Airport Travellers Time Series.

Description

A dataset containing the total number of monthly travellers recorded in an unnamed airport between January 2013 and December 2019.

Usage

airport

Format

A data frame with 84 rows and 2 variables:

Date

Months in which the traveller numbers are recorded.

Travellers

Total travellers in the airport of that month.

AvgTemp

Monthly average temperature (in °C) of the city where the airport is located.

AvgRain

Monthly average rain fall (in mm) in the city where the airport is located.

Examples

data(airport)
head(airport)

Outlier Identification

Description

The function 'is.outlier' checks whether any time series observations are outliers based on the interquartile range (IQR) rule.

Usage

is.outlier(x, method = c("iqr", "sigma", "zscore"), param = NULL)

Arguments

Details

With method = "iqr", the interquartile range rule for outlier identification is applied. An observation x_i will be identified as outlier if one of the following conditions fulfils:

x_i < q_1 - m \cdot (q_3 - q_1)

x_i > q_3 + m \cdot (q_3 - q_1)

where q_1 and q_3 are the 1st and 3rd quartiles of the time series x, respectively. m is the value specified by param. If omitted, it will be set as 1.5.

By using method = "sigma", the following criteria for outlier identification, known as the 3-sigma rule, are applied:

x_i < \mu(x) - m \cdot \sigma(x)

x_i > \mu(x) + m \cdot \sigma(x)

where \mu(x) and \sigma(x) are the mean and standard deviation of the time series x, respectively. m is the value specified by param. If omitted, it will be set as 3.

The z-score rule, specified by method = "zscore", compares the standardised observation values to a specific threshold:

\left|\dfrac{(x_i - \mu(x))}{\sigma(x)}\right| > m

where \mu(x) and \sigma(x) are the mean and standard deviation of the time series x, respectively. m is the value specified by param. If omitted, it will be set as 2. Note that 2 is the threshold for mild outliers. If checking for extreme outliers is required, the value should be set as 3.

Value

A vector indicating whether the values in x are outliers (TRUE) or not (FALSE).

Author(s)

Ka Yui Karl Wu

Examples

is.outlier(airport$Travellers, method = "zscore")

Predict Time Series Values

Description

The function 'predict' is generic and predicts past/future values of a time series.

Usage

## S3 method for class 'tsarima'
predict(
  object,
  n.ahead = 1L,
  newxreg = NULL,
  se.fit = TRUE,
  alpha = 0.05,
  log = NULL,
  ...
)

## S3 method for class 'tsesm'
predict(object, n.ahead = 1L, se.fit = TRUE, alpha = 0.05, ...)

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

Arguments

Value

An object of class "tspredict".

The function print is used to obtain and print the prediction results, including the predicted values, the corresponding standard errors, as well as the lower and upper limit of the prediction intervals.

An object of class "tspredict" is a list usually containing the following elements:

Author(s)

Ka Yui Karl Wu

References

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasting and control. Holden-Day.

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Examples

predict(tsarima(airport$Travellers, order = c(1, 1, 0), 
                seasonal = c(0, 1, 1), log = TRUE, include.const = TRUE),
        n.ahead = 6, alpha = 0.05)

predict(tsesm(airport$Travellers, order = "holt-winters"), 
        n.ahead = 6, alpha = 0.05)

Extract Information of a Time Series

Description

The function 'tsfreq' extract the frequency of a time series, while 'tstime' extract the time periods in which the series data were observed, and 'tstimegap' returns the time gap between the observations of a time series.

The function 'tsname' can be used to extract or specify the name of a time series.

Usage

## Extract frequency of a time series
tsfreq(x)

## Extract observation time periods of a time series
tstime(x)

## Extract time gaps between a time series' observations
tstimegap(t)

## Get or set the name of a time series
tsname(x, x.name = NULL)

## Copy Attributes from a Time Series to Another
tsattrcopy(x, x.orig)

Arguments

Details

To set a new name for a time series, the function must be assigned to an object. Otherwise, the new name will not be taken over.

