Topic11: Time series and trend analysis

060074

STATISTICS

Introduction

• A time series consists of a set of observations which are measured at specified (usually equal) time intervals.

• Time series analysis attempts to identify those factors that exert an influence on the values in the series. Once these factors are identified, the time series may be used for both short-term and long-term forecasting.

A several of time series

yearGDP

(100 million yuan)

Total population(year-end)

(10000 persons)

Natural Growth Rate of

Population (‰)

Household consumption ex

penditures(yuan)

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

18547.9

21617.8

26638.1

34634.4

46759.4

58478.1

67884.6

74772.4

79552.8

80471.6

114333

115823

117171

118517

119850

121121

122389

123626

124810

125924

14.39

12.98

11.60

11.45

11.21

10.55

10.42

10.06

9.53

9.48

803

896

1070

1331

1781

2311

2726

2944

3094

3130

Time series components

The four components usually identified are:• Secular trend ----the underlying

movement of the series• Seasonal variation• Cyclical variation• Irregular variation

While it is possible to break down a timeseries into these four components, the task is not always simple.

t

ind

icatio

n

Nov.1992

The value in November, 1992 was decided by four

factors in which the secular trend is more important.

Secular trend

• The secular trend is the long-term growth or decline of a series. It is decided by the property of the variable itself.

• In typical economic contexts, ‘long-term’ may mean 10 years or more. Essentially, the period should be long enough for any persistent pattern to emerge.

• Secular trends allow us to look at past patterns or trends and use these to make some prediction about the future.

• In some situations it is possible to isolate the effect of secular trends from the time series and hence make studies of the other components easier.

Actual data

Straight-line trend

Exponential trend

The longer the time, the clearer the trend

Seasonal variation

• The seasonal variation of a time series is a pattern of change that recurs regularly over time. Seasonal patterns typically are one year long; that is, the pattern starts repeating itself at a fixed time each year.

• While variations may recur every year, the concept of seasonal variation also extends to those patterns that occur monthly, weekly, daily or even hourly.

• Time series graphs may be seasonally adjusted or deseasonalized by “seasonal index” when the seasonal variation of it is very strong. Such graphs give us a true picture of genuine movements in the time series after the seasonal effects have been removed.

Examples of seasonal variation

• Air conditioner sales are greater in the summer months.

• Heater sales are greater in the winter months• The total number of people seeking work is

large at the end of each year when students leave school

• Motels, hotels and camping grounds have a greater volume of customers in holiday seasons

• Train ticket sales increase dramatically during festive seasons

• Medical practitioners report a substantial increase in the number of flu cases each winter

• Liquor outlets undergo increased sales during festive seasons

• Airline ticket sales (and price!) increase during school holidays

• The amount of electricity and water used varies within each 24-hour period

• The volume of work for tax agents increases dramatically around the time when income tax forms have to be filed.

Cyclical variation

• In a similar manner to seasonal variations, cyclical variations have recurring patterns, but have a longer and move erratic time scale.

• Unlike seasonal variation, there is no guarantee that there will be any regularly recurring pattern of cyclical variation. It is usually impossible to predict just how long these periods of expansion and contraction will be.

Examples of causes of cyclical variation

• Floods

• Earthquakes / hurricanes

• Droughts

• Wars

• Changes in interest rates

• Major increases or decreases in the population

• The opening of a new shopping complex

• The building of a new airport• Economics depressions or recessions• Major sporting events, such as the

Olympic Games• Changes in consumer spending (i.e.

lack of confidence)• Changes in government monetary

policy

Irregular variation

• Irregular variation in the time series occurs varying (usually short) periods. It follows no regular pattern and is by nature unpredictable. It usually occurs randomly and may be linked to events that also occur randomly.

• It cannot be explained mathematically. In general, if the variation in a time series cannot be accounted for by secular trend, or by seasonal or cyclical variation, then it is usually attributed to irregular variation.

Examples of events that might cause irregular variation

• The assassination (or disappearance) of a country’s leader

• Short-term variation in the weather, such as unseasonably warm winters (they may affect sales of certain products)

• Sudden changes in interest rates• The collapse of large (or even small)

companies

• Strikes (e.g. a strike by airline pilots affects many people working in the travel industry)

• A government calling an unexpected election

• Sudden shifts in government policy

• Natural disasters

• Dramatic changes to the stock market

• The effect of war in the Middle East on petrol prices around the world

Measurement of secular trend

• Measurement of secular trend can be somewhat subjective, depending on the technique used to measure it.

