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ARIMAForecast.R
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ARIMAForecast.R
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require(forecast)
require(tseries)
require(rugarch)
require(tsoutliers)
require(arfima)
is.weekend <- function(wkd) {
if((wkd == "Saturday") || (wkd == "Sunday")) {
return(1)
} else {
return(0)
}
}
mark.time <- function(start, end, discretion) {
discretion.string = paste0(discretion, " sec")
ts.raw = seq(start, end, discretion.string)
marks = weekdays(ts.raw)
marked.vector = sapply(marks, is.weekend)
return(marked.vector)
}
get.best.arima <- function(request.time.series.list, maxord = c(1,1,1,1,1,1)) {
best.aic <- 1e8
n <- length(request.time.series.list$series)
marked.days <- mark.time(request.time.series.list$start,
request.time.series.list$end,
request.time.series.list$discretion)
for(p in 0:maxord[1])
for(d in 0:maxord[2])
for(q in 0:maxord[3])
for(P in 0:maxord[4])
for(D in 0:maxord[5])
for(Q in 0:maxord[6])
{
fit <- arima(request.time.series.list$series,
order = c(p, d, q),
seasonal = list(order = c(P, D, Q),
period = compute.ts.highest.period(request.time.series.list$series)),
method = "CSS",
xreg = marked.days)
fit.aic <- -2 * fit$loglik + (log(n) + 1) * length(fit$coef)
if(fit.aic < best.aic)
{
best.aic <- fit.aic
best.fit <- fit
best.model <- c(p, d, q, P, D, Q)
}
}
return(list(AIC = best.aic, SARIMA = best.fit, coef = best.model))
}
# A function to compute the time series highest-order period in order to be used for seasonality estimates
# for 1 hour discretion
compute.ts.highest.period <- function(time.series) {
ts.spectrum <- spectrum(time.series)
period <- (1 / ts.spectrum$freq[which(ts.spectrum$spec == max(ts.spectrum$spec))]) / 3600
return(round(period))
}
# A function to fit SARIMA with the specified parameters
get.SARIMA.with.defined.parameters <- function(request.time.series.list, parameters) {
marked.days <- mark.time(request.time.series.list$start,
request.time.series.list$end,
request.time.series.list$discretion)
arima.period <- parameters[4]
fit <- forecast::Arima(request.time.series.list$series,
order = c(parameters[1], parameters[2], parameters[3]),
seasonal = list(order = c(parameters[5], parameters[6], parameters[7]),
period = arima.period),
method = "CSS",
xreg = marked.days)
return(fit)
}
#Function to fit SARIMA model adjusted for outliers
create.SARIMA.model.weekly.OutliersAdjusted <- function(request.time.series.list, parameters) {
marked.days <- mark.time(request.time.series.list$start,
request.time.series.list$end,
request.time.series.list$discretion)
arima.period <- parameters[4]
fit.tso <- tsoutliers::tso(request.time.series.list$series,
marked.days,
types = c("IO", "AO", "TC"),
maxit.oloop = 15,
tsmethod = "arima",
args.tsmethod = list(order = c(parameters[1], parameters[2], parameters[3]),
seasonal = list(order = c(parameters[5], parameters[6], parameters[7]),
period = arima.period),
method = "CSS"))
return(fit.tso)
}
# A function to create the SARIMA model either using best AIC automatic fit model or the model with the specified
# parameters. The period is derived automatically.
create.SARIMA.model.weekly <- function(request.time.series.list, auto = TRUE, parameters = c(0,0,0,0,0,0,0)) {
if(auto) {
ARIMA.model = get.best.arima(request.time.series.list, maxord = c(2,2,2,2,2,2))
return(ARIMA.model$SARIMA)
} else {
ARIMA.model.fit = get.SARIMA.with.defined.parameters(request.time.series.list, parameters)
return(ARIMA.model.fit)