Value

For 'tstime', a list will be returned to the user with two elements: time (observation time) and frequnecy (observation frequency).

For 'tsfreq', R extracts the attribute 'seasonal.cycle' from the time series object x.

For 'tstimegap', R calculates the time gap between the time periods stored in the vector t.

So, if x and t are consistent and refer to the data and time of the same time series, the results of 'tsfreq' and 'tstimegap' as well as the frequnecy element of 'tstime' must be identical.

If x.name is NULL, the attribute series.name of x will be returned. Otherwise, the series will be returned with a new value for the attribute series.name specified by x.name.

For tsattrcopy, the function does the same as attributes. However, attributes only works if both x and x.orig share the same length, whereas tsattrcopy does not require this property and returns x with all the attributes originated from the series x.orig.

Author(s)

Ka Yui Karl Wu

See Also

time, frequency, attributes

Examples

tsfreq(x = airport$Travellers)
tstime(x = airport$Travellers)
tstimegap(t = airport$Date)
airport$Travellers <- tsname(airport$Travellers, x.name = "Number of Travellers in Airport X")
tsname(airport$Travellers)
tsattrcopy(airport$Travellers[1:60], airport$Travellers)

Auto- Covariance and -Correlation Function Estimation

Description

The function 'tsacf' computes (and by default plots) estimates of the autocorrelation function. Function pacf is the function used for the partial autocorrelations, and function tsacov computes the autocovariance function.

Usage

## Autocorrelation Function (ACF)
tsacf(
  x, y = NULL, lag.max = 8, 
  type = c("correlation", "covariance", "partial", 
           "cross-correlation", "cross-covariance"), 
  show.plot = TRUE, na.action = na.omit, 
  demean = TRUE, alpha = 0.05, 
  x.name = NULL, title = NULL)

## Partial Autocorrelation Function (PACF)
tspacf(...)

## Autocovariance (ACOV)
tsacov(...)

## cross-correlation (CCF)
tsccf(...)

## cross-ovariance (CCOV)
tsccov(...)

## S3 method for class 'tsacf'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

## S3 method for class 'tsacf'
plot(x, title = NULL, ...)

Arguments

Details

For type = "correlation" and "covariance", the estimates are based on the sample covariance of x_t and x_{t-k} (lag 0 autocorrelation is fixed at 1 by convention.). For "cross-correlation" and "cross-covariance", the estimates are based on the sample covariance of x_t and y_{t-k}.

By default, no missing values are allowed. However, by default, na.action = na.omit, the covariances are computed only from complete cases. This means that the estimate computed may well not be a valid autocorrelation sequence, and may contain missing values. Missing values are not allowed when computing the PACF of a multivariate time series.

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

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

Different from acf, the lags here are not converted based on the seasonal cycle length. It simply reflects the time lag k between X_t and X_{t-k}. Furthermore, the ACF/PACF/ACOV/CCF/CCOV plots are created using ggplot2.

The generic functions plot and print have both methods for objects of class "tsacf".

For 'cross-correlation' and 'cross-covariance', positive lags indicate that y is shifted forward in time relative to x, while negative lags indicate y is shifted backward.

Value

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

Author(s)

Ka Yui Karl Wu

References

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasting and control. Holden-Day.

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

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

Examples

# Autocorrelation (ACF)
airport.acf <- tsacf(airport$Travellers, lag.max = 24, show.plot = FALSE)
print(airport.acf, digits = 4)
plot(airport.acf)

# Partial Autocorrelation (PACF)
tspacf(airport$Travellers, lag.max = 24)

# Autocovariance (ACOV)
tsacov(airport$Travellers, lag.max = 24, show.plot = FALSE)

# Cross-Correlation (CCF)
tsccf(airport$AvgRain, airport$Travellers, lag.max = 24)

# Cross-Covariance (CCOV)
tsccov(airport$AvgRain, airport$Travellers, lag.max = 24)

Fitting ARIMA Models

Description

The 'tsarima' function is used to fit an ARIMA model to a univariate time series.

