL,
  beta = NULL,
  gamma = NULL,
  phi = NULL,
  additive.only = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  lower = c(rep(1e-04, 3), 0.8),
  upper = c(rep(0.9999, 3), 0.98),
  opt.crit = c("lik", "amse", "mse", "sigma", "mae"),
  nmse = 3,
  bounds = c("both", "usual", "admissible"),
  ic = c("aicc", "aic", "bic"),
  restrict = TRUE,
  allow.multiplicative.trend = FALSE,
  use.initial.values = FALSE,
  ...
)

Arguments

Details

Based on the classification of methods as described in Hyndman et al (2008).

The methodology is fully automatic. The only required argument for ets is the time series. The model is chosen automatically if not specified. This methodology performed extremely well on the M3-competition data. (See Hyndman, et al, 2002, below.)

Value

An object of class ets.

The generic accessor functions fitted.values and residuals extract useful features of the value returned by ets and associated functions.

Author(s)

Rob J Hyndman

References

Hyndman, R.J., Koehler, A.B., Snyder, R.D., and Grose, S. (2002) "A state space framework for automatic forecasting using exponential smoothing methods", International J. Forecasting, 18(3), 439–454.

Hyndman, R.J., Akram, Md., and Archibald, B. (2008) "The admissible parameter space for exponential smoothing models". Annals of Statistical Mathematics, 60(2), 407–426.

Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. https://robjhyndman.com/expsmooth/.

See Also

stats::HoltWinters(), rwf(), Arima().

Examples

fit <- ets(USAccDeaths)
plot(forecast(fit))

Find dominant frequency of a time series

Description

findfrequency returns the period of the dominant frequency of a time series. For seasonal data, it will return the seasonal period. For cyclic data, it will return the average cycle length.

Usage

findfrequency(x)

Arguments

Details

The dominant frequency is determined from a spectral analysis of the time series. First, a linear trend is removed, then the spectral density function is estimated from the best fitting autoregressive model (based on the AIC). If there is a large (possibly local) maximum in the spectral density function at frequency f, then the function will return the period 1/f (rounded to the nearest integer). If no such dominant frequency can be found, the function will return 1.

Value

an integer value

Author(s)

Rob J Hyndman

Examples

findfrequency(USAccDeaths) # Monthly data
findfrequency(taylor) # Half-hourly data
findfrequency(lynx) # Annual data

h-step in-sample forecasts for time series models.

Description

Returns h-step forecasts for the data used in fitting the model.

Usage

## S3 method for class 'ARFIMA'
fitted(object, h = 1, ...)

## S3 method for class 'Arima'
fitted(object, h = 1, ...)

## S3 method for class 'ar'
fitted(object, ...)

## S3 method for class 'bats'
fitted(object, h = 1, ...)

## S3 method for class 'ets'
fitted(object, h = 1, ...)

## S3 method for class 'modelAR'
fitted(object, h = 1, ...)

## S3 method for class 'nnetar'
fitted(object, h = 1, ...)

## S3 method for class 'tbats'
fitted(object, h = 1, ...)

Arguments

Value

A time series of the h-step forecasts.

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

See Also

forecast.Arima(), forecast.bats(), forecast.tbats(), forecast.ets(), forecast.nnetar(), residuals.Arima(), residuals.bats() residuals.tbats(), residuals.ets(), residuals.nnetar().

Examples

fit <- ets(WWWusage)
plot(WWWusage)
lines(fitted(fit), col = "red")
lines(fitted(fit, h = 2), col = "green")
lines(fitted(fit, h = 3), col = "blue")
legend("topleft", legend = paste("h =", 1:3), col = 2:4, lty = 1)

Forecasting using Holt-Winters objects

Description

Returns forecasts and other information for univariate Holt-Winters time series models.

Usage

## S3 method for class 'HoltWinters'
forecast(
  object,
  h = if (frequency(object$x) > 1) 2 * frequency(object$x) else 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  ...
)

Arguments

Details

This function calls stats::predict.HoltWinters() and constructs an object of class forecast from the results.

It is included for completeness, but the ets() is recommended for use instead of stats::HoltWinters.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

stats::predict.HoltWinters, stats::HoltWinters().

Examples

fit <- HoltWinters(WWWusage, gamma = FALSE)
plot(forecast(fit))

Forecasting using Structural Time Series models

Description

Returns forecasts and other information for univariate structural time series models.

Usage

## S3 method for class 'StructTS'
forecast(
  object,
  h = if (object$coef["epsilon"] > 1e-10) 2 * object$xtsp[3] else 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  ...
)

Arguments

Details

This function calls predict.StructTS and constructs an object of class forecast from the results.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

stats::StructTS().

Examples

fit <- StructTS(WWWusage, "level")
plot(forecast(fit))

Forecasting using a bagged model

Description

Returns forecasts and other information for bagged models.

Usage

## S3 method for class 'baggedModel'
forecast(
  object,
  h = if (frequency(object$y) > 1) 2 * frequency(object$y) else 10,
  ...
)

Arguments

Details

Intervals are calculated as min and max values over the point forecasts from the models in the ensemble. I.e., the intervals are not prediction intervals, but give an indication of how different the forecasts within the ensemble are.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Christoph Bergmeir, Fotios Petropoulos

References

Bergmeir, C., R. J. Hyndman, and J. M. Benitez (2016). Bagging Exponential Smoothing Methods using STL Decomposition and Box-Cox Transformation. International Journal of Forecasting 32, 303-312.

See Also

baggedModel().

Examples

fit <- baggedModel(WWWusage)
fcast <- forecast(fit)
plot(fcast)

## Not run: 
fit2 <- baggedModel(WWWusage, fn = "auto.arima")
fcast2 <- forecast(fit2)
plot(fcast2)
accuracy(fcast2)

## End(Not run)

Forecasting using BATS and TBATS models

Description

Forecasts h steps ahead with a BATS model. Prediction intervals are also produced.

Usage

## S3 method for class 'bats'
forecast(object, h, level = c(80, 95), fan = FALSE, biasadj = NULL, ...)

## S3 method for class 'tbats'
forecast(
  object,
  h,
  level = c(80, 95),
  fan = FALSE,
  simulate = FALSE,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 5000,
  biasadj = NULL,
  ...
)

Arguments

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Slava Razbash and Rob J Hyndman

References

De Livera, A.M., Hyndman, R.J., & Snyder, R. D. (2011), Forecasting time series with complex seasonal patterns using exponential smoothing, Journal of the American Statistical Association, 106(496), 1513-1527.

See Also

bats(), tbats(), forecast.ets().

Examples

## Not run: 
fit <- bats(USAccDeaths)
plot(forecast(fit))

taylor.fit <- bats(taylor)
plot(forecast(taylor.fit))

## End(Not run)

Forecasts for intermittent demand using Croston's method

Description

Returns forecasts and other information for Croston's forecasts applied to y.

Usage

## S3 method for class 'croston_model'
forecast(object, h = 10, ...)

croston(y, h = 10, alpha = 0.1, type = c("croston", "sba", "sbj"), x = y)

Arguments

Details

Based on Croston's (1972) method for intermittent demand forecasting, also described in Shenstone and Hyndman (2005). Croston's method involves using simple exponential smoothing (SES) on the non-zero elements of the time series and a separate application of SES to the times between non-zero elements of the time series. The smoothing parameters of the two applications of SES are assumed to be equal and are denoted by alpha.

Note that prediction intervals are not computed as Croston's method has no underlying stochastic model.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

References

Croston, J. (1972) "Forecasting and stock control for intermittent demands", Operational Research Quarterly, 23(3), 289-303.

Shale, E.A., Boylan, J.E., & Johnston, F.R. (2006). Forecasting for intermittent demand: the estimation of an unbiased average. Journal of the Operational Research Society, 57(5), 588-592.

Shenstone, L., and Hyndman, R.J. (2005) "Stochastic models underlying Croston's method for intermittent demand forecasting". Journal of Forecasting, 24, 389-402.

Syntetos A.A., Boylan J.E. (2001). On the bias of intermittent demand estimates. International Journal of Production Economics, 71, 457–466.

See Also

ses().

Examples

y <- rpois(20, lambda = 0.3)
fcast <- croston(y)
autoplot(fcast)

Forecasting using ETS models

Description

Returns forecasts and other information for univariate ETS models.

