1. Forecasting
Purpose is to forecast, not to explain the historical pattern
Models for forecasting may not make sense as a description for ”physical” behaviour of the time series
Common sense and mathematics in a good combination produces ”optimal” forecasts
With time series regression models, forecasting (prediction) is a natural step and forecasting limits (intervals) can be constructed
With Classical decomposition, forecasting may be done, but estimation of accuracy lacks and no forecasting limits are produced
Classical decomposition is usually combined with Exponential smoothing methods

2. Exponential smoothing
Use the historical data to forecast the future
Let different parts of the history have different impact on the forecasts
Forecast model is not developed from any statistical theory

3. Single exponential smoothing
Given are historical values y1,y2,…yT
Assume data contains no trend

4. Algorithm for forecasting:
where
is a smoothing parameter with value between 0 and 1
The forecast procedure is a recursion formula
How shall we choose α?
Where should we start, i.e. which is the initial value ?

5. For long length time series:
Use a part (usually first half) of the historical data
and calculate their average:
Set
Update with the rest of the historical data
using the recursion formula

6. Example: Sales of everyday commodities
Year Sales values
Note! This time series is short but we use it for illustration purposes!

7. Calculate the average of the first 8 observations of the series:
Set
Assume first that the sales are very stable, i.e. during the period the background mean value is assumed not to change
Set α to be relatively small. This means that the latest observation plays a less role than the history in the forecasts. Thumb rule: 0.05 < α < 0.3
E.g. Set α=0.1
Update using the next 8 values of the historical data

9. Forecasts:

10. For short length time series:
Calculate the average of all historical data i.e.
Update from the beginning of the time series:
There are a lot of alternatives:
Average of all data, update from the middle of the series
Average of the first half, update from beginning
etc.

12. Analysis of example data with MINITAB

14. MTB > Name c3 "FORE1" c4 "UPPE1" c5 "LOWE1"
MTB > SES 'Sales values';
SUBC> Weight 0.1;
SUBC> Initial 8;
SUBC> Forecasts 3;
SUBC> Fstore 'FORE1';
SUBC> Upper 'UPPE1';
SUBC> Lower 'LOWE1';
SUBC> Title "SES alpha=0.1".
Single Exponential Smoothing for Sales values
Data Sales values
Length 16
Smoothing Constant
Alpha 0.1

15. Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper

16. MINITAB uses smoothing from 1st value!

17. Assume now that the sales are less stable, i. e
Assume now that the sales are less stable, i.e. during the period the background mean value is possibly changing.
(Note that a change means an occasional “level shift” , not a systematic trend)
Set α to be relatively large. This means that the latest observation becomes more important in the forecasts.
E.g. Set α=0.5 (A bit exaggerated)

18. Single Exponential Smoothing for Sales values
Data Sales values
Length 16
Smoothing Constant
Alpha 0.5
Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper

19. Slightly narrower prediction intervals

20. We can also use some adaptive procedure to continuosly evaluate the forecast ability and maybe change the smoothing parameter over time
Alt. We can run the process with different alphas and choose the one that performs best. This can be done with the MINITAB procedure.

22. Yet, narrower prediction intervals
Single Exponential Smoothing for Sales values
---
Smoothing Constant
Alpha
Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper
Yet, narrower prediction intervals

23. Exponential smoothing for times series with trend and/or seasonal variation
Double exponential smoothing (one smoothing parameter) for trend
Holt’s method (two smoothing parameters) for trend
Multiplicative Winter’s method (three smoothing parameters) for seasonal (and trend)
Additive Winter’s method (three smoothing parameters) for seasonal (and trend)

24. Modern methods The classical approach: Method Pros Cons
Time series regression
Easy to implement
Fairly easy to interpret
Covariates may be added (normalization)
Inference is possible (though sometimes questionable)
Static
Normal-based inference not generally reliable
Cyclic component hard to estimate
Decomposition
Easy to interpret
Possible to have dynamic seasonal effects
Cyclic components can be estimated
Descriptive (no inference per def)
Static in trend

25. Explanation to the static behaviour:
The classical approach assumes all components except the irregular ones (i.e. t and IRt ) to be deterministic, i.e. fixed functions or constants
To overcome this problem, all components should be allowed to be stochastic, i.e. be random variates.
A time series yt should from a statistical point of view be treated as a stochastic process.
We will interchangeably use the terms time series and process depending on the situation.

26. Stationary and non-stationary time series
Characteristics for a stationary time series:
Constant mean
Constant variance
 A time series with trend is non-stationary!

