Trend Detection in CarManufacturing Processes

Anton Andriyashin

Michal Benko

Wolfgang Hardle

Roman Timofeev

Uwe Ziegenhagen

http://ise.wiwi.hu-berlin.dehttp://www.case.hu-berlin.de

Motivation 1-2

Car Production Today

� outsourcing, different suppliers involved� painting is important cost factor� cost(painting)>cost(body)

Prozess Prototyping - OberflächentechnikBesuch Prof. Härdle, 19.10.2004

Rohbau VorreinigenVorreinigen Spülen PhosphatierenSpülen PhosphatierenSpülen Phosphatieren Passivieren Spülen Passivieren Spülen

Elektro-TauchlackierenElektro-TauchlackierenSpülenSpülenTauchspülenTauchspülenEinbrennenEinbrennen

SchleifenSchleifen AbdeckenAbdecken UnterbodenschutzUnterbodenschutz AbdeckenAbdecken Nahtabdichung (automatisch & manuell)Nahtabdichung (automatisch & manuell) AbreibenAbreiben

Füller (ESTA)Füller (ESTA)EinbrennenEinbrennenSchleifenSchleifenReinigenReinigenBC innenBC innenBC ESTABC ESTABC Air

Flash offFlash off CC innenCC innen CC ESTACC ESTA Flash offFlash off EinbrennenEinbrennen QualitätskontrolleQualitätskontrolle EndmontageEndmontage

OEM

Trend Detection

Motivation 1-3

Problem

� early detection of trend needed

� construction of statistical tests

� comparison of test procedures

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2-4

Outline

1. Motivation X

2. Dataset

3. Tests

4. Simulation

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Data 3-5

Dataset

� 7100 painted cars, 20 different colors

� luminance, a- and b- color axis from fiveangles for bonnet and wing

� explanatory variables: color lot, temperature,measuring person

� 42% silver (luminance most important component)

� customers most sensitive to changes of metallic colors

Trend Detection

Data 3-6

Used Angles

15°

25°

75°

45°

110°

δ

surface of wing/bonnet

1

δ L* a* b*

15◦ 98.94 1.58 4.4325◦ 81.35 1.17 4.2045◦ 50.98 0.73 2.5875◦ 32.30 0.48 1.29

110◦ 26.11 0.21 0.87

Figure 1: Used angles and typical Lab-values for ’silver’

Trend Detection

Data 3-7

The Lab Colorspace

� color model proposed by International Commission onIllumination

� each color defined by a tuple of luminance and the positionsbetween red-green (a-axis) and blue-green (b-axis)

Source: Wikipedia

Figure 2: Color tiles for increasing values of luminance, source: Wikipedia

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Data 3-8

Data Example

0 100 200 300 400 500

Observations

7980

8182

L*

Figure 3: Sequence of L*-values for 550 ’silver’ observations

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Tests 4-9

Proposed Test Procedures

local t-Test, 3-Slope TestH0: slope equal to zero

Change Point EstimationH0: E (xi ) = const

Mann-Kendallsign-test, H0: no trend

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Tests 4-10

Local t-test

� (αl , βl) = arg min(αl ,βl )

∑ni=1(yi − αl − βlxi )

2I(xi ∈ Ll)where Ll = [x0 + (k − 1) · ∆, x0 + (k − 1) · ∆ + I ], l = 1 . . . n/I

and ∆ indicates the step size

� no (local) linear trend, H0: βl = 0

� t-test statistics tl =bβl

SE(bβl )

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Tests 4-11

Local t-Test

Figure 4: Schematic chart of local t-test

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Tests 4-12

Change Point Estimation

H0 : µ1 = µ2 H1 : µ1 6= µ2

The different µ-values are computed as µj = n−1∑

i∈Ijyi , the

resulting test statistics is asymptotically χ2-distributed with onedegree of freedom.

n1 · n2

n1 + n2

(µ1 − µ2)2

σ2

L−→ χ2(1)

σ2 calculated by non-parametric smoothing.

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Tests 4-13

Non-Parametric Smoothing

0 100 200 300 400 500

Observations

4243

4445

4647

L*-

Val

ue

-3 -2 -1 0 1

X

00.

5

Y

Figure 5: data with smoothed curve and distributions of residuals (blue)

and fitted normal distribution pdf (red)

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Tests 4-14

Mann-Kendall Test

S =n∑

i=2

i−1∑j=1

sign(Xi − Xj)

Under the null hypothesis of no trend the test statistics followsapproximatively a standard normal distribution.

S√n(n−1)(2n+5)

18

L−→ N(0, 1)

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Tests 4-15

Example

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Tests 4-16

Integration Measure

Idea: Look at clusters of bars

Tests 4-13

Example

Color Harmonization

Control false alarms and gaps

τ number of subsequent bars necessary to start one integralκ number of allowed gaps between two groups of bars

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Tests 4-17

Example

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Tests 4-18

Example

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Simulation 5-19

Selection of l , τ and κ

� regression line with trend at 300 observations

� quasi-residuals calculated by non-parametric regression

� 50 simulation runs for each parameter setting

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6-20

Results

Mean delay (5 tests)l 25 50 75κ 3 5 3 5 3 5

τ

3 211 211 213 213 213 213

5 131 131 132 132 132 132

10 71 71 72 72 72 72

Table 1: Mean delays of all tests for γ = 2.86◦and ∆ = 5

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6-21

Conclusion

� choice of l , κ and τ

� comparison of tests:local t-test outperforms Mann-Kendallchange point estimation performs well

� threshold value to be adjusted by further analysis

Thank You for listening!

Trend Detection