Transcript Time Series Analysis - University of Technology

Time Series Forecasting– Part I
What is a Time Series ?
 Components of Time Series
 Evaluation Methods of Forecast
 Smoothing Methods of Time Series

by Duong Tuan Anh
Faculty of Computer Science and Engineering
September 2011
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What is a Time series ?
A time series is a collection of observations made
sequentially in time.
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0
50
100
150
200
250
300
350
400
450
500
Examples: Financial time series, scientific time series
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Time series models
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Regression models
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Predict the response over time of the variable
under study to changes in one or more of the
explanatory variables.
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Deterministic models of time series
 Stochastic models of time series
All the three kinds of models can be used for
forecasting.
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Components of a time series
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The pattern or behavior of the data in a time series
has several components.
Theoretically, any time series can be decomposed
into:
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Trend
Cyclical
Seasonal
Irregular
However, this decomposition is often not straightforward because these factors interact.
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Trend component
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The trend component accounts for the gradual shifting of the time
series to relatively higher or lower values over a long period of
time.
Trend is usually the result of long-term factors such as changes in
the population, demographics, technology, or consumer
preferences.
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Seasonal component
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The seasonal component accounts for regular patterns of
variability within certain time periods, such as a year.
The variability does not always correspond with the seasons of
the year (i.e. winter, spring, summer, fall).
There can be, for example, within-week or within-day “seasonal”
behavior.
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Cyclical component
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Any regular pattern of sequences of values above and
below the trend line lasting more than one year can
be attributed to the cyclical component.
Usually, this component is due to multiyear cyclical
movements in the economy.
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Evaluating Methods of forecasts
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Forecasting method is selected - many times by
intuition, previous experience, or computer resource
availability
Divide the data into two sections - an initialization part
and a test part
Use the forecast technique to determine the fitted
values for the initialization data set
Use the forecast technique to forecast the test data
set and determine the forecast errors
Evaluate errors (MAD, MPE, MSD, MAPE)
Use the technique, modify, or develop new model
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Evaluation Methods of Forecasts
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There are three measures of accuracy of the fitted
models: MAPE, MAD and MSD for each of the sample
forecasting and smoothing methods.
For all three measures, the smaller the value, the
better the fit of the model.
Use these statistics to compare the fit of the different
methods.
MAPE (Mean Absolute Percentage Error) measure the
accuracy of fitted time series values. It expresses
accuracy as a percentage.
|(yt-yt’)/yt|
MAPE = --------------  100
(yt  0)
n
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MAPE, MAD, and MSD
where yt is the actual value, yt’ is the fitted value and n is
the number of observations.
 MAD (Mean Absolute Deviation) expresses accuracy
in the same units as the data, which help
conceptualize the amount of error.
|yt-yt’|
MAD = ---------n
where yt is the actual value, yt’ is the fitted value and n is
the number of observations.
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MAPE, MAD, and MSD