• The methods used to measure it.

1. semi-averages

2. least-squares linear regression

3. moving averages

4. exponential smoothing

5. growth model

Semi-averages

year Extra income($) Semi-totals ($)Semi-averages

($)

1998 4701

29819 5963.8

1999 5298

2000 5938

2001 6673

2002 7209

2003 7422 disregard

2004 7780

44570 8914.0

2005 8476

2006 9066

2007 9363

2008 9885

5963.8

8914.0

Graph of actual data

Semi-average trend line

2000 2006

Least-squares linear regression

• A more sophisticated way of fitting a straight line to a time series is to use the method of least-squares linear regression

• In this case, the observations are the (dependent) y-variables and time is the (independent) x-variable

• Since in this case the x-variable is time units, the calculations may be simplified as follows

year Value of x Extra income-y x2 xy

1998 -5 4701 25 -23505

1999 -4 5298 16 -21192

2000 -3 5938 9 -17814

2001 -2 6673 4 -13346

2002 -1 7209 1 -7209

2003 0 7422 0 0

2004 1 7780 1 7780

2005 2 8476 4 16592

2006 3 9066 9 27198

2007 4 9363 16 37452

2008 5 9885 25 49425

total 0 81811 110 55381

n=奇数

x46.50336.7437y

36.743711

81811yxbya

74.506110

55381

x

xy

xx

yyxxb

bxay

22

Excel

year xNumber of house

yx2 xy

1995 -7 49 49 -343

1996 -5 133 25 -665

1997 -3 69 6 -207

1998 -1 170 1 -170

1999 1 133 1 133

2000 3 175 9 525

2001 5 152 25 760

2002 7 185 49 1295

total 0 1066 168 1328

n=偶数

x90.725.133y

25.1338

1066yxbya

90.7168

1328

x

xy

S

Sb

bxay

2xx

xy

Moving averages

• The method of moving averages is based on the premise that, if the values in a time series are averaged over a sufficient period, the effect of short-term variations will be reduced. That is, short-term cyclical, seasonal and irregular variations will be smoothed out, leaving an apparently smooth graph to show the overall trend.

Calculation of the 3-year moving averages for data

year Number of sales3-year moving

total3-year moving

average

1994 1011 ---- ----

1995 1031 3018 1006

1996 976 3027 1009

1997 1020 3191 1064

1998 1195 3389 1130

1999 1174 3630 1210

2000 1261 3765 1255

2001 1330 3975 1325

2002 1384 ---- ----

Calculation of the 4-year moving averages for data

year y4-year total

4-year average

4-year total

4-year average

Moving average

1992 47.6 ---- ---- ---- ---- ----

1993 48.9 ---- ---- 203.3 50.8 ----

1994 51.5 203.3 50.8 213.6 53.4 52.1

1995 55.3 213.6 53.4 226.4 56.6 55.0

1996 57.9 226.4 56.6 240.2 60.0 58.3

1997 61.7 240.2 60.0 255.1 63.8 61.9

1998 65.3 255.1 63.8 273.3 68.3 66.0

1999 70.2 273.3 68.3 296.3 74.1 71.2

2000 76.1 296.3 74.1 324.2 81.0 77.6

2001 84.7 324.2 81.0 ---- ---- ----

2002 93.2 ---- ---- ---- ---- ----

Exponential smoothing• Exponnential smoothing is a method for continuall

y revising an estimate in the light of more recent trends. It is based on averaging (or smoothing) the past values in a series in an exponential manner.

• Recurrence relation: Sx=αyx+(1－ α)Sx-1

where: Sx= the smoothed value for observation x

yx= the actual value of observation x

Sx-1= the smoothed value previously calculated for observation (x-1)

α= the smoothing constant , (1－ α) is referred to as resistant coefficient where 0≤α≤1