Usage

## S3 method for class 'ets'
forecast(
  object,
  h = if (object$m > 1) 2 * object$m else 10,
  level = c(80, 95),
  fan = FALSE,
  simulate = FALSE,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 5000,
  PI = TRUE,
  lambda = object$lambda,
  biasadj = NULL,
  ...
)

Arguments

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

ets(), ses(), holt(), hw().

Examples

fit <- ets(USAccDeaths)
plot(forecast(fit, h = 48))

Forecasting using ARIMA or ARFIMA models

Description

Returns forecasts and other information for univariate ARIMA models.

Usage

## S3 method for class 'fracdiff'
forecast(
  object,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  simulate = FALSE,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 5000,
  lambda = object$lambda,
  biasadj = attr(lambda, "biasadj"),
  ...
)

## S3 method for class 'Arima'
forecast(
  object,
  h = if (object$arma[5] > 1) 2 * object$arma[5] else 10,
  level = c(80, 95),
  fan = FALSE,
  xreg = NULL,
  simulate = FALSE,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 5000,
  lambda = object$lambda,
  biasadj = attr(lambda, "biasadj"),
  ...
)

## S3 method for class 'ar'
forecast(
  object,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  simulate = FALSE,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 5000,
  lambda = NULL,
  biasadj = FALSE,
  ...
)

Arguments

Details

For Arima or ar objects, the function calls stats::predict.Arima() or stats::predict.ar and constructs an object of class forecast from the results. For fracdiff objects, the calculations are all done within fracdiff::fracdiff() using the equations given by Peiris and Perera (1988).

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

References

Peiris, M. & Perera, B. (1988), On prediction with fractionally differenced ARIMA models, Journal of Time Series Analysis, 9(3), 215-220.

See Also

stats::predict.Arima(), stats::predict.ar(), auto.arima(), Arima(), stats::arima(), stats::ar(), arfima().

Examples

fit <- Arima(WWWusage, c(3, 1, 0))
plot(forecast(fit))

library(fracdiff)
x <- fracdiff.sim(100, ma = -0.4, d = 0.3)$series
fit <- arfima(x)
plot(forecast(fit, h = 30))

Forecast a linear model with possible time series components

Description

forecast.lm is used to predict linear models, especially those involving trend and seasonality components.

Usage

## S3 method for class 'lm'
forecast(
  object,
  newdata,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = object$lambda,
  biasadj = attr(lambda, "biasadj"),
  ts = TRUE,
  ...
)

Arguments

Details

forecast.lm is largely a wrapper for stats::predict.lm() except that it allows variables "trend" and "season" which are created on the fly from the time series characteristics of the data. Also, the output is reformatted into a forecast object.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

tslm(), stats::lm().

Examples

y <- ts(rnorm(120, 0, 3) + 1:120 + 20 * sin(2 * pi * (1:120) / 12), frequency = 12)
fit <- tslm(y ~ trend + season)
plot(forecast(fit, h = 20))

Mean Forecast

Description

Returns forecasts and prediction intervals for a Gaussian iid model. meanf() is a convenience function that combines mean_model() and forecast().

Usage

## S3 method for class 'mean_model'
forecast(
  object,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = object$lambda,
  biasadj = attr(object$lambda, "biasadj"),
  bootstrap = FALSE,
  npaths = 5000,
  ...
)

meanf(
  y,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  bootstrap = FALSE,
  npaths = 5000,
  x = y
)

Arguments

Details

The model assumes that the data are independent and identically distributed

Y_t \sim N(\mu,\sigma^2)

Forecasts are given by

Y_{n+h|n}=\mu

where \mu is estimated by the sample mean.

The function summary() is used to obtain and print a summary of the results, while the function plot() produces a plot of the forecasts and prediction intervals. The generic accessor functions stats::fitted() and stats::residuals() extract useful features of the object returned by mean_model().

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

mean_model()

Examples

fit_nile <- mean_model(Nile)
fit_nile |> forecast(h = 10) |> autoplot()
nile.fcast <- meanf(Nile, h = 10)

Forecast a multiple linear model with possible time series components

Description

forecast.mlm is used to predict multiple linear models, especially those involving trend and seasonality components.

Usage

## S3 method for class 'mlm'
forecast(
  object,
  newdata,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = object$lambda,
  biasadj = attr(object$lambda, "biasadj"),
  ts = TRUE,
  ...
)

Arguments

Details

forecast.mlm is largely a wrapper for forecast.lm() except that it allows forecasts to be generated on multiple series. Also, the output is reformatted into a mforecast object.

Value

An object of class mforecast.

The function summary is used to obtain and print a summary of the results, while the function plot produces a plot of the forecasts and prediction intervals.

The generic accessor functions fitted.values and residuals extract useful features of the value returned by forecast.lm.

An object of class mforecast is a list containing at least the following elements:

Author(s)

Mitchell O'Hara-Wild

See Also

tslm(), forecast.lm(), stats::lm().

Examples

lungDeaths <- cbind(mdeaths, fdeaths)
fit <- tslm(lungDeaths ~ trend + season)
fcast <- forecast(fit, h = 10)

carPower <- as.matrix(mtcars[, c("qsec", "hp")])
carmpg <- mtcars[, "mpg"]
fit <- lm(carPower ~ carmpg)
fcast <- forecast(fit, newdata = data.frame(carmpg = 30))

Forecasting using user-defined model

Description

Returns forecasts and other information for user-defined models.

Usage

## S3 method for class 'modelAR'
forecast(
  object,
  h = if (object$m > 1) 2 * object$m else 10,
  PI = FALSE,
  level = c(80, 95),
  fan = FALSE,
  xreg = NULL,
  lambda = object$lambda,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 1000,
  ...
)

Arguments

Details

Prediction intervals are calculated through simulations and can be slow. Note that if the model is too complex and overfits the data, the residuals can be arbitrarily small; if used for prediction interval calculations, they could lead to misleadingly small values.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman and Gabriel Caceres

See Also

nnetar().

Forecasting time series

Description

mforecast is a class of objects for forecasting from multivariate time series or multivariate time series models. The function invokes particular methods which depend on the class of the first argument.

Usage

## S3 method for class 'mts'
forecast(
  object,
  h = if (frequency(object) > 1) 2 * frequency(object) else 10,
  level = c(80, 95),
  fan = FALSE,
  robust = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  find.frequency = FALSE,
  allow.multiplicative.trend = FALSE,
  ...
)

Arguments

Details

For example, the function forecast.mlm() makes multivariate forecasts based on the results produced by tslm().

Value

An object of class mforecast.

The function summary is used to obtain and print a summary of the results, while the function plot produces a plot of the multivariate forecasts and prediction intervals.

The generic accessors functions fitted.values and residuals extract various useful features of the value returned by forecast$model.

An object of class mforecast is a list usually containing at least the following elements:

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

See Also

Other functions which return objects of class mforecast are forecast.mlm(), forecast.varest().

Forecasting using neural network models

Description

Returns forecasts and other information for univariate neural network models.

Usage

## S3 method for class 'nnetar'
forecast(
  object,
  h = if (object$m > 1) 2 * object$m else 10,
  PI = FALSE,
  level = c(80, 95),
  fan = FALSE,
  xreg = NULL,
  lambda = object$lambda,
  bootstrap = FALSE,
  npaths = 1000,
  innov = NULL,
  ...
)

Arguments

Details

Prediction intervals are calculated through simulations and can be slow. Note that if the network is too complex and overfits the data, the residuals can be arbitrarily small; if used for prediction interval calculations, they could lead to misleadingly small values. It is possible to use out-of-sample residuals to ameliorate this, see examples.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman and Gabriel Caceres

See Also

nnetar().

Examples

## Fit & forecast model
fit <- nnetar(USAccDeaths, size = 2)
fcast <- forecast(fit, h = 20)
plot(fcast)

## Not run: 
## Include prediction intervals in forecast
fcast2 <- forecast(fit, h = 20, PI = TRUE, npaths = 100)
plot(fcast2)

## Set up out-of-sample innovations using cross-validation
fit_cv <- CVar(USAccDeaths, size = 2)
res_sd <- sd(fit_cv$residuals, na.rm = TRUE)
myinnovs <- rnorm(20 * 100, mean = 0, sd = res_sd)
## Forecast using new innovations
fcast3 <- forecast(fit, h = 20, PI = TRUE, npaths = 100, innov = myinnovs)
plot(fcast3)

## End(Not run)

Naive and Random Walk Forecasts

Description

Returns forecasts and prediction intervals for a generalized random walk model. rwf() is a convenience function that combines rw_model() and forecast(). naive() is a wrapper to rwf() with drift=FALSE and lag=1, while snaive() is a wrapper to rwf() with drift=FALSE and lag=frequency(y).

