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Beltwide Cotton Conference
January 11-12, 2007
New Orleans, Louisiana
BIMODALITY OF COMPACT
YARN HAIRINESS
Jiří Militký , Sayed Ibrahim
and
Dana křemenaková
Technical University of Liberec, 46117 Liberec,
Czech Republic
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.
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
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
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).
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?
Histogram (83 columns)
Hair Diagram
Normal Dist. fit
12
8
6
4
2
0
100
200
300
400
2
3
4
5
6
7
8
9
10 11 12
2
3
4
10
11
5
Distance
0,3
Nonparametric Density
hair length
10
0,2
0,1
0
2
3
4
5
6
7
8
9
Yarn Hairiness
Gaussian curve fit (20 columns)
Smooth curve fit
12
6
7
8
9 10 11 12
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:
CN (t j 1, t j )
f H ( x) 
N hj
•Where
CN (t j 1, t j ) is the number of sample
elements in this interval
and
h j (t j  t j 1)
of this interval.
Number of classes
h  3.49*(min(s, Rq) /1.34) / n1/ 3
is the length
Rq  upper quartile  lower quartile
Rq  x(0.75)  x(0.25)
M  int[2.46 (N-1)0.4 ]
•For all samples is N= 18458 and M=125
h = 0.133
Kernel density
function
The Kernel type nonparametric of
sample probability density function
N
 x  xi 
1
fˆ ( x) 
K
N i 1  h 


Kernel function K x : bi-quadratic
- symmetric around zero
- properties of PDF
Optimal bandwidth : h
1. Based on the assumptions of near normality
2. Adaptive smoothing
1/ 5
h

0.9*(min(
s
,
Rq
)
/1.34)
/
n
3. Exploratory (local hj ) requirement
of equal probability in all classes
h = 0.1278
Bi-modal distribution
Two Gaussian Distribution
MATLAB 7.1 RELEASE 14
The bi-modal distribution can be approximated by two
Gaussian distributions,
2
2
 ( xi  B1 
 ( xi  B2 
fG ( xi )  A1*exp  
 A2*exp  


C
1
C
2




Where A1 , A2 are proportions of shorter and longer
hair distribution respectively, B1 , B 2 are the means
and C1 , C 2
are the standard deviations.
H-yarn Program written in Matlab code, using the least
square method is used for estimating these parameters.
Bi-modality of Yarn Hairiness
Mixed Gaussian Distribution
The frequency distribution and fitted bimodal distribution curve
Analysis of Results
Check the type of Distribution
Bimodality parametric
• Mixture of distributions
Rankit plot
5
4.5
4
estimation and likelihood
ratio test
• Test of significant distance
between modes (Separation)
3.5
3
3.5
4
4.5
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
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.
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
Analysis of Results I
Mixture of Gauss distributions
CDF plots
Probability density function (PDF) f (x),
Cumulative Distribution Function (CDF) F
(x),
and Empirical CDF (ECDF) Fn(x)
Unimodal CDF: convex in (−∞, m), concave
in [m, ∞)
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)|
1
rectangular
empirical
normal
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
4
4.5
8
CDF plots
0.5
0.4
rectangular
empirical
normal
0.3
0.2
Dip Statistic (for n= 18500): 0.0102
Critical value (n = 1000): 0.017
Critical value (n = 2000): 0.0112
0.1
0
2
2.5
3
3.5
5
Analysis 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:
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 III
Likelihood 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:
Analysis of Results V
Significance of difference of means
• Two sample t test of equality of means
• T1 equal variances
• T2 different variances
Analysis of Results VI
PDF and CDF
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 )
hairness histogram
0.45
h = 0.33226
0.4
0.35
Rel. Freq.
0.3
0.25
0.2
0.15
0.1
0.05
0
MATLAB
AKDEST
1Devaluates
the
univariate
Adaptive Kernel
Density
Estimate with kernel
0
1
2
3
4
i
cdf ( j ) 
x
i 1
N
(i )
x
i 1
(i )
, x(i )  x(i 1)
5
6
7
8
9
Parameter estimation of
mixture of two Gaussians model
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.
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.
Stationarity test
Periodogram
System A
1.2
0.6
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
rel. frequency [-]
0.4
0.5
System B
Cumulative periodogram
1.2
1
cumul. periodogram [-]
cumul. periodogram [-]
1
0.8
0.8
0.6
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
rel. frequency [-]
0.4
0.5
System C
Cumulative periodogram
1.2
1
cumul. periodogram [-]
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.
14.6 tex
Cumulative periodogram
0.8
0.6
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
rel. frequency [-]
0.4
0.5
h40sussen.txt
Autocorrelation
Autocorrelation
Time Domain Analysis
Autocorrelation
1
0.99
0.99
0.98
0.98
0.97
0.97
0.96
0.96
0.95
0.95
0.94
Autocorrelation
1
0.94
0.93
0
50
100
0.93
150
Lag
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
N 1
 ( y( j )  y
R(1) 
) * ( y( j  1)  y )
j
[ s 2 ( N  1)]
ACF
ACF
-0.05
-0.1
20
40
60
80
Lag
System A
100
0.05
0
Autocorrelation function
0
-0.15
0
ACF
0.05
Autocorrelation function
Autocorrelation function
0.05
-0.05
-0.1
-0.15
0
20
40
60
Lag
System B
80
100
0
-0.05
-0.1
-0.15
0
20
40
60
80
Lag
System C
For sufficiently high L is first order autocorrelation equal to zero
100
Fourier Frequency Spectrum
1.75
5
5
2.5
2.5
0.0062095
1
0.5
0
10.505
11.609
10.478
0.2324
0.5
10.492
0
10.481
-0.5
-0.5
-1
-1.5
0
100
200
300
-1
-1.5
500
400
1.25
PSD TISA
0
1
1.75
1.5
1.5
50
1.25
1
1
90
0.75
0.75
0.5
PSD TISA
7.5
Hair Sussen
10
Hair Sussen
Hair Sussen
10
7.5
0
Hair Sussen
Frequency
domain
h40sussen.txt
h40sussen.txt
Parametric Reconstruction [7 Sine]
r^2=1e-08 SE=0.994675 F=9.2185e-06
0.5
95
99
99.9
0.25
0
dis tance
0
5
10
15
20
0.25
0
25
Frequency
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.
Power spectal density Welsch
Power spectal density Welsch
Power spectal density Welsch
500
300
200
250
400
150
100
PSD Welsch
PSD Welsch
PSD Welsch
200
150
100
50
5
10
15
frequency [1/m]
System A
20
25
0
0
200
100
50
0
0
300
5
10
15
frequency [1/m]
System B
20
25
0
0
5
10
15
frequency [1/m]
System C
20
25
Fractal Dimension
Hurst 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 R / S  K * (n obs) ^ H . 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.
100
10
10
1
10
100
n obs
1000
System A
1
10000
1000
1000
R/S
10
1
1
10
100
n obs
System B
1000
1000
100
100
R/S
100
R/S
R/S
100
1000
100
R/S
1000
H=628487, SD H=0.00214796, r2=0.93009
R/S
h40zinser.txt
Hurst Exponent
H=0.659309, Sd H=0.000808122, r2=0.995684
1000
1
h40sussen.txt
Hurst Exponent
h40rieter.txt
Hurst Exponent
H=0.6866, SH H = 0.001768, r2= 0.9739
10
10
1
10000
1
10
1
10
100
n obs
System C
1000
1
10000
Summery of results of ACF, Power
Spectrum and Hurst Exponent
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.