CHAPTER 1: Decomposition Methodspersonal.cb.cityu.edu.hk/msawan/teaching/ms6215/MS6215Ch1.pdf · CHAPTER 1: Decomposition Methods Prof. Alan Wan 1/48. 1. Data Types and Causal vs.Time

1. Data Types and Causal vs.Time Series Models2. Classical Decomposition of Time Series

3. Multiplicative Decomposition Model4. Measuring Forecast Accuracy and Forecast Classification

5. Additive Decomposition Model

CHAPTER 1: Decomposition Methods

Prof. Alan Wan

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Table of contents

1. Data Types and Causal vs.Time Series Models

2. Classical Decomposition of Time Series

3. Multiplicative Decomposition Model

4. Measuring Forecast Accuracy and Forecast Classification


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Types of Data

I Time series data: a sequence of observations measured overtime, usually at equally spaced intervals, e.g., weekly, monthlyand annually).Examples of time series data include: Quarterly GrossDomestic Product (GDP), Annual rainfall volume, daily stockmarket index, etc.

I Cross sectional data: data on one or more variables collectedat the same point in time.

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Types of Data

I Time series data: a sequence of observations measured overtime, usually at equally spaced intervals, e.g., weekly, monthlyand annually).Examples of time series data include: Quarterly GrossDomestic Product (GDP), Annual rainfall volume, daily stockmarket index, etc.

I Cross sectional data: data on one or more variables collectedat the same point in time.

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Causal vs. Time Series Models

I Causal (regression) models: the investigator specifies somebehaviourial relationship and estimates the unknownparameters using regression techniques.

I Time series models: the investigator uses past data of thetarget variable to forecast the present and future values of thevariable.

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I Causal (regression) models: the investigator specifies somebehaviourial relationship and estimates the unknownparameters using regression techniques.

I Time series models: the investigator uses past data of thetarget variable to forecast the present and future values of thevariable.

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I Causal models provide information on the causal relationshipbetween the target variable and its determinants (theregressors).

I On the other hand, there are many instances when onecannot, or prefers not to, construct causal models due toreasons such as

1. insufficient information on the behaviourial relationship2. lack of, or conflicting, theories3. insufficient data on the explanatory variables4. superior forecasts produced by time series models

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I Causal models provide information on the causal relationshipbetween the target variable and its determinants (theregressors).

I On the other hand, there are many instances when onecannot, or prefers not to, construct causal models due toreasons such as

1. insufficient information on the behaviourial relationship2. lack of, or conflicting, theories3. insufficient data on the explanatory variables4. superior forecasts produced by time series models

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I Here are some of the direct benefits of using time seriesmodels:

1. little storage capacity is needed2. some time series models are automatic in that user

intervention is not required to update the forecasts each period3. some time series models are evolutionary in that the models

adapt as new information is received

I This course is mainly concerned with forecasting using timeseries models

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I Here are some of the direct benefits of using time seriesmodels:

1. little storage capacity is needed2. some time series models are automatic in that user

intervention is not required to update the forecasts each period3. some time series models are evolutionary in that the models

adapt as new information is received

I This course is mainly concerned with forecasting using timeseries models

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Classical Decomposition of Time Series

I Trend (TC) - does not necessarily imply a monotonicallyincreasing or decreasing series but simply a lack of constantmean, although in practice, we often use a linear or quadraticfunction to predict the trend.

I Cycle (CL) - refers to patterns or waves in the data that arerepeated after approximately equal intervals withapproximately equal intensity. For example, some economistsbelieve that business cycles repeat themselves every 4 or 5years.

I Seasonal (SN) - refers to a cycle of one year’s duration.

I Random (Irregular) (IR) - refers to the (unpredictable)variations not covered by the three components above.

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We are concerned with two types of Decomposition Models:

I Multiplicative Model:

Yt = TCt × SNt × CLt × IRt

I Additive Model:

Yt = TCt + SNt + CLt + IRt

The goal is to find estimates of the four components.

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I Additive Model:



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I Additive Model:



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Multiplicative Decomposition Model

Example 1: U.S. Retail and Food Services Sales from 1996Q1 ro2008Q1:

US Retail & Food Services Sales

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

500,000

Q1-96

Q3-96

Q1-97

Q3-97

Q1-98

Q3-98

Q1-99

Q3-99

Q1-00

Q3-00

Q1-01

Q3-01

Q1-02

Q3-02

Q1-03

Q3-03

Q1-04

Q3-04

Q1-05

Q3-05

Q1-06

Q3-06

Q1-07

Q3-07

Q1-08

Time

Sale

s Y

(t)

(in

MN

US

$)

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Example 2: Quarterly Number of Visitor Arrivals in Hong Kongfrom 2002Q1 to 2008Q1:

Number of Visitor Arrivals in Hong Kong

0

500000

1000000

1500000

2000000

2500000

3000000

Q1- 02

Q3- 02

Q1- 03

Q3- 03

Q1- 04

Q3- 04

Q1- 05

Q3- 05

Q1- 06

Q3- 06

Q1- 07

Q3- 07

Q1- 08

Time

Nu

mb

er o

f V

isit

ors

Y(t

)

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I Cycles are often difficult to identify with a short time series.

