212.87.118.245: /* Overriding parameters */

2016-09-25T19:16:34Z

Overriding parameters

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[[Category:Intro to R|092]]
{{Cat|Forecasting}}
== Approaches to Forecasting ==
* ARIMA (AutoRegresive Integrated Moving Average)
* ETS (Exponential smoothing state space model)

We will discuss how those methods work and how to use them.

== Forecast package overview ==
<source lang = "rsplus">
# Install Libraries
install.packages("forecast")
library("forecast")

rawdata <- read.table("http://training-course-material.com/images/1/19/Sales-time-series.txt",h=T)
rawdata$Date <- as.Date(rawdata$Date)
head(rawdata)
plot(rawdata)

# Using build in ts object
sts <- ts(rawdata$Sales,start=2001,frequency=12)
plot(sts)

#par(mfrow = c(2, 2))

# Forecast using ETS method
fc.ets = forecast(sts)
plot(fc.ets)
plot(fc.ets$residuals)
plot(fc.ets$fitted)

# Forecast using ARIMA method
ar = auto.arima(sts)
ar
fc.arima = forecast(ar)
fc.arima
plot(fc.arima)

accuracy(fc.ets)
accuracy(fc.arima)
</source>
:[[File:ClipCapIt-160306-165547.PNG]]

== Exponential Smoothing ==
; Names
* AKA: exponentially weighted moving average (EWMA)
* Equivalent to ARIMA (0,1,1) model with no constant term

;Used for
* smoothed data for presentation
* make forecasts
* simple moving average: past observations are weighted equally
* exponential smoothing: assigns exponentially decreasing weights over time

; Formula
: {''x''''t''} - raw data sequence
: {''s''''t''} - output of the exponential smoothing algorithm (estimate of the next value of ''x'')
: α - ''smoothing factor'', 0 < α < 1.

<math>
\begin{align}
s_1& = x_0\\
s_{t}& = \alpha x_{t-1} + (1-\alpha)s_{t-1}, t>1
\end{align}
</math>

; Choosing right α
* no formal way of choosing α
* statistical technique may be used to optimize the value of α (e.g. OLS)
* the bigger the α the close it gets to naive forecasting (the same ports as original series with one period lag)

== Double Exponential Smoothing ==
* Simple exponential smoothing does not do well when there is a trend (there will be always bias)
* '''Double exponential smoothing''' is a group of methods dealing with the problem

=== Holt-Winters double exponential smoothing ===
; Input
* {''x''''t''} - raw data sequence of observations
* ''t'' = 0

; Model
* {''s''''t''} - smoothed value for time ''t''
* {''b''''t''} - best estimate of the trend at time ''t''

:<math>
\begin{align}
s_1& = x_0\\
b_1& = x_1 - x_0\\
\end{align}

</math>

And for t > 1 by

:<math>
\begin{align}
s_{t}& = \alpha x_{t} + (1-\alpha)(s_{t-1} + b_{t-1})\\
b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\
\end{align}

</math>
where α is the ''data smoothing factor'', 0 < α < 1, and β is the ''trend smoothing factor'', 0 < β < 1.

; Output
* ''F''''t+m'' - an estimate of the value of ''x'' at time ''t+m, m>0'' based on the raw data up to time ''t''

To forecast beyond x''t''
:<math>
\begin{align}
F_{t+m}& = s_t + mb_t
\end{align}
</math>

==Triple exponential smoothing==
* takes into account '''seasonal changes''' as well as trends
* first suggested by Holt's student, Peter Winters, in 1960

; Input
* {''x''''t''} - raw data sequence of observations
* ''t'' = 0
* ''L'' length a cycle of seasonal change

The method calculates:
* a trend line for the data
* seasonal indices that weight the values in the trend line based on where that time point falls in the cycle of length ''L''.

* {''s''''t''} represents the smoothed value of the constant part for time ''t''.
* {''b''''t''} represents the sequence of best estimates of the linear trend that are superimposed on the seasonal changes
* {''c''''t''} is the sequence of seasonal correction factors
* ''c''''t'' is the expected proportion of the predicted trend at any time ''t mod L'' in the cycle that the observations take on
* To initialize the seasonal indices ''c''''t-L'' there must be at least one complete cycle in the data

The output of the algorithm is again written as ''F''''t+m'', an estimate of the value of ''x'' at time ''t+m, m>0'' based on the raw data up to time ''t''. Triple exponential smoothing is given by the formulas

:<math>
\begin{align}
s_0& = x_0\\
s_{t}& = \alpha \frac{x_{t}}{c_{t-L}} + (1-\alpha)(s_{t-1} + b_{t-1})\\
b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\
c_{t}& = \gamma \frac{x_{t}}{s_{t}}+(1-\gamma)c_{t-L}\\
F_{t+m}& = (s_t + mb_t)c_{t-L+((m-1)\pmod L)},
\end{align}
</math>

where α is the ''data smoothing factor'', 0 < α < 1, β is the ''trend smoothing factor'', 0 < β < 1, and γ is the ''seasonal change smoothing factor'', 0 < γ < 1.

The general formula for the initial trend estimate ''b''''0'' is:
:<math>
\begin{align}
b_0& = \frac{1}{L} (\frac{x_{L+1}-x_1}{L} + \frac{x_{L+2}-x_2}{L} + \ldots + \frac{x_{L+L}-x_L}{L})
\end{align}
</math>

Setting the initial estimates for the seasonal indices ''c''''i'' for ''i'' = 1,2,...,L is a bit more involved. If ''N'' is the number of complete cycles present in your data, then:
:<math>
\begin{align}
\\
c_i& = \frac{1}{N} \sum_{j=1}^{N} \frac{x_{L(j-1)+i}}{A_j} \quad \forall i& = 1,2,\ldots,L \\
\end{align}
</math>
where
:<math>
\begin{align}
A_j& = \frac{\sum_{i=1}^{L} x_{L(j-1)+i}}{L} \quad \forall j& = 1,2,\ldots,N
\end{align}
</math>
Note that ''A''''j'' is the average value of ''x'' in the ''j''th cycle of your data.

== ETS ==
* Error, Trend, Seasonality

== Overriding parameters ==
<source lang="rsplus">
rawdata <- read.table("http://training-course-material.com/images/1/19/Sales-time-series.txt",h=T)
rawdata$Date <- as.Date(rawdata$Date)
head(rawdata)
plot(rawdata)

# Using build in ts object
sts <- ts(rawdata$Sales,start=2001,frequency=12)
plot(sts)

# Forecast using ETS method
model.ets = ets(sts,model="ANA")
fc.ets = forecast(model.ets)

model.ets1 = ets(sts,model="AAA",beta=0.2)
fc.ets1 = forecast(model.ets1)
plot(fc.ets1)

accuracy(fc.ets)
accuracy(fc.ets1)
</source>
:[[File:ClipCapIt-160306-165819.PNG]]

== See Also ==
* http://www.jstatsoft.org/v27/i03/paper
* http://robjhyndman.com/talks/MelbourneRUG.pdf

R - Forecasting - Revision history

212.87.118.245: /* Overriding parameters */