Package 'EXPAR'

Fitting of EXPAR model

Description

Searches for the best EXPAR model among many orders (defaults upto 5), compares them using information criterion and returns the best fit.

Usage

best_EXPAR(ts_data, max.p = 5, ic = "AIC", opt_method = "BFGS")

Arguments

ts_data	A univarite time series data, to which an EXPAR model is to be fitted.
max.p	The maximum order upto which models are to be searched for comparison.
ic	Information criterion to be used for model selection: Akaike information criterion ("AIC"), corrected Akaike information criterion ("AIC_c") or Bayesian information criterion ("BIC").
opt_method	The optimization algorithm to be used for RSS minimization. Corresponds to arguments from optim() in stats. Defaults to the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm.

Details

Fits max.p number of EXPAR models to the given dataset by minimisation of RSS using optimise_EXPAR() and returns the best model among the evaluated ones. Model selection is based on the information critera given in ic.

The various information criterion are calculated (estimated) from RSS as,

$\textup{AIC} = 2k + n\log(\frac{\textup{RSS}}{n})$

$\textup{AIC}_\textup{c} = \textup{AIC} + \frac{2k(k+1)}{n-k-1}$

$\textup{BIC} = k\log(n) + n\log(\frac{\textup{RSS}}{n})$

where, $n,k$ are the number of observations and the number of parameters, respectively.

Value

Returns the fitted EXPAR model as a list with the following components,

series	The data used for fitting the model.
order	Order $p$ of the fitted EXPAR model.
n	Number of observations in series.
k	Number of parameters in the model.
par	Parameters of the fitted model.
Fitted	Fitted values obtained from the model.
Residuals	Residuals of the fitted model.
RSS	The residual sum of squares.
AIC	Akaike information criterion, evaluated from RSS.
AIC_c	Corrected Akaike information criterion, evaluated from RSS.
BIC	Bayesian information criterion, evaluated from RSS.
counts	counts returned by optim()
convergence	convergence returned by optim()
message	message returned by optim()

Author(s)

Saikath Das, Bishal Gurung, Achal Lama and KN Singh

References

Haggan and Ozaki (1981). Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika, 68(1):189-199. <doi:10.1093/biomet/68.1.189>.

Gurung (2015). An exponential autoregressive (EXPAR) model for the forecasting of all India annual rainfall. Mausam, 66(4):847-849. <doi:10.54302/mausam.v66i4.594>.

Examples

datats <- ts(egg_price_index[,3], start = c(2013, 1), frequency = 12)
best_EXPAR(datats)

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Price Index of Eggs in Urban Ares of India

Description

This data set contains observations of Consumer Price Index (CPI) of eggs for urban areas in India. It consists of 86 monthly observations, ranging from January, 2013 till February, 2020. Some observations after this point of time were also available, but were discarded due to their questionable reliability and missingness pertaining to the onset of COVID-19 pandemic from March, 2020.

Usage

egg_price_index

Format

A data frame containing 86 observations. Observations are contained in the "egg_price_index" column, while the first two columns pertain to year and month of observation, repectively.

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Fitting of EXPAR model with known parameters

Description

Fits an EXPAR model to the dataset for given parameter values, without any optimisation.

Usage

fit_EXPAR(ts_data, par)

Arguments

ts_data	A univariate data to which the model is to be fitted, preferably (but not limited to) a ts object.
par	The parameter vector. Any value from the parametric space of EXPAR model is permissible. Number of required parameters is $2p+1$, where $p$ is order of the model. Entries are arranged consecutively, for example, for order 2, the form of input is c(Phi1,Phi2,Pi1,Pi2,Gamma).

Details

This function uses the form of EXPAR model to generate consecutive fitted observations as theoretical values obtained from the model. The form of EXPAR model employed is given by,

$y_{t+1}=\sum_{i=1}^{p}[({\phi_i+\pi_i e^{-\gamma y_t^2}})\ y_{t-i+1}\ ]+\epsilon_{t+1}$

where, $\gamma>0$ and $\{\epsilon_t\}$ is white noise process with zero mean and constant variance $\sigma_e^2$

The process of fitting involves using actual observations in the dataset in the RHS of above equation and obtaining the fitted values $y_{t+1}$. No optimization is done as parameter estimation is omitted. It is useful for simulation of data with given order and parameters.

