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Legion package

Description

Package contains functions for multivariate time series forecasting

Details

The following functions areincluded in the package:

• ves - Vector Exponential Smoothing.

• vets - Vector ETS-PIC model.

• oves - Multivariate occurrence ETS model.

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

Kandrika Pritularga

References

• de Silva A., Hyndman R.J. and Snyder, R.D. (2010). The vectorinnovations structural time series framework: a simple approach tomultivariate forecasting. Statistical Modelling, 10 (4), pp.353-374

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

• Lütkepohl, H. (2005). New Introduction to Multiple Time SeriesAnalysis. New introduction to Multiple Time Series Analysis. Berlin,Heidelberg: Springer Berlin Heidelberg.doi:10.1007/978-3-540-27752-1

• Svetunkov, I., Chen, H., & Boylan, J. E. (2023).A new taxonomy for vector exponential smoothing and its applicationto seasonal time series. European Journal of Operational Research,304(3), 964–980.doi:10.1016/j.ejor.2022.04.040

See Also

forecast,es,adam

Examples

## Not run: y <- cbind(rnorm(100,10,3),rnorm(100,10,3))ves(y,h=20,holdout=TRUE)## End(Not run)

Vector ETS-PIC model

Description

Function constructs vector ETS model based on VETS-PIC taxonomy and returnsforecast, fitted values, errors and matrix of states along with other useful variables.

Usage

auto.vets(data, model = "PPP", lags = c(frequency(data)),  loss = c("likelihood", "diagonal", "trace"), ic = c("AICc", "AIC", "BIC",  "BICc"), h = 10, holdout = FALSE, occurrence = c("none", "fixed",  "logistic"), bounds = c("admissible", "usual", "none"), silent = TRUE,  parallel = FALSE, ...)vets(data, model = "PPP", lags = c(frequency(data)),  parameters = c("level", "trend", "seasonal", "damped"),  initials = c("seasonal"), components = c("none"),  loss = c("likelihood", "diagonal", "trace"), ic = c("AICc", "AIC", "BIC",  "BICc"), h = 10, holdout = FALSE, occurrence = c("none", "fixed",  "logistic"), bounds = c("admissible", "usual", "none"), silent = TRUE,  ...)

Arguments

Details

Function estimates vector ETS in the form of the Single Source of Error state spacemodel of the following type:

\mathbf{y}_{t} = \mathbf{o}_{t} (\mathbf{W} \mathbf{v}_{t-l} + \mathbf{x}_t\mathbf{a}_{t-1} + \mathbf{\epsilon}_{t})

\mathbf{v}_{t} = \mathbf{F} \mathbf{v}_{t-l} + \mathbf{G}\mathbf{\epsilon}_{t}

\mathbf{a}_{t} = \mathbf{F_{X}} \mathbf{a}_{t-1} + \mathbf{G_{X}}\mathbf{\epsilon}_{t} / \mathbf{x}_{t}

Wherey_{t} is the vector of time series on observationt,o_{t}is the vector of Bernoulli distributed random variable (in case of normal data itbecomes unit vector for all observations),\mathbf{v}_{t} is the matrix ofstates andl is the matrix of lags,\mathbf{x}_t is the vector ofexogenous variables.\mathbf{W} is the measurement matrix,\mathbf{F}is the transition matrix and\mathbf{G} is the persistence matrix.Finally,\epsilon_{t} is the vector of error terms.

Conventionally we formulate values as:

\mathbf{y}'_t = (y_{1,t}, y_{2,t}, \dots, y_{m,t})

wherem is the number of series in the group.

\mathbf{v}'_t = (v_{1,t}, v_{2,t}, \dots, v_{m,t})

wherev_{i,t} is vector of components for i-th time series.

\mathbf{W}' = (w_{1}, \dots , 0;\vdots , \ddots , \vdots;0 , \vdots , w_{m})

is matrix of measurement vectors.

The main idea of the function is in imposing restrictions on parameters / initials /components of the model in order to capture the common dynamics between series.

