Simulation Functions

The smooth package provides simulation functions for generating synthetic time series data from various state-space models. These are useful for Monte Carlo experiments, model validation, and understanding model behaviour.

Note: Not yet implemented in Python.

Overview

Function	Model	Description
`sim.es()`	ETS	Simulates from Exponential Smoothing models
`sim.ssarima()`	ARIMA	Simulates from State-Space ARIMA models
`sim.ces()`	CES	Simulates from Complex Exponential Smoothing
`sim.gum()`	GUM	Simulates from Generalised Univariate Models
`sim.sma()`	SMA	Simulates from Simple Moving Average models
`sim.oes()`	oETS	Simulates occurrence probabilities
`simulate()`	Any	Simulates from a fitted model

sim.es() - Exponential Smoothing Simulation

Simulates data from the ETS framework with predefined or randomly generated parameters.

R Usage

library(smooth)

# Basic simulation: ETS(A,N,N) with random parameters
set.seed(41)
sim <- sim.es("ANN", frequency=12, obs=120)

# With specific parameters
sim <- sim.es("MAdM", frequency=12, obs=120,
              phi=0.95, persistence=c(0.1, 0.05, 0.01))

# Multiple time series
sim <- sim.es("MNN", frequency=12, obs=120, nsim=50,
              probability=0.2, initial=10, persistence=0.1)

# Time series with pre-generated errors
randErrors <- function(...){
    # A silly example with whatever
    return(sample(c(1:12), 120, replace=T))
}

sim <- sim.es("MNN", frequency=12, obs=120, randomizer="randErrors")

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`model`	character	-	"ANN"	ETS model type (e.g., "ANN", "MAM", "MAdM")
`obs`	integer	-	10	Number of observations
`nsim`	integer	-	1	Number of series to simulate
`frequency`	integer	-	1	Seasonal frequency
`persistence`	numeric vector	-	NULL	Smoothing parameters (alpha, beta, gamma). If NULL, values are generated
`phi`	numeric	-	1	Damping parameter. Ignored if no trend
`initial`	numeric vector	-	NULL	Initial states for level and trend (max length 2). If NULL, generated
`initialSeason`	numeric vector	-	NULL	Initial seasonal coefficients. Length should equal `frequency`. If NULL, generated
`bounds`	character	-	"usual"	Bounds for persistence: "usual", "admissible", or "restricted"
`randomizer`	character	-	"rnorm"	Random number generator function name (e.g., "rnorm", "rlnorm", "rt", "rlaplace", "rs")
`probability`	numeric	-	1	Probability of occurrence (can be vector for time-varying probability)
`...`	-	-	-	Additional parameters passed to randomizer (e.g., `mean`, `sd` for `rnorm()`)

Output

Element	Description
`model`	Model name
`data`	Simulated time series (vector or matrix if nsim > 1)
`states`	State matrix (or array if nsim > 1). States in columns, time in rows
`persistence`	Smoothing parameters used (vector or matrix if nsim > 1)
`phi`	Damping parameter value used
`initial`	Initial values (vector or matrix)
`initialSeason`	Initial seasonal coefficients (vector or matrix)
`profile`	The final profile produced in the simulation
`probability`	Vector of probabilities used
`intermittent`	Type of intermittent model used
`residuals`	Error terms used (vector or matrix)
`occurrence`	Occurrence variable values (vector or matrix)
`logLik`	True log-likelihood value

Examples

# Simple ETS(A,N,N)
sim <- sim.es("ANN", frequency=12, obs=120)
plot(sim)

# Multiplicative trend with damping
sim <- sim.es("MAdM", frequency=12, obs=120,
              phi=0.95, persistence=c(0.1, 0.05, 0.01))
plot(sim)

# Using Log-Normal errors for multiplicative models
sim <- sim.es("MAdM", frequency=12, obs=120,
              phi=0.95, persistence=c(0.1, 0.05, 0.01),
              randomizer="rlnorm", meanlog=0, sdlog=0.015)

# Intermittent demand
sim <- sim.es("MNN", frequency=12, obs=120,
              probability=0.2, initial=10, persistence=0.1)
plot(sim)

# With initial seasonal values
sim <- sim.es("MMM", persistence=c(0.1, 0.1, 0.1), initial=c(2000, 1),
              initialSeason=c(1.1, 1.05, 0.9, 0.95), frequency=4, obs=80,
              mean=0, sd=0.01)

sim.ssarima() - ARIMA Simulation

Simulates data from State-Space ARIMA models.

