8.1. Introduction
8.1.6. What are the basic lifetime distribution models used for non-repairable populations?
8.1.6.4. |
Lognormal |
The lognormal life distribution, like the Weibull, is a very flexible
model that can empirically fit many types of failure data. The two parameter
form has parameters 
= the shape parameter and T50 = the median
(a scale
parameter).
Note: If time to failure, tf, has a lognormal
distribution, then the (natural) logarithm of time to failure has a normal
distribution with mean µ = ln T50 and standard deviation .
This makes lognormal data convenient to work with; just take natural logarithms
of all the failure times and censoring times and analyze the resulting
normal data. Later on, convert back to real time and lognormal parameters
using as the
lognormal shape and T50 = eµ as the (median)
scale parameter.
Below is a summary of the key formulas for the lognormal.
))*EXP[-(1/2*sigma**2)*(LN(t)-LN(T50)**2 = PHI[(LN(t)/LN(T50))/sigma] where PHI(z) is the standard normal CDF; RELIABILITY: R(T) = 1 - F(t); FAILURE RATE: h(t) = f(t)/R(t); MEAN: T50*EXP(sigma**2/2); MEDIAN: T50; VARIANCE: T50**2*EXP(sigma**2)*(EXP(sigma**2) - 1)](eqns/lnform.gif)
Note: A more general 3-parameter form of the lognormal includes
an additional waiting time parameter 
(sometimes called a shift or location parameter). The formulas
for the 3-parameter lognormal are easily obtained from the above formulas
by replacing t by (t - )
wherever t appears. No failure can occur before
hours, so the time scale starts at
and not 0. If a shift parameter
is known (based, perhaps, on the physics of the failure mode), then all
you have to do is subtract
from all the observed failure times and/or readout times and analyze the
resulting shifted data with a 2-parameter lognormal.
Examples of lognormal PDF and failure rate plots are shown below. Note
that lognormal shapes for small sigmas are very similar to Weibull shapes
when the shape parameter 
is large and large sigmas give plots similar to small Weibull 's.
Both distributions are very flexible and it is often difficult to choose
which to use based on empirical fits to small samples of (possibly censored)
data.
- As shown in the preceding plots, the lognormal PDF and failure rate shapes are flexible enough to make the lognormal a very useful empirical model. In addition, the relationship to the normal (just take natural logarithms of all the data and time points and you have "normal" data) makes it easy to work with mathematically, with many good software analysis programs available to treat normal data.
- The lognormal model can be theoretically derived under assumptions matching many failure degradation processes common to electronic (semiconductor) failure mechanisms. Some of these are: corrosion, diffusion, migration, crack growth, electromigration, and, in general, failures resulting from chemical reactions or processes. That does not mean that the lognormal is always the correct model for these mechanisms, but it does perhaps explain why it has been empirically successful in so many of these cases.
A brief sketch of the theoretical arguments leading to a lognormal
model follows.
Applying the Central Limit Theorem to small additive errors in the log domain and justifying a normal model is equivalent to justifying the lognormal model in real time when a process moves towards failure based on the cumulative effect of many small "multiplicative" shocks. More precisely, if at any instant in time a degradation process undergoes a small increase in the total amount of degradation that is proportional to the current total amount of degradation, then it is reasonable to expect the time to failure (i.e., reaching a critical amount of degradation) to follow a lognormal distribution (Kolmogorov, 1941).A more detailed description of the multiplicative degradation argument appears in a later section.
The following commands in Dataplot will evaluate the PDF, the CDF, and
the failure rate (PDF/(1-CDF)) of a lognormal at time T, with shape
and median life (scale parameter) T50
(the 0 in the commands below is the location parameter):
LET PDF = LGNPDF(T,For example, if T = 5000 and
LET CDF = LGNCDF(T,
LET HAZ = LGNHAZ(T,
= .5 and T50 = 20,000, the above commands will
produce a PDF of .34175E-5, a CDF of .002781, and a failure rate of
3427 %/K.
To generate 100 lognormal random numbers from a lognormal with shape .5 and median life 20,000, use the following commands:
LET SIGMA = 0.5Next, to see how well these random lognormal data points are fit by a lognormal, we plot them using the lognormal probability plot command. First we have to set
LET SAMPLE = LOGNORMAL RANDOM NUMBERS ...
FOR I = 1 1 100
LET SAMPLE = 20000*SAMPLE
= SD to .5
(see PPCC PLOT
for how to estimate the value of SD from actual data).
LET SIGMA = .5The resulting plot is below. Points that line up approximately on a straight line indicates a good fit to a lognormal (with shape SD = .5). The time that corresponds to the (normalized) x-axis T50 of 1 is the estimated T50 according to the data. In this case it is close to 20,000, as expected.
X1LABEL EXPECTED (NORMALIZED) VALUES
Y1LABEL TIME
LOGNORMAL PROBABILITY PLOT SAMPLE
Finally, we note that EXCEL has a built in function to calculate the
lognormal CDF. The command is =LOGNORMDIST(5000,9.903487553,0.5) to evaluate
the CDF of a lognormal at time T = 5000 with
= .5 and T50 = 20,000 and ln T50 =
9.903487553. The answer returned is .002781. There is no lognormal PDF
function in EXCEL. The normal PDF can be used as follows:
=(1/5000)*NORMDIST(8.517193191,9.903487553,0.5,FALSE)
where 8.517193191 is ln 5000 and "FALSE" is needed to get PDF's instead of CDF's. The answer returned is 3.42E-06.
