javier garcia - verdugo sanchez - six sigma training - w4 autocorrelation and cross correlation

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Page 1/31 11b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler Week 4 Page 2/31 11b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler If we analyze processes we use samples in accordance to an in advance prepared sampling plan. These samples were mostly taken in time intervals. For a clear analysis, the assumption is that the observed values are independent from each other. A deviation from this assumption can cause misleading conclusion which results in the initiation of wrong actions. The statistic offers the possibility to determine the degree of relations within a set of data. The evaluation method calls autocorrelation. If a significant autocorrelation is detected, than it can be included in the mathematical model. Therefore the effect of it for the data set can calculated and corrected. Introduction

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Page 1: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 1/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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Week 4

Page 2/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

If we analyze processes we use samples in accordanc e to an in advance prepared sampling plan. These samples were mostly taken in time intervals.

For a clear analysis, the assumption is that the ob served values are independent from each other. A deviation from this assumption can cause misleading conclusion which results in the in itiation of wrong actions.

The statistic offers the possibility to determine t he degree of relations within a set of data. The evaluation method calls a utocorrelation. If a significant autocorrelation is detected, than it ca n be included in the mathematical model. Therefore the effect of it for the data set can calculated and corrected.

Introduction

Page 2: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 3/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Autocorrelation is a value of correlation between t wo observation in a time series.

The probability of an active autocorrelation is lar ger, the smaller the time difference between the two samples is.

Disturbances due to noise variables occurs over a t ime period, which may longer than the sample interval. In addition, many technical processes tend to level out slowly after changes or disturbances. In this case, the probability for a correlation of observations over one hour is higher than over ten hours.

If observation were made in short time periods, so that they correlate strongly with each other, they don’t deliver indepe ndent information.

Autocorrelation is a good tool for this evaluation.

Introduction

Page 4/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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File: Autocorrelation1.mtw

Example: Daily Body WeightWe have measured the

weight daily over a time frame of 4 weeks.

The data are normal distributed.

The analysis shows an unpleasant trend.

Is that everything?

Is there a relation within the values?

We check this with Autocorrelation.

Week Day Weight1 1 77,21 2 77,11 3 76,91 4 76,81 5 77,11 6 77,21 7 77,32 1 77,42 2 77,32 3 77,12 4 77,22 5 77,42 6 77,42 7 77,63 1 77,63 2 77,53 3 77,33 4 77,43 5 77,53 6 77,53 7 77,84 1 77,94 2 77,84 3 77,64 4 77,74 5 77,64 6 77,84 7 77,9

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Page 3: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 5/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

The graphic displays the correlation coefficients o ver 7 time series lags combined with a confidence interval of 95 %. It shows a large posit ive, significant spike at lag 1 with a

subsequent positive autocorrelation.

Example: Daily Body Weight

Stat

>Time Series…

>Autocorrelation…

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Lag ACF T LBQ

1 0,805006 4,26 20,16

2 0,581408 2,03 31,08

3 0,428218 1,31 37,24

4 0,301430 0,87 40,42

5 0,304455 0,86 43,81

6 0,363586 1,00 48,86

7 0,328108 0,87 53,16

Page 6/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Final Estimates of Parameters

Type Coef SE Coef T P

AR 1 0,9155 0,1042 8,79 0,000

Constant 6,54385 0,02951 221,73 0,000

Mean 77,4802 0,3494

In the second step we determine the autoregressive model (ARIMA) via autocorrelation. With this option we can calculate uninfluenced theoretical values and perform the residual diagnostic. If the residual are normal distributed and without a trend then we can accept the measurement values.

Example: Daily Body Weight

Stat

>Time Series…

>ARIMA…

Page 4: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 7/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Example: Daily Body Weight

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Page 8/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Below you see the evaluation of a viscosity measure ment of a chemical process in a one hour frequency. File: Autocorrelation2.mtw. Ple ase note that number of observation outside of the +/-3Sigma control limits , indicates that the process is not in control! For viscosity we can assume no sudden chan ges, that means that these observation are auto correlated! Lets evaluate this .