}
}
# A function that makes the actual forecast depending on whether we have outliers embedded in the model or not.
forecast.requests <- function(request.time.series.list, ARIMA.model, n.predicted.values, outliers = NULL) {
start = request.time.series.list$end
discretion = request.time.series.list$discretion
duration = n.predicted.values * discretion
marked.ts.for.prediction = mark.time(start + discretion, start + duration, discretion)
newxreg <- marked.ts.for.prediction
if((!is.null(outliers)) && (nrow(outliers) > 0)) {
marked.days <- mark.time(request.time.series.list$start,
request.time.series.list$end,
request.time.series.list$discretion)
newxreg <- outliers.effects(outliers, length(request.time.series.list$series) + n.predicted.values, pars = coefs2poly(ARIMA.model))
newxreg <- cbind(c(marked.days, marked.ts.for.prediction), newxreg)
colnames(newxreg)[1] <- "xreg"
newxreg <- ts(newxreg[-seq_along(request.time.series.list$series),], start = as.numeric(start))
ARIMA.model$xreg <- ARIMA.model$call$xreg
}
prediction = forecast::forecast(ARIMA.model, h = n.predicted.values, xreg = newxreg)
# https://stats.stackexchange.com/questions/169468/how-to-do-forecasting-with-detection-of-outliers-in-r-time-series-analysis-pr
return(prediction)
}
# Function to compute the limits for autocorrelation values
corellogram.limits <- function(acf.obj) {
n <- acf.obj$n.used
ret <- list()
ret$lb <- - 1 / n - 2 / sqrt(n)
ret$ub <- - 1 / n + 2 / sqrt(n)
return(ret)
}
# Function to estimate the MA/AR coefficients for ARIMA model based on ACF analysis
determine.coefficient <- function(lags, interval.width) {
fraction.of.interval.width <- 0.55 # raw estimate
important.lags <- abs(lags) > (fraction.of.interval.width * interval.width)
important.lags.shifted <- c(important.lags[1], important.lags[-length(important.lags)])
difference.in.important.lags <- abs(important.lags.shifted - important.lags)
intervals.of.important.lags <- split(important.lags, cumsum(difference.in.important.lags))
first.interval <- intervals.of.important.lags$`0`
c <- 0
if(first.interval[1] == TRUE) {
c <- length(first.interval)
}
return(c)
}
# Function to derive the GARCH model
create.GARCH.model.weekly <- function(train.timeseries, arima.residuals, coefs, pred.steps) {
marked.days <- mark.time(train.timeseries$start,
train.timeseries$end,
train.timeseries$discretion)
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(coefs[1], coefs[2]),
submodel = NULL, external.regressors = as.matrix(marked.days), variance.targeting = FALSE),
distribution.model = "norm", start.pars = list(), fixed.pars = list())
garch <- ugarchfit(spec = spec,
data = arima.residuals,
out.sample = pred.steps,
solver = "hybrid",
solver.control=list(trace=0))
}
# Function to derive the forecasts using the ANN models
arima.forecast <- function(example.ts, pred.steps, model.type) {
Sys.setlocale("LC_TIME", "English") # This is very important in case the computer locale for time is different from English - it is needed for weekends marking. TODO: workaround?
# Supported time series models:
# - SARIMA - single SARIMA model
# - SARIMA+GARCH - SARIMA for forecasting the mean and GARCH for forecasting the variance
# - SOAARIMA - outliers-adjusted SARIMA for forecasting the mean and GARCH for forecasting the variance
# - SOAARIMA+GARCH - single outliers-adjusted SARIMA model
# - SARFIMA - single seasonal ARFIMA model
# - SARFIMA+GARCH - seasonal ARFIMA for forecasting the mean and GARCH for forecasting the variance
resulting.model <- NULL
# Measuring the time of parameters selction process start (ARIMA)
parameters.selection.start <- Sys.time()
# Preliminary analysis to choose ARIMA parameters
# I. Getting rid of the seasonality
test.set.length.days <- 7
train.timeseries <- list()
train.timeseries$series <- window(example.ts$series, as.numeric(example.ts$start), as.numeric(example.ts$end - test.set.length.days * 24 * 3600))
train.timeseries$start <- example.ts$start
train.timeseries$end <- example.ts$end - test.set.length.days * 24 * 3600
train.timeseries$discretion <- example.ts$discretion
test.ts <- list()
test.ts$series <- window(example.ts$series, as.numeric(example.ts$end - test.set.length.days * 24 * 3600 + 1), as.numeric(example.ts$end))
test.ts$start <- example.ts$end - test.set.length.days * 24 * 3600 + 1
test.ts$end <- example.ts$end
test.ts$discretion <- example.ts$discretion
time.series <- train.timeseries$series
# s
s <- compute.ts.highest.period(time.series)
# To avoid errors occuring due to large period value (defaulting to day-period)
if(s > 24) {
s <- 24
}
unseasoned.ts <- diff(time.series, s)
# II. Checking for trend-stationarity with KPSS test
border.kpss.p.val <- 0.05
trend.stationary.ts <- unseasoned.ts
kpss.stat <- kpss.test(trend.stationary.ts, null = "Trend")
# d
d <- 0
if(length(grep("SARIMA", model.type)) > 0) {
d.limit <- 12 # stop-condition on d value
while((kpss.stat$p.value <= border.kpss.p.val) & (d <= d.limit)) { #non-stationarity in trend, needs more differencing
trend.stationary.ts <- diff(trend.stationary.ts)
d <- d + 1
kpss.stat <- kpss.test(trend.stationary.ts, null = "Trend")