Usage

## S3 method for class 'rw_model'
forecast(
  object,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  simulate = FALSE,
  bootstrap = FALSE,
  npaths = 5000,
  innov = NULL,
  lambda = object$lambda,
  biasadj = FALSE,
  ...
)

rwf(
  y,
  h = 10,
  drift = FALSE,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  lag = 1,
  ...,
  x = y
)

naive(
  y,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  ...,
  x = y
)

snaive(
  y,
  h = 2 * frequency(x),
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  ...,
  x = y
)

Arguments

Details

The model assumes that

Y_t = Y_{t-p} + c + \varepsilon_{t}

where p is the lag parameter, c is the drift parameter, and \varepsilon_t\sim N(0,\sigma^2) are iid.

The model without drift has c=0. In the model with drift, c is estimated by the sample mean of the differences Y_t - Y_{t-p}.

If p=1, this is equivalent to an ARIMA(0,1,0) model with an optional drift coefficient. For p>1, it is equivalent to an ARIMA(0,0,0)(0,1,0)p model.

The forecasts are given by

Y_{T+h|T}= Y_{T+h-p(k+1)} + ch

where k is the integer part of (h-1)/p. For a regular random walk, p=1 and c=0, so all forecasts are equal to the last observation. Forecast standard errors allow for uncertainty in estimating the drift parameter (unlike the corresponding forecasts obtained by fitting an ARIMA model directly).

The generic accessor functions stats::fitted() and stats::residuals() extract useful features of the object returned.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

rw_model(), Arima()

Examples

# Three ways to do the same thing
gold_model <- rw_model(gold)
gold_fc1 <- forecast(gold_model, h = 50)
gold_fc2 <- rwf(gold, h = 50)
gold_fc3 <- naive(gold, h = 50)

# Plot the forecasts
autoplot(gold_fc1)

# Drift forecasts
rwf(gold, drift = TRUE) |> autoplot()

# Seasonal naive forecasts
snaive(wineind) |> autoplot()

Returns local linear forecasts and prediction intervals using cubic smoothing splines estimated with spline_model().

Description

The cubic smoothing spline model is equivalent to an ARIMA(0,2,2) model but with a restricted parameter space. The advantage of the spline model over the full ARIMA model is that it provides a smooth historical trend as well as a linear forecast function. Hyndman, King, Pitrun, and Billah (2002) show that the forecast performance of the method is hardly affected by the restricted parameter space.

Usage

## S3 method for class 'spline_model'
forecast(
  object,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = object$lambda,
  biasadj = attr(lambda, "biasadj"),
  simulate = FALSE,
  bootstrap = FALSE,
  innov = NULL,
  npaths = 5000,
  ...
)

splinef(
  y,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  method = c("gcv", "mle"),
  x = y
)

Arguments

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

References

Hyndman, King, Pitrun and Billah (2005) Local linear forecasts using cubic smoothing splines. Australian and New Zealand Journal of Statistics, 47(1), 87-99. https://robjhyndman.com/publications/splinefcast/.

See Also

spline_model()

Examples

fit <- spline_model(uspop)
fcast <- forecast(fit)
autoplot(fcast)
summary(fcast)

Forecasting using stl objects

Description

Forecasts of STL objects are obtained by applying a non-seasonal forecasting method to the seasonally adjusted data and re-seasonalizing using the last year of the seasonal component.

Usage

## S3 method for class 'stl'
forecast(
  object,
  method = c("ets", "arima", "naive", "rwdrift"),
  etsmodel = "ZZN",
  forecastfunction = NULL,
  h = frequency(object$time.series) * 2,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  xreg = NULL,
  newxreg = NULL,
  allow.multiplicative.trend = FALSE,
  ...
)

## S3 method for class 'stlm'
forecast(
  object,
  h = 2 * object$m,
  level = c(80, 95),
  fan = FALSE,
  lambda = object$lambda,
  biasadj = attr(lambda, "biasadj"),
  newxreg = NULL,
  allow.multiplicative.trend = FALSE,
  ...
)

stlf(
  y,
  h = frequency(x) * 2,
  s.window = 7 + 4 * seq(6),
  t.window = NULL,
  robust = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  x = y,
  ...
)

Arguments

Details

forecast.stlm forecasts the seasonally adjusted data, then re-seasonalizes the results by adding back the last year of the estimated seasonal component.

stlf combines stlm() and forecast.stlm. It takes a ts argument, applies an STL decomposition, models the seasonally adjusted data, reseasonalizes, and returns the forecasts. However, it allows more general forecasting methods to be specified via forecastfunction.

forecast.stl is similar to stlf except that it takes the STL decomposition as the first argument, instead of the time series.

Note that the prediction intervals ignore the uncertainty associated with the seasonal component. They are computed using the prediction intervals from the seasonally adjusted series, which are then reseasonalized using the last year of the seasonal component. The uncertainty in the seasonal component is ignored.

The forecasting method for the seasonally adjusted data can be specified in stlf and forecast.stl using either method or forecastfunction. The method argument provides a shorthand way of specifying forecastfunction for a few special cases. More generally, forecastfunction can be any function with first argument a ts object, and other h and level, which returns an object of class forecast(). For example, forecastfunction = thetaf uses the thetaf() function for forecasting the seasonally adjusted series.

Value

stlm returns an object of class stlm. The other functions return objects of class forecast.

There are many methods for working with forecast() objects including summary to obtain and print a summary of the results, while plot produces a plot of the forecasts and prediction intervals. The generic accessor functions fitted.values and residuals extract useful features.

Author(s)

Rob J Hyndman

See Also

stats::stl(), forecast.ets(), forecast.Arima().

Examples

tsmod <- stlm(USAccDeaths, modelfunction = ar)
plot(forecast(tsmod, h = 36))

decomp <- stl(USAccDeaths, s.window = "periodic")
plot(forecast(decomp))

plot(stlf(AirPassengers, lambda = 0))

Theta method forecasts.

Description

Returns forecasts and prediction intervals for a theta method forecast. thetaf() is a convenience function that combines theta_model() and forecast.theta_model(). The theta method of Assimakopoulos and Nikolopoulos (2000) is equivalent to simple exponential smoothing with drift (Hyndman and Billah, 2003). The series is tested for seasonality using the test outlined in A&N. If deemed seasonal, the series is seasonally adjusted using a classical multiplicative decomposition before applying the theta method. The resulting forecasts are then reseasonalized. Prediction intervals are computed using the underlying state space model.

Usage

## S3 method for class 'theta_model'
forecast(
  object,
  h = if (frequency(object$y) > 1) 2 * frequency(object$y) else 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = object$lambda,
  biasadj = FALSE,
  ...
)

thetaf(
  y,
  h = if (frequency(y) > 1) 2 * frequency(y) else 10,
  level = c(80, 95),
  fan = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  x = y,
  ...
)

Arguments

Details

More general theta methods are available in the forecTheta package.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

References

Assimakopoulos, V. and Nikolopoulos, K. (2000). The theta model: a decomposition approach to forecasting. International Journal of Forecasting 16, 521-530.

Hyndman, R.J., and Billah, B. (2003) Unmasking the Theta method. International J. Forecasting, 19, 287-290.

See Also

stats::arima(), meanf(), rwf(), ses()

Examples

nile_fit <- theta_model(Nile)
forecast(nile_fit) |> autoplot()

Forecasting time series

Description

forecast is a generic function for forecasting from time series or time series models. The function invokes particular methods which depend on the class of the first argument.

Usage

## S3 method for class 'ts'
forecast(
  object,
  h = if (frequency(object) > 1) 2 * frequency(object) else 10,
  level = c(80, 95),
  fan = FALSE,
  robust = FALSE,
  lambda = NULL,
  biasadj = FALSE,
  find.frequency = FALSE,
  allow.multiplicative.trend = FALSE,
  model = NULL,
  ...
)

## Default S3 method:
forecast(object, ...)

## S3 method for class 'forecast'
print(x, ...)

Arguments

Details

For example, the function forecast.Arima() makes forecasts based on the results produced by stats::arima().