• Generally, we choose: S1=y1 , so S2=αy2+ (1-α) S1

year x Observation yx Sx-1 ( 1-α) Sx-1 αyx Sx

1992 1 47.6 47.60

1993 2 48.9 47.60 28.56 19.56 48.12

1994 3 51.5 48.12 28.87 20.60 49.47

1995 4 55.3 49.47 29.68 22.12 51.80

1996 5 57.9 51.80 31.08 23.16 54.24

1997 6 61.7 54.24 32.54 24.68 57.22

1998 7 65.3 57.22 34.33 26.12 60.45

1999 8 70.2 60.45 36.27 28.08 64.35

2000 9 76.1 64.35 38.61 30.44 69.05

2001 10 84.7 69.05 41.43 33.88 75.31

2002 11 93.2 75.31 45.19 37.28 82.47

α=0.40S1=y1, S2= αy2+(1-α) S1 ,S3= αy3+(1-α)S2,-----

Sx=αyx+(1－ α)Sx-1

Actual data

Exponential smoothing trend curve (α=0.40)

Excel

The exponential model uses the current smoothed estimate as a forecast for future years. In this case, we would therefore forecast average daily sales of milk to be 82.47L in 2003

The smoothing constant ----α

• The selection of the most suitable value of α is not easy. The greater α is the more important recent trends are. Generally the value of α is chosen rather subjectively and However, the following criteria are useful:

1. suppose that the time series has strong irregular variation, or a seasonal variation causing wide swings, which it is desired to suppress. Then we might want to take more account of past trends of the series than recent trends. In this case, the value of α could be set small (say, =0.1) so that the history dominates the value of the smoothed observation.

2. suppose that the time series has little variation. Then we might want to take more account of recent observations than those in the past. In this case, the value of α could be set large (say, =0.9). Recent observations will dominate the value of the smoothed observation, with previous values providing merely a kind of background stability.

Sx=αyx+ (1-α) Sx-1

Growth model

• Suppose that we note from a graph of the data that the trend appears to be exponnential. In this case, a growth model may be appropriate. A growth model is one that takes account of this exponential trend.

• Suppose that we have a time series in which time is represented by the variable x and the corresponding observations are represented by the variable y. Further, suppose that we feel that the values of y are rising exponentially in relation to x. Then we may fit the model:

y of value

predicted the is y and constants are b and a :where ˆ

aey so ey bxerrorx

Constants----a and b

bxaey

ˆ

xz errorxy

2

c

21

cb

ea

b and a of values the then 3.

xccz say

x, on z of line regression squares-least the find 2.

lnlnlnz

that such z variable

aform to values-y the of logarithms natural the take 1.

:are

b and a of values eappropriat mostthe find to steps The

1

sample (z, x)

Actual data

Growth curve

year 1997 1998 1999 2000 2001 2002

Sales (y) 127 130 148 160 185 220

x 1 2 3 4 5 6

Z=lny 4.844 4.868 4.997 5.075 5.220 5.394

The least-squares regression line of z on x

z=4.678+0.111x

c1=4.678 c2=0.111

a=e4.678 =107.55 b=0.111

)(111.0

111.0

55.107ˆ

55.107ˆyear

x

eyor

ey

Homework:S368 11.3, 11.6, 11.8, 11.19, 11.21

Excel

Class workOutput of automobile made in China from

1991 to 2008

yearOutput

(10 thousands)year

Output

(10 thousands)

1991

1992

1993

1994

1995

1996

1997

1998

1999

17.56

19.63

23.98

31.64

43.72

36.98

47.18

64.47

58.35

2000

2001

2002

2003

2004

2005

2006

2007

2008

51.40

71.42

106.67

129.85

136.69

145.27

147.52

158.25

163.00

1.Find the 3-year moving average for output of auto in the table2.Find the least-squares regression line of output of auto in the table3.Use the exponential smoothing model in the table to forecast the average output of auto in 2009 (α=0.4)

The average retail price of one dozen eggs in Hobart at 30 June is shown below at each of the 5-year intervals between 1971 and 1996. Use the growth model (use the last two digits of the year, i.e. 71, 76, etc.) to predict the price of eggs (to the nearest cent) in Hobart on 30 June 2001

Year 1971 1976 1981 1986 1991 1996

Price($) 0.70 1.08 1.63 2.02 2.39 2.75

Answer

z=-4.038+0.05394(year)

c1=-4.038 c2=0.05394

a=e-4.038 =0.01763 b=0.05394

10.4$e01763.0y

101year Let

e01763.0y

101*05394.0

)year(05394.0

The key of multiple choice in pre-topic and this topic

S.288

• d, e, b, d, a

• b, e, b, e, e

S.367

• e, a, b, d, d

• d, b, b, e, b