Example: Viscosity Measurement

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Page 5: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 9/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Stat

>Time Series…

>Autocorrelation…

Minitab generates an autocorrelation at each time l ag of 1 from the original measurement data set. The significance of the correlation is sh own by the coefficient and the T value. The graphic display the 95% confidence interval as an additional information. In this case we have a significant autocorrelation up to ap proximately 4 lags.

Example: Viscosity Measurement

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Lag ACF T LBQ

1 0,820897 8,21 69,43

2 0,699538 4,57 120,36

3 0,601390 3,30 158,39

4 0,489573 2,43 183,86

5 0,410502 1,93 201,95

6 0,366249 1,66 216,51

7 0,349833 1,54 229,93

8 0,352668 1,52 243,72

Page 10/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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Description with Fitted Line Plot

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Therefore generate a new column with a lag of 1 hour

versus the original observations.

Please note the strong positive correlation with

coefficient of 0,8.

Similar diagrams can be generated for the lags 2, 3, etc. in order to display the

correlation.

Page 6: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 11/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

We notice , that the observation with a 1 hour lag are correlated with correlation coefficient of r 1=0,82 and the observations with a 2 hour lag with a correlation coefficient of r 2=0,70.

For the 3 and 4 hour lags we have a coefficient of r3=0,60 and r 4=0,49.

Because all these observation falls outside the +/- 2 sigma limits (red lines) we can say that these delayed autocorrelation are statisticall y significant.

A regressive decrease of the correlation can be obs erved. This pattern is typically of an autoregressive process.

For the clarification of the correlation between th e 1 hour lag observations lets show the viscosity at the time t-1, call that Yt-1, versus the viscosity at the time t, call that Yt . We receive the best model, if we use the moving ave rage of all changes versus the previous values. We can have 1st order or higher or der models. In this case we have 1st order model for autoregressive processes (AR 1) :

t1ttYcY ε+ϕ+=

C is a constant which results out of the average ch ange of the moving average and εεεε is the residual error, which we call white noise.

The Autoregressive Model

Page 12/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

ARIMA = Autoregressive Integrated Moving Average

Final Estimates of Parameters

Type Coef SE Coef T P

AR 1 0,8467 0,0549 15,42 0,000

Constant 13,1219 0,3806 34,48 0,000

Mean 85,588 2,482

The fitted model:

Y = 13,12 + 0,847 Y t-1 +ε

The ARIMA Model in Minitab

Stat

>Time Series…

>ARIMA…

Page 7: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 13/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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The residual are independent and normal distributed which indicates that the model is valid.

The Residual Analysis

The fitted model:

Y = 13,12 + 0,847 Y t-1 +ε

Page 14/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Notice the process does not show any evidence of be ing out of control once the auto correlative structure in the data due to dependency among successive observations is modeled .Such dependency is often due to the slowness of the process to change relative to the sampling frequency. That is, the samplinginterval is much shorter than the time constant for the process dynamics.

Here is an opportunity to reduce the sampling rate withoutcompromising the control of the process. This may h ave little importance if there is an on-line viscometer or an APC system where data gathering is cheap and control is automa tic. But it could be a major saving if viscosity is read in the lab and overcorrection (i.e. false alarms) is done leading to g reater process variation

Interpretation of the Results

Page 8: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 15/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

• Observations taken in time series are frequently au to correlatedand hence are not independently distributed

• In this case, assumptions for using conventional co ntrol charts are violate

• This can impact the number of false out-of-control signals if the process is being monitored via an SPC chart

• The autocorrelation can be modeled and the control chart can be applied to the residuals to more correctly identify out-of-control situations

• In practice, if the process is under manual control rather than APC, it is often better to alter the sampling plan so th e sampling interval exceeds the time constant

• This saves the costs of sampling and the costs of i ntroducing more variation caused by unnecessary process interf erence.

Results & Learning's

Page 16/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

This is the correlation between two variables, e.g. one input like xt at the time t and a output like yt+k observed at time t+k. In this case the lag is k – times between the observations.

