BIMODALITY OF COMPACT BIMODALITY OF COMPACT YARN HAIRINESSYARN HAIRINESS

Jiří Militký , Sayed Ibrahim Jiří Militký , Sayed Ibrahim andand

Dana kDana křemenakovářemenakováTechnical University of Liberec, 46117 Liberec, Technical University of Liberec, 46117 Liberec,

Czech RepublicCzech Republic

Beltwide Cotton ConferenceJanuary 11-12, 2007

New Orleans, Louisiana

IntroductionIntroduction

Hairiness is considered as sum of the fibre ends and loops standing out Hairiness is considered as sum of the fibre ends and loops standing out from the main compact yarn bodyfrom the main compact yarn body

The most popular instrument is the The most popular instrument is the UsterUster hairiness system, which hairiness system, which characterizes the hairiness by H value, and is defined as the total length characterizes the hairiness by H value, and is defined as the total length of all hairs within one centimeter of yarn.of all hairs within one centimeter of yarn.

The system introduced by The system introduced by ZweigleZweigle, counts the number of hairs of , counts the number of hairs of defined lengths. The S3 gives the number of hairs of 3mm and longer.defined lengths. The S3 gives the number of hairs of 3mm and longer.

The information obtained from both systems are limited, and the The information obtained from both systems are limited, and the available methods either compress the data into a single vale H or S3, available methods either compress the data into a single vale H or S3, convert the entire data set into a spectrogram deleting the important convert the entire data set into a spectrogram deleting the important spatial information.spatial information.

Other less known instruments such as Other less known instruments such as ShirleyShirley hairiness meter or hairiness meter or F-HairF-Hair meter give very poor information about the distribution characteristics meter give very poor information about the distribution characteristics of yarn hairiness.of yarn hairiness.

Some laboratory systems dealing with Some laboratory systems dealing with image processingimage processing, decomposing , decomposing the hairiness into two exponential functions (Neckar,s Model), this the hairiness into two exponential functions (Neckar,s Model), this method is time consuming, dealing with very short lengths. method is time consuming, dealing with very short lengths.

Outlines

• Investigating the possibility of approximating the Investigating the possibility of approximating the yarn hairiness distribution by a mixture of two yarn hairiness distribution by a mixture of two Gaussian distributions.Gaussian distributions.

• Complex characterization of yarn hairiness data in Complex characterization of yarn hairiness data in time and frequency domain i.e. describing the time and frequency domain i.e. describing the hairiness by:hairiness by:-  -  periodic componentsperiodic components- - Random variationRandom variation

- Chaotic behavior- Chaotic behavior

RingRing-Compact -Compact SpinningSpinning

1)Draft arrangement1a) Condensing element1b) Perforated apronVZ Condensing zone

2) Yarn Balloon with new Structure3) Traveler, 4) Ring5) Spindle, 6) Ring carriage 7) Cop, 8) Balloon limiter9) Yarn guide, 10) Roving E) Spinning triangle of compact spinning

Experimental Part Experimental Part &&

Method of EvaluationMethod of Evaluation

• Three cotton combed yarnThree cotton combed yarnss of count of countss 14.6 14.6, 20 , 20 and 30 and 30 tex tex were were produced on produced on three commercial three commercial compact compact ring ring sspinning machinespinning machines. . The yarns The yarns were tested on Uster Tester 4 for 1 minute at were tested on Uster Tester 4 for 1 minute at 400 m/min.400 m/min.

• The raw data from Uster tester 4 were extracted The raw data from Uster tester 4 were extracted and converted to individual readings and converted to individual readings corresponding to yarn hairiness, i.e. the total corresponding to yarn hairiness, i.e. the total hair length per unit length (centimeter).hair length per unit length (centimeter).

Investigation of Investigation of

Bimodality of yarn HairinessBimodality of yarn Hairiness

2

4

6

8

10

12

hair le

ngth

0 100 200 300 400

Distance

2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12

0

0,1

0,2

0,3

Nonpara

metr

ic D

ensity

2 3 4 5 6 7 8 9 10 11 12

Yarn Hairiness

Hair Diagram Histogram (83 columns) Normal Dist. fit

Smooth curve fit

Number and width of bars affect the shape of the probability distribution Number and width of bars affect the shape of the probability distribution

The question is how to optimize the width of bars for better evaluationThe question is how to optimize the width of bars for better evaluation??

