SADC Course in Statistics Trends in time series (Session 02)

SADC Course in Statistics

Trends in time series

(Session 02)

2To put your footer here go to View > Header and Footer

Learning ObjectivesBy the end of this session, you will be able to

• explain the main components of a time series

• use graphical procedures to examine a time series for possible trends

• describe the trend in the series– using a moving average– by fitting a line to the trend

• use the trend to do simple forms of forecasting


General approach to analysis

Step 1: Plot the data with time on x-axis

Step 2: Study the pattern over time– Is there a trend? – Is there a seasonality effect? – Are there any long term cycles?– Are there any sharp changes in behaviour?– Can such changes be explained?– Are there any outliers, i.e. observations that

differ greatly from the general pattern


General approach to analysis

Step 3: State clearly your objectives for study of the time series

Step 4: Paying attention to the objectives, analyse the data, using descriptive analyses first, and moving later to other forms of analyses

Step 5: Write a report giving the key results and the main findings and conclusions


Main components of a time series

Trend: the general direction in which the series is running during a long period

Seasonal effects: Short-term fluctuations that occur regularly – often associated with months or quarters

Cyclical effects: Long-term fluctuations that occur regularly in the series

Residual: Whatever remains after the above have been taken into account, i.e. the unexplained (random) components of variation


Examining trends

Following slides include:

• Identifying if there appears to be a trend

• Describing and examining the trend using– Moving averages– Fitting a straight line to the data

• Simple approaches to forecasts, and likely dangers

Note: Cyclical effects will not be covered since theyare usually difficult to identify and need long series


An ExampleData below show the number of unemployed school leavers in the UK (in ‘000)Source: Employment Gazette

Year Jan-Mar Apr-Jun Jul-Sep Oct-Dec

1979 22 12 110 31

1980 21 26 150 70

1981 50 36 146 110

Graphing the data is needed, but first need to put data in list format


Putting the data in list formatYear Quarter Leavers

1 1979.00 22

1 1979.25 12

1 1979.50 110

1 1979.75 31

2 1980.00 21

2 1980.25 26

2 1980.50 150

2 1980.75 70

3 1981.00 50

3 1981.25 36

3 1981.50 146

3 1981.75 110

The decimals for the quarters are only intended to give a rough approximation of the time point


Clear seasonal and trend effects:


Estimating trend - moving average

We will ignore the seasonal pattern for now, and concentrate on estimating the trend…

• Purpose: to ‘smooth’ out the local variation and possibly seasonal effects

• Method: Replace each observation by a weighted average of observations around (and including) the particular observation.

• Results will depend on– number of observations used– weights used


An example

Suppose we have the series

Y1, Y2, ……….. , Yn

Then a moving average with weights1/8 , ¼ , ¼ , ¼ , 1/8

Would have the ith value equal to

Yi* = (1/8)Yi-2 + (¼)Yi-1 + (¼)Yi

+ (¼)Yi+1 + (1/8)Yi+2


Points to note…

• Weights should always add to 1.

• To find a moving average, need to decide what order to use, i.e. how many to average. For data in quarters, sensible to use order=4

• Where order=r, and r is odd, moving average (m.a.) values cannot be defined for the first (r-1)/2 and last (r-1)/2 obsns in the series

• If r is even, m.a. values will lie midway between times of observation - nicer to have it coinciding with the time points of observed data


Possible action when r is even If data are in quarters (say) could use a 4-point moving average first with equal weights,

i.e. ¼, ¼, ¼, ¼

Then use a further moving average of order 2 on the first computed moving average, again with equal weights, i.e. ½, ½

This is equivalent to using weights1/8 , ¼ , ¼ , ¼ , 1/8 on the original data

series! i.e. it is equivalent to a 5-point m.a. See next slide last column for an example…


Calculating the moving average

Year Quarter Leavers Three-point meanFive-point

moving average

1 1979.00 22

1 1979.25 12 48.00

1 1979.50 110 51.00 43.63

1 1979.75 31 54.00 45.25

2 1980.00 21 26.00 52.00

2 1980.25 26 65.7 61.88

2 1980.50 150 82.0 70.38

2 1980.75 70 90.0 75.25

3 1981.00 50 52.0 76.00

3 1981.25 36 77.3 80.50

3 1981.50 146 97.3

3 1981.75 110

Verify a few of the values in the last 2 columns…


Time Series & the fitted trend

0

40

80

120

160

1 2 3 4 5 6 7 8 9 10 11 12

Time (quarters)

Un

emp

loye

d S

cho

ol

Lea

vers

('0

00)

leavers maverage

The increasing trend is now clearly visible


Estimating trend – fitting a line

A second method for estimating trend is to find an equation giving the “best fit” to the trend.

The simplest is a straight line, i.e.

Y =a + bX

Here “a” is called an intercept and “b” is called the slope of the line.

Formulae for a and b, which minimise the squared distances of each data point from the line, are given on the final slide.


Interpreting the line

020406080100120140160

1979 1980 1981 1982

Time (Years)

Un

emp

loye

d S

cho

ol

Lea

vers

('0

00)

pq

a

The intercept “a” is shown on the graph.

The slope “b” is given by

b = p/q


Equation of the line

0

40

80

120

160

0 4 8 12

Time (quarters)

Un

em

plo

ye

d S

ch

oo

l Le

av

ers

('

00

0)

Equation of line is y = 19.4 + 7.07x


Forecasting – simple approach

Return to the equation of the line of “best fit” to the data, i.e.

y = 19.4 + 7.07x

For a future value of x, say at quarter 13, we may find y as

y = 19.4 + 7.07 (13) = 111.3

However, there are many dangers involved with doing this – and it is not advisable to do without recognising these


Some dangers in predictionsFitting a straight line as done here is part of a larger topic called “Regression Analysis” (contents of Module H8)

One basic assumption is that repeat observations taken in time are assumed to be independent. This is rarely the case.

Another is that the validity of the prediction becomes poorer as the value of x used (in the prediction) moves away from the time values used in finding the equation.

DO NOT MAKE FORECASTS without further knowledge of many statistical and other practical issues involved in doing so.


Practical work• Use data on temperature records, to find a

moving average, study its pattern, and write a short report of the results and conclusions

• Go through parts of Section 2 of

CAST for SADC : Higher Level

for improving your knowledge of the basic ideas of this session.

Details of both exercises are given in thehandout entitled “Estimating Trend in a TimeSeries” (Practical 02).


Calculations for “best fit” line

• Let the equation of line be: Y = a + b*i, where i refers to the time points, incrementing by 1.

• Then

• and

n(n+1)a Y - × b

2

2

( )i n i=n i n

i ii 1 i=1 i 1

i=n i n2

i=1 i 1

1 i Y - i ) ( Y

nb1

i - ( i)n


Some practical work follows…

SADC Course in Statistics Trends in time series (Session 02)

Documents

view header

time series trend

series y

time point slide

time series step

trend effects

time series session

original data series