Example:

In multi-step chemical process where 5 hours separa tes when the reactor temperature is changed and when the yie ld is impacted and measured, the correlation is strongest when yield lags temperature by 5 hours

Time Lagged Cross Correlation

Page 9: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 17/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Time Lagged Cross Correlation

• Some reasons why we might need to know the time lagged relationships between KPIV’s and one or more KPOV’s

– Analyze multi-vari type data

• see what input and noise variables show some relationshipwith output variables in order to better control the process

• allow for the possibility of a time difference or lag between change in one versus the effect on the other

– To better know when a change in an input variable will impact the output variable

– To identify and develop a transfer function model that can be used for Automatic Process Control (APC)

Page 18/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

In a chemical process the CO 2 content shall be controlled at a level of 47-53% with the methane feed rate. An experiment wi ll be conducted to determine the control parameter. After the evaluati on, it seams that additional other factors effect the CO 2 content. The data have been colleted in minute frequency. File Cross Correlation .mtw

Only 31% of the variation explained??

Example: Time Lagged Cross Correlation

Methane Feed CO2 Conc.0,37 53,4-0,18 52

-1,302 54,90,435 55,70,987 51,61,866 49,20,79 47,5

0,645 51,12,812 501,239 460,535 47,91,019 50,61,223 50,10,255 49,2

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Page 10: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 19/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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The time series plot looks like expected:

Negative correlations result in mirror images.

A detailed look discovers that the plots are shifte d by 1!

Example: Time Lagged Cross Correlation

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Page 20/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Minitab shows the correlation coefficient for every lag. The lag of –1min has the most significant effect.

Stat

>Time Series…

>Cross Correlation…

Cross Correlation Function: CO2 Conc.; Methane Feed

CCF - correlates CO2 Conc.(t) and Methane Feed(t+k)

-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0+----+----+----+----+----+----+----+----+----+----+

-15 0,109 XXXX-14 0,081 XXX-13 0,164 XXXXX-12 0,362 XXXXXXXXXX-11 0,261 XXXXXXXX-10 0,133 XXXX-9 -0,004 X-8 0,006 X-7 0,044 XX-6 0,060 XXX-5 0,049 XX-4 -0,018 X-3 -0,215 XXXXXX-2 -0,527 XXXXXXXXXXXXXX-1 -0,965 XXXXXXXXXXXXXXXXXXXXXXXXX0 -0,556 XXXXXXXXXXXXXXX1 -0,185 XXXXXX2 -0,068 XXX3 0,094 XXX4 0,087 XXX5 0,047 XX6 -0,014 X7 -0,049 XX8 0,142 XXXXX9 0,282 XXXXXXXX10 0,382 XXXXXXXXXXX11 0,198 XXXXXX12 0,072 XXX13 0,113 XXXX14 0,053 XX15 0,067 XXX

Cross Correlation in Minitab

Page 11: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 21/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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Generate a new column with a lag of 1 min.

Now we receive the expected good correlation between

input and output.

Creation of the Time Lagged Model

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>Lag…

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>Fitted Line Plot…

Page 22/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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The residuals are normal distributed without trends. Therefore the model can be used.

In which area must the methane feed rate be controlled, if the CO 2portion shall be kept between 47-53%?

Doe you think that there are other items which should be considered?

The Residual AnalysisStat

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>Fitted Line Plot…

>Graphs

>Four in one

Page 12: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 23/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

The values are independent from each other?

Also, if we control the process inputs in the futur e with more tide limits, we have to watch always the results.

The Values of the CO 2 Content

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Final Estimates of ParametersType Coef SE Coef T PAR 1 0,6409 0,1450 4,42 0,000Constant 18,8337 0,4761 39,55 0,000Mean 52,448 1,326

Autocorrelation Function: CO2 Conc.