Usage

tsarima(
  x,
  order = c(0L, 0L, 0L),
  seasonal = list(order = c(0L, 0L, 0L), period = NA),
  xreg = NULL,
  include.const = TRUE,
  log = FALSE,
  train.prop = 1,
  arch.test = FALSE,
  transform.pars = TRUE,
  fixed = NULL,
  init = NULL,
  method = c("CSS-ML", "ML", "CSS"),
  SSinit = c("Gardner1980", "Rossignol2011"),
  optim.method = "BFGS",
  optim.control = list(),
  kappa = 1e+06
)

## S3 method for class 'tsarima'
print(
  x,
  digits = max(3L, getOption("digits") - 3L),
  se = TRUE,
  signif.stars = TRUE,
  ...
)

## S3 method for class 'tsarima'
summary(
  object,
  digits = max(3L, getOption("digits") - 3L),
  se = TRUE,
  signif.stars = TRUE,
  ...
)

Arguments

Details

Different definitions of ARMA models have different signs for the AR and/or MA coefficients. The definition used here is the original Box & Jenkins (1970) formulation:

x_t=\phi_1 x_{t-1}+\ldots+\phi_p x_{t-p}+\varepsilon_t-\theta_1\varepsilon_{t-1}-\ldots-\theta_p\varepsilon_{t-q}

and so the MA coefficients differ in sign from the output of stats::arima. Further, if include.const is TRUE (the default for an ARMA model), this formula applies to x_t-\mu rather than x_t. For ARIMA models with differencing, the differenced series usually follows a zero-mean ARMA model, but include.const is still available in case a constant term is required for the model. However, the estimation of the coefficients' standard error may not be successful. If an xreg term is included, a linear regression (with a constant term if include.mean is TRUE and there is no differencing) is fitted with an ARMA model for the error term.

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

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

If train.prop is smaller than 1, the function will only treat the training part of the series as past data. When applying 'tsforecast' or 'predict', the forecast will start after the end of the training part of the original series.

Value

A list of class 'tsarima' with components:

Fitting methods

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

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

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

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

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

Author(s)

Ka Yui Karl Wu

References

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasting and control. Holden-Day.

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Hyndman, R. J., Athanasopoulos, G., Bergmeir, C., Caceres, G., Chhay, L., O'Hara-Wild, M., Petropoulos, F., Razbash, S., Wang, E., & Yasmeen, F. (2025). forecast: Forecasting functions for time series and linear models. R package version 8.24.0,
https://pkg.robjhyndman.com/forecast/.

Hyndman, R. J., & Khandakar, Y. (2008). Automatic time series forecasting: the forecast package for R. Journal of Statistical Software, 27(3), 1-22. doi:10.18637/jss.v027.i03.

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

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

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

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

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

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

See Also

arima

Examples

tsarima(airport$Travellers, 
        order = c(1, 1, 0), seasonal = c(0, 1, 1),
        log = TRUE, include.const = TRUE)

Box Plots

Description

Produce box-and-whisker plot of a given univariate time series.

Usage

tsboxplot(
  x,
  title = NULL,
  x.name = NULL,
  x.col = "darkgrey",
  mean.col = "steelblue4"
)

Arguments

Details

Since x is a univariate time series, no parallel box plots can be plotted here.

Missing values are ignored when forming boxplots.

Value

A boxplot of x will be displayed with no further values or objects returned.

Author(s)

Ka Yui Karl Wu

References

Chambers, J. M., Cleveland, W. S., Kleiner, B., & Tukey, P. A. (1983). Graphical Methods for Data Analysis. Wadsworth & Brooks/Cole.

Examples

tsboxplot(airport$Travellers)

Convert One-Dimensional Data to Time Series

Description

The function 'tsconvert' is used to convert any one-dimensional vector/list into a time-series objects.