R Usage

# Default: ARIMA(0,1,1)
sim <- sim.ssarima(frequency=12, obs=120, nsim=10)

# SARIMA(0,1,1)(1,0,2)_12 with drift
sim <- sim.ssarima(orders=list(ar=c(0,1), i=c(1,0), ma=c(1,2)),
                   lags=c(1,12), constant=TRUE, obs=120)

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`orders`	list	-	list(ar=0, i=1, ma=1)	ARIMA orders per lag. List with `ar`, `i`, `ma` vectors
`lags`	numeric vector	-	1	Seasonal lags. Length must match length of order vectors
`obs`	integer	-	10	Number of observations
`nsim`	integer	-	1	Number of series to simulate
`frequency`	integer	-	1	Time series frequency
`arma`	list	-	NULL	Fixed AR/MA parameters (lag-wise order). Can be passed from an estimated ssarima
`constant`	logical/numeric	-	FALSE	Include constant/drift. Can be TRUE or a specific value
`initial`	numeric vector	-	NULL	Initial state values. If NULL, generated from uniform distribution
`bounds`	character	-	"admissible"	Bounds for AR/MA: "admissible" or "none"
`randomizer`	character	-	"rnorm"	Random number generator function name
`probability`	numeric	-	1	Occurrence probability (can be vector)
`...`	-	-	-	Additional parameters passed to randomizer

Output

Element	Description
`model`	ARIMA model specification
`arma`	List of AR/MA parameters (list of matrices if nsim > 1)
`constant`	Constant values (vector if nsim > 1)
`initial`	Initial state values (matrix if nsim > 1)
`profile`	The final profile produced in the simulation
`data`	Simulated time series (vector or matrix)
`states`	State array
`residuals`	Generated errors (vector or matrix)
`occurrence`	Occurrence variable values
`logLik`	True log-likelihood

Examples

# ARIMA(0,1,1)
sim <- sim.ssarima(frequency=12, obs=120)
plot(sim)

# Seasonal ARIMA with drift
sim <- sim.ssarima(orders=list(ar=c(0,1), i=c(1,0), ma=c(1,2)),
                   lags=c(1,12), constant=TRUE, obs=120)
plot(sim)

# Intermittent SARIMA
sim <- sim.ssarima(orders=list(ar=c(1,0), i=c(0,1), ma=c(2,1)),
                   lags=c(1,7), obs=120, probability=0.2)
plot(sim)

# High frequency SARIMA(1,0,2)_1(0,1,1)_7(1,0,1)_30
sim <- sim.ssarima(orders=list(ar=c(1,0,1), i=c(0,1,0), ma=c(2,1,1)),
                   lags=c(1,7,30), obs=360, nsim=10)

sim.ces() - Complex Exponential Smoothing Simulation

Simulates data from CES models.