}
# Trying to evaluate the value of d based on try-evaluate approach.
# Stopping evaluation when the differencing leads to more autocorellations to be outside of the acceptable interval.
previous.d <- d
d <- d + 1 # Information contained in d could also be captured using other model parameters, e.g. q
trend.stationary.ts.pacf <- pacf(trend.stationary.ts)
trend.stationary.ts.pacf.lags <- trend.stationary.ts.pacf$acf
fraction.of.interval.width <- 0.5 # raw estimate
trend.stationary.ts.limits.pacf <- corellogram.limits(trend.stationary.ts.pacf)
trend.stationary.ts.pacf.interval.width <- trend.stationary.ts.limits.pacf$ub - trend.stationary.ts.limits.pacf$lb
important.lags <- abs(trend.stationary.ts.pacf.lags) > (fraction.of.interval.width * trend.stationary.ts.pacf.interval.width)
fraction.of.important.lags <- sum(important.lags) / length(important.lags)
trend.stationary.ts.new <- diff(trend.stationary.ts)
trend.stationary.ts.new.pacf <- pacf(trend.stationary.ts.new)
trend.stationary.ts.new.pacf.lags <- trend.stationary.ts.new.pacf$acf
trend.stationary.ts.new.limits.pacf <- corellogram.limits(trend.stationary.ts.new.pacf)
trend.stationary.ts.new.pacf.interval.width <- trend.stationary.ts.new.limits.pacf$ub - trend.stationary.ts.new.limits.pacf$lb
important.lags.new <- abs(trend.stationary.ts.new.pacf.lags) > (fraction.of.interval.width * trend.stationary.ts.new.pacf.interval.width)
fraction.of.important.lags.NEW <- sum(important.lags.new) / length(important.lags.new)
previous.trend.stationary.ts <- trend.stationary.ts
while((d <= d.limit) && (fraction.of.important.lags.NEW <= fraction.of.important.lags)) {
previous.trend.stationary.ts <- trend.stationary.ts
previous.d <- d
d <- d + 1
trend.stationary.ts.pacf <- pacf(trend.stationary.ts)
trend.stationary.ts.pacf.lags <- trend.stationary.ts.pacf$acf
fraction.of.interval.width <- 0.5 # raw estimate
trend.stationary.ts.limits.pacf <- corellogram.limits(trend.stationary.ts.pacf)
trend.stationary.ts.pacf.interval.width <- trend.stationary.ts.limits.pacf$ub - trend.stationary.ts.limits.pacf$lb
important.lags <- abs(trend.stationary.ts.pacf.lags) > (fraction.of.interval.width * trend.stationary.ts.pacf.interval.width)
fraction.of.important.lags <- sum(important.lags) / length(important.lags)
trend.stationary.ts.new <- diff(trend.stationary.ts)
trend.stationary.ts.new.pacf <- pacf(trend.stationary.ts.new)
trend.stationary.ts.new.pacf.lags <- trend.stationary.ts.new.pacf$acf
trend.stationary.ts.new.limits.pacf <- corellogram.limits(trend.stationary.ts.new.pacf)
trend.stationary.ts.new.pacf.interval.width <- trend.stationary.ts.new.limits.pacf$ub - trend.stationary.ts.new.limits.pacf$lb
important.lags.new <- abs(trend.stationary.ts.new.pacf.lags) > (fraction.of.interval.width * trend.stationary.ts.new.pacf.interval.width)
fraction.of.important.lags.NEW <- sum(important.lags.new) / length(important.lags.new)
trend.stationary.ts <- trend.stationary.ts.new
}
d <- previous.d
trend.stationary.ts <- previous.trend.stationary.ts
}
# III. Estimating the trend
trended.data <- data.frame(time = time(trend.stationary.ts), value = trend.stationary.ts)
ts.trend <- lm(value ~ time, data = trended.data)
#detrend.stationary <- trend.stationary.ts - ts.trend$fitted.values
detrend.stationary <- trend.stationary.ts
# IV. Estimating the MA component coefficients for ARIMA
acf.detrend <- acf(detrend.stationary)
