Forecasting and Statistical Process Control MBA Statistics 51-651-02 COURSE #5
Jan 11, 2016
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
0
50
100
150
200
250
300
350
400
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
0
50
100
150
200
250
300
350
400
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
)
9
A cycle is a regular pattern repeating periodically with a long period (more than one year).
Cyclical effect
-1.5
-1
-0.5
0
0.5
1
1.5
1 3 5 7 9
11
13
15
17
19
21
23
25
27
29
Time
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Seasonal effect is similar to cyclical effect but with shorter period (less than 1 year).
Seasonal effect
-10
-5
0
5
10
1 3 5 7 9
11
13
15
17
19
21
23
25
27
29
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
-10
0
10
20
30
40
50
60
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
-10
0
10
20
30
40
50
60
0 10 20 30 40
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
-10
0
10
20
30
40
50
60
0 10 20 30 40
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
-10
0
10
20
30
40
50
60
0 10 20 30 40 50
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
-10
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50
Sales
Pred
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Model 1 (without Seasons)
-10
-5
0
5
10
0 5 10 15 20 25 30 35
Quarter
Res
idu
als
Model 2 (with seasons)
-5
0
5
0 5 10 15 20 25 30 35
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
0
5
10
15
20
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
ber
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
0
10
20
30
40
50
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