I Classical decomposition typically combines cycles and trend asone entity, that is,

Yt = TCt × SNt × IRt

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Consider the following 4-year quarterly time series on sales volume:

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A plot of the series reveals the following pattern:

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I We first estimate the seasonal component (SNt).

I Note that Yt = TCt × SNt × IRt

∴ SNt = YtTCt×IRt

I

Moving Average for periods 1− 4 =72 + 110 + 117 + 172

4= 117.75


4= 118.75

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I


4= 117.75


4= 118.75

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I


4= 117.75


4= 118.75

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Assuming that the average of the observations is also the medianof the observations, the moving average (MA) forperiods 1-4, 2-5, 3-6 are centered at t = 2.5, 3.5 and 4.5 respectively.

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I To obtain averages that center at periods 3, 4, 5, etc. wecalculate the mean of every two consecutive moving averagesas follows:

I

Centered Moving Average for period 3 =117.75 + 118.75

2= 118.25


2= 119

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I To obtain averages that center at periods 3, 4, 5, etc. wecalculate the mean of every two consecutive moving averagesas follows:

I


2= 118.25


2= 119

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I Because the centered moving average (CMA) contains noseasonality and no or little irregularity, the seasonalcomponent may be estimated by

SNt = YtCMAt

I For example,

SN3 = 117118.25 = 0.989

SN4 = 172119 = 1.445

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I Because the centered moving average (CMA) contains noseasonality and no or little irregularity, the seasonalcomponent may be estimated by

SNt = YtCMAt

I For example,

SN3 = 117118.25 = 0.989

SN4 = 172119 = 1.445

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I After all the SNt′s have been computed, they are further

averaged to eliminate irregularities in the series.

I We also adjust the seasonal indices so that they sum to thenumber of seasons in a year, i.e., 4 for quarterly data, 12 formonthly data. (Why?)

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Multiplicative Decomposition ModelJ~ ... ., ~r r :-.. ·.·-~::.~;if~":~· "';i;:; -~;''.;~1:}~{'.'

~ I\ Period (t) Year Quarter Sales MA(t) CMA(t) SN(t) SN(t)

1 1 1 72 0.606312789 2 2 110 0.919068591 3 3 117 117.75 118.25 0.989429175 0.992121312 4 4 172 118.75 119 1.445378151 1.482497308 5 2 1 76 119.25 120.875 0.628748707 0.606312789 6 2 112 122.5 125.25 0.894211577 0.919068591 7 3 130 128 128.25 1.013645224 0.992121312 8 4 194 128.5 129.375 1.499516908 1.482497308 9 3 1 78 130.25 130 0.6 0.606312789

10 2 119 129.75 130.625 0.911004785 0.919068591 11 3 128 131.5 131.875 0.970616114 0.992121312 12 4 201 132.25 134.125 1.49860205 1.482497308 13 4 1 81 136 137.625 0.588555858 0.606312789 14 2 134 139.25 141.125 0.949512843 0.919068591 15 3 141 143 0.992121312 16 4 216 1.482497308

A. Quarter Average Final SN(t)

1 (0.628748707 + 0.6 + 0.588555858)/3 = 0.605768 0.606312789 2 f0.894211577 + 0.911004185 + 6.94951284~ = 0.918243 0.919968591 ----~--

3 (0.989429175 + 1.013645224 + 0.970616114)/3 = 0.99123 0.992121312 4 (1.445378151+1.499516908 + 1.49860205)/3 = 1.481166 1.482497308

Sum= 3.996407 4

Normalizing Factor: 4/3.996407 = 1.000899

'~~""-- i --~~~ ·~ --,----diiiil!il1ii i' iiiBiiiilli'iiil·'· :l

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I We next estimate the trend (TCt).

I Define the deseasonalized or seasonally adjusted series as:

Dt = Yt

SNt

I For example,

D1 = 720.6063 = 118.7506

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I We next estimate the trend (TCt).

I Define the deseasonalized or seasonally adjusted series as:

Dt = Yt

SNt

I For example,

D1 = 720.6063 = 118.7506

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100

105

110

115

120

125

130

135

140

145

150

0 2 4 6 8 10 12 14 16 18

Plot of D(t) against t

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I TCt may be estimated by regression based on a linear trend.Write

Dt = β0 + β1t + ε, t = 1, 2, · · · , n.

I Then the estimated trend is

TCt = Dt = b0 + b1t,

where b0 and b1 are the least squares estimators of β0 and β1respectively.

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I TCt may be estimated by regression based on a linear trend.Write

Dt = β0 + β1t + ε, t = 1, 2, · · · , n.

I Then the estimated trend is

TCt = Dt = b0 + b1t,

where b0 and b1 are the least squares estimators of β0 and β1respectively.

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I For this data set,

TCt = 113.6997914 + 1.854638009t

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I The predicted values of TC may be computed by substitutingthe relevant values of t into the estimated trend equation.