In case of multiplicative model, instead of the vector y_t we use its logarithms.As a result the multiplicative model is much easier to work with.

For some more information about the model and its implementation, see thevignette:vignette("vets","legion")

Value

Object of class "legion" is returned. It contains the following list ofvalues:

• model - The name of the fitted model;

• timeElapsed - The time elapsed for the construction of the model;

• states - The matrix of states with components in columns and time in rows;

• persistence - The persistence matrix;

• transition - The transition matrix;

• measurement - The measurement matrix;

• phi - The damping parameter value;

• B - The vector of all the estimated coefficients;

• lagsAll - The vector of the internal lags used in the model;

• nParam - The number of estimated parameters;

• occurrence - The occurrence model estimated with VETS;

• data - The matrix with the original data;

• fitted - The matrix of the fitted values;

• holdout - The matrix with the holdout values (ifholdout=TRUE inthe estimation);

• residuals - The matrix of the residuals of the model;

• Sigma - The covariance matrix of the errors (estimated with the correctionfor the number of degrees of freedom);

• forecast - The matrix of point forecasts;

• ICs - The values of the information criteria;

• logLik - The log-likelihood function;

• lossValue - The value of the loss function;

• loss - The type of the used loss function;

• lossFunction - The loss function if the custom was used in the process;

• accuracy - the values of the error measures. Currently not available.

• FI - Fisher information if user asked for it usingFI=TRUE.

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

References

• de Silva A., Hyndman R.J. and Snyder, R.D. (2010). The vectorinnovations structural time series framework: a simple approach tomultivariate forecasting. Statistical Modelling, 10 (4), pp.353-374

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

• Lütkepohl, H. (2005). New Introduction to Multiple Time SeriesAnalysis. New introduction to Multiple Time Series Analysis. Berlin,Heidelberg: Springer Berlin Heidelberg.doi:10.1007/978-3-540-27752-1

• Svetunkov, I., Chen, H., & Boylan, J. E. (2023).A new taxonomy for vector exponential smoothing and its applicationto seasonal time series. European Journal of Operational Research,304(3), 964–980.doi:10.1016/j.ejor.2022.04.040

See Also

ves,es,adam

Examples

Y <- ts(cbind(rnorm(100,100,10),rnorm(100,75,8)),frequency=12)# The simplest model applied to the data with the default valuesvets(Y,model="ANN",h=10,holdout=TRUE)# Multiplicative damped trend model with common parameters# and initial seasonal indicesvets(Y,model="MMdM",h=10,holdout=TRUE,parameters=c("l","t","s","d"),     initials="seasonal")# Automatic selection of ETS componentsvets(Y, model="PPP", h=10, holdout=TRUE, initials="seasonal")

legion classes checkers

Description

Functions to check if an object is of the specified class

Usage

is.legion(x)is.oves(x)is.legion.sim(x)

Arguments

Details

The list of methods includes:

• is.legion() tests if the object was produced by a vector model (e.g.ves);

• is.oves() tests if the object was produced byovesfunction;

• is.legion.sim() tests if the object was produced by the functionssim.ves;

Value

TRUE if this is the specified class andFALSE otherwise.

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

Examples

ourModel <- ves(cbind(rnorm(100,100,10),rnorm(100,100,10)))is.legion(ourModel)

Occurrence part of Vector State Space

Description

Function calculates the probability for the occurrence part of vector state space model.This is needed in order to forecast intermittent demand using other functions.

Usage

oves(data, occurrence = c("logistic", "none", "fixed"), ic = c("AICc",  "AIC", "BIC", "BICc"), h = 10, holdout = FALSE,  probability = c("dependent", "independent"), model = "ANN",  persistence = NULL, transition = NULL, phi = NULL, initial = NULL,  initialSeason = NULL, xreg = NULL, ...)

Arguments

Details

The function estimates probability of demand occurrence, using one of the VESstate-space models.

