FORECASTING Statistics For Business

MATH0201

The components of time series

• A time series is a series of figures or values recorded over time.

• Any pattern found in the data is then assumed to continue into the future and an extrapolative forecast is produced.

• There are four components of a time series:

▫ trend,

▫ seasonal variations,

▫ cyclical variations and

▫ random variations.

Standard time series models

1. The time series additive model: y = t + c + s + I 2. The time series multiplicative model: y = t × c × s × I Where y is a given time series value t is the trend component c is the cyclic component s is the seasonal component i is the irregular component

Description of time series components

1. Trend This is a long term movement 2. Cyclical variation Repeating up and down movements due to interactions of factors influencing economy.

3. Seasonal variation The effect of seasons – spring, summer. Autumn and winter – on the series. 4. Irregular or random variation These are disturbances due to ‘everyday’ unpredictable influences, such as weather conditions, illness and so on.

Based on the data, plot the time series.

Numbers of car sell in XYZ Company

Qtr 1 Qtr 2 Qtr 3 Qtr 4

Year 1 73 90 121 98

Year 2 69 92 145 107

Year 3 86 111 157 122

Year 4 88 109 159 131

0

20

40

60

80

100

120

140

160

180

Qtr 1 Qtr 2 Qtr 3 Qtr 4 Qtr 1 Qtr 2 Qtr 3 Qtr 4 Qtr 1 Qtr 2 Qtr 3 Qtr 4 Qtr 1 Qtr 2 Qtr 3 Qtr 4

Year 1 Year 2 Year 3 Year 4

Fr

eq

ue

nc

y

Year

Numbers of car sell in XYZ Company

The trend

• The trend is the underlying long-term movement over time in the values of the data recorded.

A

B

C

Example: Moving averages of an odd number of

results

Required

• Take a moving average of the annual sales over a period of three years.

Note the following points. (i) The moving average series has five figures relating to the years from 20X1 to 20X5. The original series had seven figures for the years from 20X0 to 20X6. (ii) There is an upward trend in sales, which is more noticeable from the series of moving averages than from the original series of actual sales each year.

Example: Moving averages over an even number of

periods

Finding the seasonal component using

the additive model

Step 1 : The additive model for time series analysis is Y = T + S + R Step 2 : If we deduct the trend from the additive model, we get Y - T = S + R . Step 3 : If we assume that R, the random, component of the time series is relatively small and therefore negligible, then S = Y - T • Therefore, the seasonal component, S = Y - T

(the de-trended series).

Example: The trend and seasonal

variations Output at a factory appears to vary with the day of the week. Output over the last three weeks has been as follows Required • Find the seasonal variation for each of the 15 days,

and the average seasonal variation for each day of the week using the moving averages method.

Actual results fluctuate up and down according to the day of the week and so a moving average of five will be used.

These might be rounded up or down as follows.

Monday –13; Tuesday +16; Wednesday +1; Thursday +28; Friday –32; Total 0.

Finding the seasonal component using the

multiplicative (proportional) model

Step 1 Calculate the moving total for an appropriate period.

Step 2 Calculate the moving average (the trend) for the period. (Calculate the mid-point of two moving averages if there are an even number of periods.)

Step 3 Calculate the seasonal variation. For a multiplicative model, this is Y/T.

Step 4 Calculate an average of the seasonal variations.

Step 5 Adjust the average seasonal variations so that they add up to an average of 1.

• Therefore, the seasonal component, S = Y/T (the de-trended series)

• Should sum (in this case) to 5 (an average of 1).

• They actually sum to 5.0045 so 0.0009 has to be deducted from each one. This is too small to make a difference to the figures above.

Adjusted average 0.8636 1.1636 1.0141 1.2996 0.6591

Forecasting

Step 1 Plot a trend line: use the line of best fit method, linear regression analysis or the moving averages method. Step 2 Extrapolate the trend line. This means extending the trend line outside the range of known data and forecasting future results from historical data. Step 3 Adjust forecast trends by the applicable average seasonal variation to obtain the actual forecast. • (a) Additive model – add positive variations to and

subtract negative variations from the forecast trends. • (b) Multiplicative model – multiply the forecast

trends by the seasonal variation.

• forecast sales in week 4.

0.8645

1.1645

1.0150

1.3005

0.6600

86.9

117.7

103.2

132.9

67.8

Additive model

Multiplication model