acf.ts.limits.detrend <- corellogram.limits(acf.detrend)
acf.interval.width <- acf.ts.limits.detrend$ub - acf.ts.limits.detrend$lb
non.seasonal.acf <- acf.detrend$acf[2:(s-1)]
seasonal.acf <- acf.detrend$acf[(s+1):length(acf.detrend$acf)]
# q
q <- determine.coefficient(non.seasonal.acf, acf.interval.width)
# Q
Q <- determine.coefficient(seasonal.acf, acf.interval.width)
# V. Estimating the AR component coefficients for ARIMA
pacf.detrend <- pacf(detrend.stationary)
pacf.ts.limits.detrend <- corellogram.limits(pacf.detrend)
pacf.interval.width <- pacf.ts.limits.detrend$ub - pacf.ts.limits.detrend$lb
non.seasonal.pacf <- pacf.detrend$acf[2:(s-1)]
seasonal.pacf <- pacf.detrend$acf[(s+1):length(pacf.detrend$acf)]
# p
p <- determine.coefficient(non.seasonal.pacf, pacf.interval.width)
# P
P <- determine.coefficient(seasonal.pacf, pacf.interval.width)
# VI. Estimating the number of seasonal differences
# D
acf.seasonality <- acf(detrend.stationary, 5 * s)
acf.seasonality.lags <- acf.seasonality$acf
acf.seasonality.lags.selected <- c(acf.seasonality.lags[2 * s], acf.seasonality.lags[3 * s], acf.seasonality.lags[4 * s], acf.seasonality.lags[5 * s])
D <- sum(acf.seasonality.lags.selected > acf.interval.width)
# Measuring the time of parameters selection process end (ARIMA)
parameters.selection.end <- Sys.time()
parameters.selection.duration.1 <- difftime(parameters.selection.end, parameters.selection.start, units = "secs")
# Measuring the time of model fitting process start (ARIMA)
model.fitting.start <- Sys.time()
# VII. Deriving ARIMA model based on the derived parameters
ARIMA.model <- NULL
outliers <- NULL
if(length(grep("SARIMA", model.type)) > 0) {
ARIMA.model <- create.SARIMA.model.weekly(train.timeseries, FALSE, c(p,d,q,s,P,D,Q))
} else if(length(grep("SOAARIMA", model.type)) > 0) { # Adjusting the acquired model for possible outliers.
ARIMA.model.full <- create.SARIMA.model.weekly.OutliersAdjusted(train.timeseries, c(p,d,q,s,P,D,Q))
ARIMA.model <- ARIMA.model.full$fit
outliers <- ARIMA.model.full$outliers
if(is.null(outliers) || (nrow(outliers) == 0)) {
ARIMA.model <- create.SARIMA.model.weekly(train.timeseries, FALSE, c(p,d,q,s,P,D,Q))
}
} else if(length(grep("SARFIMA", model.type)) > 0) {
marked.days <- mark.time(train.timeseries$start,
train.timeseries$end,
train.timeseries$discretion)
ARIMA.model <- arfima::arfima(train.timeseries$series,
order = c(p, 0 , q),
seasonal = list(order = c(P, 0, Q),
period = s),
xreg = as.matrix(marked.days))
}
# VIII. Forecasting using ARIMA model and adjusting it using the forecast
# Forecasting using basic ARIMA model
ARIMA.forecast <- NULL
if((length(grep("SARIMA", model.type)) > 0) || (length(grep("SOAARIMA", model.type)) > 0)) {
ARIMA.forecast <- forecast.requests(train.timeseries, ARIMA.model, pred.steps, outliers)
} else if(length(grep("SARFIMA", model.type)) > 0) {
start = train.timeseries$end
discretion = train.timeseries$discretion
duration = pred.steps * discretion
marked.ts.for.prediction = mark.time(start + discretion, start + duration, discretion)
pred.ARFIMA <- predict(ARIMA.model, pred.steps, newxreg = as.matrix(marked.ts.for.prediction))
sd.ARFIMA <- pred.ARFIMA[[1]]$exactSD
mean.ARFIMA <- pred.ARFIMA[[1]]$Forecast
lower.95 <- mean.ARFIMA - 1.96 * sd.ARFIMA
upper.95 <- mean.ARFIMA + 1.96 * sd.ARFIMA
ARIMA.forecast <- list(mean = mean.ARFIMA,
lower = lower.95,
upper = upper.95)
}
resulting.model <- ARIMA.forecast