If model = NULL,the function forecast.ts() makes forecasts using ets() models (if the data are non-seasonal or the seasonal period is 12 or less) or stlf() (if the seasonal period is 13 or more).

If model is not NULL, forecast.ts will apply the model to the object time series, and then generate forecasts accordingly.

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

Other functions which return objects of class forecast are forecast.ets(), forecast.Arima(), forecast.HoltWinters(), forecast.StructTS(), meanf(), rwf(), splinef(), thetaf(), croston(), ses(), holt(), hw().

Examples

WWWusage |> forecast() |> plot()
fit <- ets(window(WWWusage, end = 60))
fc <- forecast(WWWusage, model = fit)

Fourier terms for modelling seasonality

Description

fourier returns a matrix containing terms from a Fourier series, up to order K, suitable for use in Arima(), auto.arima(), or tslm().

Usage

fourier(x, K, h = NULL)

fourierf(x, K, h)

Arguments

Details

fourierf is deprecated, instead use the h argument in fourier.

The period of the Fourier terms is determined from the time series characteristics of x. When h is missing, the length of x also determines the number of rows for the matrix returned by fourier. Otherwise, the value of h determines the number of rows for the matrix returned by fourier, typically used for forecasting. The values within x are not used.

Typical use would omit h when generating Fourier terms for training a model and include h when generating Fourier terms for forecasting.

When x is a ts object, the value of K should be an integer and specifies the number of sine and cosine terms to return. Thus, the matrix returned has 2*K columns.

When x is a msts object, then K should be a vector of integers specifying the number of sine and cosine terms for each of the seasonal periods. Then the matrix returned will have 2*sum(K) columns.

Value

Numerical matrix.

Author(s)

Rob J Hyndman

See Also

seasonaldummy()

Examples

library(ggplot2)

# Using Fourier series for a "ts" object
# K is chosen to minimize the AICc
deaths.model <- auto.arima(
  USAccDeaths,
  xreg = fourier(USAccDeaths, K = 5),
  seasonal = FALSE
)
deaths.fcast <- forecast(
  deaths.model,
  xreg = fourier(USAccDeaths, K = 5, h = 36)
)
autoplot(deaths.fcast) + xlab("Year")

# Using Fourier series for a "msts" object
taylor.lm <- tslm(taylor ~ fourier(taylor, K = c(3, 3)))
taylor.fcast <- forecast(
  taylor.lm,
  data.frame(fourier(taylor, K = c(3, 3), h = 270))
)
autoplot(taylor.fcast)

Australian monthly gas production

Description

Australian monthly gas production: 1956–1995.

Usage

gas

Format

Time series data

Source

Australian Bureau of Statistics.

Examples

plot(gas)
seasonplot(gas)
tsdisplay(gas)

Get response variable from time series model.

Description

getResponse is a generic function for extracting the historical data from a time series model (including Arima, ets, ar, fracdiff), a linear model of class lm, or a forecast object. The function invokes particular methods which depend on the class of the first argument.

Usage

getResponse(object, ...)

## Default S3 method:
getResponse(object, ...)

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

## S3 method for class 'Arima'
getResponse(object, ...)

## S3 method for class 'fracdiff'
getResponse(object, ...)

## S3 method for class 'ar'
getResponse(object, ...)

## S3 method for class 'tbats'
getResponse(object, ...)

## S3 method for class 'bats'
getResponse(object, ...)

## S3 method for class 'mforecast'
getResponse(object, ...)

## S3 method for class 'baggedModel'
getResponse(object, ...)

Arguments

Value

A numerical vector or a time series object of class ts.

Author(s)

Rob J Hyndman

Histogram with optional normal and kernel density functions

Description

Plots a histogram and density estimates using ggplot.

Usage

gghistogram(
  x,
  add.normal = FALSE,
  add.kde = FALSE,
  add.rug = TRUE,
  bins,
  boundary = 0
)

Arguments

Value

None.

Author(s)

Rob J Hyndman

See Also

graphics::hist(), ggplot2::geom_histogram()

Examples

gghistogram(lynx, add.kde = TRUE)

Time series lag ggplots

Description

Plots a lag plot using ggplot.

Usage

gglagplot(
  x,
  lags = if (frequency(x) > 9) 16 else 9,
  set.lags = 1:lags,
  diag = TRUE,
  diag.col = "gray",
  do.lines = TRUE,
  colour = TRUE,
  continuous = frequency(x) > 12,
  labels = FALSE,
  seasonal = TRUE,
  ...
)

gglagchull(
  x,
  lags = if (frequency(x) > 1) min(12, frequency(x)) else 4,
  set.lags = 1:lags,
  diag = TRUE,
  diag.col = "gray",
  ...
)

Arguments

Details

"gglagplot" will plot time series against lagged versions of themselves. Helps visualising 'auto-dependence' even when auto-correlations vanish.

"gglagchull" will layer convex hulls of the lags, layered on a single plot. This helps visualise the change in 'auto-dependence' as lags increase.

Value

None.

Author(s)

Mitchell O'Hara-Wild

See Also

stats::lag.plot()

Examples

gglagplot(woolyrnq)
gglagplot(woolyrnq, seasonal = FALSE)

lungDeaths <- cbind(mdeaths, fdeaths)
gglagplot(lungDeaths, lags = 2)
gglagchull(lungDeaths, lags = 6)

gglagchull(woolyrnq)

Create a seasonal subseries ggplot

Description

Plots a subseries plot using ggplot. Each season is plotted as a separate mini time series. The blue lines represent the mean of the observations within each season.

Usage

ggmonthplot(x, labels = NULL, times = time(x), phase = cycle(x), ...)

ggsubseriesplot(x, labels = NULL, times = time(x), phase = cycle(x), ...)

Arguments

Details

The ggmonthplot function is simply a wrapper for ggsubseriesplot as a convenience for users familiar with stats::monthplot().

Value

Returns an object of class ggplot.

Author(s)

Mitchell O'Hara-Wild

See Also

stats::monthplot()

Examples

ggsubseriesplot(AirPassengers)
ggsubseriesplot(woolyrnq)

Seasonal plot

Description

Plots a seasonal plot as described in Hyndman and Athanasopoulos (2014, chapter 2). This is like a time plot except that the data are plotted against the seasons in separate years.

Usage

ggseasonplot(
  x,
  season.labels = NULL,
  year.labels = FALSE,
  year.labels.left = FALSE,
  type = NULL,
  col = NULL,
  continuous = FALSE,
  polar = FALSE,
  labelgap = 0.04,
  ...
)

seasonplot(
  x,
  s,
  season.labels = NULL,
  year.labels = FALSE,
  year.labels.left = FALSE,
  type = "o",
  main,
  xlab = NULL,
  ylab = "",
  col = 1,
  labelgap = 0.1,
  ...
)

Arguments

Value

None.

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

References

Hyndman and Athanasopoulos (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. https://otexts.com/fpp2/

See Also

stats::monthplot()

Examples

ggseasonplot(AirPassengers, col = rainbow(12), year.labels = TRUE)
ggseasonplot(AirPassengers, year.labels = TRUE, continuous = TRUE)

seasonplot(AirPassengers, col = rainbow(12), year.labels = TRUE)

Time series display

Description

Plots a time series along with its acf and either its pacf, lagged scatterplot or spectrum.

Usage

ggtsdisplay(
  x,
  plot.type = c("partial", "histogram", "scatter", "spectrum"),
  points = TRUE,
  smooth = FALSE,
  lag.max,
  na.action = na.contiguous,
  theme = NULL,
  ...
)

tsdisplay(
  x,
  plot.type = c("partial", "histogram", "scatter", "spectrum"),
  points = TRUE,
  ci.type = c("white", "ma"),
  lag.max,
  na.action = na.contiguous,
  main = NULL,
  xlab = "",
  ylab = "",
  pch = 1,
  cex = 0.5,
  ...
)

Arguments

Details

ggtsdisplay will produce the equivalent plot using ggplot graphics.

Value

None.