R Usage

# CES(n) - non-seasonal
sim <- sim.ces(frequency=12, obs=120)

# CES(s) - simple seasonal
sim <- sim.ces("simple", frequency=24, obs=240)

# CES(p) - partial seasonal
sim <- sim.ces("partial", b=0.2, frequency=12, obs=240, nsim=10)

# CES(f) - full seasonal
sim <- sim.ces("full", frequency=12, obs=240, nsim=10)

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`seasonality`	character	-	"none"	Seasonality type: "none", "simple", "partial", "full"
`obs`	integer	-	10	Number of observations
`nsim`	integer	-	1	Number of series to simulate
`frequency`	integer	-	1	Seasonal frequency (must be > 1 for seasonal models)
`a`	complex	-	NULL	First complex smoothing parameter. CES is sensitive to these values
`b`	complex/numeric	-	NULL	Second complex smoothing parameter. Real for "partial", complex for "full". NULL for "none"/"simple"
`initial`	complex matrix	-	NULL	Initial states. For "simple"/"partial"/"full", first two columns contain non-seasonal components
`randomizer`	character	-	"rnorm"	Random number generator function name
`probability`	numeric	-	1	Occurrence probability (can be vector)
`...`	-	-	-	Additional parameters passed to randomizer

Output

Element	Description
`model`	CES model type
`a`	Complex parameter a (vector if nsim > 1)
`b`	Complex parameter b (NULL for "none"/"simple", vector if nsim > 1)
`initial`	Initial state values (array if nsim > 1)
`profile`	The final profile produced in the simulation
`data`	Simulated time series (vector or matrix)
`states`	State matrix/array
`residuals`	Generated errors (vector or matrix)
`occurrence`	Occurrence variable values
`logLik`	Log-likelihood of the constructed model

Examples

# Non-seasonal CES
sim <- sim.ces(frequency=12, obs=120)
plot(sim)

# Simple seasonal (solar-like patterns)
set.seed(41)
sim <- sim.ces("simple", frequency=24, obs=240)
plot(sim)

# Full seasonal
sim <- sim.ces("full", frequency=12, obs=240, nsim=10)
plot(sim)

sim.gum() - Generalised Univariate Model Simulation

Simulates data from GUM. Best practice is to simulate from a fitted model.

R Usage

# Generate complex multi-frequency data
set.seed(41)
sim <- sim.gum(orders=c(1,1,1), lags=c(1,48,336), nsim=1,
               frequency=336, obs=3360,
               measurement=rep(1,3), transition=diag(3),
               persistence=c(0.045, 0.162, 0.375),
               initial=initialMatrix)  # Provide initial values

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`orders`	numeric vector	-	c(1)	Orders for each lag (number of states per lag type)
`lags`	numeric vector	-	c(1)	Lags for components. Length must match `orders`
`obs`	integer	-	10	Number of observations
`nsim`	integer	-	1	Number of series to simulate
`frequency`	integer	-	1	Time series frequency
`measurement`	numeric vector	-	NULL	Measurement vector w. If NULL, randomly generated (0-1)
`transition`	matrix/vector	-	NULL	Transition matrix F. If vector, formed as matrix(transition, nc, nc)
`persistence`	numeric vector	-	NULL	Persistence vector g (smoothing parameters). If NULL, randomly generated
`initial`	matrix	-	NULL	Initial states. If NULL, generated from uniform distribution
`randomizer`	character	-	"rnorm"	Random number generator function name
`probability`	numeric	-	1	Occurrence probability (can be vector)
`...`	-	-	-	Additional parameters passed to randomizer

Output

Element	Description
`model`	GUM model name
`measurement`	Measurement matrix w
`transition`	Transition matrix F
`persistence`	Persistence vector (smoothing parameters)
`initial`	Initial state values (array if nsim > 1)
`profile`	The final profile produced in the simulation
`data`	Simulated time series (vector or matrix)
`states`	State array
`residuals`	Generated errors (vector or matrix)
`occurrence`	Occurrence variable values
`logLik`	Log-likelihood of the constructed model

Examples

# GUM(1[1]) - simple model
sim <- sim.gum(orders=c(1), lags=c(1), obs=120, nsim=100)
plot(sim)

# GUM(1[1],1[4]) - seasonal model
sim <- sim.gum(orders=c(1,1), lags=c(1,4), frequency=4, obs=80,
               nsim=100, transition=c(1, 0, 0.9, 0.9))
plot(sim)

sim.sma() - Simple Moving Average Simulation

Simulates data from SMA models.