Usage

tsconvert(x, t, frequency = NULL, format = NULL, x.name = NULL)

Arguments

Details

The function tsconvert is used to convert vectors/lists into time-series objects. These are vectors or matrices which inherit from class "ts" (and have additional attributes) and represent data sampled at equispaced points in time. Time series must have at least one observation, and although they need not be numeric there is very limited support for non-numeric series.

Argument frequency indicates the sampling frequency of the time series, with the default value 1 indicating one sample in each unit time interval. For example, one could use a value of 7 for frequency when the data are sampled daily, and the natural time period is a week, or 12 when the data are sampled monthly and the natural time period is a year. Values of 4 and 12 are assumed in (e.g.) print methods to imply a quarterly and monthly series respectively. Note that frequency does not need to be a whole number: for example, frequency = 0.2 would imply sampling once every five time units.

Value

x is returned as a 'ts' object with the attributes 'tsp' (start, end, and frequency of x), 'series.name' (series name, optional), and 'seasonal.cycle' (time gap between series observations).

Author(s)

Ka Yui Karl Wu

See Also

attributes

Examples

travellers <- tsconvert(x = airport$Travellers, t = airport$Date)

Decompose a Time Series

Description

Decompose a time series into trend, cyclical, seasonal and irregular components. Deals with additive or multiplicative components.

Usage

tsdecomp(
  x,
  type = c("additive", "multiplicative"),
  trend.method = c("lm", "loess"),
  tcc.order = 3,
  x.name = NULL,
  show.plot = TRUE
)

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

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

Arguments

Details

The additive model used is:

Y_t = T_t + C_t + S_t + I_t

The multiplicative model used is:

Y_t = T_t \cdot C_t \cdot S_t \cdot I_t

The function first determines the trend-cycle component using a moving average, and removes it from the time series. Then, the seasonal figure is computed by averaging, for each time unit, over all periods. The seasonal figure is then centred. Finally, the error component is determined by removing trend and seasonal figure (recycled as needed) from the original time series.

This only works well if 'x' covers an integer number of complete periods.

The function 'plot' generates the following plots:

Value

An object of class "tsdecomp" with following components:

Author(s)

Ka Yui Karl Wu

References

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Kendall, M., & Stuart, A. (1983) The Advanced Theory of Statistics, Vol.3, Griffin. pp. 410-414.

Examples

tsdecomp(airport$Travellers, type = "multiplicative")

Difference a Time Series

Description

The function 'tsdiff' generates the differenced series of a non-stationary time series.

Usage

tsdiff(x, lag = 1L, order = 1L, lag.D = 0L, order.D = 0L)

Arguments

Details

The parameters lag and lag.D are only necessary if the lag difference for differencing is not 1 or \ell, the seasonal cycle length, respectively. If order.D > 0 but lag.D is omitted, R will use the frequency of the series for this parameter.

Value

The differences of x will be returned with all the attributes being carried over. Unlike the function diff, the output series by tsdiff has the same length as the original series by adding NA to those observations at the beginning of the series for which no differencing can be carried out.

Author(s)

Ka Yui Karl Wu

See Also

diff

Examples

travellers_d <- tsdiff(x = airport$Travellers, order = 1, order.D = 1)
tsexplore(travellers_d, lag.max = 60)

Exponential Smoothing Forecasts

Description

The 'tsesm' function forecasts future values of a univariate time series using exponential smoothing.

Usage

tsesm(
  x,
  order = c("simple", "holt", "holt-winters"),
  damped = FALSE,
  initial = c("optimal", "simple"),
  type = c("additive", "multiplicative"),
  alpha = NULL,
  beta = NULL,
  gamma = NULL,
  lambda = NULL,
  phi = NULL,
  biasadj = FALSE,
  exp.trend = FALSE,
  seasonal.period = NULL,
  train.prop = 1
)

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

## S3 method for class 'tsesm'
summary(object, ...)

Arguments

Details

The 'tsesm' function uses the same mechanism for exponential smoothing like 'forecast::ses', 'forecast::holt', and 'forecast::hw'. Instead of using three different functions, all of them are integrated in tsesm, and user can choose the method by specifying the value of the order parameter.

