msarima | R Documentation |
Function constructs Multiple Seasonal State Space ARIMA, estimating AR, MA terms and initial states. It is a wrapper of adam function.
msarima(y, orders = list(ar = c(0), i = c(1), ma = c(1)), lags = c(1),
constant = FALSE, AR = NULL, MA = NULL, model = NULL,
initial = c("optimal", "backcasting", "complete"), ic = c("AICc", "AIC",
"BIC", "BICc"), loss = c("likelihood", "MSE", "MAE", "HAM", "MSEh", "TMSE",
"GTMSE", "MSCE"), h = 10, holdout = FALSE, bounds = c("usual",
"admissible", "none"), silent = TRUE, xreg = NULL,
regressors = c("use", "select", "adapt"), initialX = NULL, ...)
auto.msarima(y, orders = list(ar = c(3, 3), i = c(2, 1), ma = c(3, 3)),
lags = c(1, frequency(y)), initial = c("optimal", "backcasting",
"complete"), ic = c("AICc", "AIC", "BIC", "BICc"), loss = c("likelihood",
"MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE", "MSCE"), h = 10,
holdout = FALSE, bounds = c("usual", "admissible", "none"),
silent = TRUE, xreg = NULL, regressors = c("use", "select", "adapt"),
initialX = NULL, ...)
msarima_old(y, orders = list(ar = c(0), i = c(1), ma = c(1)), lags = c(1),
constant = FALSE, AR = NULL, MA = NULL, initial = c("backcasting",
"optimal"), ic = c("AICc", "AIC", "BIC", "BICc"), loss = c("likelihood",
"MSE", "MAE", "HAM", "MSEh", "TMSE", "GTMSE", "MSCE"), h = 10,
holdout = FALSE, cumulative = FALSE, interval = c("none", "parametric",
"likelihood", "semiparametric", "nonparametric"), level = 0.95,
bounds = c("admissible", "none"), silent = c("all", "graph", "legend",
"output", "none"), xreg = NULL, regressors = c("use", "select"),
initialX = NULL, ...)
y |
Vector or ts object, containing data needed to be forecasted. |
orders |
List of orders, containing vector variables |
lags |
Defines lags for the corresponding orders (see examples above).
The length of |
constant |
If |
AR |
Vector or matrix of AR parameters. The order of parameters should be lag-wise. This means that first all the AR parameters of the firs lag should be passed, then for the second etc. AR of another msarima can be passed here. |
MA |
Vector or matrix of MA parameters. The order of parameters should be lag-wise. This means that first all the MA parameters of the firs lag should be passed, then for the second etc. MA of another msarima can be passed here. |
model |
Previously estimated MSARIMA model. |
initial |
Can be either character or a vector of initial states.
If it is character, then it can be |
ic |
The information criterion used in the model selection procedure. |
loss |
The type of Loss Function used in optimization. There are also available analytical approximations for multistep functions:
Finally, just for fun the absolute and half analogues of multistep estimators
are available: |
h |
Length of forecasting horizon. |
holdout |
If |
bounds |
What type of bounds to use in the model estimation. The first letter can be used instead of the whole word. |
silent |
If |
xreg |
The vector (either numeric or time series) or the matrix (or
data.frame) of exogenous variables that should be included in the model. If
matrix included than columns should contain variables and rows - observations.
Note that |
regressors |
The variable defines what to do with the provided xreg:
|
initialX |
The vector of initial parameters for exogenous variables.
Ignored if |
... |
Other non-documented parameters. see adam for details.
|
cumulative |
If |
interval |
Type of interval to construct. This can be:
The parameter also accepts |
level |
Confidence level. Defines width of prediction interval. |
The model, implemented in this function differs from the one in ssarima function (Svetunkov & Boylan, 2019), but it is more efficient and better fitting the data (which might be a limitation).
The basic ARIMA(p,d,q) used in the function has the following form:
(1 - B)^d (1 - a_1 B - a_2 B^2 - ... - a_p B^p) y_[t] = (1 + b_1 B +
b_2 B^2 + ... + b_q B^q) \epsilon_[t] + c
where y_[t]
is the actual values, \epsilon_[t]
is the error term,
a_i, b_j
are the parameters for AR and MA respectively and c
is
the constant. In case of non-zero differences c
acts as drift.
This model is then transformed into ARIMA in the Single Source of Error State space form (based by Snyder, 1985, but in a slightly different formulation):
y_{t} = o_{t} (w' v_{t-l} + x_t a_{t-1} + \epsilon_{t})
v_{t} = F v_{t-l} + g \epsilon_{t}
a_{t} = F_{X} a_{t-1} + g_{X} \epsilon_{t} / x_{t}
Where o_{t}
is the Bernoulli distributed random variable (in case of
normal data equal to 1), v_{t}
is the state vector (defined based on
orders
) and l
is the vector of lags
, x_t
is the
vector of exogenous parameters. w
is the measurement
vector,
F
is the transition
matrix, g
is the persistence
vector, a_t
is the vector of parameters for exogenous variables,
F_{X}
is the transitionX
matrix and g_{X}
is the
persistenceX
matrix. The main difference from ssarima
function is that this implementation skips zero polynomials, substantially
decreasing the dimension of the transition matrix. As a result, this
function works faster than ssarima on high frequency data,
and it is more accurate.