R Usage

# SMA(10)
sim <- sim.sma(order=10, obs=240, frequency=12)
plot(sim)

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`order`	integer	-	NULL	SMA order. If NULL, random order from 1 to 100 is selected
`obs`	integer	-	10	Number of observations
`nsim`	integer	-	1	Number of series to simulate
`frequency`	integer	-	1	Time series frequency
`initial`	numeric vector	-	NULL	Initial state values. If NULL, generated
`randomizer`	character	-	"rnorm"	Random number generator function name
`probability`	numeric	-	1	Occurrence probability (can be vector)
`...`	-	-	-	Additional parameters passed to randomizer

Output

Element	Description
`model`	SMA model name
`data`	Simulated time series (vector or matrix)
`states`	State matrix (or array if nsim > 1)
`initial`	Initial values (vector or matrix)
`probability`	Vector of probabilities used
`intermittent`	Type of intermittent model used
`residuals`	Generated errors (vector or matrix)
`occurrence`	Occurrence variable values
`logLik`	Log-likelihood of the constructed model

Examples

# SMA(10) with normal errors
sim <- sim.sma(order=10, obs=40, randomizer="rnorm", mean=0, sd=100)
plot(sim)

sim.oes() - Occurrence ETS Simulation

Simulates occurrence probabilities using ETS for intermittent demand modelling. The function calls for sim.es() and then does internal transformations of the simulated data to align with the occurrence models.

R Usage

# Simple occurrence model
sim <- sim.oes("MNN", frequency=12, obs=60, initial=1,
               randomizer="rlnorm", meanlog=0, sdlog=0.1)

# General occurrence with model A and model B
sim <- sim.oes("MNN", frequency=12, obs=60, occurrence="general",
               modelB="MNN", persistence=0.2, persistenceB=0.1)

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`model`	character	-	"MNN"	ETS model type for model A. "MZZ" models recommended for positive errors
`obs`	integer	-	10	Number of observations
`nsim`	integer	-	1	Number of series to simulate
`frequency`	integer	-	1	Seasonal frequency
`occurrence`	character	-	"odds-ratio"	Occurrence type: "odds-ratio", "inverse-odds-ratio", "direct", "general"
`bounds`	character	-	"usual"	Bounds for persistence: "usual", "admissible", "restricted"
`randomizer`	character	-	"rlnorm"	Random number generator (rlnorm, rinvgauss, rgamma, rnorm recommended)
`persistence`	numeric vector	-	NULL	Smoothing parameters for model A
`phi`	numeric	-	1	Damping parameter for model A
`initial`	numeric vector	-	NULL	Initial states for model A
`initialSeason`	numeric vector	-	NULL	Initial seasonal coefficients for model A
`modelB`	character	-	model	ETS model type for model B (used in "general" and "inverse-odds-ratio")
`persistenceB`	numeric vector	-	persistence	Smoothing parameters for model B
`phiB`	numeric	-	phi	Damping parameter for model B
`initialB`	numeric vector	-	initial	Initial states for model B
`initialSeasonB`	numeric vector	-	initialSeason	Initial seasonal coefficients for model B
`...`	-	-	-	Additional parameters passed to randomizer (shared by model A and B)

Output

Element	Description
`model`	Model name
`modelA`	Model A object, generated using sim.es()
`modelB`	Model B object, generated using sim.es()
`probability`	The probability values generated by the model
`occurrence`	Type of occurrence used in the model
`logLik`	Log-likelihood of the constructed model

Examples

# Log-normal errors
sim <- sim.oes("MNN", frequency=12, obs=60, initial=1,
               randomizer="rlnorm", meanlog=0, sdlog=0.1)
plot(sim$probability)

# Using inverse Gaussian (requires statmod package)
sim <- sim.oes("MNN", frequency=12, obs=60,
               randomizer="rinvgauss", mean=1, dispersion=0.5)

# Generate iETS(MNN) with TSB style probabilities
oETSMNN <- sim.oes("MNN", obs=50, occurrence="d", persistence=0.2, initial=1,
                   randomizer="rlnorm", meanlog=0, sdlog=0.3)
iETSMNN <- sim.es("MNN", obs=50, frequency=12, persistence=0.2, initial=4,
                  probability=oETSMNN$probability)

simulate() - Simulate from Fitted Model

Simulates new data from a fitted smooth model using its estimated parameters.