The option 'simple‘ is the lowest exponential smoothing order, or the first exponential smoothing order (that’s why the parameter is called 'order' and not 'method'). It can be used to forecast series with only level (\ell) component. The corresponding forecasting formula is given by:

\tilde{x}_{t+h|t} = \ell_t = \alpha x_{t-1} + (1-\alpha)\ell_{t-1},

where \alpha is the smoothing parameter for the level component, and h is the number of periods ahead for forecasting.

With the 'holt' option, the second exponential smoothing order can be chosen. It is suitable for the forecasting of series with level (\ell) and trend (b) components.

\ell_t = \alpha x_{t} + (1-\alpha)(\ell_{t-1} + b_{t-1})

b_t = \beta(\ell_{t}-\ell_{t-1}) + (1-\beta)b_{t-1}

\tilde{x}_{t+h|t} = \ell_{t}+hb_{t}

where \alpha and \beta are the smoothing parameters for the level and trend components, respectively, h is the number of periods ahead for forecasting.

The third exponential smoothing order can be specified by the 'holt-winters' option. It can be used to forecast time series with level (\ell), trend (b), and seasonal (s) components.

\ell_t = \alpha (x_{t} - s_{t-m}) + (1-\alpha)(\ell_{t-1} + b_{t-1})

b_t = \beta(\ell_{t}-\ell_{t-1}) + (1-\beta)b_{t-1}

s_t = \gamma(x_{t}-\ell_{t-1}-b_{t-1}) + (1-\gamma)s_{t-m}

\tilde{x}_{t+h|t} = \ell_{t}+hb_{t}+s_{t+h-m(k+1)}

where \alpha, \beta, and \gamma are the smoothing parameters for the level, trend, and seasonal components, respectively, h is the number of periods ahead for forecasting, m is the number of periods within a seasonal cycle, and k is the integer part of (h-1)/m, which ensures that the estimates of the seasonal indices used for forecasting come from the final period of the series.

If train.prop is smaller than 1, the function will only treat the training part of the series as past data. When applying 'tsforecast' or 'predict', the forecast will start after the end of the training part of the original series.

Value

A list of class "tsesm" with components:

Author(s)

Ka Yui Karl Wu

References

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasting and control. Holden-Day.

Hyndman, R. J., Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Hyndman, R. J., Athanasopoulos, G., Bergmeir, C., Caceres, G., Chhay, L., O'Hara-Wild, M., Petropoulos, F., Razbash, S., Wang, E., Yasmeen, F. (2025).
forecast: Forecasting functions for time series and linear models. R package version 8.24.0,
https://pkg.robjhyndman.com/forecast/.

Hyndman, R. J., Khandakar, Y. (2008). Automatic time series forecasting: the forecast package for R. Journal of Statistical Software, 27(3), 1-22.

Examples

tsesm(airport$Travellers, order = "simple")
tsesm(airport$Travellers, order = "holt")
tsesm(airport$Travellers, order = "holt-winters")

Explore a Time Series Numerically and Graphically

Description

The function 'tsexplore' generates statistics, various tests, and graphs regarding the location, deviation, distribution of a time series.

Usage

tsexplore(
  x,
  show.plot = TRUE,
  x.name = NULL,
  mu = 0,
  adf.lag = 0,
  lag.max = 8,
  ...
)

## S3 method for class 'tsexplore'
plot(
  x,
  trend = c("linear", "smooth", "none"),
  histbin = 15,
  lwidth = 0.7,
  pwidth = 0.7,
  x.col = "darkgrey",
  extra.col = "red",
  ...
)

## S3 method for class 'tsexplore'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

Value

An object of class "tsexplore" with following components:

Details of the 'stats' component

The following statistics and test results are stored in the component 'stats' of the 'tsexplore' object:

Author(s)

Ka Yui Karl Wu

References

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Examples

tsexplore(airport$Travellers, lag.max = 24, histbin = 10)

Forecast Time Series based on Fitted Models

Description

The generic function 'tsforecast' is used for forecasting time series or time series models.