Gaussian curve fit (20 columns)

Basics Basics of of Probability density function IProbability density function I

1

j

( , )( )

hjN j

H

C t tf x

N

1( , )jN jC t t

1jh ( )j jt t

0.4int[2.46 (N-1) ]M

•The area of a column in a histogram represents a The area of a column in a histogram represents a piecewise constant estimator piecewise constant estimator of sample probability density. Its height is estimated by:of sample probability density. Its height is estimated by:

•Where is the number of sample Where is the number of sample

elements in this intervalelements in this interval

and is the length and is the length

of this interval.of this interval.

Number of classesNumber of classes

•For all samples is N= 18458 and M=125

1/33.49*(min( , ) /1.34) /h s Rq n

(0.75) (0.25)

Rq upper quartile lower quartile

Rq x x

h = 0.133h = 0.133

KernelKernel density density functionfunction

The Kernel type nonparametric of The Kernel type nonparametric of ssample probability density functionample probability density function

1

1ˆ( )N

i

i

x xf x K

N h

K x

OptimalOptimal bandwidthbandwidth : h : h1. Based on the assumptions of near normality1. Based on the assumptions of near normality2. Adaptive smoothing2. Adaptive smoothing3. Exploratory (local 3. Exploratory (local hhj j ) ) requirementrequirement

of equal probability in all classesof equal probability in all classes

Kernel function : bi-quadratic- symmetric around zero- properties of PDF

1/50.9*(min( , ) /1.34) /h s Rq n

h = 0.1278h = 0.1278

Bi-modal distributionBi-modal distribution

Two Gaussian DistributionTwo Gaussian Distribution

The bi-modal distribution can be approximated by two Gaussian distributions,

2 2( 1 ( 2

( ) 1*exp 2*exp1 2

i iiG

x B x Bf x A A

C C

Where , are proportions of shorter and longer hair distribution respectively, , are the means and , are the standard deviations.

H-yarn Program written in Matlab code, using the least square method is used for estimating these parameters.

1A 2A1B 2B

1C

MATLAB MATLAB 77..11 RELEASE 1 RELEASE 144

2C

Bi-modality of Yarn HairinessBi-modality of Yarn HairinessMixed Gaussian DistributionMixed Gaussian Distribution

The frequency distribution and fitted bimodal distribution curve

Analysis of Results Analysis of Results Check the type of DistributionCheck the type of Distribution

Bimodality parametricBimodality parametric • Mixture of distributions Mixture of distributions estimation and likelihood estimation and likelihood ratio testratio test• Test of significant distance Test of significant distance between modebetween modess (Separation) (Separation)

Bimodality nonparametric:Bimodality nonparametric:• kernel density (Silverman test)

• CDF (DIP, Kolmogorov tests)

• Rankit plot

3.5 4 4.53

3.5

4

4.5

5Rankit plot

unimodal Gaussian smoother closest to theunimodal Gaussian smoother closest to the x and the closest bimodal Gaussian smootherx and the closest bimodal Gaussian smoother

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

In general, the Dip test is for bimodality. However, mixture of two distributions does not necessarily result in a bimodal distribution.

Basic Distribution function definitions

ProbabilityProbability density function (PDF) density function (PDF) f f ((xx), ), Cumulative Distribution Function (CDF) Cumulative Distribution Function (CDF) F F ((xx), ), and Empirical CDF (ECDF) and Empirical CDF (ECDF) FnFn((xx) )

Unimodal CDF: convex in (−∞, m), concave Unimodal CDF: convex in (−∞, m), concave in [m, ∞) in [m, ∞) Bimodal CDF: one bumpBimodal CDF: one bumpLet Let G ∗G ∗ = arg min sup= arg min supx |Fnx |Fn((xx) ) − G− G((xx))||, , wherewhere GG((xx) is a unimode CDF.) is a unimode CDF.Dip Statistic: Dip Statistic: d d = sup= supx |Fnx |Fn((xx) ) − G∗− G∗((xx))||

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

CDF plots

rectangularempiricalnormal

2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5CDF plots

rectangularempiricalnormal

Dip Statistic (for n= 18500): 0.0102Dip Statistic (for n= 18500): 0.0102Critical value (n = 1000): Critical value (n = 1000): 0.0170.017

Critical value (n = 2000): 0.0112Critical value (n = 2000): 0.0112

Analysis of Results IAnalysis of Results IMixture of Gauss distributions

Analysis of Results IIAnalysis of Results II Dip Test

Points A and B are modes, shaded areas C,D are bumps, area E and F is a shoulder point

Dip test statistics:Dip test statistics:

It is the largest vertical difference between the empirical cumulative distribution FE and the Uniform distribution FU

This test is actually identification of mixed mixture of normal distribution, is only rejecting unimodality

Analysis of Results IIIAnalysis of Results IIILikelihood ratio testLikelihood ratio test