Author(s)

Rob J Hyndman

References

Hyndman and Athanasopoulos (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. https://otexts.com/fpp2/

See Also

stats::plot.ts(), Acf(), stats::spec.ar()

Examples

library(ggplot2)
ggtsdisplay(USAccDeaths, plot.type = "scatter", theme = theme_bw())

tsdisplay(diff(WWWusage))
ggtsdisplay(USAccDeaths, plot.type = "scatter")

Daily morning gold prices

Description

Daily morning gold prices in US dollars. 1 January 1985 – 31 March 1989.

Usage

gold

Format

Time series data

Examples

tsdisplay(gold)

Is an object a particular model type?

Description

Returns true if the model object is of a particular type

Usage

is.acf(x)

is.Arima(x)

is.baggedModel(x)

is.bats(x)

is.ets(x)

is.modelAR(x)

is.stlm(x)

is.nnetar(x)

is.nnetarmodels(x)

Arguments

Is an object constant?

Description

Returns true if the object's numerical values do not vary.

Usage

is.constant(x)

Arguments

Is an object a particular forecast type?

Description

Returns true if the forecast object is of a particular type

Usage

is.forecast(x)

is.mforecast(x)

is.splineforecast(x)

Arguments

Moving-average smoothing

Description

ma computes a simple moving average smoother of a given time series.

Usage

ma(x, order, centre = TRUE)

Arguments

Details

The moving average smoother averages the nearest order periods of each observation. As neighbouring observations of a time series are likely to be similar in value, averaging eliminates some of the randomness in the data, leaving a smooth trend-cycle component.

\hat{T}_{t} = \frac{1}{m} \sum_{j=-k}^k y_{t+j}

where k=\frac{m-1}{2}.

When an even order is specified, the observations averaged will include one more observation from the future than the past (k is rounded up). If centre is TRUE, the value from two moving averages (where k is rounded up and down respectively) are averaged, centering the moving average.

Value

Numerical time series object containing the simple moving average smoothed values.

Author(s)

Rob J Hyndman

See Also

stats::decompose()

Examples

plot(wineind)
sm <- ma(wineind, order = 12)
lines(sm, col = "red")

Mean Forecast Model

Description

Fits a Gaussian iid model to a univariate time series.

Usage

mean_model(y, lambda = NULL, biasadj = FALSE)

Arguments

Details

The model assumes that the data are independent and identically distributed

Y_t \sim N(\mu,\sigma^2)

Forecasts are given by

Y_{n+h|n}=\mu

where \mu is estimated by the sample mean.

The function summary() is used to obtain and print a summary of the results, while the function plot() produces a plot of the forecasts and prediction intervals. The generic accessor functions stats::fitted() and stats::residuals() extract useful features of the object returned by mean_model().

Value

An object of class mean_model.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

See Also

forecast.mean_model(), meanf()

Examples

fit_nile <- mean_model(Nile)
fit_nile |> forecast(h = 10) |> autoplot()

Time Series Forecasts with a user-defined model

Description

Experimental function to forecast univariate time series with a user-defined model

Usage

modelAR(
  y,
  p,
  P = 1,
  FUN,
  predict.FUN,
  xreg = NULL,
  lambda = NULL,
  model = NULL,
  subset = NULL,
  scale.inputs = FALSE,
  x = y,
  ...
)

Arguments

Details

This is an experimental function and only recommended for advanced users. The selected model is fitted with lagged values of ⁠y as inputs. The inputs are for lags 1 to ⁠p⁠, and lags ⁠mtomPwherem = frequency(y)⁠. If ⁠xreg⁠is provided, its columns are also used as inputs. If there are missing values in⁠yorxreg', the corresponding rows (and any others which depend on them as lags) are omitted from the fit. The model is trained for one-step forecasting. Multi-step forecasts are computed recursively.

Value

Returns an object of class modelAR.

The function summary is used to obtain and print a summary of the results.

The generic accessor functions fitted.values and residuals extract useful features of the value returned by modelAR.

Author(s)

Rob J Hyndman and Gabriel Caceres

Examples

## Set up functions
my_lm <- function(x, y) {
 structure(lsfit(x,y), class = "lsfit")
}
predict.lsfit <- function(object, newdata = NULL) {
  n <- length(object$qr$qt)
  if(is.null(newdata)) {
    z <- numeric(n)
    z[seq_len(object$qr$rank)] <- object$qr$qt[seq_len(object$qr$rank)]
    as.numeric(qr.qy(object$qr, z))
  } else {
    sum(object$coefficients * c(1, newdata))
  }
}
# Fit an AR(2) model
fit <- modelAR(
  y = lynx,
  p = 2,
  FUN = my_lm,
  predict.FUN = predict.lsfit,
  lambda = 0.5,
  scale.inputs = TRUE
)
forecast(fit, h = 20) |> autoplot()

Compute model degrees of freedom

Description

Compute model degrees of freedom

Usage

modeldf(object, ...)

Arguments

Number of days in each season

Description

Returns number of days in each month or quarter of the observed time period.

Usage

monthdays(x)

Arguments

Details

Useful for month length adjustments

Value

Time series

Author(s)

Rob J Hyndman

See Also

bizdays()

Examples

par(mfrow = c(2, 1))
plot(
  ldeaths,
  xlab = "Year",
  ylab = "pounds",
  main = "Monthly deaths from lung disease (UK)"
)
ldeaths.adj <- ldeaths / monthdays(ldeaths) * 365.25 / 12
plot(
  ldeaths.adj,
  xlab = "Year",
  ylab = "pounds",
  main = "Adjusted monthly deaths from lung disease (UK)"
)

Multiple seasonal decomposition

Description

Decompose a time series into seasonal, trend and remainder components. Seasonal components are estimated iteratively using STL. Multiple seasonal periods are allowed. The trend component is computed for the last iteration of STL. Non-seasonal time series are decomposed into trend and remainder only. In this case, stats::supsmu() is used to estimate the trend. Optionally, the time series may be Box-Cox transformed before decomposition. Unlike stats::stl(), mstl is completely automated.

Usage

mstl(
  x,
  lambda = NULL,
  biasadj = FALSE,
  iterate = 2,
  s.window = 7 + 4 * seq(6),
  ...
)

Arguments

See Also

stats::stl(), stats::supsmu()

Examples

library(ggplot2)
mstl(taylor) |> autoplot()
mstl(AirPassengers, lambda = "auto") |> autoplot()

Multi-Seasonal Time Series

Description

msts is an S3 class for multi seasonal time series objects, intended to be used for models that support multiple seasonal periods. The msts class inherits from the ts class and has an additional "msts" attribute which contains the vector of seasonal periods. All methods that work on a ts class, should also work on a msts class.

Usage

msts(data, seasonal.periods, ts.frequency = floor(max(seasonal.periods)), ...)

Arguments

Value

An object of class c("msts", "ts"). If there is only one seasonal period (i.e., length(seasonal.periods) == 1), then the object is of class ts.

Author(s)

Slava Razbash and Rob J Hyndman

Examples

x <- msts(taylor, seasonal.periods = c(2 * 24, 2 * 24 * 7, 2 * 24 * 365), start = 2000 + 22 / 52)
y <- msts(USAccDeaths, seasonal.periods = 12, start = 1949)

Interpolate missing values in a time series

Description

By default, uses linear interpolation for non-seasonal series. For seasonal series, a robust STL decomposition is first computed. Then a linear interpolation is applied to the seasonally adjusted data, and the seasonal component is added back.

Usage

na.interp(
  x,
  lambda = NULL,
  linear = (frequency(x) <= 1 || sum(!is.na(x)) <= 2 * frequency(x))
)

Arguments

Details

A more general and flexible approach is available using na.approx in the zoo package.

Value

Time series

Author(s)

Rob J Hyndman

See Also

tsoutliers()

Examples

data(gold)
plot(na.interp(gold))

Number of differences required for a stationary series

Description

Functions to estimate the number of differences required to make a given time series stationary. ndiffs estimates the number of first differences necessary.

