R fitting and forecasting daily time series - r

Question

I am working with a daily time serie and I need to build a forecast for 90 days (or maybe more) based on my history - The current time serie has roughly 298 data points.
The issue I have is the famous flat line in the final forecast - and yes I might not have a seasonality but I am trying to work this out. Another issue is how to find the best model and adapt it from here on for this kind of behaviour.
I created a test case to investigate this further and any help is appreciated.
Thanks,
To start with
x <- day_data # My time serie
z <- 90 # Days to forecast
low_bound_date <- as.POSIXlt(min(x$time), format = "%m/%d/%Y") # oldest date in the DF.
> low_bound_date
[1] "2015-12-21 PST"
low_bound_date$yday
> low_bound_date$yday # Day in Julian
[1] 354
lbyear <- as.numeric(substr(low_bound_date, 1, 4))
> lbyear
[1] 2015
This is my time serie content
> ts
Time Series:
Start = c(2065, 4)
End = c(2107, 7)
Frequency = 7
[2] 20.73 26.19 27.51 26.11 26.28 27.58 26.84 27.00 26.30 28.75 28.43 39.03 41.36 45.42 44.80 45.33 47.79 44.70 45.17
[20] 34.90 32.54 32.75 33.35 34.76 34.11 33.59 33.60 38.08 30.45 29.66 31.09 31.36 31.96 29.30 30.04 30.85 31.13 25.09
[39] 17.88 23.73 25.31 31.30 35.18 34.13 34.96 35.12 27.36 38.33 38.59 38.14 38.54 41.72 37.15 35.92 37.37 32.39 30.64
[58] 30.57 30.66 31.16 31.50 30.68 32.21 32.27 32.55 33.61 34.80 33.53 33.09 20.90 6.91 7.82 15.78 7.25 6.19 6.38
[77] 38.06 39.82 35.53 38.63 41.91 39.76 37.26 38.79 37.74 35.61 39.70 35.79 35.36 29.63 22.07 35.39 35.99 37.35 38.82
[96] 25.80 21.31 18.85 9.52 20.75 36.83 44.12 37.79 34.45 36.05 16.39 21.84 31.39 34.26 31.50 30.87 28.88 42.83 41.52
[115] 42.34 47.35 44.47 44.10 44.49 26.89 18.17 40.44 43.93 41.56 39.98 40.31 40.59 40.17 40.22 40.50 32.68 35.89 36.06
[134] 34.30 22.67 12.56 13.29 12.34 28.00 35.27 36.57 33.78 32.15 33.58 34.62 30.96 32.06 33.05 30.66 32.47 30.42 32.83
[153] 31.74 29.39 22.39 12.58 16.46 5.36 4.01 15.32 32.79 31.66 32.02 27.60 31.47 31.61 34.96 27.77 31.91 33.94 33.43
[172] 26.94 28.38 21.42 24.51 23.82 31.71 26.64 27.96 29.29 29.25 28.70 27.02 27.62 30.90 27.46 27.37 26.46 27.77 13.61
[191] 5.87 12.18 5.68 4.15 4.35 4.42 16.42 25.18 26.06 27.39 27.57 28.86 15.18 5.19 5.61 8.28 7.78 5.13 4.90
[210] 5.02 5.27 16.31 25.01 26.19 25.96 24.93 25.53 25.56 26.39 26.80 26.73 26.00 25.61 25.90 25.89 13.80 6.66 6.41
[229] 5.28 5.64 5.71 5.38 5.76 7.20 7.27 5.55 5.31 5.94 5.75 5.93 5.77 6.57 5.52 5.51 5.47 5.69 19.75
[248] 29.22 30.75 29.63 30.49 29.48 31.83 30.42 29.27 30.40 29.91 32.00 30.09 28.93 14.54 7.75 5.63 17.17 22.27 24.93
[267] 35.94 37.42 33.13 25.88 24.27 37.64 37.42 38.33 35.20 21.32 7.32 4.81 5.17 17.49 23.77 23.36 27.60 26.53 24.99
[286] 24.22 23.76 24.10 24.22 27.06 25.53 23.40 37.07 26.52 25.19 28.02 28.53 26.67
First step, I get my data in ts
day_data_ts <- ts(x$avg_day, start = c(lbyear,low_bound_date$yday), frequency=7)
plot(day_data_ts)
plot_ts
acf(day_data_ts)
acf_ts
Second step, I get my data in msts
day_data_msts <- msts(x$avg_day, seasonal.periods=c(7,365.25), start = c(lbyear,low_bound_date$yday))
plot(day_data_msts)
acf(day_data_msts)
I did several fitting iterations to try and figure out the best fit and forecast model.
First fitting test is with the ts only.
fit1 <- HoltWinters(day_data_ts)
> fit1
Holt-Winters exponential smoothing with trend and additive seasonal component.
Call: HoltWinters(x = day_data_ts)
Smoothing parameters: alpha: 1 beta : 0.006757112 gamma: 0
Coefficients:
[,1]
a 28.0922449
b 0.1652477
s1 0.6241837
s2 1.9084694
s3 0.9913265
s4 0.8198980
s5 -1.7015306
s6 -1.2201020
s7 -1.4222449
fit2 <- tbats(day_data_ts)
> fit2
BATS(1, {0,0}, 0.8, -)
Parameters: Alpha: 1.309966 Beta: -0.3011143 Damping Parameter: 0.800001
Seed States:
[,1]
[1,] 15.282259
[2,] 2.177787
Sigma: 5.501356 AIC: 2723.911
fit3 <- ets(day_data_ts)
> fit3
ETS(A,N,N)
Smoothing parameters: alpha = 0.9999
Initial states: l = 25.2275
sigma: 5.8506
AIC AICc BIC
2756.597 2756.678 2767.688
fit4 <- auto.arima(day_data_ts)
> fit4
ARIMA(1,1,2)
Coefficients:
ar1 ma1 ma2
0.7396 -0.6897 -0.2769
s.e. 0.0545 0.0690 0.0621
sigma^2 estimated as 30.47: log likelihood=-927.9
AIC=1863.81 AICc=1863.94 BIC=1878.58
Second test is using msts. I also changed the ets model to MAM.
fit5 <- tbats(day_data_msts)
> fit5
BATS(1, {0,0}, 0.8, -)
Parameters: Alpha: 1.309966 Beta: -0.3011143 Damping Parameter: 0.800001
Seed States:
[,1]
[1,] 15.282259
[2,] 2.177787
Sigma: 5.501356 AIC: 2723.911
fit6 <- ets(day_data_msts, model="MAN")
> fit6
ETS(M,A,N)
Smoothing parameters: alpha = 0.9999 beta = 9e-04
Initial states: l = 52.8658 b = 3.9184
sigma: 0.3459
AIC AICc BIC
3042.744 3042.949 3061.229
fit7 <- auto.arima(day_data_msts)
> fit7
ARIMA(1,1,2)
Coefficients:
ar1 ma1 ma2
0.7396 -0.6897 -0.2769
s.e. 0.0545 0.0690 0.0621
sigma^2 estimated as 30.47: log likelihood=-927.9
AIC=1863.81 AICc=1863.94 BIC=1878.58