Value

The object of class "oves" is returned. It contains following list ofvalues:

• model - the type of the estimated ETS model;

• fitted - fitted values of the constructed model;

• forecast - forecast forh observations ahead;

• states - values of states (currently level only);

• variance - conditional variance of the forecast;

• logLik - likelihood value for the model

• nParam - number of parameters used in the model;

• residuals - residuals of the model;

• data - actual values of probabilities (zeros and ones).

• persistence - the vector of smoothing parameters;

• initial - initial values of the state vector;

• initialSeason - the matrix of initials seasonal states;

• occurrence - type of occurrence model used;

• probability - type of probability used;

• issModel - intermittent state-space model used forcalculations. Useful only in the case ofoccurrence="l" andprobability="d".

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

See Also

oes,ves

Examples

Y <- cbind(c(rpois(25,0.1),rpois(25,0.5),rpois(25,1),rpois(25,5)),           c(rpois(25,0.1),rpois(25,0.5),rpois(25,1),rpois(25,5)))oves(Y, occurrence="l")oves(Y, occurrence="l", probability="i")

Plots for the fit and states

Description

The function produces diagnostics plots for alegion model

Usage

## S3 method for class 'legion'plot(x, which = c(1, 2, 4, 6), level = 0.95,  legend = FALSE, ask = prod(par("mfcol")) < length(which) * nvariate(x) &&  dev.interactive(), lowess = TRUE, ...)

Arguments

Details

The list of produced plots includes:

1. Actuals vs Fitted values. Allows analysing, whether there are any issues in the fit.Does the variability of actuals increase with the increase of fitted values? Is the relationwell captured? They grey line on the plot corresponds to the perfect fit of the model.

2. Standardised residuals vs Fitted. Plots the points and the confidence bounds(red lines) for the specified confidencelevel. Useful for the analysis of outliers;

3. Studentised residuals vs Fitted. This is similar to the previous plot, but with theresiduals divided by the scales with the leave-one-out approach. Should be more sensitiveto outliers;

4. Absolute residuals vs Fitted. Useful for the analysis of heteroscedasticity;

5. Squared residuals vs Fitted - similar to (3), but with squared values;

6. Q-Q plot with the specified distribution. Can be used in order to see if theresiduals follow the assumed distribution. The type of distribution depends on the one usedin the estimation (seedistribution parameter inalm);

7. ACF of the residuals. Are the residuals autocorrelated? Seeacf fordetails;

8. Fitted over time. Plots actuals (black line), fitted values (purple line), point forecast(blue line) and prediction interval (grey lines). Can be used in order to make sure that the modeldid not miss any important events over time;

9. Standardised residuals vs Time. Useful if you want to see, if there is autocorrelation orif there is heteroscedasticity in time. This also shows, when the outliers happen;

10. Studentised residuals vs Time. Similar to previous, but with studentised residuals;

11. PACF of the residuals. No, really, are they autocorrelated? See pacf function from statspackage for details;

12. Plot of the states of the model. It is not recommended to produce this plot together withthe others, because there might be several states, which would cause the creation of a differentcanvas. In case of "msdecompose", this will produce the decomposition of the series into stateson a different canvas.

Which of the plots to produce, is specified via thewhich parameter.Currently onlywhich=c(1,4:7) are supported.

Value

The function produces the number of plots, specified in the parameterwhich.

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

See Also

plot.greybox

Examples

ourModel <- es(c(rnorm(50,100,10),rnorm(50,120,10)), "ANN", h=10)plot(ourModel, c(1:11))plot(ourModel, 12)

Simulate Vector Exponential Smoothing

Description

Function generates data using VES model as a data generating process.

Usage

sim.ves(model = "ANN", obs = 10, nsim = 1, nvariate = 2,  frequency = 1, persistence = NULL, phi = 1, transition = NULL,  initial = NULL, initialSeason = NULL,  seasonal = c("individual, common"), weights = rep(1/nvariate, nvariate),  bounds = c("usual", "admissible", "restricted"), randomizer = c("rnorm",  "rt", "rlaplace", "rs"), ...)