Usage

ndiffs(
  x,
  alpha = 0.05,
  test = c("kpss", "adf", "pp"),
  type = c("level", "trend"),
  max.d = 2,
  ...
)

Arguments

Details

ndiffs uses a unit root test to determine the number of differences required for time series x to be made stationary. If test = "kpss", the KPSS test is used with the null hypothesis that x has a stationary root against a unit-root alternative. Then the test returns the least number of differences required to pass the test at the level alpha. If test = "adf", the Augmented Dickey-Fuller test is used and if test = "pp" the Phillips-Perron test is used. In both of these cases, the null hypothesis is that x has a unit root against a stationary root alternative. Then the test returns the least number of differences required to fail the test at the level alpha.

Value

An integer indicating the number of differences required for stationarity.

Author(s)

Rob J Hyndman, Slava Razbash & Mitchell O'Hara-Wild

References

Dickey DA and Fuller WA (1979), "Distribution of the Estimators for Autoregressive Time Series with a Unit Root", Journal of the American Statistical Association 74:427-431.

Kwiatkowski D, Phillips PCB, Schmidt P and Shin Y (1992) "Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root", Journal of Econometrics 54:159-178.

Osborn, D.R. (1990) "A survey of seasonality in UK macroeconomic variables", International Journal of Forecasting, 6:327-336.

Phillips, P.C.B. and Perron, P. (1988) "Testing for a unit root in time series regression", Biometrika, 72(2), 335-346.

Said E and Dickey DA (1984), "Testing for Unit Roots in Autoregressive Moving Average Models of Unknown Order", Biometrika 71:599-607.

See Also

auto.arima() and ndiffs()

Examples

ndiffs(WWWusage)
ndiffs(diff(log(AirPassengers), 12))

Neural Network Time Series Forecasts

Description

Feed-forward neural networks with a single hidden layer and lagged inputs for forecasting univariate time series.

Usage

nnetar(
  y,
  p,
  P = 1,
  size = NULL,
  repeats = 20,
  xreg = NULL,
  lambda = NULL,
  model = NULL,
  subset = NULL,
  scale.inputs = TRUE,
  parallel = FALSE,
  num.cores = 2,
  x = y,
  ...
)

Arguments

Details

A feed-forward neural network is fitted with lagged values of y as inputs and a single hidden layer with size nodes. The inputs are for lags 1 to p, and lags m to mP where m = frequency(y). If xreg is provided, its columns are also used as inputs. If there are missing values in y or xreg, the corresponding rows (and any others which depend on them as lags) are omitted from the fit. A total of repeats networks are fitted, each with random starting weights. These are then averaged when computing forecasts. The network is trained for one-step forecasting. Multi-step forecasts are computed recursively.

For non-seasonal data, the fitted model is denoted as an NNAR(p,k) model, where k is the number of hidden nodes. This is analogous to an AR(p) model but with nonlinear functions. For seasonal data, the fitted model is called an NNAR(p,P,k)[m] model, which is analogous to an ARIMA(p,0,0)(P,0,0)[m] model but with nonlinear functions.

Value

Returns an object of class nnetar.

The function summary is used to obtain and print a summary of the results.

The generic accessor functions fitted.values and residuals extract useful features of the value returned by nnetar.

Author(s)

Rob J Hyndman and Gabriel Caceres

Examples

fit <- nnetar(lynx)
fcast <- forecast(fit)
plot(fcast)

## Arguments can be passed to nnet()
fit <- nnetar(lynx, decay = 0.5, maxit = 150)
plot(forecast(fit))
lines(lynx)

## Fit model to first 100 years of lynx data
fit <- nnetar(window(lynx, end = 1920), decay = 0.5, maxit = 150)
plot(forecast(fit, h = 14))
lines(lynx)

## Apply fitted model to later data, including all optional arguments
fit2 <- nnetar(window(lynx, start = 1921), model = fit)

Number of differences required for a seasonally stationary series

Description

Functions to estimate the number of differences required to make a given time series stationary. nsdiffs estimates the number of seasonal differences necessary.

Usage

nsdiffs(
  x,
  alpha = 0.05,
  m = frequency(x),
  test = c("seas", "ocsb", "hegy", "ch"),
  max.D = 1,
  ...
)

Arguments

Details

nsdiffs uses seasonal unit root tests to determine the number of seasonal differences required for time series x to be made stationary (possibly with some lag-one differencing as well).

Several different tests are available:

• If test = "seas" (default), a measure of seasonal strength is used, where differencing is selected if the seasonal strength (Wang, Smith & Hyndman, 2006) exceeds 0.64 (based on minimizing MASE when forecasting using auto.arima on M3 and M4 data).

• If test = "ch", the Canova-Hansen (1995) test is used (with null hypothesis of deterministic seasonality)

• If test = "hegy", the Hylleberg, Engle, Granger & Yoo (1990) test is used.

• If test = "ocsb", the Osborn-Chui-Smith-Birchenhall (1988) test is used (with null hypothesis that a seasonal unit root exists).

Value

An integer indicating the number of differences required for stationarity.

Author(s)

Rob J Hyndman, Slava Razbash and Mitchell O'Hara-Wild

References

Wang, X, Smith, KA, Hyndman, RJ (2006) "Characteristic-based clustering for time series data", Data Mining and Knowledge Discovery, 13(3), 335-364.

Osborn DR, Chui APL, Smith J, and Birchenhall CR (1988) "Seasonality and the order of integration for consumption", Oxford Bulletin of Economics and Statistics 50(4):361-377.

Canova F and Hansen BE (1995) "Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability", Journal of Business and Economic Statistics 13(3):237-252.

Hylleberg S, Engle R, Granger C and Yoo B (1990) "Seasonal integration and cointegration.", Journal of Econometrics 44(1), pp. 215-238.

See Also

auto.arima(), ndiffs(), ocsb.test(), uroot::hegy.test(), and uroot::ch.test()

Examples

nsdiffs(AirPassengers)

Osborn, Chui, Smith, and Birchenhall Test for Seasonal Unit Roots

Description

An implementation of the Osborn, Chui, Smith, and Birchenhall (OCSB) test.

Usage

ocsb.test(x, lag.method = c("fixed", "AIC", "BIC", "AICc"), maxlag = 0)

Arguments

Details

The regression equation may include lags of the dependent variable. When lag.method = "fixed", the lag order is fixed to maxlag; otherwise, maxlag is the maximum number of lags considered in a lag selection procedure that minimises the lag.method criterion, which can be AIC or BIC or corrected AIC, AICc, obtained as AIC + (2k(k+1))/(n-k-1), where k is the number of parameters and n is the number of available observations in the model.

Critical values for the test are based on simulations, which has been smoothed over to produce critical values for all seasonal periods.

Value

ocsb.test returns a list of class "OCSBtest" with the following components:

• statistics the value of the test statistics.

• pvalues the p-values for each test statistics.

• method a character string describing the type of test.

• data.name a character string giving the name of the data.

• fitted.model the fitted regression model.

References

Osborn DR, Chui APL, Smith J, and Birchenhall CR (1988) "Seasonality and the order of integration for consumption", Oxford Bulletin of Economics and Statistics 50(4):361-377.

See Also

nsdiffs()

Examples

ocsb.test(AirPassengers)

Plot characteristic roots from ARIMA model

Description

Produces a plot of the inverse AR and MA roots of an ARIMA model. Inverse roots outside the unit circle are shown in red.

Usage

## S3 method for class 'Arima'
plot(
  x,
  type = c("both", "ar", "ma"),
  main,
  xlab = "Real",
  ylab = "Imaginary",
  ...
)

## S3 method for class 'ar'
plot(x, main, xlab = "Real", ylab = "Imaginary", ...)

## S3 method for class 'Arima'
autoplot(object, type = c("both", "ar", "ma"), ...)

## S3 method for class 'ar'
autoplot(object, ...)

Arguments

Details

autoplot will produce an equivalent plot as a ggplot object.

Value

None. Function produces a plot

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

See Also

Arima(), stats::ar()

Examples

library(ggplot2)

fit <- Arima(WWWusage, order = c(3, 1, 0))
plot(fit)
autoplot(fit)

fit <- Arima(woolyrnq, order = c(2, 0, 0), seasonal = c(2, 1, 1))
plot(fit)
autoplot(fit)

plot(ar.ols(gold[1:61]))
autoplot(ar.ols(gold[1:61]))

Plot components from BATS model

Description

Produces a plot of the level, slope and seasonal components from a BATS or TBATS model. The plotted components are Box-Cox transformed using the estimated transformation parameter.