Answer 1

You can forecast on previously estimated model as follows (use built in timeseries LakeHuron):
library(forecast)
y <- LakeHuron
tsdisplay(y)
# estimate ARMA(1,1)
mod_2 <- Arima(y, order = c(1, 0, 1))
#make forecast for 5 periods (years in this case)
fHuron <- forecast(mod_2, h = 5)
#show results in table
fHuron
#plot results
plot(fHuron)
This will give you:
Pay attention that ARIMA model bases its forecast on previous values, so if we make prediction on many periods the model will use already predicted values to predict next. Which will reduce accuracy.
To fit optimal ARIMA model use this function:
library(R.utils) #for the function 'withTimeout'
fitARIMA<-function(timeseriesObject, timout)
{
final.aic <- Inf
final.order <- c(0,0,0)
for (p in 0:5) for (q in 0:5) {
if ( p == 0 && q == 0) {
next
}
arimaFit = tryCatch(
withTimeout(arima(timeseriesObject
,order=c(p, 0, q))
,timeout = timeout)
,error=function( err ) FALSE
,warning=function( err ) FALSE )
if( !is.logical( arimaFit ) ) {
current.aic <- AIC(arimaFit)
if (current.aic < final.aic) {
final.aic <- current.aic
final.order <- c(p, 0, q)
final.arima <- arima(timeseriesObject, order=final.order)
}
} else {
next
}
}
final.order<-c(final.order,final.aic)
final.order
}