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The Distribution Wizard in
Weibull++ 7
When performing life data
analysis, Weibull++ 7's
Distribution Wizard can provide guidance in selecting a
distribution based on statistical tests. The Distribution Wizard
uses three factors in order to rank distributions: the
Kolmogorov-Smirnov (K-S) test, a normalized correlation
coefficient and the likelihood value. This article will show how
these rankings are calculated.
The
Distribution Wizard
The
Distribution Wizard in Weibull++ 7 ranks the selected
distributions in terms of the fit to the data entered, as shown
in Figure 1.
Figure 1: Distribution Wizard
In order
to determine the ranking, the three tests are used in
conjunction with weights assigned to each test.
Detailed
results of the calculations can be found on the Initial sheet of
the Analysis Details page, as shown in Figure 2.
Figure 2: Analysis Details Initial Results
The second column, AVGOF, contains
values obtained using the Kolmogorov-Smirnov (K-S) test. The
third column, AVPLOT, provides the results of the second test,
which is a normalized correlation coefficient (rho). The fourth
column, LKV, contains the likelihood values.
On the
Intermediate sheet of the Analysis Details page, these values
are then weighted and combined into one overall value, DESV, as
shown in Figure 3.
Figure 3: Analysis Details Intermediate Results
The weight
(or importance) assigned to each test can be defined by the
user. Clicking the
Setup button opens the Advanced Setup window, as shown in
Figure 4.
Figure 4: Distribution Wizard: Advanced Setup Window
The weights defined in this window are used in the DESV
calculation. Note that the user can specify different weights
depending on whether the parameter estimation method is rank
regression or MLE.
Once DESV values have been calculated for each
distribution, they are then used to determine
overall rankings for the selected distributions.
Example Using the Distribution Wizard
Assume
the following data are available:
|
Time-to-failure,
hrs |
| 10 |
| 30 |
| 50 |
| 60 |
The Distribution Wizard will calculate
AVGOF, AVPLOT and LKV for each distribution selected for
consideration and then obtain an overall rank for each one
of them. As an example, lets calculate these values for the
exponential distribution when using least squares or rank
regression on X (RRX). For more information on these
parameter estimation methods, see
http://www.weibull.com/LifeDataWeb/least_squares.htm and
http://www.weibull.com/LifeDataWeb/mle_for_complete_data.htm,
respectively. Note that the parameters obtained via these
methods are different and therefore the values of AVGOF,
AVPLOT and LKV are different in each case.
Results using RRX
Given the data available, estimation of the exponential
distribution parameter using rank regression on X results in
Lambda equal to 0.02613.
Figure 5: Data Folio
The K-S statistical test can be performed such that the null and
alternative hypotheses are:
The K-S test statistic (D) is the maximum
difference between the observed and predicted probability:
where:
For this example:
Time-to-failure,
hrs |
Observed Probability,
 |
Predicted Probability,
 |
Absolute Difference |
| 10
|
0.15910
|
0.22996 |
0.07086 |
| 30
|
0.38573 |
0.54339 |
0.15766 |
| 50
|
0.61427 |
0.72925
|
0.11498 |
| 60
|
0.84090 |
0.79151 |
0.04939 |
Note that observed probability is calculated
using median ranks. For more details on median ranks, refer
to
http://www.weibull.com/LifeDataWeb/least_squares.htm.
The predicted probability is calculated using the
distribution selected and the parameter(s) estimated
(exponential with Lambda = 0.02613). The difference between
those two values is calculated and the largest absolute
difference is D. From the calculations above: 
In many statistical textbooks, tables are available that
tabulate critical values for the K-S test for different
distributions [1, Appendix G]. For example,
for a significance level of a = 0.1 and four data points:
Since D < Dcrit, then at a
significance level of 0.10, H0 cannot be
rejected.
Weibull++ calculates the critical probability at
which we cannot reject H0:
where d is a random variable that follows the distribution
of D. Note that AVGOF = 1
p value.
Large values of AVGOF, close to 1, indicate that there is a
significant difference between the theoretical distribution
(the one we are trying to test) and the data set.
For the exponential distribution (from Figure 2):
The plot fit, AVPLOT, is given by:
Using the values calculated previously:
The last term is the log of the likelihood value obtained
with the estimated parameters (see Figure 5). More
information on the likelihood function can be found at
http://www.weibull.com/hotwire/issue33/relbasics33.htm
and
http://www.weibull.com/LifeDataWeb/appendix_a_parameter_estimation.htm.
Once all test results have been calculated for each
distribution, distributions are ranked for each test, as
shown in Figure 2. In this example, the exponential
distribution ranks 8th when using AVGOF, 10th when using the
AVPLOT and 11th when using LKV. A weighted solution is then
obtained. Using the weights assigned to each of the tests,
as shown in Figure 4, the weighted average can then be
calculated, as shown in Figure 3.
Distributions
are then ranked by values of DESV, the lowest value being ranked
as number 1. In this example, the number 1 ranking distribution
is the generalized gamma distribution.
References
[1] Kececioglu, Dimitri, "Reliability
and Life Testing Handbook Vol. 1", 1993.
[2] ReliaSoft Corporation,
Life
Data (Weibull) Analysis Reference, ReliaSoft Publishing,
Tucson, AZ, 2008.
|
= observed
probability
=
predicted probability based on the distribution