Time Series- A collection of measurements recorded at specific intervals of time.

1. Short term featuresNoise:Spike/Outlier:

Minor variation about a general trendAn obvious difference from the surrounding values

e.g.

time

2. Long term featuresTrend:

Seasonal Variation:

Often there is a trend for measurements to remain steady, or show a definite increase or decrease over time.

Fairly regular up/down patterns (called cyclical movement if over very long periods)

2

4

6

8

10

12

e.g.

Long Term Trend:Over long term, sales are increasing overall

Seasonal Variation:Sales peak in summer and are lowest in winter. Sales rise again in spring and these are higher than in autumn

Ice Cream Sales (in 000’s)

Smoothing Techniques- Used when averaging out random variations to see if there is an overall trend.- Done by averaging all of the data over the period of any natural cycle.

The number of values used to form a moving average is called ‘the order of the moving average’

(e.g. we use a 5 point moving mean (order of 5) if the natural cycle is a working week)

e.g. Use 4 point moving means to smooth the data for Elliot’s Fish and Chip shop. Then graph the raw data and the mean of means.

Season Quarterly sales

Moving mean

Mean of means

Seasonal Difference

Sept. 9040Dec. 8650

Mar. 96 8370June 9250Sept. 9033Dec. 8578

Mar. 97 8495June 9407Sept. 9209Dec. 8740

Mar. 98 8618June 9504Sept. 9246Dec. 8929

Mar. 99 8670

882888268808883988788922896389949018902790749087

88278817882488598900894389799006902390519081

Mean of means are used so there is a 1 to 1 correspondence between the raw and smoothed data.

Quarterly Sales for Elliots's Fish and Chips Shop

8300

8500

8700

8900

9100

9300

9500

Sal

es ($

)

Sep

t.

Dec

.M

ar. 9

6Ju

ne

June

Mar

. 97

Dec

.

Sep

t.

June

Mar

. 98

Dec

.S

ept.

Mar

. 99

Dec

.

Sep

t.

Quarter Years

9700

9900

Sep

t.D

ec.

Mar

. 00

June

June

Sep

t.

Seasonal EffectsSeasonal Difference =data value – moving meanSeasonal Effect =averaging off all of the seasonal

differencesMaking Predictions- Extend the trend line to find the smoothed data value then add/subtract the average seasonal difference

e.g. Using the data and graph of ‘Elliot’s Fish and Chip Shop’ a) Find the seasonal effects for June and Decemberb)Use the long term trend line to predict the turnover in December 1999 and June 2000.

e.g. Use 4 point moving means to smooth the data for Elliot’s Fish and Chip shop. Then graph the raw data and the mean of means.

Season Quarterly sales

Moving mean

Mean of means

Seasonal Difference

Sept. 9040Dec. 8650

Mar. 96 8370June 9250Sept. 9033Dec. 8578

Mar. 97 8495June 9407Sept. 9209Dec. 8740

Mar. 98 8618June 9504Sept. 9246Dec. 8929

Mar. 99 8670

882888268808883988788922896389949018902790749087

88278817882488598900894389799006902390519081

-457433209-281-405464230-266-405453165

e.g. Using the data and graph of ‘Elliot’s Fish and Chip Shop’ a) Find the seasonal effects for June and Decemberb)Use the long term trend line to predict the turnover in December 1999 and June 2000.Seasonal Effect for June =

433 + 464 + 453 3

= 1350 = $450 3

Seasonal Effect for December =

-281 + -266 2

= -547 = -$273.5 2

Quarterly Sales for Elliots's Fish and Chips Shop

8300

8500

8700

8900

9100

9300

9500

Sal

es ($

)

Sep

t.

Dec

.M

ar. 9

6Ju

ne

June

Mar

. 97

Dec

.

Sep

t.

June

Mar

. 98

Dec

.S

ept.

Mar

. 99

Dec

.

Sep

t.

Quarter Years

9700

9900

Sep

t.D

ec.

Mar

. 00

June

June

Sep

t.

Dec ‘99 = 9200

June ‘00 = 9250

e.g. Using the data and graph of ‘Elliot’s Fish and Chip Shop’ a) Find the seasonal effects for June and Decemberb)Use the long term trend line to predict the turnover in December 1999 and June 2000.Seasonal Effect for June =

433 + 464 + 453 3

= 1350 = $450 3

Seasonal Effect for December =

-281 + -266 2

= -547 = -$273.5 2

Prediction for December 1999 =

9200 + -273.5 = $8926.50

Prediction for June 2000 = 9250 + 450 = $9700