# Adjusting by trend
if(length(grep("SARIMA", model.type)) > 0) {
trend.predictions <- predict(ts.trend, data.frame(time = seq(start(resulting.model$mean), end(resulting.model$mean), 3600)))
SARIMA.forecast.adjusted.by.trend <- resulting.model
SARIMA.forecast.adjusted.by.trend$mean <- SARIMA.forecast.adjusted.by.trend$mean + ts(trend.predictions,
start = start(resulting.model$mean),
end = end(resulting.model$mean),
frequency = frequency(resulting.model$mean))
resulting.model <- SARIMA.forecast.adjusted.by.trend
}
# Measuring the time of model fitting process end (ARIMA)
model.fitting.end <- Sys.time()
model.fitting.duration.1 <- difftime(model.fitting.end, model.fitting.start, units = "secs")
# Measuring the time of parameters selection process start (GARCH)
parameters.selection.duration.2 <- difftime(parameters.selection.start, parameters.selection.start, units = "secs")
model.fitting.duration.2 <- difftime(model.fitting.start, model.fitting.start, units = "secs")
parameters.selection.start <- Sys.time()
if( length(grep("GARCH", model.type)) > 0 ) { #GARCH model included
# IX. Fitting GARCH models to the variance if necessary
residuals.ARIMA <- residuals(ARIMA.model)
if(class(residuals.ARIMA) == "list") {
residuals.ARIMA <- residuals.ARIMA[[1]]
names(residuals.ARIMA) <- c()
}
squared.residuals <- residuals.ARIMA ^ 2
squared.residuals.acf <- acf(squared.residuals)
squared.residuals.acf.lags <- squared.residuals.acf$acf
squared.residuals.limits.acf <- corellogram.limits(squared.residuals.acf)
garch.acf.interval.width <- squared.residuals.limits.acf$ub - squared.residuals.limits.acf$lb
q.garch <- determine.coefficient(squared.residuals.acf.lags, garch.acf.interval.width)
sGARCH.model <- NULL
if(q.garch > 0) { # Variance could be described by some model
squared.residuals.pacf <- pacf(squared.residuals)
squared.residuals.pacf.lags <- squared.residuals.pacf$acf
squared.residuals.limits.pacf <- corellogram.limits(squared.residuals.pacf)
garch.pacf.interval.width <- squared.residuals.limits.pacf$ub - squared.residuals.limits.pacf$lb
p.garch <- determine.coefficient(squared.residuals.pacf.lags, garch.pacf.interval.width)
if(p.garch == 0) {
p.garch <- 1 # Otherwise the model won't work
}
# Measuring the time of parameters selection process end (GARCH)
parameters.selection.end <- Sys.time()
parameters.selection.duration.2 <- difftime(parameters.selection.end, parameters.selection.start, units = "secs")
# Measuring the time of model fitting process start (GARCH)
model.fitting.start <- Sys.time()
sGARCH.model <- create.GARCH.model.weekly(example.ts, residuals.ARIMA, c(p.garch, q.garch), pred.steps)
# Checking GARCH model whether it captures all the information about variance
check.garch.acf <- acf(sGARCH.model@fit$residuals / sGARCH.model@fit$sigma)
check.garch.acf.lags <- check.garch.acf$acf
check.garch.limits.acf <- corellogram.limits(check.garch.acf)
check.garch.acf.interval.width <- check.garch.limits.acf$ub - check.garch.limits.acf$lb
fraction.of.interval.width <- 0.6 # raw estimate
important.lags <- abs(check.garch.acf.lags) > (fraction.of.interval.width * check.garch.acf.interval.width)
fraction.of.important.lags <- sum(important.lags) / length(important.lags)
if(fraction.of.important.lags > 0.05) {
#print("We need to adjust GARCH model because some information was not captured by it. Source: ACF of standardized residuals.")