The single normal distribution model (μ,σ), the likelihood function is:

Where the data set contains n observations.The mixture of two normal distributions, assumes that each data point belongs to one of tow sub-population. The likelihood of this function is given as:

The likelihood ratio can be calculated from Lu and Lb as follows:

Significance of difference of meansSignificance of difference of means

• Two sample Two sample tt test of equality of means test of equality of means• T1 equal variancesT1 equal variances

• T2 different variancesT2 different variances

Analysis of Results VAnalysis of Results V

PDF and CDFPDF and CDF

0 1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45hairness histogram

Rel

. F

req.

h = 0.33226

Analysis of Results VIAnalysis of Results VI

Kernel density estimator: Kernel density estimator: Adaptive Kernel Density Adaptive Kernel Density Estimator for univariate data. Estimator for univariate data. (choice of band width (choice of band width hh determines the amount of determines the amount of smoothing. If a long tailed smoothing. If a long tailed distribution, fixed band width distribution, fixed band width suffer from constant width suffer from constant width across the entire sample. For across the entire sample. For very small band width an over very small band width an over smoothing may occur )smoothing may occur )

MATLAB AKDEST 1D- MATLAB AKDEST 1D- evaluates the univariate evaluates the univariate Adaptive Kernel Density Adaptive Kernel Density Estimate with kernelEstimate with kernel

( )1

( ) ( 1)

( )1

( ) ,

i

ii

i iN

ii

xcdf j x x

x

 

Parameter estimatParameter estimationion of of

mixture of two Gaussians modelmixture of two Gaussians model

Complex Characterization of Complex Characterization of Yarn HairinessYarn Hairiness

The yarn hairiness can be The yarn hairiness can be also also described described according to the:according to the:

- Random variationRandom variation - Periodic componentsPeriodic components - Chaotic behaviorChaotic behavior - The H-yarn program provides all calculations The H-yarn program provides all calculations

and offers graphs dealing with the analysis of and offers graphs dealing with the analysis of yarn hairiness as Stochastic Process.yarn hairiness as Stochastic Process.

Basic definitions of Basic definitions of

Time Series Time Series

• Since, the yarn hairiness is measured at equal-distance, the data Since, the yarn hairiness is measured at equal-distance, the data obtained could be analyzed on the base of time series.obtained could be analyzed on the base of time series.

•A time series A time series is a sequence of observations taken sequentially in time. is a sequence of observations taken sequentially in time. The nature of the dependence among observations of a time series is of The nature of the dependence among observations of a time series is of considerable practical interest.considerable practical interest.

•First of all, one should investigate the stationarity of the system. First of all, one should investigate the stationarity of the system.

•Stationary model assumes that the process remains in Stationary model assumes that the process remains in equilibrium equilibrium about a about a constant mean level. constant mean level. The random process is strictly stationary The random process is strictly stationary if all statistical characteristics and distributions are independent on if all statistical characteristics and distributions are independent on ensemble location.ensemble location. •Many tests such as Many tests such as nonparametric test, run test, variability (difference nonparametric test, run test, variability (difference test), cumulative periodogramtest), cumulative periodogram construction construction are provided to explore the are provided to explore the stationarity of the processstationarity of the process..

Stationarity testPeriodogram

For characterization of independence hypothesis against periodicity alternative the cumulative periodogram C(fi) can be constructed.For white noise series (i.i.d normally distributed data), the plot of C(fi) against fi

would be scattered about a straight line joining the points (0,0) and (0.5,1). Periodicities would tend to produce a series of neighboring values of I(fi) which were

large. The result of periodicities therefore bumps on the expected line. The limit lines for 95 % confidence interval of C(fi) are

drawn at distances.

0 0.1 0.2 0.3 0.4 0.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

rel. frequency [-]

cu

mu

l. p

erio

do

gra

m [-]

Cumulative periodogram

0 0.1 0.2 0.3 0.4 0.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

rel. frequency [-]

cu

mu

l. p

erio

do

gra

m [-]

Cumulative periodogram

0 0.1 0.2 0.3 0.4 0.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

rel. frequency [-]

cu

mu

l. p

erio

do

gra

m [-]

Cumulative periodogram

System A 14.6 tex

System B

System C

Time Domain Analysis Autocorrelation

Simply the Autocorrelation function is a comparison of a signal with itself as a function of time shift.