MSD(Mean Squared Deviation) is a more sensitive
measure of an unusually large forecast error than
MAD.
(yt-yt’)2
MSD = ---------n
where yt is the actual value, yt’ is the fitted value and n is
the number of observations.
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Methods of smoothing time series
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Arithmetic Moving Average
Exponential Smoothing Methods
Holt-Winters method for Exponential Smoothing
Smoothing a time series: to eliminate some of short-term
fluctuations.
Smoothing also can be done to remove seasonal
fluctuations, i.e., to deseasonalize a time series.
These models are deterministic in that no reference is
made to the sources or nature of the underlying
randomness in the series.
The models involves extrapolation techniques.
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Averaging Methods
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Simple Averages - quick, inexpensive (should only be
used on stationary data)
 Moving Average method consists of computing an
average of the most recent n data values for the series
and using this average for forecasting the value of the
time series for the next period.
 Moving averages are useful if one can assume item to
be forecast will stay steady over time.
 Series of arithmetic means – used only for smoothing,
provides overall impression of data over time
 (most recent n data items)
Moving Average = -----------------------------------------
n
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Moving average methods
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Works best with stationary data.
The smaller the number, the more weight given to
recent periods.
A smaller number is desirable when there are
sudden shifts in the level of the series.
The greater the number, less weight is given to
more recent periods.
The larger the order of the moving average, the
greater the smoothing effect. Larger n when there
are wide, infrequent fluctuations in the data.
By smoothing recent actual values, removes
randomness.
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Weighted Moving Averages
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Weighted Moving Average - place more weight on
recent observations. Sum of the weights needs to
equal 1.
Used when trend is present
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Older data usually less important
(weight for period n)(Value in period n)
WMA = --------------------------------------------------------
weights
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Notes on Moving Averages
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MA models do not provide information about forecast
confidence.
We can not calculate standard errors.
We can not explain the stochastic component of the
time series. This stochastic component creates the
error in our forecast.
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Exponential Smoothing Methods
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Single Exponential Smoothing (Averaging)
 Double Exponential Smoothing & Holt’s Method
 Winter’s Model.
Note:
- Single Exponential Smoothing is for series without
trend and without seasonal component.
- Double Exponential Smoothing is for series with trend
and without seasonal component.
- Winter’s model is for for series with trend and
seasonal component.
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Single Exponential Smoothing
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Continually revising a forecast in light of more recent
experiences. Averaging (smoothing) past values of a
series in a decreasing (exponential) manner. The
observations are weighted with more weight being
given to the more recent observations
At = αYt-1 + (1 – α) At-1
(S1)
New forecast = α  (old observation) + (1- α)  old
forecast
Here we denote the original series by yt and the
smoothed series by At.
The equation can be rewritten as:
At = At-1 + α(Yt –At-1)
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Single Exponential Smoothing
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When looking at the formula – new forecast is really
the old forecast plus a times the error in the old
forecast
To get started, we need a smoothing constant a, an
initial forecast, and an actual value. We can use the
first actual as the forecast value or we can average the
first n observations.
The smoothing constant serves as the weighting
factor. When a is close to 1, the new forecast will
include a substantial adjustment for any error that
occurred in the preceding forecast. When a is close to
0, the new forecast is very similar to the old forecast.
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Single Exponential Smoothing (cont.)
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The smoothing constant α is not an arbitrary choice but generally falls between 0.1 and 0.5. If we want
predictions to be stable and random variation
smoothed, use a small a. If we want a rapid response,
a larger a value is required.
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Why Exponential?
At = aYt-1 + (1- a)At-1
At-1 = aYt-2 + (1- a)At-2
At-2 = aYt-3 + (1- a)At-3
…
At = aYt-1 + (1- a) aYt-2 + (1- a) a2Yt-3 +
…. + (1 - a) akYt-k+1
ak decreases exponentially.
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Sales data Single Exponential Smoothing .1
Actual
850
Sales
Smoothed
750
Forecast
650
Actual
Smoothed
Forecast
550
450
Smoothing Constant
350
Alpha:
0.100
250
MAPE:
37.0
MAD:
134.9
MSD:
27735.5
150
0
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20
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Time
The small a here smooths the data.
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Sales data Single Exponential Smoothing .6
940
Actual
840
Smoothed
Forecast
740
Actual
Smoothed
Forecast
Sales
640
540
440
Smoothing Constant
340
240
140
0
5
10
15
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Alpha:
0.600
MAPE:
36.5
MAD:
134.5
MSD:
22248.4
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Time
The large a in this example responds quickly to the data.
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Tracking
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Use a tracking signal (measure of errors over time) and
setting limits. For example, if we forecast 10 periods, count
the number of negative and positive errors. If the number
of positive errors is substantially less or greater than n/2,
then the process is out of control.
 Can also use 95% prediction interval (1.96 * sqrt (MSE)). If
the forecast error is outside of the interval, use a new
optimal a.
 Looking back at the .1 single exponential smoothing:
1.96*sqrt(24261) = +-305 Observation #21 is out-of-control. We
need to re-evaluate alpha level because this technique is
biased.
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Exponential Smoothing Adjusted for Trend:
Holt’s method
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In some situations, the observed data are trending and
contain information that allows the anticipation of
future upward movement.
In that case, a linear trend forecast function is needed.
Holt’s smoothing method allows for evolving local
linear trend in a time series and can be used to
forecast.
When there is a trend, an estimate of the current slope
and the current level is required.
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Holt’s Method
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Holt’s method uses two coefficients.
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a is the smoothing constant for the level
b is the trend smoothing constant - used to remove
random error.
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Advantage of Holt’s method: it provides
flexibility in selecting the rates at which the
level and trend are tracked.
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Equations in Holt’s method
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The exponentially smoothed series, or the current level estimate:
At = aYt + (1- a)(At-1 + Tt-1)
(S2)
The trend estimate:
Tt = b(At – At-1)+(1- b)Tt-1
(S3)
Forecast p periods into the future:
Y’t+p = At + pTt
where
At = new smoothed value (estimate of current level)
Yt = new actual value at time t.
Tt = trend estimate
Y’t+p = forecast for p periods into the future.
a = smoothing constant for the level
b = smoothing constant for trend estimate
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How to initiate Holt’s method
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To get started, initial values for A and T in equation
(S2) and (S3) must be determined.
One approach is to set A1 to Y1 and T1 to zero.
The second approach is to use the average of the first
five or six observations as A1. T1 is then estimated by
the slope of a line that is fit to these five or six
observations.
Holt’s method
Holt exponential smoothing with parameters a = 1.0 and b = 0.099 for time
series of electricity consumption.
Winter’s Method
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Winters’ method is an easy way to account for
seasonality when data have a seasonal pattern.
It extends Holt’s Method to include an estimate for
seasonality.
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a is the smoothing constant for the level
b is the trend smoothing constant - used to remove random
error.
g smoothing constant for seasonality
This formula removes seasonal effects. The forecast
is modified by multiplying by a seasonal index.
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Winter’s Method
The four equations used in Winters’
(multiplication) smoothing are:
 The smoothed series or level estimate:
At = aYt /St-s+ (1- a)(At-1 + Tt-1)
where
 The trend estimate:
At = new smoothed value (estimate of
Tt = b(At – At-1)+(1- b) Tt-1
current level)
 The seasonality estimate:
Yt = new actual value at time t.
St = gYt/At + (1- g)St-s
 Forecast p periods into the future: Tt = trend estimate
Y’t+p = forecast for p periods into the future.
Y’t+p = (At + pTt)St-s+p
Tt = trend estimate
a = smoothing constant for the level
WINTERS’ METHOD
b = smoothing constant for trend estimate
Is also called TRIPLE
g = smoothing constant for seasonality
EXPONENTIAL SMOOTHING )
estimate
p = periods to be forecast into the future
s = length of seasonality
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How to initiate Winter’s method
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To begin the Winter’s method, the initial values for the
smoothed series At, the trend Tt and the seasonal
indices St must be set.
One approach is to set the first estimate of At to Y1.
The trend is estimated to 0 and the seasonal indices
are each set to 1.0.
Winter’s Method
Winter's Method Sales Data
Sales
Actual
900
Smoothed
800
Forecast
700
Actual
Smoothed
Forecast
600
500
Smoothing Constants
Alpha (level):
0.400
Gamma (trend): 0.100
Delta (season): 0.300
400
300
200
MAPE:
MAD:
MSD:
100
0
5
10
15
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15.21
63.55
7636.86
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Time
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