}
check.garch.pacf <- pacf(sGARCH.model@fit$residuals / sGARCH.model@fit$sigma)
check.garch.pacf.lags <- check.garch.pacf$acf
check.garch.limits.pacf <- corellogram.limits(check.garch.pacf)
check.garch.pacf.interval.width <- check.garch.limits.pacf$ub - check.garch.limits.pacf$lb
fraction.of.interval.width <- 0.6 # raw estimate
important.lags <- abs(check.garch.pacf.lags) > (fraction.of.interval.width * check.garch.pacf.interval.width)
fraction.of.important.lags <- sum(important.lags) / length(important.lags)
if(fraction.of.important.lags > 0.05) {
#print("We need to adjust GARCH model because some information was not captured by it. Source: PACF of standardized residuals.")
}
}
# X. Using ARIMA/GARCH combination for forecasts of mean/variance plus trend forecast
# External regressors - marked weekends
start = train.timeseries$end
discretion = train.timeseries$discretion
duration = pred.steps * discretion
marked.ts.for.prediction = mark.time(start + discretion, start + duration, discretion)
garch.forecast <- ugarchforecast(sGARCH.model,
n.ahead = 1,
n.roll = pred.steps - 1,
out.sample = pred.steps,
external.forecasts = list(mregfor = as.matrix(marked.ts.for.prediction), vregfor = as.matrix(marked.ts.for.prediction)))
# Adjusting by GARCH forecast of mean/variance for residuals
ARIMA.forecast.adjusted.by.GARCH <- resulting.model
ARIMA.forecast.adjusted.by.GARCH$mean <- ARIMA.forecast.adjusted.by.GARCH$mean + ts(as.vector(garch.forecast@forecast$seriesFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$mean),
end = end(ARIMA.forecast.adjusted.by.GARCH$mean),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$mean))
if((length(grep("SARIMA", model.type)) > 0) || (length(grep("SOAARIMA", model.type)) > 0)) {
ARIMA.forecast.adjusted.by.GARCH$lower[,1] <- ARIMA.forecast.adjusted.by.GARCH$lower[,1] - ts(as.vector(garch.forecast@forecast$sigmaFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$lower[,1]),
end = end(ARIMA.forecast.adjusted.by.GARCH$lower[,1]),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$lower[,1]))
ARIMA.forecast.adjusted.by.GARCH$lower[,2] <- ARIMA.forecast.adjusted.by.GARCH$lower[,2] - ts(as.vector(garch.forecast@forecast$sigmaFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$lower[,2]),
end = end(ARIMA.forecast.adjusted.by.GARCH$lower[,2]),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$lower[,2]))
ARIMA.forecast.adjusted.by.GARCH$upper[,1] <- ARIMA.forecast.adjusted.by.GARCH$upper[,1] + ts(as.vector(garch.forecast@forecast$sigmaFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$upper[,1]),
end = end(ARIMA.forecast.adjusted.by.GARCH$upper[,1]),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$upper[,1]))
ARIMA.forecast.adjusted.by.GARCH$upper[,2] <- ARIMA.forecast.adjusted.by.GARCH$upper[,2] + ts(as.vector(garch.forecast@forecast$sigmaFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$upper[,2]),
end = end(ARIMA.forecast.adjusted.by.GARCH$upper[,2]),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$upper[,2]))
} else if(length(grep("SARFIMA", model.type)) > 0) {
ARIMA.forecast.adjusted.by.GARCH$lower <- ARIMA.forecast.adjusted.by.GARCH$lower - ts(as.vector(garch.forecast@forecast$sigmaFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$lower),
end = end(ARIMA.forecast.adjusted.by.GARCH$lower),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$lower))
ARIMA.forecast.adjusted.by.GARCH$upper <- ARIMA.forecast.adjusted.by.GARCH$upper + ts(as.vector(garch.forecast@forecast$sigmaFor),
start = start(ARIMA.forecast.adjusted.by.GARCH$upper),
end = end(ARIMA.forecast.adjusted.by.GARCH$upper),
frequency = frequency(ARIMA.forecast.adjusted.by.GARCH$upper))
}
resulting.model <- ARIMA.forecast.adjusted.by.GARCH
}
# Measuring the time of model fitting process end (GARCH)
model.fitting.end <- Sys.time()
model.fitting.duration.2 <- difftime(model.fitting.end, model.fitting.start, units = "secs")
# Summing durations
parameters.selection.duration <- parameters.selection.duration.1 + parameters.selection.duration.2
model.fitting.duration <- model.fitting.duration.1 + model.fitting.duration.2
resulting.model$parameters.selection.duration <- parameters.selection.duration
resulting.model$model.fitting.duration <- model.fitting.duration
return(resulting.model)
}