Arguments

Value

List of the following values is returned:

• model - Name of ETS model.

• data - The matrix (or an array ifnsim>1) of thegenerated series.

• states - The matrix (or array ifnsim>1) of states.States are in columns, time is in rows.

• persistence - The matrix (or array ifnsim>1) ofsmoothing parameters used in the simulation.

• transition - The transition matrix (or array ifnsim>1).

• initial - Vector (or matrix) of initial values.

• initialSeason - Vector (or matrix) of initial seasonalcoefficients.

• residuals - Error terms used in the simulation. Either matrixor array, depending onnsim.

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

References

• de Silva A., Hyndman R.J. and Snyder, R.D. (2010). The vectorinnovations structural time series framework: a simple approach tomultivariate forecasting. Statistical Modelling, 10 (4), pp.353-374

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

• Lütkepohl, H. (2005). New Introduction to Multiple Time SeriesAnalysis. New introduction to Multiple Time Series Analysis. Berlin,Heidelberg: Springer Berlin Heidelberg.doi:10.1007/978-3-540-27752-1

• Svetunkov, I., Chen, H., & Boylan, J. E. (2023).A new taxonomy for vector exponential smoothing and its applicationto seasonal time series. European Journal of Operational Research,304(3), 964–980.doi:10.1016/j.ejor.2022.04.040

See Also

ves,Distributions

Examples

# Create 40 observations of quarterly data using AAA model with errors# from normal distributionVESAAA <- sim.ves(model="AAA",frequency=4,obs=40,nvariate=3,                   randomizer="rnorm",mean=0,sd=100)# You can also use mvrnorm function from MASS package as randomizer,# but you need to provide mu and Sigma explicitly## Not run: VESANN <- sim.ves(model="ANN",frequency=4,obs=40,nvariate=2,                   randomizer="mvrnorm",mu=c(100,50),sigma=matrix(c(40,20,20,30),2,2))## End(Not run)# When generating the data with multiplicative model a more diligent definitiion# of parameters is needed. Here's an example with MMM model:VESMMM <- sim.ves("AAA", obs=120, nvariate=2, frequency=12, initial=c(10,0),          initialSeason=runif(12,-1,1), persistence=c(0.06,0.05,0.2), mean=0, sd=0.03)VESMMM$data <- exp(VESMMM$data)# Note that smoothing parameters should be low and the standard diviation should# definitely be less than 0.1. Otherwise you might face the explosions.

Vector Exponential Smoothing in SSOE state space model

Description

Function constructs vector ETS model and returns forecast, fitted values, errorsand matrix of states along with other useful variables.

Usage

ves(data, model = "PPP", lags = c(frequency(data)),  persistence = c("common", "individual", "dependent"),  transition = c("common", "individual", "dependent"), phi = c("common",  "individual"), initial = c("individual", "common"),  initialSeason = c("common", "individual"), loss = c("likelihood",  "diagonal", "trace"), ic = c("AICc", "AIC", "BIC", "BICc"), h = 10,  holdout = FALSE, occurrence = c("none", "fixed", "logistic"),  bounds = c("admissible", "usual", "none"), silent = TRUE, ...)

Arguments

Details

Function estimates vector ETS in a form of the Single Source of Error state spacemodel of the following type:

\mathbf{y}_{t} = (\mathbf{W} \mathbf{v}_{t-l} + \mathbf{x}_t\mathbf{a}_{t-1} + \mathbf{\epsilon}_{t})

\mathbf{v}_{t} = \mathbf{F} \mathbf{v}_{t-l} + \mathbf{G}\mathbf{\epsilon}_{t}

\mathbf{a}_{t} = \mathbf{F_{X}} \mathbf{a}_{t-1} + \mathbf{G_{X}}\mathbf{\epsilon}_{t} / \mathbf{x}_{t}

Wherey_{t} is the vector of time series on observationt,\mathbf{v}_{t} is the matrix ofstates andl is the matrix of lags,\mathbf{x}_t is the vector ofexogenous variables.\mathbf{W} is the measurement matrix,\mathbf{F}is the transition matrix and\mathbf{G} is the persistence matrix.Finally,\epsilon_{t} is the vector of error terms.