I For example,

TC1 = 113.6997914 + 1.854638009(1) = 115.5544294

TC2 = 113.6997914 + 1.854638009(2) = 117.4090674

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I One can then compute the forecasted values of Yt by:

Yt = TCt × SNt

I In-sample fitted values:

Y1 = 115.5544× 0.6063 = 70.0621..Y16 = 143.3740× 1.4825 = 212.5516

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I One can then compute the forecasted values of Yt by:

Yt = TCt × SNt

I In-sample fitted values:

Y1 = 115.5544× 0.6063 = 70.0621..Y16 = 143.3740× 1.4825 = 212.5516

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Out-of-sample forecasts:

Y17 = TC17 × SN17

= [113.670 + 1.855(17)]× 0.6063

= 145.2286× 0.6063

= 88.054

Y18 = TC18 × SN18

= [113.670 + 1.855(18)]× 0.9191

= 147.0833× 0.9191

= 135.1796

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Measuring Forecast Accuracy

Let et = Yt − Yt be the forecast error.

I Mean Squared Error (MSE)

MSE =∑n

t=1 e2t /n

RMSE =√MSE

I Mean Absolute Deviation (MAD)

MAD =∑n

t=1 |et |/n

RMAD =√MAD

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MSE =∑n

t=1 e2t /n

RMSE =√MSE


MAD =∑n

t=1 |et |/n

RMAD =√MAD

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MSE =∑n

t=1 e2t /n

RMSE =√MSE


MAD =∑n

t=1 |et |/n

RMAD =√MAD

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Method A Method Bet = -2 -4

1.5 0.7-1 0.52.1 1.40.7 0.1

Method A: MSE = 2.43, MAD = 1.46

Method B: MSE = 3.742, MAD = 1.34

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I Naive prediction - use the last period actual value to predictthe next period’s unknown, i.e.,use Yt−1 to predict Yt .

I Theil’s U Statistic:

U =

√ ∑(Yt−Yt)2/n∑

(Yt−Yt−1)2/n

I if U = 1⇒ forecasts produced are no better than naiveforecasts;if U = 0⇒ forecasts produced perfect fit

I U is expected to lie between 0 and 1 - the smaller the value ofU, the better the forecasts

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U =

√ ∑(Yt−Yt)2/n∑

(Yt−Yt−1)2/n



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U =

√ ∑(Yt−Yt)2/n∑

(Yt−Yt−1)2/n



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U =

√ ∑(Yt−Yt)2/n∑

(Yt−Yt−1)2/n



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For the model used in our last example, MSE = 11.932, MAD =2.892 and U = 0.0546.

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Types of Forecasts

I Expost forecast - Prediction for the period in which theactual observation is available

I Exante forecast - Prediction for the period in which theactual observation is not available

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Types of Forecasts

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Additive Decomposition Model

The diagrams in the top and bottom panels depict situations ofmultiplicative seasonality and additive seasonality respectively.

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Multiplicative decomposition (Yt = TCt × SNt × IRt) is used whenthe time series exhibits seasonal variations that follow the trend(multiplicative seasonality). For example,

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Additive decomposition (Yt = TCt + SNt + IRt) is used when thetime series exhibits seasonal variations that are constant and donot follow the trend (additive seasonality). For example,

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I To construct the additive model, we first calculate MAt andCMAt as per multiplicative decomposition.

I The initial seasonal component may be estimated by

SNt = Yt − CMAt .

I For example, using our previous data set,

SN3 = 117− 118.25 = −1.25SN4 = 172− 119 = 53

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I To construct the additive model, we first calculate MAt andCMAt as per multiplicative decomposition.

I The initial seasonal component may be estimated by

SNt = Yt − CMAt .

I For example, using our previous data set,

SN3 = 117− 118.25 = −1.25SN4 = 172− 119 = 53

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The initial seasonal indices are then averaged and adjusted so thatthey sum to zero (Why?)

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I The seasonally adjusted series is Dt = Yt − SNt .

I TCt may be estimated by regression as per multiplicativedecomposition, i.e.,

Dt = β0 + β1t + ε, t = 1, 2, · · · , n.

and

TCt = Dt = b0 + b1t

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I The seasonally adjusted series is Dt = Yt − SNt .

I TCt may be estimated by regression as per multiplicativedecomposition, i.e.,

Dt = β0 + β1t + ε, t = 1, 2, · · · , n.

and

TCt = Dt = b0 + b1t

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I So, TCt = 113.2270833 + 1.980637255t

and

Yt = TCt + SNt

I For example,

TC1 = 113.2270833 + 1.980637255(1) = 115.2077206

and

Y1 = 115.2077206− 50.80208333 = 64.40563725

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MSE = 27.911 and MAD = 4.477

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Documents

CHAPTER 1: Decomposition Methodspersonal.cb.cityu.edu.hk/msawan/teaching/ms6215/MS6215Ch1.pdf · CHAPTER 1: Decomposition Methods Prof. Alan Wan 1/48. 1. Data Types and Causal vs.Time