MBA Statistics 51-651-0 2 COURSE # 5

Forecasting and Statistical Process Control

MBAStatistics 51-651-02

COURSE #5

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Part I: Forecasting

Part II: Statistical Process Control

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Forecasting

Uncertainty means we have to anticipate future events

Good forecasting results from a combination of good technical skills and informed judgement

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Insulator Sales DataData sets of chapter 10

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50

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Jan-96

Apr-96

Jul-96

Oct-96

Jan-97

Apr-97

Jul-97

Oct-97

Jan-98

Apr-98

Jul-98

Sa

les

(in

00

0s

)

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Time Series

Data measured over time is called a time series.

Usually such data are collected at regular time periods.

Aim is to detect patterns that will enable us to forecast future values.

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Forecasting Process

Choose a forecasting model Apply the model retrospectively, and

obtain fitted values and residuals Use the residuals to examine the

adequacy of the model If model acceptable, use it to forecast

future observations Monitor the performance of the model

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Time Series Components Long term trend

– Fundamental rise or fall in the data over a long period of time.

Seasonal effect– Regular and repeating patterns occurring over

some period of time Cyclical effect

– Regular underlying swings in the data

Random variation– Irregular and unpredictable variations in the

data

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Identifying the Trend

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Jan-96

Apr-96

Jul-96

Oct-96

Jan-97

Apr-97

Jul-97

Oct-97

Jan-98

Apr-98

Jul-98

Sa

les

(in

00

0s

)

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A cycle is a regular pattern repeating periodically with a long period (more than one year).

Cyclical effect

-1.5

-1

-0.5

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0.5

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1.5

1 3 5 7 9

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Time

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Seasonal effect is similar to cyclical effect but with shorter period (less than 1 year).

Seasonal effect

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1 3 5 7 9

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Time

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Random effect Random variations (also called noise)

include all irregular changes not due to other effects (trend, cyclical, seasonal). 

The noise is like a fog, often hiding the other components.

One of the goal is to try to get rid of the effect (using smoothing).

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Models

 additive model

yt = Tt + Ct + St + Rt

 multiplicative model

yt = Tt Ct St Rt

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Illustration: Sales vs Quarter (ts.xls)

Sales

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0 5 10 15 20 25 30 35

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Moving Averages

Used to smooth data so we can see the trend or seasonality– removes random variation

We can take moving averages of any number time periods (preferable to take an odd number)

How much smoothing?– too little: random variation not removed– too much: trend may also be eliminated

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Smoothing of Sales

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Sales

MA(3)

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Remarks

Considering MA over 3 periods, one can see a linear trend and seasonality of order 4, looking at peaks.

The MA series over 5 periods is too smooth and seasonality almost disappeared.

It is preferable to center the smoothed series with respect to the original one.

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Smoothing of Sales

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Sales

MA(3)

MA(5)

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Exponential Smoothing

Smoothing aims to remove random so as to reveal the underlying trend and seasonality.

Moving averages use only the last few figures, and give them equal weight. We are loosing data.

Exponential smoothing uses all the data giving less and less weight to data further back in time.

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Updating Procedure

New Forecast

= × Latest Actual Value

+ (1 – ) × Previous Forecast

damping factor

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Exponential Smoothing in Excel

In Excel we use the damping factor (1-)

For = 0.8, we use 0.2 in Excel The best value of is found by trial and

error, and is the one that gives the smallest MSE.

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Exponential smoothing for Sales Data

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Sales

Exp Smooth

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Using Regression for estimating trend and seasonal effects Can fit a linear regression model to the

time series. Use dummy variables corresponding to

seasonality. More complicated for multiplicative

effects. Desaisonalized series corresponds to

residuals + constant!

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Regression approach

What happens if the only explanatory variable is the quarter? Look at the residuals.

Introduce 3 dummy variables S1, S2, S3, corresponding to the seasonality of order 4.

Look at residuals now. What are the predictions for the next 10

quarters?

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Prediction of the next 10 quarters

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Sales

Pred

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Model 1 (without Seasons)

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Quarter

Res

idu

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Model 2 (with seasons)

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Quarter

Re

sid

ua

ls

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Part II: Statistical Process Control (SPC)

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Statistical Process Control

Statistical process control (SPC) is a collection of management and statistical techniques whose objective is to bring a process into a state of stability or control

And then to maintain this state All processes are variable and being in

control is not a natural state. SPC is an effective way to improve product

and service quality

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Five Stage Improvement Plan

Understand the

Process

Eliminate Errors

Remove Slack

Reduce Variation

Plan for Improveme

nt

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Benefits of reducing variation Effect of tampering Common cause highway Special and common causes Construction and use of control charts Establishment and monitoring Specifications and capability Strategies for reducing variation

Aspects of SPC

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Processes

PeopleMaterialEquipmentMethodEnvironment

PeopleMaterialEquipmentMethodEnvironment

INPUTSPROCESSING

SYSTEM OUPUTS

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Process Variability

Process

Inputs Outputs

Collect and analyse dataReduce variation

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Improved Process: less variability in input => less variability in output

Process

Inputs Outputs

Collect and analyse dataReduce variation

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Common Cause Highway

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Experiment number

Num

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of r

ed b

eads

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The Key to Reducing Variation

To distinguish between data that fall within the common cause highway, and data that falls outside the highway.

Common cause variation indicates a systemic problem.

Special cause variation is almost certainly worthy of separate investigation.

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Epic Video Sales

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1 3 5 7 9 11 13 15 17 19 21 23 25

Month

Vid

eo S

ales

($0

0)

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Localised in nature Not part of the overall system Not always present in the process Abnormalities, unusual, non-random Contribute greatly to variation Can often be fixed by people working on the

process

Special Causes of Variation

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Common Causes of Variation

In the system Always present in the process Common to all machines, operators,

and all parts of the process Random fluctuations Events that individually have a small

effect, but collectively can add up to quite a lot of variation

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Three Sigma Limits

The arithmetic mean gives the centre line of the common cause highway

The mean plus three standard deviations gives the upper boundary of the highway. This boundary is called the upper control limit (UCL)

The mean minus three standard deviations gives the lower boundary of the highway. This boundary is called the lower control limit (LCL)

If a point falls outside the 3-sigma limits it is almost certainly a special cause.

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Why 3-Sigma Limits? In trying to distinguish between

common and special causes there are two mistakes that we can make.

Interfering too often in the process. Thinking that the problem is a special cause when in fact it belongs to the system.

Missing important events. Saying that a result belongs to the system when in fact it is a special cause.

too narrow; 2-sigma

too wide; 4-sigma

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Patterns

Specific patterns on a control chart also indicate a lack of randomness

We need rules to help us decide when we have a pattern – to avoid seeing patterns when none really

exist A pattern would indicate that special

causes could be present

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9 Points Below the Mean

Mean

UCL

LCL

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Stability and Predictability

Stable Process

time??

??

????

???

?????

?

time

Unstable process Source: Ford Motor Company

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Stability and Predictability

A stable process is predictable in the long run.

In contrast, with an unstable process special causes dominate.

Nothing is gained by adjusting a stable process

A stable process can only be improved by fundamental changes to the system.

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Implementing SPC

There are two stages involved in implementing SPC

The establishment of control charts– scpe.xls