Usage

## S3 method for class 'bats'
plot(x, main = "Decomposition by BATS model", ...)

## S3 method for class 'tbats'
autoplot(object, range.bars = FALSE, ...)

## S3 method for class 'bats'
autoplot(object, range.bars = FALSE, ...)

## S3 method for class 'tbats'
plot(x, main = "Decomposition by TBATS model", ...)

Arguments

Value

None. Function produces a plot

Author(s)

Rob J Hyndman

See Also

bats()], tbats()

Examples

## Not run: 
fit <- tbats(USAccDeaths)
plot(fit)
autoplot(fit, range.bars = TRUE)

## End(Not run)

Plot components from ETS model

Description

Produces a plot of the level, slope and seasonal components from an ETS model.

Usage

## S3 method for class 'ets'
plot(x, ...)

## S3 method for class 'ets'
autoplot(object, range.bars = NULL, ...)

Arguments

Details

autoplot will produce an equivalent plot as a ggplot object.

Value

None. Function produces a plot

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

See Also

ets()

Examples

fit <- ets(USAccDeaths)
plot(fit)
plot(fit, plot.type = "single", ylab = "", col = 1:3)

library(ggplot2)
autoplot(fit)

Forecast plot

Description

Plots historical data with forecasts and prediction intervals.

Usage

## S3 method for class 'forecast'
plot(
  x,
  include,
  PI = TRUE,
  showgap = TRUE,
  shaded = TRUE,
  shadebars = (length(x$mean) < 5),
  shadecols = NULL,
  col = 1,
  fcol = 4,
  pi.col = 1,
  pi.lty = 2,
  ylim = NULL,
  main = NULL,
  xlab = "",
  ylab = "",
  type = "l",
  flty = 1,
  flwd = 2,
  ...
)

## S3 method for class 'forecast'
autoplot(
  object,
  include,
  PI = TRUE,
  shadecols = c("#596DD5", "#D5DBFF"),
  fcol = "#0000AA",
  flwd = 0.5,
  ...
)

## S3 method for class 'splineforecast'
autoplot(object, PI = TRUE, ...)

## S3 method for class 'forecast'
autolayer(object, series = NULL, PI = TRUE, showgap = TRUE, ...)

## S3 method for class 'splineforecast'
plot(x, fitcol = 2, type = "o", pch = 19, ...)

Arguments

Details

autoplot will produce a ggplot object.

plot.splineforecast autoplot.splineforecast

Value

None.

Author(s)

Rob J Hyndman & Mitchell O'Hara-Wild

References

Hyndman and Athanasopoulos (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. https://otexts.com/fpp2/

See Also

stats::plot.ts()

Examples

library(ggplot2)

wine.fit <- hw(wineind, h = 48)
plot(wine.fit)
autoplot(wine.fit)

fit <- tslm(wineind ~ fourier(wineind, 4))
fcast <- forecast(fit, newdata = data.frame(fourier(wineind, 4, 20)))
autoplot(fcast)

fcast <- splinef(airmiles, h = 5)
plot(fcast)
autoplot(fcast)

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

generics

accuracy, forecast

ggplot2

autoplot

magrittr

%>%

Residuals for various time series models

Description

Returns time series of residuals from a fitted model.

Usage

## S3 method for class 'forecast'
residuals(object, type = c("innovation", "response"), ...)

## S3 method for class 'ar'
residuals(object, type = c("innovation", "response"), ...)

## S3 method for class 'Arima'
residuals(object, type = c("innovation", "response", "regression"), h = 1, ...)

## S3 method for class 'bats'
residuals(object, type = c("innovation", "response"), h = 1, ...)

## S3 method for class 'tbats'
residuals(object, type = c("innovation", "response"), h = 1, ...)

## S3 method for class 'ets'
residuals(object, type = c("innovation", "response"), h = 1, ...)

## S3 method for class 'ARFIMA'
residuals(object, type = c("innovation", "response"), ...)

## S3 method for class 'nnetar'
residuals(object, type = c("innovation", "response"), h = 1, ...)

## S3 method for class 'stlm'
residuals(object, type = c("innovation", "response"), ...)

## S3 method for class 'tslm'
residuals(object, type = c("innovation", "response", "deviance"), ...)

Arguments

Details

Innovation residuals correspond to the white noise process that drives the evolution of the time series model. Response residuals are the difference between the observations and the fitted values (equivalent to h-step forecasts). For functions with no h argument, h = 1. For homoscedastic models, the innovation residuals and the response residuals for h = 1 are identical. Regression residuals are available for regression models with ARIMA errors, and are equal to the original data minus the effect of the regression variables. If there are no regression variables, the errors will be identical to the original series (possibly adjusted to have zero mean). arima.errors is a deprecated function which is identical to residuals.Arima(object, type="regression"). For nnetar objects, when type = "innovations" and lambda is used, a matrix of time-series consisting of the residuals from each of the fitted neural networks is returned.

Value

A ts object.

Author(s)

Rob J Hyndman

See Also

fitted.Arima(), checkresiduals().

Examples

fit <- Arima(lynx, order = c(4, 0, 0), lambda = 0.5)

plot(residuals(fit))
plot(residuals(fit, type = "response"))

Random walk model

Description

Fit a generalized random walk with Gaussian errors (and optional drift) to a univariate time series.

Usage

rw_model(y, lag = 1, drift = FALSE, lambda = NULL, biasadj = FALSE)

Arguments

Details

The model assumes that

Y_t = Y_{t-p} + c + \varepsilon_{t}

where p is the lag parameter, c is the drift parameter, and \varepsilon_t\sim N(0,\sigma^2) are iid.

The model without drift has c=0. In the model with drift, c is estimated by the sample mean of the differences Y_t - Y_{t-p}.

If p=1, this is equivalent to an ARIMA(0,1,0) model with an optional drift coefficient. For p>1, it is equivalent to an ARIMA(0,0,0)(0,1,0)p model.

The forecasts are given by

Y_{T+h|T}= Y_{T+h-p(k+1)} + ch

where k is the integer part of (h-1)/p. For a regular random walk, p=1 and c=0, so all forecasts are equal to the last observation. Forecast standard errors allow for uncertainty in estimating the drift parameter (unlike the corresponding forecasts obtained by fitting an ARIMA model directly).

The generic accessor functions stats::fitted() and stats::residuals() extract useful features of the object returned.

Value

An object of class rw_model.

See Also

forecast.rw_model(), rwf(), naive(), snaive()

Examples

model <- rw_model(gold)
forecast(model, h = 50) |> autoplot()

Seasonal adjustment

Description

Returns seasonally adjusted data constructed by removing the seasonal component.

Usage

seasadj(object, ...)

## S3 method for class 'stl'
seasadj(object, ...)

## S3 method for class 'mstl'
seasadj(object, ...)

## S3 method for class 'decomposed.ts'
seasadj(object, ...)

## S3 method for class 'tbats'
seasadj(object, ...)

## S3 method for class 'seas'
seasadj(object, ...)

Arguments

Value

Univariate time series.

Author(s)

Rob J Hyndman

See Also

stats::stl(), stats::decompose(), tbats().

Examples

plot(AirPassengers)
lines(seasadj(decompose(AirPassengers, "multiplicative")), col = 4)

Extract components from a time series decomposition

Description

Returns a univariate time series equal to either a seasonal component, trend-cycle component or remainder component from a time series decomposition.

Usage

seasonal(object)

trendcycle(object)

remainder(object)

Arguments

Value

Univariate time series.

Author(s)

Rob J Hyndman

See Also

stats::stl(), stats::decompose(), tbats(), seasadj().

Examples

plot(USAccDeaths)
fit <- stl(USAccDeaths, s.window = "periodic")
lines(trendcycle(fit), col = "red")

library(ggplot2)
autoplot(
  cbind(
    Data = USAccDeaths,
    Seasonal = seasonal(fit),
    Trend = trendcycle(fit),
    Remainder = remainder(fit)
  ),
  facets = TRUE
) +
  labs(x = "Year", y = "")

Seasonal dummy variables

Description

seasonaldummy returns a matrix of dummy variables suitable for use in Arima(), auto.arima() or tslm(). The last season is omitted and used as the control.

Usage

seasonaldummy(x, h = NULL)

seasonaldummyf(x, h)

Arguments

Details

seasonaldummyf is deprecated, instead use the h argument in seasonaldummy.

