EWMA card

An EWMA card (of English. E xponentially w eighted m oving a verage ) is a process control chart for the preparation of exponentially weighted moving average values .

use

EWMA cards are suitable for monitoring processes with regard to their constant quality over time by evaluating test data. They are used for statistical process control to optimize production and service processes. With them, slight shifts in the process means ( trends ) can be identified, which stand out due to the lower weighting of older samples . All data of the monitored business or industrial process that has ever been collected is used. While other control charts deal with rational subsets of samples individually, the EWMA chart tracks the exponentially weighted moving average of all previous sampling means. It is particularly suitable for small sample sizes and the sample size n = 1, since it can also detect small changes, since the exponential weighting of the mean values ​​leads to tighter control limits than with the usual mean value map.

diagram

A diagram point can either stand for individual observations or on subgroups , whereby the value of the subgroup is calculated from the mean value of the individual values ​​belonging to the subgroup. For the calculation of the individual points, information from the previous points is used and then weighted by the user . The advantage here is that the result is only slightly influenced by high or low values ​​entering the calculation. Almost all trends can be found by changing the weight , generally smaller weights are given to identify smaller trends. Exponentially weighted motion variance can be used to obtain a significance score or limits that automatically adjust to the data observed.

calculation

The calculation is based on a time series . The goal is to predict the value by a value at the point in time . The value observed at the point in time and the forecast value created at the point in time are used for this forecast . To make a forecast, the forecast is corrected for the forecast error : ${\displaystyle x_{1},x_{2},x_{3},\ldots }$ ${\displaystyle t}$ ${\displaystyle x_{t+1}}$ ${\displaystyle {\ddot {x}}_{t+1}}$ ${\displaystyle t}$ ${\displaystyle x_{t}}$ ${\displaystyle t-1}$ ${\displaystyle {\ddot {x}}_{t}}$ ${\displaystyle x_{t+1}}$ ${\displaystyle {\ddot {x}}_{t}}$ ${\displaystyle x_{t}-{\ddot {x}}_{t}}$

${\displaystyle {\ddot {x}}_{t+1}={\ddot {x}}_{t}+(x_{t}-{\ddot {x}}_{t})}$

You can see that through can be found out. Previous calculations show that only part λ · (x _t - ẍ _t ) of the forecast error should be taken into account when making the correction . This leads to the following forecast: ${\textstyle x_{t+1}}$ ${\displaystyle x_{t}}$

${\displaystyle {\ddot {x}}_{t+1}={\ddot {x}}_{t}+\lambda (x_{t}-{\ddot {x}}_{t})}$

If one uses the expression in the previous equation for the forecast, one speaks of exponential smoothing of the first order. You can also transform the expression as follows:

${\displaystyle {\begin{aligned}{\ddot {x}}_{t+1}&={\ddot {x}}_{t}+\lambda (x_{t}-{\ddot {x}}_{t})\\&={\ddot {x}}_{t}+\lambda x_{t}-\lambda {\ddot {x}}_{t}\\&=\lambda x_{t}+(1-\lambda ){\ddot {x}}_{t}\end{aligned}}}$

So we calculate from and using a convex linear combination . Where is the weight of the current observation . The EWMA map is based on first-order exponential smoothing. Starting from a starting value , an estimated value of is determined at each point in time with the aid of exponential smoothing of the first order . This is called . We replace in equation so through and through and get the following equation ${\displaystyle {\ddot {x}}_{t+1}}$ ${\displaystyle x_{t}}$ ${\displaystyle {\ddot {x}}_{t}}$ ${\displaystyle \lambda }$ ${\displaystyle x_{t}}$ ${\displaystyle z_{0}}$ ${\displaystyle t}$ ${\displaystyle x_{t+1}}$ ${\displaystyle z_{t}}$ ${\displaystyle {\ddot {x}}_{t+1}=\lambda x_{t}+(1-\lambda ){\ddot {x}}_{t}}$ ${\displaystyle {\ddot {x}}_{t+1}}$ ${\displaystyle z_{t}}$ ${\displaystyle {\ddot {x}}_{t}}$ ${\displaystyle z_{t-1}}$

${\displaystyle z_{t}=\lambda x_{t}+(1-\lambda )z_{t-1}}$

The following applies . Usually you bet . ${\displaystyle 0\leq \lambda \leq 1}$ ${\displaystyle z_{0}=\mu _{0}}$

The following applies:

${\displaystyle {\begin{aligned}z_{t}&=\lambda x_{t}+(1-\lambda )\lambda x_{t-1}+\ldots +(1-\lambda )^{t-1}\lambda x_{1}+(1-\lambda )^{t}z_{0}\\&=\lambda \sum _{j=0}^{t-1}(1-\lambda )^{j}x_{t-j}+(1-\lambda )^{t}z_{0}\end{aligned}}}$