Autocorrelation coefficient of first order R(1) can be evaluated as1

2

( ( ) ) * ( ( 1) )

(1)[ ( 1)]

N

j

y j y y j y

Rs N

0 20 40 60 80 100-0.15

-0.1

-0.05

0

0.05ACF

Lag

Au

toco

rre

latio

n fu

nctio

n

0 20 40 60 80 100-0.15

-0.1

-0.05

0

0.05ACF

Lag

Au

toco

rre

latio

n fu

nct

ion

0 20 40 60 80 100-0.15

-0.1

-0.05

0

0.05ACF

Lag

Au

toco

rre

latio

n fu

nct

ion

System A System B System C

For sufficiently high L is first order autocorrelation equal to zero

h40sussen.txtAutocorrelation

0 50 100 150Lag

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Au

toco

rre

latio

n

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Au

toco

rre

latio

n

Frequency domain

h40sussen.txtFourier Frequency Spectrum

50

90

95

99 99.9

0 5 10 15 20 25Frequency

0

0.25

0.5

0.75

1

1.25

1.5

1.75

PS

D T

ISA

0

0.25

0.5

0.75

1

1.25

1.5

1.75

PS

D T

ISA

The Fast Fourier Transformation is used to transform from time domain to frequency domain and back again is based on Fourier transform and its inverse. There are many types of spectrum analysis, PSD, Amplitude spectrum, Auto regressive frequency spectrum, moving average frequency spectrum, ARMA freq. Spectrum and many other types are included in Hyarn program.

0 5 10 15 20 250

50

100

150

200

250

300

frequency [1/m]

PS

D W

els

ch

Power spectal density Welsch

0 5 10 15 20 250

50

100

150

200

frequency [1/m]

PS

D W

els

ch

Power spectal density Welsch

System A System CSystem B

0 5 10 15 20 250

100

200

300

400

500

frequency [1/m]

PS

D W

els

ch

Power spectal density Welsch

h40sussen.txtParametric Reconstruction [7 Sine]

r^2=1e-08 SE=0.994675 F=9.2185e-06

0.0062095

0.2324 10.478

10.481

10.492

10.50511.609

0 100 200 300 400 500distance

-1.5

-1

-0.5

0

0.5

1

Ha

ir S

usse

n

-1.5

-1

-0.5

0

0.5

1

Ha

ir S

usse

n

0

2.5

5

7.5

10

Ha

ir S

usse

n

0

2.5

5

7.5

10

Ha

ir S

usse

n

Fractal DimensionFractal DimensionHurst ExponentHurst Exponent

The cumulative of white Identically Distribution noise is known as Brownian motion or a random walk. The Hurst exponent is a good estimator for measuring the fractal dimension. The Hurst equation is given as . The parameter H is the Hurst exponent.

The fractal dimension can be measured by 2-H. In this case the cumulative of white noise will be 1.5. More useful is expressing the fractal dimension 1/H using probability space rather than geometrical space.

*/ ( ) ^R S K n obs H

h40sussen.txtHurst Exponent

H=628487, SD H=0.00214796, r2=0.93009

1 10 100 1000 10000n obs

1

10

100

1000

R/S

1

10

100

1000

R/S

h40zinser.txtHurst Exponent

H=0.659309, Sd H=0.000808122, r2=0.995684

1 10 100 1000 10000n obs

1

10

100

1000

R/S

1

10

100

1000

R/S

h40rieter.txtHurst Exponent

H=0.6866, SH H = 0.001768, r2= 0.9739

1 10 100 1000 10000n obs

1

10

100

1000

R/S

1

10

100

1000

R/S

System A System B System C

Summery of results of ACF, Power Summery of results of ACF, Power Spectrum and Hurst ExponentSpectrum and Hurst Exponent

ConclusionsConclusions

• Preliminary investigation shows that the yarn hairiness Preliminary investigation shows that the yarn hairiness distribution can be fitted to a bimodal model distribution.distribution can be fitted to a bimodal model distribution.

• The yarn Hairiness can be described by two mixed Gaussian The yarn Hairiness can be described by two mixed Gaussian distributions, the portion, mean and the standard deviation of each distributions, the portion, mean and the standard deviation of each component leads to deeper understanding and evaluation of component leads to deeper understanding and evaluation of hairiness.hairiness.

• This method is quick compared to image analysis system, beside This method is quick compared to image analysis system, beside that, the raw data is obtained from world wide used instrument that, the raw data is obtained from world wide used instrument “Uster Tester”. “Uster Tester”.

• The Hyarn system is a powerful program for evaluation and The Hyarn system is a powerful program for evaluation and analysis of yarn hairiness as a dynamic process, in both time and analysis of yarn hairiness as a dynamic process, in both time and frequency domain. frequency domain.

• Hyarn program is capable of estimating the short and long term Hyarn program is capable of estimating the short and long term dependency.dependency.