The number of dummy variables is determined from the time series characteristics of x. When h is missing, the length of x also determines the number of rows for the matrix returned by seasonaldummy. the value of h determines the number of rows for the matrix returned by seasonaldummy, typically used for forecasting. The values within x are not used.

Value

Numerical matrix.

Author(s)

Rob J Hyndman

See Also

fourier()

Examples

plot(ldeaths)

# Using seasonal dummy variables
month <- seasonaldummy(ldeaths)
deaths.lm <- tslm(ldeaths ~ month)
tsdisplay(residuals(deaths.lm))
ldeaths.fcast <- forecast(
  deaths.lm,
  data.frame(month = I(seasonaldummy(ldeaths, 36)))
)
plot(ldeaths.fcast)

# A simpler approach to seasonal dummy variables
deaths.lm <- tslm(ldeaths ~ season)
ldeaths.fcast <- forecast(deaths.lm, h = 36)
plot(ldeaths.fcast)

Exponential smoothing forecasts

Description

Returns forecasts and other information for exponential smoothing forecasts applied to y.

Usage

ses(
  y,
  h = 10,
  level = c(80, 95),
  fan = FALSE,
  initial = c("optimal", "simple"),
  alpha = NULL,
  lambda = NULL,
  biasadj = FALSE,
  x = y,
  ...
)

holt(
  y,
  h = 10,
  damped = FALSE,
  level = c(80, 95),
  fan = FALSE,
  initial = c("optimal", "simple"),
  exponential = FALSE,
  alpha = NULL,
  beta = NULL,
  phi = NULL,
  lambda = NULL,
  biasadj = FALSE,
  x = y,
  ...
)

hw(
  y,
  h = 2 * frequency(x),
  seasonal = c("additive", "multiplicative"),
  damped = FALSE,
  level = c(80, 95),
  fan = FALSE,
  initial = c("optimal", "simple"),
  exponential = FALSE,
  alpha = NULL,
  beta = NULL,
  gamma = NULL,
  phi = NULL,
  lambda = NULL,
  biasadj = FALSE,
  x = y,
  ...
)

Arguments

Details

ses, holt and hw are simply convenient wrapper functions for forecast(ets(...)).

Value

An object of class forecast.

forecast class

An object of class forecast is a list usually containing at least the following elements:

model

A list containing information about the fitted model

method

The name of the forecasting method as a character string

mean

Point forecasts as a time series

lower

Lower limits for prediction intervals

upper

Upper limits for prediction intervals

level

The confidence values associated with the prediction intervals

x

The original time series.

residuals

Residuals from the fitted model. For models with additive errors, the residuals will be x minus the fitted values.

fitted

Fitted values (one-step forecasts)

The function summary can be used to obtain and print a summary of the results, while the functions plot and autoplot produce plots of the forecasts and prediction intervals. The generic accessors functions fitted.values and residuals extract various useful features from the underlying model.

Author(s)

Rob J Hyndman

References

Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag: New York. https://robjhyndman.com/expsmooth/.

Hyndman and Athanasopoulos (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. https://otexts.com/fpp2/

See Also

ets(), stats::HoltWinters(), rwf(), stats::arima().

Examples

fcast <- holt(airmiles)
plot(fcast)
deaths.fcast <- hw(USAccDeaths, h = 48)
plot(deaths.fcast)

Simulation from a time series model

Description

Returns a time series based on the model object object.

Usage

## S3 method for class 'ets'
simulate(
  object,
  nsim = length(object$x),
  seed = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  ...
)

## S3 method for class 'Arima'
simulate(
  object,
  nsim = length(object$x),
  seed = NULL,
  xreg = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  lambda = object$lambda,
  ...
)

## S3 method for class 'ar'
simulate(
  object,
  nsim = object$n.used,
  seed = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  ...
)

## S3 method for class 'rw_model'
simulate(
  object,
  nsim = length(object$x),
  seed = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  lambda = object$lambda,
  ...
)

## S3 method for class 'fracdiff'
simulate(
  object,
  nsim = object$n,
  seed = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  lambda = object$lambda,
  ...
)

## S3 method for class 'nnetar'
simulate(
  object,
  nsim = length(object$x),
  seed = NULL,
  xreg = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  lambda = object$lambda,
  ...
)

## S3 method for class 'modelAR'
simulate(
  object,
  nsim = length(object$x),
  seed = NULL,
  xreg = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  lambda = object$lambda,
  ...
)

## S3 method for class 'tbats'
simulate(
  object,
  nsim = length(object$y),
  seed = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  ...
)

## S3 method for class 'spline_model'
simulate(
  object,
  nsim = length(object$y),
  seed = NULL,
  future = TRUE,
  bootstrap = FALSE,
  innov = NULL,
  lambda = object$lambda,
  ...
)

Arguments

Details

With simulate.Arima(), the object should be produced by Arima() or auto.arima(), rather than stats::arima(). By default, the error series is assumed normally distributed and generated using stats::rnorm(). If innov is present, it is used instead. If bootstrap = TRUE and innov = NULL, the residuals are resampled instead.

When future = TRUE, the sample paths are conditional on the data. When future = FALSE and the model is stationary, the sample paths do not depend on the data at all. When future = FALSE and the model is non-stationary, the location of the sample paths is arbitrary, so they all start at the value of the first observation.

Value

An object of class ts.

Author(s)

Rob J Hyndman

See Also

ets(), Arima(), auto.arima(), ar(), arfima(), nnetar().

Examples

fit <- ets(USAccDeaths)
plot(USAccDeaths, xlim = c(1973, 1982))
lines(simulate(fit, 36), col = "red")

Forecast seasonal index

Description

Returns vector containing the seasonal index for h future periods. If the seasonal index is non-periodic, it uses the last values of the index.

Usage

sindexf(object, h)

Arguments

Value

Time series

Author(s)

Rob J Hyndman

Examples

uk.stl <- stl(UKDriverDeaths, "periodic")
uk.sa <- seasadj(uk.stl)
uk.fcast <- holt(uk.sa, 36)
seasf <- sindexf(uk.stl, 36)
uk.fcast$mean <- uk.fcast$mean + seasf
uk.fcast$lower <- uk.fcast$lower + cbind(seasf, seasf)
uk.fcast$upper <- uk.fcast$upper + cbind(seasf, seasf)
uk.fcast$x <- UKDriverDeaths
plot(uk.fcast, main = "Forecasts from Holt's method with seasonal adjustment")

Cubic spline stochastic model

Description

Fits a state space model based on cubic smoothing splines. The cubic smoothing spline model is equivalent to an ARIMA(0,2,2) model but with a restricted parameter space. The advantage of the spline model over the full ARIMA model is that it provides a smooth historical trend as well as a linear forecast function. Hyndman, King, Pitrun, and Billah (2002) show that the forecast performance of the method is hardly affected by the restricted parameter space.

Usage

spline_model(y, method = c("gcv", "mle"), lambda = NULL, biasadj = FALSE)

Arguments

Value

An object of class spline_model.

Author(s)

Rob J Hyndman

References

Hyndman, King, Pitrun and Billah (2005) Local linear forecasts using cubic smoothing splines. Australian and New Zealand Journal of Statistics, 47(1), 87-99. https://robjhyndman.com/publications/splinefcast/.

See Also

stats::smooth.spline(), stats::arima(), holt().

Examples

fit <- spline_model(uspop)
fit
fit |> forecast() |> autoplot()

Forecasting model using STL with a generative time series model

Description

Forecasts of STL objects are obtained by applying a non-seasonal forecasting model to the seasonally adjusted data and re-seasonalizing using the last year of the seasonal component. stlm takes a time series y, applies an STL decomposition, and models the seasonally adjusted data using the model passed as modelfunction or specified using method. It returns an object that includes the original STL decomposition and a time series model fitted to the seasonally adjusted data. This object can be passed to the forecast.stlm for forecasting.

Usage

stlm(
  y,
  s.window = 7 + 4 * seq(6),
  t.window = NULL,
  robust = FALSE,
  method = c("ets", "arima"),
  modelfunction = NULL,
  model = NULL,
  etsmodel = "ZZN",
  lambda = NULL,
  biasadj = FALSE,
  xreg = NULL,
  allow.multiplicative.trend = FALSE,
  x = y,
  ...
)

Arguments