1. A Study of Tapered Beard Sampling Method as Used in HVI
Yiyun Cai, Xiaoliang Leon Cui, James Rodgers
USDA, ARS, Southern Regional Research Center, New Orleans, LA
Vikki Martin, Mike Watson
Cotton Incorporated, Cary, NC

2. Outline Background Earlier Research Experimental Observations
Theoretical Investigation
Modeling
Conclusion Remarks
Acknowledgements

3. Background
Beard method is used for sampling cotton fibers to generate fibrograms that can give length parameters. It is the sampling method used by HVI
HVI is extensive used to measure fiber length parameters
The purpose is to investigate the bias of the beard method
There have been different assumptions regarding the bias of the beard sampling method for measuring cotton fiber length
We have seen discrepancies in measurements that can not be explained by using completely length-biased or non-biased assumptions
Bias in length measurement especially affects the short fiber region

4. Earlier Research
Hertel assumed that the probability of a fiber being sampled for fibrogram is proportional to its length (Length-biased sampling)
Zeidman and Batra also provided a derivation that tried to use the unit step function to prove that sampling for Fibrograph is length biased

5. Earlier Research
Chu and Riley’s later found that sampling using HVI Fibrosampler did not agree with the length biased measurement assumption
The assumption was then revised to that each fiber has equal probability of being sampled

6. Experimental Observations
We tested eight sets of cotton
Original fibers were tested by AFIS to measure the length of each individual fiber, at least 35,000 fibers were measured for each set, data were used to calculate the entire length distribution
Fibers were then sampled by using the HVI fibrosampler and sampled specimen were tested by AFIS
Results indicated that the length distributions of the specimen sampled by HVI Fibrosampler can not be explained as length-biased

7. Experimental Observations
If the measurement is length-biased, the length biased mean length by number should be always longer than the original mean length by number:
MLl=MLn+Varn/MLn

8. Experimental Observations
Mean lengths of the original sample and the fibers picked by the HVI Fibrosampler

9. Theoretical Investigation
The characteristics of measurement bias can be obtained by analyzing the probability of fiber sampling
The completely length-biased assumption holds when there is only one pin
It is the same as the result from Zeidman and Batra’s derivation based on the unit step function

10. Theoretical Investigation
Fiber Length Li
Sampling Position
Sampling Width W
If the ith group of fibers has Ni fibers of length Li, then the probability of one fiber in this group being sampled is Li /W. The number of fibers being sampled is
ni=Ni X Li / W
For Li < W, This is length biased

11. Theoretical Investigation
The distance between pins/needles of the Fibrosampler clamp is ~ 2 mm. There are more than 30 pins in the clamp
Length biased assumption does not hold considering the small distance between pins/needles
Length bias still has its influence in the short fiber region

12. Theoretical Investigation
Fiber Length Li
Sampling Positions
Distance between pins (ΔW)
Sampling Width (W)
If the ith group of fibers has Ni fibers of length Li, then the probability of one fiber in this group being sampled is Li / ΔW for Li < ΔW,. The number of fibers being sampled is ni=Ni X Li / ΔW, which is length biased. For Li < ΔW, the probability is 1, which means it is unbiased.

13. Theoretical Investigation
In real situations, fiber orientation is random, which projects more “short” fibers in the direction that is perpendicular to the sampling direction
The analysis indicates that for short fibers, the measurement is length biased, but for fibers longer than certain length, the measurement is unbiased.

14. Modeling
At first, We simulated the sampling procedure of 50, cm long and 50, cm long fibers that have randomly distributed orientations and positions on a plane
Results agree with the theoretical analysis, the measured distribution changes from completely length biased to less length biased when the distance between needles decreases

15. Modeling

16. Modeling
We then simulated the sampling length distributions of the original fibers that had their lengths measured by AFIS
The parameters of simulations adopted the configurations of a fiber comb of HVI Fibrosampler, which has 36 needles and the distance between two needles is 2 mm.
Results show discrepancies between the simulated sampling distribution and the distribution of the original fibers especially in the short fiber region

17. Modeling

18. Modeling

19. Conclusion Remarks
Our research shows that the beard sampling method as used in HVI is not completely length biased, it is nearly an unbiased method.
The results also indicate that though the sampling method used in HVI is not completely length biased, original fiber’s length distribution still impacts sampled fibers’ length distribution, especially in the short fiber region.
Short fiber content by number of the sampled fiber is lower than that of the original fiber, and this difference is inherently introduced by the sampling method.

20. Acknowledgements
The authors would like to thank Cotton Incorporated for financially supporting this research

21. Thank You