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The relationships between the Weibull and Lognormal distributions

By Tammy Bell,2014-04-16 21:46
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The relationships between the Weibull and Lognormal distributions

THE APPROXIMATE EQUATIONS BETWEEN THE WEIBULL

    AND LOGNORMAL DISTRIBUTIONS BY USING MRR

     Chi-Chao Liu Phillip C. T. Willey

    Department of Business Administration, The Institution of Engineering Takming University of Science & Technology, and Technology,

    No. 56, Sec. 1 Huan-Shan Rd., Ettrick 1 North Drive, Chilwell,

    Neihu District Taipei 11451, Taiwan, ROC Nottingham NG9 4DY, UK

     allenliu@takming.edu.tw phillipwilley@theiet.org

    ABSTRACT

    Reliability and failure data, both from life testing and from in-service records, are often modeled by the Weibull or Lognormal distributions so as to be able to interpolate and/or extrapolate results. As the Weibull and Lognormal distributions are not from the same mathematical family, we are unable to derive mathematical relationships between their shape and scale parameters directly. This study tries to use a simulation method to determine their approximate relationship. The simulation was carried out using WeibullSMITH? software which uses the Monte Carlo method

    to generate samples from the Weibull (or the Lognormal) distribution and can then fit the Lognormal (or the Weibull) line to the sample points. The paper found approximate equations of shape parameters (or scale parameters) for complete sample sizes varies from 3 to 99 and from 100 to 1000 between both distributions. For η=1,

    β=0.5, 1, 3, 5 and n=10, 25, 50,100, the residuals between true ρ (shape parameter

    of the Lognormal) and estimated ρ are less than ?0.003. Again in the same

    conditions as above, the residuals between true θ (scale parameter of the Lognormal)

     are less than ?0.0005. The approximate equations are simple, and estimated θ

    direct, and their accuracy is acceptable. They are, therefore, recommended for use.

    Keywords: Weibull distribution, Lognormal distribution, Monte Carlo simulation, Median Rank Regression (MRR), approximate equations

    INTRODUCTION

    In statistics, there are many distributions which can be used in various areas. Such distributions include the Normal, Chi-squared, Exponential, Rayleigh, Weibull, Erlang, Gamma, Extreme-Value, Lognormal and others. The relationships between various distributions are shown in Fig. 1 where the direction of each arrow represents going from the general to a special case.

    Here we focus only on the Weibull and Lognormal distributions, both distributions can, interestingly, have skewed frequency curves (the Weibull can be positively or negatively skewed but the Lognormal can only be positively skewed). These two

    distributions are not in the same mathematical family, yet each can be made to fit life data with acceptable accuracy. Several writers in the field mention that the Lognormal is a distribution which competes against the Weibull distribution in the area of reliability (Abernethy, 2005; Adams, 1962; Johnson & Kotz, 1970).

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    Lognormal Normal Chi-Squared

    Distribution Distribution Distribution

    Rayleigh

    Distribution

    Weibull Gamma Exponential

    Distribution Distribution Distribution

    Erlang Extreme-Value

    Distribution Distribution

    Fig. 1 The relationship between various distributions (Dey, 1983)

    MATHEMATICAL MODEL

    The Weibull probability distribution has three parameters, ηβ, and t. It can be 0

    used to represent the failure probability density function (PDF) with time, so that:

    (????tt0(1?(?(?(????tt)?0?? (1) ;;(?ftew(?)?

    for η > 0, β > 0, t > 0, -? < t0 < t,

    where β is the shape parameter (determining what the Weibull PDF looks like) and is positive and η is a scale parameter (representing the characteristic life at which 63.2% of the population can be expected to have failed) which is also positive. t is 0

    a location or shift or threshold parameter (sometimes called a guarantee time, failure-free time or minimum life). t can be any real number. If t = 0 then the 00

    Weibull distribution is said to be two-parameter.

Fig. 2 shows the diverse shape of the Weibull PDF with t = 0 and various values of 0

    η and β (=0.5, 1, 2, 3, 5). Note that figures are all based on the assumption that t 0

    = 0.

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    Fig. 2 The Weibull PDF

    The cumulative distribution function (CDF), denoted by F(t), is:

    (????tt0?(?(??)???;;Ft1e (2) w

    Fig. 3 shows the Weibull CDF with t = 0 and various values of η and β (=0.5, 1, 2, 0

    3, 5). All curves intersect at the point of (1, 0.632), the characteristic point for the

    Weibull CDF.