Value

Returns the fitted EXPAR model as a list with the following components,

Fitted	Fitted values obtained from LHS of the aforementioned model. Due to obvious reasons, fits of the first $p$ observations are not obtained.
Residuals	The residuals $\epsilon_{t+1}$, computed by iterative fitting.
RSS	The residual sum of squares.
AIC	Akaike information criterion, evaluated from RSS.
AIC_c	Corrected Akaike information criterion, evaluated from RSS.
BIC	Bayesian information criterion, evaluated from RSS.

Examples

datats <- ts(egg_price_index[,3], start = c(2013, 1), frequency = 12)
fit_EXPAR(datats, par = c(0.45,0.68,0.48))

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Forecasting from EXPAR model

Description

Generates out-of-sample forecasts from any fitted EXPAR model.

Usage

forecast_EXPAR(EXPAR_model, h = 1)

Arguments

EXPAR_model	A fitted EXPAR model.
h	Number of periods for forecasting. Defaults to one-step ahead forecast.

Details

Forecasts from EXPAR model are generated iteratively using mathematical form of EXPAR model, considering fitted values and residuals while putting future error terms as zero.

Value

Vector of forecasted values

Examples

datats <- ts(egg_price_index[,3], start = c(2013, 1), frequency = 12)
best_model = best_EXPAR(datats)
forecast_EXPAR(best_model, h = 5)

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Initial values for fitting EXPAR model

Description

Determines the initial values to be used during parameter estimation.

Usage

inital_val(ts_data, order)

Arguments

ts_data	A univarite time series data, to which an EXPAR model is to be fitted.
order	Order $p$ of the EXPAR model to be considered.

Details

This function determines the initial values required for optimising the parameters of EXPAR model. The initial values are extracted from an ARIMA$(p,d,0)$ model with the same order and appropriate differencing. These will act as default initial values in the optimise_EXPAR() and best_EXPAR() functions, unless specified by user. Without these values, the optimisation process may fail.

Value

It returns the initial values of the EXPAR model over which optimisation is to be done parameter estimation.

Examples

datats <- ts(egg_price_index[,3], start = c(2013, 1), frequency = 12)
inital_val(datats, 2)

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Fitting of EXPAR model with given order

Description

Fits an EXPAR model of given order to the data by RSS minimisation

Usage

optimise_EXPAR(ts_data, order, init, opt_method = "BFGS")

Arguments

ts_data	A univarite time series data, to which an EXPAR model of given order is to be fitted.
order	Order $p$ of the EXPAR model to be considered.
init	The initial values for optimisation. If omitted, initial values are decided using initial_val().
opt_method	The optimization algorithm to be used for RSS minimization. Corresponds to arguments from optim() in stats library. Defaults to the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm.

Details

This function estimates the parameters an EXPAR$(p)$ model by minimizing RSS and returns a fit with the optimised parameters. The fit is returned using the function fit_EXPAR(), with the parameters inputted being the optimized ones. Available optimisation techniques include possible arguments from method in optim().

Value

Returns the fitted EXPAR model as a list with the following components,

series	The data used for fitting the model.
order	Order $p$ of the fitted EXPAR model.
n	Number of observations in series.
k	Number of parameters in the model.
par	Parameters of the fitted model.
Fitted	Fitted values obtained from the model.
Residuals	Residuals of the fitted model.
RSS	The residual sum of squares.
AIC	Akaike information criterion, evaluated from RSS.
AIC_c	Corrected Akaike information criterion, evaluated from RSS.
BIC	Bayesian information criterion, evaluated from RSS.
counts	counts returned by optim()
convergence	convergence returned by optim()
message	message returned by optim()

Examples

datats <- ts(egg_price_index[,3], start = c(2013, 1), frequency = 12)
optimise_EXPAR(datats, 2, opt_method = "BFGS")

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