R Usage

# Fit model
model <- adam(AirPassengers, "MMM", lags=12)

# Simulate 50 series of 100 observations
simData <- simulate(model, nsim=50, obs=100)

# Compare original and simulated
par(mfcol=c(1,2))
plot(model,7)
plot(simData)

Parameters

Parameter	Type (R)	Type (Python)	Default	Description
`object`	adam/smooth	-	-	Fitted model
`nsim`	integer	-	1	Number of series to simulate
`seed`	integer	-	NULL	Random seed
`obs`	integer	-	nobs(object)	Number of observations

Examples

# From ADAM
model <- adam(BJsales)
simData <- simulate(model, nsim=50, obs=100)

# From ETS
model <- es(BJsales)
simData <- simulate(model, nsim=50, obs=100)

# From SSARIMA
model <- auto.ssarima(BJsales)
simData <- simulate(model, nsim=50, obs=100)

# From CES
model <- auto.ces(BJsales)
simData <- simulate(model, nsim=50, obs=100)

# From GUM
model <- auto.gum(BJsales)
simData <- simulate(model, nsim=50, obs=100)

# From SMA
model <- sma(BJsales)
simData <- simulate(model, nsim=50, obs=1000)

Best Practices

Parameter Selection

For multiplicative models, use lower variance for random errors:

# Recommended for multiplicative models
sim <- sim.es("MAdM", frequency=12, obs=120,
              randomizer="rlnorm", meanlog=0, sdlog=0.015)

Note on multiplicative errors: In case of multiplicative error models, the randomizer generates 1+e_t, not e_t. This means the mean should typically be 1, not zero.

Simulation from Fitted Models

When possible, simulate from fitted models to ensure realistic parameters:

# Fit to real data first
model <- adam(realData, "ZXZ", lags=12)

# Then simulate
simulated <- simulate(model, nsim=100, obs=500)

Monte Carlo Studies

# Generate many replications for Monte Carlo
nsim <- 1000
results <- matrix(NA, nsim, 2)

for(i in 1:nsim) {
  sim <- sim.es("AAN", frequency=12, obs=120)
  fit <- adam(sim$data, "ZXZ", lags=12, silent=TRUE)
  results[i,] <- coef(fit)[c("alpha","beta")]
}

# Analyse estimation properties
colMeans(results)
apply(results, 2, sd)

Bounds for Parameter Generation

Bound Type	Description
`"usual"`	Standard bounds from Hyndman et al. (2008) p.156
`"restricted"`	Similar to "usual" but with upper bound equal to 0.3
`"admissible"`	Bounds from tables 10.1 and 10.2 of Hyndman et al. (2008)

For SSARIMA, bounds ensure stability and stationarity when set to "admissible".

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Simulation Functions

Simulation Functions

Overview

sim.es() - Exponential Smoothing Simulation

R Usage

Parameters

Output

Examples

sim.ssarima() - ARIMA Simulation

R Usage

Parameters

Output

Examples

sim.ces() - Complex Exponential Smoothing Simulation

R Usage

Parameters

Output

Examples

sim.gum() - Generalised Univariate Model Simulation

R Usage

Parameters

Output

Examples

sim.sma() - Simple Moving Average Simulation

R Usage

Parameters

Output

Examples

sim.oes() - Occurrence ETS Simulation

R Usage

Parameters

Output

Examples

simulate() - Simulate from Fitted Model

R Usage

Parameters

Examples

Best Practices

Parameter Selection

Simulation from Fitted Models

Monte Carlo Studies

Bounds for Parameter Generation

See Also

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