    Fig. 3 The Weibull CDF

The PDF for 3-parameter Lognormal distribution is:

    2?~??tt????0?ln(???)???????2????~???? (3) ;;fteL;;2tt0

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for θ > 0, ρ > 0, t > 0, -? < t0 < t,

    where ρ is the shape parameter, θ is the scale parameter and t is the location 0

    parameter. The units of ρ, θ and t are the same as in the Weibull case. The 0

    Lognormal is said to be a 2-parameter distribution when t = 0. Fig. 4 shows the 0

    diverse shape of the Lognormal PDF with t = 0 and various values of θ and ρ (=0.4, 0

    0.8, 1.6, 2.5, 4).

    Fig. 4 The Lognormal PDF

    The corresponding Lognormal CDF is the integral of the PDF from 0 to time-to-failure t. It can be written in terms of the standard Normal CDF as:

    ~??tt??0 (4) ;;Ftln(??L)????

    where Φ() is the CDF of the standard Normal distribution defined as:

    2??(?z(?12)?;; (5) zed??2

    where μ=lnθ. Φ() is tabulated in many publications. Fig. 5 shows the Lognormal CDF with t = 0 and various values of θ and ρ (=0.4, 0.8, 1.6, 2.5, 4). 0

    It is clear that all curves intersect at the point of (1, 0.5), the characteristic point of the

    Lognormal CDF.

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    Fig. 5 The Lognormal CDF

    DATA FITTING METHOD

    The linear form of the resulting Weibull CDF can be represented by a rearranged version of equation (2):

    ??11(? (6) lntlnlnln(?;;1Ft(W)?

    Comparing this equation with the linear form y=Bx+A, leads to y= ln t and x= lnln{1/[1-F(t)]}. W

    If we minimize it by using the least squares method we obtain:

    2nn??2nxx(???ii,,11)?iiˆ (7) (,nnn

    nxyxy???iiii,,,111iii

    and

    nn??(?yxii??(?11ii(?ˆnn((?(?)?ˆ (8) e

    where n is the sample size and ^ indicates an estimate. The mathematical expressions for x and y are: ii

    ??1 (9) xlnlni?;;1FtWi??

    and

     (10) ylntii

    F(t) can be estimated by using Benard’s formula, , which is a good approximation to i

    the median rank estimator (Abernethy, 2005; Tobias, 1986). This paper uses the

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    Bernard’s median rank because it shows the best performance and it is the most widely used to estimate F(t). i

    The procedure for ranking complete data is the following (Mitchell, 1967; O’connor,

    1991; Willey, 1992):

    1. List the time-to-failure data from small to large.

    2. Use Benard’s formula to assign median ranks to each failure.

    3. Estimate the β and η by equations (7) and (8).

    The Lognormal CDF, when plotting against appropriate probability axes, appears linear and, therefore, can be represented by a rearranged version of equation (3) as:

    zp (11) lnlnt~

    Comparing this with the linear form y=Bx+A, leads to y=lnt and x=z. p

    The same least squares, as was used for the Weibull distribution, yields:

    2nn??2nxx(???ii,,11)?iiˆ (12) ~,nnn

    nxyxy???iiii,,,111iii

    and

    nn??(?yxii??(?11ii(?ˆnn~(?(?)?ˆ (13) e-1where x=z and y=lnt and z =Φ(z) is the percentile of the standard Normal CDF ipiipp

    which is widely tabulated (Neave, 1989). Again F(t)=Φ(z) can be estimated by ip

    using Benard’s formula. The same ranking procedure as was used for the Weibull distribution was also used for the Lognormal distribution. The only difference is that step 3 in the ranking procedures above should be replaced by ‘estimate the ρ and θ

    by equations (12) and (13).’

THE RELATIONSHIP BETWEEN THE SHAPE PARAMETERS OF THE

    WEIBULL AND LOGNORMAL DISTRIBUTIONS

    Since the Weibull and Lognormal distributions are not from the same mathematical family, we are unable to derive mathematical relationships between their shape and scale parameters directly. However, we can use a simulation method to determine their approximate relationship. The simulation was carried out using WeibullSMITH software (Fulton, 2005) which uses the Monte Carlo method to generate samples from the Weibull (or the Lognormal) distribution and can then fit the Lognormal (or the Weibull) line to the sample points. For example, 25 random samples were generated from W(1, 1) and the resulting data were fitted to both the Weibull and Lognormal lines using MRR as shown in Figures 6 and 7 respectively. As we can see from these two graphs, W(1, 1) corresponds to L(0.577, 0.822) where the 0.822 is the value σof ρ, the reciprocal of σ, and muAL= θ and sdF=e =3.377. Therefore, the

    approximate relationships between the shape parameters and scale parameters of the Weibull and Lognormal distributions can be determined.