Usage

## S3 method for default
tsforecast(object, n.ahead = 1, 
  alpha = 0.05, 
  show.plot = TRUE, 
  forecast.incl = c("all", "forecast", "predict"), 
  nobs.incl = NULL, 
  log = NULL, 
  newxreg = NULL, 
  x.name = NULL, pred.name = "Forecasts", ...)

## S3 method for class 'tsarima'
tsforecast(object, ...)

## S3 method for class 'tsesm'
tsforecast(object, ...)

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

## S3 method for class 'tsforecast'
plot(
  x,
  ylim = NULL,
  title = NULL,
  x.lwidth = 0.7,
  pred.lwidth = 0.7,
  x.col = "darkgrey",
  pred.col = "red",
  ci.col = "royalblue",
  ...
)

Arguments

Value

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

An object of class "tsforecast" is a list usually containing the following elements:

Author(s)

Ka Yui Karl Wu

References

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasting and control. Holden-Day.

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Examples

tsforecast(tsarima(airport$Travellers, order = c(1, 1, 0), 
                   seasonal = c(0, 1, 1), log = TRUE, include.const = TRUE), 
           n.ahead = 6, forecast.incl = "all", nobs.incl = 12, 
           title = "Forecast of Travellers by ARIMA")

tsforecast(tsesm(airport$Travellers, order = "holt-winters"), 
           n.ahead = 6, title = "Forecast of Travellers by Holt-Winters")

Histograms

Description

Produce a histogram of a given univariate time series.

Usage

tshistogram(
  x,
  bins = NULL,
  density = FALSE,
  density.lwidth = 0.7,
  title = NULL,
  x.name = NULL,
  x.col = "darkgrey",
  density.col = "steelblue4"
)

Arguments

Value

A histogram of x will be displayed with no further values or objects returned.

Author(s)

Ka Yui Karl Wu

References

Venables, W. N., & Ripley. B. D. (2002) Modern Applied Statistics with S. Springer.

Examples

tshistogram(airport$Travellers, density = TRUE)

Lag a Time Series

Description

The function 'tslag' back- or foreshifts a time series and generates the lagged version of it.

Usage

tslag(x, lag = 1L)

Arguments

Details

Note the sign of lag: a series lagged by a positive lag starts earlier.

Value

The same time series object as x after being back- or foreshifted for the number of periods specified in lag.

Author(s)

Ka Yui Karl Wu

See Also

lag

Examples

tslag(airport$Travellers, lag = 12)

Time Series Line Plots

Description

Produce line plot of a given univariate time series.

Usage

tslineplot(
  x,
  t = NULL,
  pred = NULL,
  pred.t = NULL,
  cil = NULL,
  ciu = NULL,
  ci.t = NULL,
  trend = c("none", "linear", "smooth"),
  title = NULL,
  x.name = NULL,
  pred.name = "Predicted",
  x.lwidth = 0.7,
  pred.lwidth = 0.7,
  x.col = "darkgrey",
  pred.col = "steelblue4",
  ci.col = "royalblue",
  t.name = "Date/Time",
  t.text.angle = 90,
  t.numbreak = 10,
  ylim = NULL
)

Arguments

Details

If x is a ts object, parameter t can be omitted. You can convert a vector, a matrix or data frame column using tsconvert to a ts object.

Value

A line plot of x will be displayed with no further values or objects returned.

Author(s)

Ka Yui Karl Wu

Examples

tslineplot(airport$Travellers, trend = "linear")

McLeod-Li Test for ARCH Effect

Description

The function 'tsmltest' applies the McLeod-Li test to examine whether ARCH effect exists in the squared residuals of an ARIMA model.

Usage

tsmltest(object, lag.max = NULL)

Arguments

Value

A list with two elements:

Author(s)

Ka Yui Karl Wu

References

McLeod, A. I., & Li, W. K. (1983). Diagnostic Checking ARMA Time Series Models Using Squared-Residual Autocorrelations. Journal of Time Series Analysis, 4(4), 269-273.
doi:10.1111/j.1467-9892.1983.tb00373.x.