Due to the flexibility of the model, multiple seasonalities can be used. For
example, something crazy like this can be constructed:
SARIMA(1,1,1)(0,1,1)[24](2,0,1)[24*7](0,0,1)[24*30], but the estimation may
take some time... Still this should be estimated in finite time (not like
with ssarima
).
The auto.msarima
function constructs several ARIMA models in Single
Source of Error state space form based on adam
function (see
adam documentation) and selects the best one based on the
selected information criterion.
For some additional details see the vignettes: vignette("adam","smooth")
and vignette("ssarima","smooth")
Object of class "adam" is returned. It contains the list of the following values:
model
- the name of the estimated model.
timeElapsed
- time elapsed for the construction of the model.
states
- the matrix of the fuzzy components of msarima, where
rows
correspond to time and cols
to states.
transition
- matrix F.
persistence
- the persistence vector. This is the place, where
smoothing parameters live.
measurement
- measurement vector of the model.
AR
- the matrix of coefficients of AR terms.
I
- the matrix of coefficients of I terms.
MA
- the matrix of coefficients of MA terms.
constant
- the value of the constant term.
initialType
- Type of the initial values used.
initial
- the initial values of the state vector (extracted
from states
).
nParam
- table with the number of estimated / provided parameters.
If a previous model was reused, then its initials are reused and the number of
provided parameters will take this into account.
fitted
- the fitted values.
forecast
- the point forecast.
lower
- the lower bound of prediction interval. When
interval="none"
then NA is returned.
upper
- the higher bound of prediction interval. When
interval="none"
then NA is returned.
residuals
- the residuals of the estimated model.
errors
- The matrix of 1 to h steps ahead errors. Only returned when the
multistep losses are used and semiparametric interval is needed.
s2
- variance of the residuals (taking degrees of freedom into
account).
interval
- type of interval asked by user.
level
- confidence level for interval.
cumulative
- whether the produced forecast was cumulative or not.
y
- the original data.
holdout
- the holdout part of the original data.
xreg
- provided vector or matrix of exogenous variables. If
regressors="s"
, then this value will contain only selected exogenous
variables.
initialX
- initial values for parameters of exogenous
variables.
ICs
- values of information criteria of the model. Includes
AIC, AICc, BIC and BICc.
logLik
- log-likelihood of the function.
lossValue
- Cost function value.
loss
- Type of loss function used in the estimation.
FI
- Fisher Information. Equal to NULL if FI=FALSE
or when FI
is not provided at all.
accuracy
- vector of accuracy measures for the holdout sample.
In case of non-intermittent data includes: MPE, MAPE, SMAPE, MASE, sMAE,
RelMAE, sMSE and Bias coefficient (based on complex numbers). In case of
intermittent data the set of errors will be: sMSE, sPIS, sCE (scaled
cumulative error) and Bias coefficient. This is available only when
holdout=TRUE
.
B
- the vector of all the estimated parameters.
Ivan Svetunkov, ivan@svetunkov.ru
Snyder, R. D., 1985. Recursive Estimation of Dynamic Linear Models. Journal of the Royal Statistical Society, Series B (Methodological) 47 (2), 272-276.
Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-540-71918-2")}.
Svetunkov, I., & Boylan, J. E. (2019). State-space ARIMA for supply-chain forecasting. International Journal of Production Research, 0(0), 1–10. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00207543.2019.1600764")}
adam, orders,
es, auto.ssarima
x <- rnorm(118,100,3)
# The simple call of ARIMA(1,1,1):
ourModel <- msarima(x,orders=c(1,1,1),lags=1,h=18,holdout=TRUE)
# Example of SARIMA(2,0,0)(1,0,0)[4]
msarima(x,orders=list(ar=c(2,1)),lags=c(1,4),h=18,holdout=TRUE)
# SARIMA of a peculiar order on AirPassengers data
ourModel <- msarima(AirPassengers,orders=list(ar=c(1,0,3),i=c(1,0,1),ma=c(0,1,2)),
lags=c(1,6,12),h=10,holdout=TRUE)
# ARIMA(1,1,1) with Mean Squared Trace Forecast Error
msarima(x,orders=c(1,1,1),lags=1,h=18,holdout=TRUE,loss="TMSE")
plot(forecast(ourModel, h=18, interval="approximate"))
# The best ARIMA for the data
ourModel <- auto.msarima(x,orders=list(ar=c(2,1),i=c(1,1),ma=c(2,1)),lags=c(1,12),
h=18,holdout=TRUE)
# The other one using optimised states
auto.msarima(x,orders=list(ar=c(3,2),i=c(2,1),ma=c(3,2)),lags=c(1,12),
h=18,holdout=TRUE)
# And now combined ARIMA
auto.msarima(x,orders=list(ar=c(3,2),i=c(2,1),ma=c(3,2)),lags=c(1,12),
combine=TRUE,h=18,holdout=TRUE)
plot(forecast(ourModel, h=18, interval="simulated"))
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