In order to be able to set up the quality control chart, we need the expected value and the variance . The limits of the EWMA card are then calculated through ${\displaystyle \operatorname {E} (Z_{t})}$ ${\displaystyle \operatorname {Var} (Z_{t})}$

upper limit = ${\displaystyle \operatorname {E} (Z_{t})+L{\sqrt {\operatorname {Var} (Z_{t})}}}$

lower limit = ${\displaystyle \operatorname {E} (Z_{t})-L{\sqrt {\operatorname {Var} (Z_{t})}}}$

If the process is under control, it applies to everyone . It follows from this ${\displaystyle \operatorname {E} (X_{t})=\mu _{0}}$ ${\displaystyle t}$

${\displaystyle \operatorname {E} (Z_{t})=\mu _{0}}$

If one assumes that they are independent with identical variance , then applies ${\displaystyle X_{t}}$ ${\displaystyle \sigma ^{2}}$

${\displaystyle \operatorname {Var} (Z_{t})=\sigma ^{2}{\frac {\lambda \left(1-(1-\lambda )^{2t}\right)}{2-\lambda }}}$

So the limits of the EWMA card are

upper limit = ${\displaystyle \mu _{0}+L\sigma {\sqrt {{\frac {\lambda }{2-\lambda }}\left(1-(1-\lambda )^{2t}\right)}}}$

Center line = ${\displaystyle \mu _{0}}$

lower limit = ${\displaystyle \mu _{0}-L\sigma {\sqrt {\frac {\lambda \left(1-(1-\lambda )^{2t}\right)}{2-\lambda }}}}$

You can see that the limits of the EWMA card become more and more stable as it grows . From follows ${\displaystyle t}$ ${\displaystyle 0\leq \lambda \leq 1}$

${\displaystyle \lim _{t\to \infty }(1-\lambda )^{2t}=0}$

So:

${\displaystyle \lim _{t\to \infty }\operatorname {Var} (Z_{t})=\lim _{t\to \infty }\sigma ^{2}{\frac {\lambda \left(1-(1-\lambda )^{2t}\right)}{2-\lambda }}=\sigma ^{2}{\frac {\lambda }{2-\lambda }}}$

The approximate limits of the EWMA card are therefore

LCL = ${\displaystyle \mu _{0}-L\sigma {\sqrt {\frac {\lambda }{2-\lambda }}}}$

CL = ${\displaystyle \mu _{0}}$

UCL = ${\displaystyle \mu _{0}+L\sigma {\sqrt {\frac {\lambda }{2-\lambda }}}}$

swell

Web links

• http://support.minitab.com/de-de/minitab/17/topic-library/quality-tools/control-charts/understanding-time-weighted-control-charts/what-is-an-ewma-chart/ (December 24, 2016)
• https://www.statsoft.de/glossary/E/ExponentiallyWeightedMovingAverageEWMAChart.htm (December 24, 2016)
• https://www.wiwi.europa-uni.de/de/lehrstuhl/fact/statistik/forschung/forschungsbericht/lsprojekt-ewma_grenzen/index.html (December 24, 2016)
• https://www.wiwi.europa-uni.de/de/lehrstuhl/fact/statistik/forschung/forschungsbericht/lsprojekt-ewma_und_cusum/index.html (December 24, 2016)
• http://statmath.wu-wien.ac.at/stat4/hackl/stp0106.htm (December 24, 2016)
• 6.3.2.4. EWMA Control Charts. In: itl.nist.gov. Retrieved December 30, 2016 . 

literature

• Norbert Vetter: Sensitivity comparison of quality control cards of the CUSUM and EWMA types. diplom.de, 1999, ISBN 978-3-832-41744-4 , ( limited preview in the Google book search)
• Jens Hitzinger: Quality control cards for monitoring the production situation with short and long-term memory. diplom.de, 2001, ISBN 978-3-832-43350-5 , p. 27 ( limited preview in the Google book search)
• German Society for Quality e. V. (DGQ): SPC 2 - quality control card technology , elaborated by the working group 162 "quality control cards" of the German Society for Quality. 4th edition, Beuth, Berlin, Cologne 1992, ISBN 3-410-32827-0 .
• K. Bernecker. German Ges. Für Qualität e. V. (DGQ): SPC 3 - Instructions for statistical process control (SPC): Quality control cards, process capability assessments (Cp, Cpk), error collection card . 1st edition, Beuth, Berlin 1990, ISBN 3-410-32821-1 .

Individual evidence

1. ↑ ^{a b c} What is an EWMA card? In: support.minitab.com. Retrieved December 30, 2016 . 
2. ↑ Exponentially weighted mean value map (EWMA). Retrieved January 23, 2017 . 
3. ↑ Peter Hackl : Process Control: Other methods EWMA card. May 7, 2001. Retrieved January 23, 2017 .