27. ARIMA – models Box-Jenkins models
A stationary times series can be modelled on basis of the serial correlations in it.
A non-stationary time series can be transformed into a stationary time series, modelled and back-transformed to original scale (e.g. for purposes of forecasting)
ARIMA – models
Auto Regressive,
Integrated,
Moving Average
This part has to do with the transformation
These parts can be modelled on a stationary series

28. Different types of transformation
1. From a series with linear trend to a series with no trend:
First-order differences zt = yt – yt – 1
MTB > diff c1 c2

29. Note that the differenced series varies around zero.

30. 2. From a series with quadratic trend to a series with no trend:
Second-order differences
wt = zt – zt – 1 = (yt – yt – 1) – (yt – 1 – yt – 2) = yt – 2yt – 1 + yt – 2
MTB > diff 2 c3 c4

32. 3. From a series with non-constant variance (heteroscedastic) to a series with constant variance (homoscedastic):
Box-Cox transformations (per def 1964)
Practically  is chosen so that yt +  is always > 0
Simpler form: If we know that yt is always > 0 (as is the usual case for measurements)

33. The log transform (ln yt ) usually also makes the data ”more” normally distributed
Example: Application of root (yt ) and log (ln yt ) transforms

34. AR-models (for stationary time series)
Consider the model
yt = δ + ·yt –1 + at
with {at } i.i.d with zero mean and constant variance = σ2
and where δ (delta) and  (phi) are (unknown) parameters
Set δ = 0 by sake of simplicity  E(yt ) = 0
Let R(k) = Cov(yt,yt-k ) = Cov(yt,yt+k ) = E(yt ·yt-k ) = E(yt ·yt+k )
 R(0) = Var(yt) assumed to be constant

35. Now:
R(0) = E(yt ·yt ) = E(yt ·( ·yt-1 + at ) =  · E(yt ·yt-1 ) + E(yt ·at ) =
=  ·R(1) + E(( ·yt-1 + at ) ·at ) =  ·R(1) +  · E(yt-1 ·at ) + E(at ·at )=
=  ·R(1) σ (for at is independent of yt-1 )
R(1) = E(yt ·yt+1 ) = E(yt ·( ·yt + at+1 ) =  · E(yt ·yt ) + E(yt ·at+1 ) =
=  ·R(0) (for at+1 is independent of yt )
R(2) = E(yt ·yt+2 ) = E(yt ·( ·yt+1 + at+2 ) =  · E(yt ·yt+1 ) +
+ E(yt ·at+2 ) =  ·R(1) (for at+1 is independent of yt ) 

36. R(0) =  ·R(1) + σ2
R(1) =  ·R(0) Yule-Walker equations
R(2) =  ·R(1) …
 R(k ) =  ·R(k – 1) =…=  k·R(0)
R(0) =  2 ·R(0) + σ2 

37. Note that for R(0) to become positive and finite (which we require from a variance) the following must hold:
This in effect the condition for an AR(1)-process to be weakly stationary
Now, note that

38. ρk is called the Autocorrelation function (ACF) of yt
”Auto” because it gives correlations within the same time series.
For pairs of different time series one can define the Cross correlation function which gives correlations at different lags between series.
By studying the ACF it might be possible to identify the approximate magnitude of 

39. Examples:

42. The look of an ACF can be similar for different kinds of time series, e.g. the ACF for an AR(1) with  = 0.3 could be approximately the same as the ACF for an Auto-regressive time series of higher order than 1 (we will discuss higher order AR-models later)
To do a less ambiguous identification we need another statistic:
The Partial Autocorrelation function (PACF):
υk = Corr (yt ,yt-k | yt-k+1, yt-k+2 ,…, yt-1 )
i.e. the conditional correlation between yt and yt-k given all observations in-between.
Note that –1  υk  1

43. A concept sometimes hard to interpret, but it can be shown that
for AR(1)-models with  positive the look of the PACF is
and for AR(1)-models with  negative the look of the PACF is

44. Assume now that we have a sample y1, y2,…, yn from a time series assumed to follow an AR(1)-model.
Example:

45. The ACF and the PACF can be estimated from data by their sample counterparts:
Sample Autocorrelation function (SAC):
if n large, otherwise a scaling
might be needed
Sample Partial Autocorrelation function (SPAC)
Complicated structure, so not shown here

46. The variance function of these two estimators can also be estimated
 Opportunity to test
H0: k = 0 vs. Ha: k  0 or
H0: k = 0 vs. Ha: k  0
for a particular value of k.
Estimated sample functions are usually plotted together with critical limits based on estimated variances.

47. Example (cont) DKK/USD exchange:
SAC:
SPAC:
Critical limits

48. Ignoring all bars within the red limits, we would identify the series as being an AR(1) with positive .
The value of  is approximately 0.9 (ordinate of first bar in SAC plot and in SPAC plot)

49. Higher-order AR-models
AR(2): or
yt-2 must be present
AR(3):
or other combinations with  3 yt-3
AR(p):
i.e. different combinations with  p yt-p

50. Stationarity conditions:
For p > 2, difficult to express on closed form.
For p = 2:
The values of 1 and 2 must lie within the blue triangle in the figure below:

51. Typical patterns of ACF and PACF functions for higher order stationary AR-models (AR( p )):
ACF: Similar pattern as for AR(1), i.e. (exponentially) decreasing
bars, (most often) positive for  1 positive and alternating for 1 negative.
PACF: The first p values of k are non-zero with decreasing
magnitude. The rest are all zero (cut-off point at p )
(Most often) all positive if  1 positive and alternating if  negative

52. Examples:
AR(2),  1 positive:
AR(5),  1 negative: