BIMODALITY OF COMPACT YARN HAIRINESS

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BIMODALITY OF COMPACT YARN HAIRINESS
Beltwide Cotton Conference January 11-12,
2007 New Orleans, Louisiana

• Jirí Militký , Sayed Ibrahim
• and
• Dana kremenaková
• Technical University of Liberec, 46117 Liberec,
• Czech Republic

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Introduction
Hairiness is considered as sum of the fibre ends
and loops standing out from the main compact yarn
body The most popular instrument is the Uster
hairiness system, which characterizes the
hairiness by H value, and is defined as the total
length of all hairs within one centimeter of
yarn. The system introduced by Zweigle, counts
the number of hairs of defined lengths. The S3
gives the number of hairs of 3mm and longer. The
information obtained from both systems are
limited, and the available methods either
compress the data into a single vale H or S3,
convert the entire data set into a spectrogram
deleting the important spatial information. Other
less known instruments such as Shirley hairiness
meter or F-Hair meter give very poor information
about the distribution characteristics of yarn
hairiness. Some laboratory systems dealing with
image processing, decomposing the hairiness into
two exponential functions (Neckar,s Model), this
method is time consuming, dealing with very short
lengths.
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Outlines

• Investigating the possibility of approximating
  the yarn hairiness distribution by a mixture of
  two Gaussian distributions.
• Complex characterization of yarn hairiness data
  in time and frequency domain i.e. describing the
  hairiness by
• -  periodic components
• - Random variation
• - Chaotic behavior

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Ring-Compact Spinning
1)Draft arrangement 1a) Condensing element 1b)
Perforated apron VZ Condensing zone 2) Yarn
Balloon with new Structure 3) Traveler, 4)
Ring
5) Spindle, 6) Ring carriage 7) Cop, 8) Balloon
limiter 9) Yarn guide, 10) Roving E) Spinning
triangle
of compact
spinning
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Experimental Part Method of Evaluation

• Three cotton combed yarns of counts 14.6, 20 and
  30 tex were produced on three commercial compact
  ring spinning machines. The yarns were tested on
  Uster Tester 4 for 1 minute at 400 m/min.
• The raw data from Uster tester 4 were extracted
  and converted to individual readings
  corresponding to yarn hairiness, i.e. the total
  hair length per unit length (centimeter).

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Investigation of Bimodality of yarn Hairiness
Number and width of bars affect the shape of the
probability distribution The question is how to
optimize the width of bars for better evaluation?

Hair Diagram
Histogram (83 columns)
Normal Dist. fit
Gaussian curve fit (20 columns)
Smooth curve fit
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Basics of Probability density function I

• The area of a column in a histogram represents a
  piecewise constant estimator of sample
  probability density. Its height is estimated by
• Where is the
  number of sample
• elements in this interval
• and is the
  length
• of this interval.
• Number of classes
• For all samples is N 18458 and M125

h 0.133
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Kernel density function
The Kernel type nonparametric of sample
probability density function
Kernel function bi-quadratic -
symmetric around zero - properties of PDF
Optimal bandwidth h 1. Based on the
assumptions of near normality 2. Adaptive
smoothing 3. Exploratory (local hj ) requirement
of equal probability in all classes
h 0.1278
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Bi-modal distributionTwo Gaussian Distribution
MATLAB 7.1 RELEASE 14
The bi-modal distribution can be approximated by
two Gaussian distributions,
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.
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Bi-modality of Yarn HairinessMixed Gaussian
Distribution
The frequency distribution and fitted bimodal
distribution curve
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Analysis of Results Check the type of
Distribution

• Bimodality parametric
• Mixture of distributions
• estimation and likelihood
• ratio test
• Test of significant distance
• between modes (Separation)
• Bimodality nonparametric
• kernel density (Silverman test)
• CDF (DIP, Kolmogorov tests)
• Rankit plot

unimodal Gaussian smoother closest to the x
and the closest bimodal Gaussian smoother
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Basic Distribution function definitions
In general, the Dip test is for bimodality.
However, mixture of two distributions does not
necessarily result in a bimodal distribution.
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Analysis of Results IMixture of Gauss
distributions
Probability density function (PDF) f (x),
Cumulative Distribution Function (CDF) F (x),
and Empirical CDF (ECDF) Fn(x) Unimodal
CDF convex in (-8, m), concave in m, 8)
Bimodal CDF one bump Let G arg min supx
Fn(x) - G(x), where G(x) is a unimode
CDF. Dip Statistic d supx Fn(x) - G(x)
Dip Statistic (for n 18500) 0.0102 Critical
value (n 1000) 0.017 Critical value (n
2000) 0.0112
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Analysis of Results II Dip Test
Dip test statistics It is the largest vertical
difference between the empirical cumulative
distribution FE and the Uniform distribution FU
Points A and B are modes, shaded areas C,D are
bumps, area E and F is a shoulder point
This test is actually identification of mixed
mixture of normal distribution, is only rejecting
unimodality
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Analysis of Results IIILikelihood ratio test
The single normal distribution model (µ,s), 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
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Significance of difference of means
Analysis of Results V

• Two sample t test of equality of means
• T1 equal variances
• T2 different variances

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PDF and CDF
Analysis of Results VI
Kernel density estimator Adaptive Kernel Density
Estimator for univariate data. (choice of band
width h determines the amount of smoothing. If a
long tailed distribution, fixed band width suffer
from constant width across the entire sample. For
very small band width an over smoothing may occur
) MATLAB AKDEST 1D- evaluates the univariate
Adaptive Kernel Density Estimate with kernel
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Parameter estimation of mixture of two Gaussians
model
 
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Complex Characterization of Yarn Hairiness

• The yarn hairiness can be also described
  according to the
• Random variation
• Periodic components
• Chaotic behavior
• The H-yarn program provides all calculations and
  offers graphs dealing with the analysis of yarn
  hairiness as Stochastic Process.

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Basic definitions of Time Series

• Since, the yarn hairiness is measured at
  equal-distance, the data obtained could be
  analyzed on the base of time series.
• A time series is a sequence of observations taken
  sequentially in time. The nature of the
  dependence among observations of a time series is
  of considerable practical interest.
• First of all, one should investigate the
  stationarity of the system.
• Stationary model assumes that the process remains
  in equilibrium about a constant mean level. The
  random process is strictly stationary if all
  statistical characteristics and distributions are
  independent on ensemble location.
• Many tests such as nonparametric test, run test,
  variability (difference test), cumulative
  periodogram construction are provided to explore
  the stationarity of the process.

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Stationarity testPeriodogram
System A
14.6 tex
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.
System B
System C
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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 as
System A
System B
System C
For sufficiently high L is first order
autocorrelation equal to zero
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Frequency domain
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.
System A
System C
System B
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Fractal DimensionHurst 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.
System A
System B
System C
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Summery of results of ACF, Power Spectrum and
Hurst Exponent
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Conclusions

• Preliminary investigation shows that the yarn
  hairiness distribution can be fitted to a bimodal
  model distribution.
• The yarn Hairiness can be described by two mixed
  Gaussian distributions, the portion, mean and the
  standard deviation of each component leads to
  deeper understanding and evaluation of
  hairiness.
• This method is quick compared to image analysis
  system, beside that, the raw data is obtained
  from world wide used instrument Uster Tester.
• The Hyarn system is a powerful program for
  evaluation and analysis of yarn hairiness as a
  dynamic process, in both time and frequency
  domain.
• Hyarn program is capable of estimating the short
  and long term dependency.