Conventionally we formulate values as:

\mathbf{y}'_t = (y_{1,t}, y_{2,t}, \dots, y_{m,t})

wherem is the number of series in the group.

\mathbf{v}'_t = (v_{1,t}, v_{2,t}, \dots, v_{m,t})

wherev_{i,t} is vector of components for i-th time series.

\mathbf{W}' = (w_{1}, \dots , 0;\vdots , \ddots , \vdots;0 , \vdots , w_{m})

is matrix of measurement vectors.

For the details on the additive model see Hyndman et al. (2008),chapter 17.

In case of multiplicative model, instead of the vector y_t we use its logarithms.As a result the multiplicative model is much easier to work with.

For some more information about the model and its implementation, see thevignette:vignette("ves","legion")

Value

Object of class "legion" is returned. It contains the following list ofvalues:

• model - The name of the fitted model;

• timeElapsed - The time elapsed for the construction of the model;

• states - The matrix of states with components in columns and time in rows;

• persistence - The persistence matrix;

• transition - The transition matrix;

• measurement - The measurement matrix;

• phi - The damping parameter value;

• lagsAll - The vector of the internal lags used in the model;

• B - The vector of all the estimated coefficients;

• initial - The initial values of the non-seasonal components;

• initialSeason - The initial values of the seasonal components;

• nParam - The number of estimated parameters;

• occurrence - The occurrence part of the model estimated with VES;

• data - The matrix with the original data;

• fitted - The matrix of the fitted values;

• holdout - The matrix with the holdout values (ifholdout=TRUE inthe estimation);

• residuals - The matrix of the residuals of the model;

• Sigma - The covariance matrix of the errors (estimated with the correctionfor the number of degrees of freedom);

• forecast - The matrix of point forecasts;

• ICs - The values of the information criteria;

• logLik - The log-likelihood function;

• lossValue - The value of the loss function;

• loss - The type of the used loss function;

• lossFunction - The loss function if the custom was used in the process;

• accuracy - the values of the error measures. Currently not available.

• FI - Fisher information if user asked for it usingFI=TRUE.

Author(s)

Ivan Svetunkov,ivan@svetunkov.com

References

• de Silva A., Hyndman R.J. and Snyder, R.D. (2010). The vectorinnovations structural time series framework: a simple approach tomultivariate forecasting. Statistical Modelling, 10 (4), pp.353-374

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

• Lütkepohl, H. (2005). New Introduction to Multiple Time SeriesAnalysis. New introduction to Multiple Time Series Analysis. Berlin,Heidelberg: Springer Berlin Heidelberg.doi:10.1007/978-3-540-27752-1

• Svetunkov, I., Chen, H., & Boylan, J. E. (2023).A new taxonomy for vector exponential smoothing and its applicationto seasonal time series. European Journal of Operational Research,304(3), 964–980.doi:10.1016/j.ejor.2022.04.040

See Also

vets,es,adam

Examples

Y <- ts(cbind(rnorm(100,100,10),rnorm(100,75,8)),frequency=12)# The simplest model applied to the data with the default valuesves(Y,model="ANN",h=10,holdout=TRUE)# Damped trend model with the dependent persistenceves(Y,model="AAdN",persistence="d",h=10,holdout=TRUE)# Multiplicative damped trend model with individual phives(Y,model="MMdM",persistence="i",h=10,holdout=TRUE,initialSeason="c")# Automatic selection between pure modelsves(Y,model="PPP",persistence="i",h=10,holdout=TRUE,initialSeason="c")# Intermittent demand vector modelY <- cbind(c(rpois(25,0.1),rpois(25,0.5),rpois(25,1),rpois(25,5)),           c(rpois(25,0.1),rpois(25,0.5),rpois(25,1),rpois(25,5)))ves(Y,model="MNN",h=10,holdout=TRUE,occurrence="l")