Lag ACF T LBQ

1 0,638884 3,50 13,51

2 0,258750 1,05 15,81

3 0,048163 0,19 15,89

4 -0,060800 -0,24 16,03

5 -0,072229 -0,28 16,23

6 -0,048721 -0,19 16,32

7 -0,011212 -0,04 16,33

8 0,000434 0,00 16,33

The Check for Autocorrelation

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Lag 1 shows a significant autocorrelation. This ind icates the next step.

We also generate here the autoregressive model and analyze the residuals.

Stat

>Time Series…

>ARIMA…

Page 13: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 25/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

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Without trend & normal distributed... The AR 1 mode l fits.

This short evaluation can be important for the future check frequency or for the display of the CO2 content.

The Residual Analysis

P-Value = 0,934

Page 26/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

• Often important correlations are missed between variables because the time lag is not considered

• Time lagged cross correlation analysis is a valuable tool to use with multi-vari data to discover and quantify relationships

• Cross correlation studies are an important step in developing control plans, as well as controller or transfer function relationships for APC

Results & Learning's

Page 14: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

Page 27/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

Appendix for Time Series Analysis

•Autocorrelation

•Time Lagged Cross Correlation

Page 28/3111b BB W4 Auto & Cross Correlation, 04, D. Szemkus/H. Winkler

• Use basic data over 5 – 10 days with more than 25 ob servation to gather the typically process variation.

• Display the effect or the KPOV with an original dat a control chart, check for observation out of the “3 sigma limits” a nd “9 points above or below the center line”.

• Create an autocorrelations diagram: Stat>Time Series >Autocorrelation .

• If the autocorrelation decreases exponential with s ome significant values at the initial lags, we assume that the data auto correlate and that we can express it with a 1st order autoregress ive model

• Create an 1st order autoregressive model: Stat>Time Series>ARIMA(Autoregressive Integrated Moving Average) and use the original KPOV for "series" and enter "1" at the menu "autoregres sive". Store the residuals and fits.

Steps for the check of Autocorrelation and Creation of a Control Chart

Page 15: Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Autocorrelation and Cross Correlation

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• Analyze the residual with the residual plots, check if the data are normal distributed and randomly distributed. No or little pattern shall be noticeable. (This is a check if the 1st order au tocorrelation model is adequate for the description of the auto correlativ e structure.)

• Create now a control chart for the residuals. If th e “out of control” signals (test 1 and 2) are differs significantly fr om the original data, use the residual for the output variable on the control chart.

• An alternative is the extension of the sampling int erval. A rough rule is, to use a new interval in a manual SPC system (that mean not in an APC system) so that the first lagged autocorrelation is < 0,5. In the viscosity example we could change the sampling interval from 1 hour to about 4 -5 hours. For some reason it could be necessary to k eep the existing interval, e.g. if another critical KPOV has to be c ontrolled often and accurate.

Steps for the check of Autocorrelation and Creation of a Control Chart

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• If the autocorrelation looks like a sinus wave, tha n we can use the 2nd order autocorrelation model. Enter “2” in the menu "autoregressive” at ARIMA. (Another possibility to check for the 2nd or der auto regression is the use of the partial autocorrelation function: Stat>Time Series>Partial Autocorrelation . (If 2 significant spikes occur we have a 2nd order model, if only 1 spike occurs it is a 1st order model). Analyze the residual ( e) with a control chart and compare it with the chart of the original measurements ( yt).

• Decide which output variable (original measurements or residuals) and which sample interval you want to use for your SPC plan.

Steps for the check of Autocorrelation and Creation of a Control Chart

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• Montgomery, ‘Intro to SQC”, John Wiley and Sons, 3rd Edition, 1996, Seite 374-398.

• Box, Hunter und Hunter, “Statistics for Experimenters” , John Wiley and Sons, 1978, Kapitel 18.

• Box, Jenkins und Reinsel, “Time Series Analysis, For ecasting and Control”, Prentice Hall, 3rd Edition, 1994.

Note: There is a whole family of autoregressive integrated moving average (ARIMA) time series models that are discussed in the above books. In this module we only covered in detail the simplest situations. Those seriously interested in the subject should read / refer to the above texts.

Literature