Examples

tsmltest(tsarima(airport$Travellers, order = c(1, 1, 0), 
                 seasonal = c(0, 1, 1), log = TRUE, include.const = TRUE))

Goodness of Fit of a Time Series Model

Description

The function 'tsmodeleval' can be used to evaluate the goodness of fit of a time series model.

Usage

tsmodeleval(object)

Arguments

Value

A list with the following model evaluation criteria:

Author(s)

Ka Yui Karl Wu

References

Hyndman, R. J., Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Examples

tsmodeleval(tsarima(airport$Travellers, 
                    order = c(1, 1, 0), seasonal = c(0, 1, 1),
                    log = TRUE, include.const = TRUE))

Generate Moving Averages of a Time Series

Description

The function 'tsmovav' calculates the moving averages of a time series.

Usage

tsmovav(
  x,
  order = 3,
  type = c("backward", "center"),
  n.ahead = 0,
  x.name = NULL,
  show.plot = TRUE
)

## S3 method for class 'tsmovav'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

## S3 method for class 'tsmovav'
plot(x, title = NULL, ...)

Arguments

Details

Centred moving averages are better suited for smoothing a time series than for forecasting. By definition, each moving average is aligned with the midpoint of its averaging window. When the number of periods in the averaging window (i.e., the moving average order) is even, the averages calculated for the two central positions must be combined. Specifically, the mean of these two middle moving averages is assigned to the central period that lies closest to the true midpoint of the series, forming its final centred moving average.

Mathematically, for odd number order r:

\tilde{y}_t = \dfrac{y_{t-\frac{r-1}{2}}+\ldots+y_{t+\frac{r+1}{2}}}{r}

For even number order r:

\tilde{y}_t = \dfrac{0.5y_{t-\frac{r}{2}}+y_{t-\frac{r}{2}+1}+\ldots+y_{t+\frac{r}{2}-1}+0.5y_{t+\frac{r}{2}}}{2r}

Backward moving average is a forecasting method that assigns each computed average to the period immediately following the observation window. This approach works the same regardless of whether the moving average order is odd or even.

\tilde{y}_t = \dfrac{y_{t-1}+\ldots+x_{y-r}}{r}

Value

Author(s)

Ka Yui Karl Wu

References

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.
https://otexts.com/fpp3/

Examples

## Backward Moving Average
tsmovav(airport$Travellers, order = 12, type = "backward", n.ahead = 6, show.plot = TRUE)

## Centered Moving Average
tsmovav(airport$Travellers, order = 12, type = "center")

Quantile-Quantile Plots

Description

'tsqqplot' is a function to produce a normal QQ plot of the values in y.

Usage

tsqqplot(
  x,
  title = NULL,
  x.name = NULL,
  qq.pwidth = 0.7,
  qq.lwidth = 0.7,
  qq.col = "black",
  qqline.col = "red"
)

Arguments

Value

A QQ plot of x will be displayed with no further values or objects returned.

Author(s)

Ka Yui Karl Wu

References

Switzer, P. (1976). Confidence procedures for two-sample problems. Biometrika, 63(1), 13-25. doi:10.1093/biomet/63.1.13.

Examples

tsqqplot(airport$Travellers)

Scatter Plot

Description

Produce a scatter plot of two given univariate time series.

Usage

tsscatterplot(
  x,
  y,
  reg = FALSE,
  title = NULL,
  x.name = NULL,
  y.name = NULL,
  pwidth = 1,
  pcol = "steelblue4",
  regwidth = 0.7,
  regcol = "red"
)

Arguments

Value

A scatter plot of x (values on the x-axis) and y (values on the y-axis) will be displayed with no further values or objects returned.

Author(s)

Ka Yui Karl Wu

Examples

tsscatterplot(airport$AvgRain, airport$Travellers)