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    Fig. 6 Samples from the Weibull (1, 1) and fitted to the Weibull line

    Fig. 7 Samples from the Weibull (1, 1) and fitted to the Lognormal line

    Following this procedure, we can find other sample sizes, from 3(1)100(10)1000, the corresponding values of β, ρ and η, θ. The relationship between the shape

    parameters of the two distributions is studied first. Fig. 8 shows the relationship between ρ and n for different values of β using MRR. It is clear that for all β the

    relationship between ρ and n is nonlinear.

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Fig. 8 The relationship between ρ and n for different values of β using MRR

    With the WeibullSMITH Software (using the MRR method) and by trial and error, general equations for the relationships between ρ and β for different sample sizes

    can be obtained.

    For complete sample sizes, n, from 3 to 99, the relationship is represented by:

    ( (14) ~0.546524??(?n5.23201)?2.235939e

    For complete sample sizes, n, from 100 to 1000, the relationship is represented by:

    ( (15) ~1.522754??(?n58.32657)?2.241233e

    Equations (14) and (15) show that for a given sample size, n, the approximate relationship between ρ and β is a straight line passing through the origin, both equations being of the form ρ = c β + 0 where c is a constant.

    The residuals for estimating ρ using equation (14) and (15) are shown in Figures 9 and 10 respectively.

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    Fig. 9 The residuals for estimating ρ using equation (14)

    Fig. 10 The residuals for estimating ρ using equation (15)

    Particular values of the residuals of ρ for different sample sizes and shape parameters used for the Monte Carlo study are shown in Table 1. From Figures 9, 10 and Table

    1, it is clear that the maximum residual for estimating ρ is less than ?0.003. Table 1

    gives corresponding values of β and ρ, together with their residuals, for sample sizes of n = 10, 25, 50 and 100.

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     Table 1 The residuals between true ρ and estimated ρ (using MRR)

    η β ρ estimatedρ n residual

    10 1 0.5 0.4166 0.4169 -0.0003

    25 1 0.5 0.4108 0.4106 0.0002

    50 1 0.5 0.4080 0.4078 0.0001

    100 1 0.5 0.4060 0.4060 0.0000

    10 1 1 0.8332 0.8337 -0.0006

    25 1 1 0.8216 0.8212 0.0004

    50 1 1 0.8159 0.8157 0.0003

    100 1 1 0.8120 0.8120 0.0001

    10 1 3 2.4995 2.5012 -0.0017

    25 1 3 2.4648 2.4637 0.0012

    50 1 3 2.4477 2.4470 0.0008

    100 1 3 2.4361 2.4359 0.0002

    10 1 5 4.1658 4.1687 -0.0029

    25 1 5 4.1081 4.1061 0.0020

    50 1 5 4.0796 4.0783 0.0013

    100 1 5 4.0602 4.0599 0.0003

However, equations (14) and (15) were derived from an ‘ideal’ situation. That is, the

    life data came from the Weibull distribution and fitted the Weibull line exactly. Such ‘ideal’ data was then fitted to the Lognormal line to produce the corresponding

    Lognormal parameters. In practice, this ‘ideal’ situation is unlikely to happen. So, it is essential to check how accurate these equations are when used in real situation. Here, a small Monte Carlo simulation was run to check the accuracy of equations (14) and (15).

    For 4 different sample sizes (without any suspensions or censoring), points which had been randomly generated from a Weibull distribution were fitted to both a Weibull line and a Lognormal line. For the Weibull line, values of β and η were computed;

    for the Lognormal line, values of ρ and θ were also computed using the software.

    Then plotting the values of β on the x axis and ρ on the y axis, their relationship

    was a straight line when the sample size was constant.

    The curve-fitting method MRR was used. For each of the sample sizes the MRR of fitting 100 replications were made. The data of β vs ρ is linear so that the least

    squares method could be used to find the best fit line. Table 2 gives the least squares equations of β vs ρ for 100 replication data. Table 3 shows the SE’s of the LS equation and of equation (14) or (15) for 100 replication data.

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