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KALIF
was developed at the Institute for Reliability and Risk Analysis,
and forms a part of the suite of programs designed to work in the Microsoft
Windows environment.
The proposed software is based on the expository papers of Meinhold and
Singpurwalla [1] and [2]. The first paper presents Kalman-Filter technique
in the light of Bayesian inference. The second one outlines an elegant
forward and backward smoothing approach for reliability assessment in
certain Dose-Response, Damage-Assessment and Accelerated-Life-Testing
Studies.
INTRODUCTION
The software allows to perform two types of analysis:
i) General Kalman Filter extrapolation and smoothing.
ii)Reliability assessment and smoothing in Dose-Response, Damage-Assessment
and Accelerated Life Testing Studies.
The later is a special case of the former.
General Kalman Filter extrapolation
and smoothing.
Suppose, we have observed some data Yt,Yt-1,...,Y1, which can be either
scalars or vectors. We want to predict (estimate) value of the next
observation, Yt+1. Kalman Filter technique solves this problem using the
model defined under the following assumptions. Observable values of Yt are
assumed to be dependent on some unobservable quantities Xt. These
quantities, often referred as a state of nature, are related to Yt through
observation equation Yt=Ft Xt+vt, where Ft is a known quantity, describing
relationship between Yt and Xt, and vt is an observation error, which is
assumed to be normally distributed with mean zero and a known variance Vt.
In the simple case, when Xt itself is being measured, Ft can be set to one,
when Xt is a scalar, or to identity matrix, when Xt is a vector. Also,
suppose, we know the relationship between every Xt and its predecessor
Xt-1, which is represented by, so called, system equation Xt=Gt Xt-1+wt,
wherein Gt being known quantity, and the system equation error wt being
normally distributed with mean zero and known variance Wt. Usually, Gt
represents by itself laws, governing the described process, and this is
assumed to be known to a practitioner or engineer, who defines the model of
interest. Thus, in order to perform the analysis the user should specify
the following quantities: Yt,Yt-1,...,Y1, X0, Ft and Gt at the very least,
and if any of Ft, Vt, Gt or Wt changes with time, values for the
corresponding Ft-1,...,F1, Vt-1,...,V1, Gt-1,...,G1 or/and Wt-1,...,W1 also
need to be specified. Note, that only the starting value X0 for the state
of nature is given. During its course Kalman Filter, using forward and
backward smoothing techniques, estimates values of the state of nature for
all corresponding points in time, including time t+1. And from the last
one, using the observation equation, value for Yt+1 is evaluated.
Reliability Assessment in
Dose-Response, Damage-Assessment and Accelerated-Life-Testing Studies.
This part of the
software deals with the reliability assessment problem with stress levels
that are presumably very small and at which testing is not feasible. Kalman
Filter model is used to update parameters of the survival function of the
system of interest. The survival function is assumed to be of the form of
Weibull distribution function. This assumption is not very restrictive
because of the flexibility of the Weibull distribution. On the other hand
properties of a Weibull distribution function are being greatly utilized in
order to transform reliability testing procedure into Kalman Filter
technique. Unlike in the case of general Kalman Filtering, here input
extends only to the pair of initial parameters of Weibull distribution and
the set of observed dose levels with their corresponding responses. To have
the analysis performed correctly these pairs have to be ordered by the dose
level in descending order, starting from the highest one. Of course, the
dose level for which extrapolation is to be performed has to be specified
as well. As in many cases of bayesian inference specification of initial
parameters may not be an easy task. In our case the Weibull distribution
has a form: exp{-axb}. Thus, as before, the user has to specify the initial
parameters a and b, corresponding to the prior subjective beliefs of the
researcher, assuming the survival function to be a Weibull distribution
function. All above remains true if the order of dose levels is reversed.
Namely, if we are interested in predicting reliability under a very large stress,
our starting point would be the pair corresponding to the smallest dose
level and input has to be ordered in ascending order with respect to the
dose level.
DATA INPUT
To enter data for a new analysis, select "New" from the
"File" menu and choose among two types of analysis. This will
bring up a Data Input Pad, appropriate for the particular analysis. It
contains messages, specifying parameters and an order, in which they have
to be entered. After completion of data entry, the file containing the data
has to be saved by selecting "Save" from the "File"
menu and specifying a name for the destination file. This will also allows
to run the same analysis again latter.To modify existing data file, simply
select "Open" from the "File" menu and select location
and the name of the file. This will open data file in the Data Input Pad.
After completion of editing, the file has to be saved.
RUNNING THE ANALYSIS
Select the "Application" option from the "Run" menu and
select the name of the input file. Program will automatically run forward
and backward smoothing recursion Kalman-Filter algorithms and extrapolate
an unknown value. If the data entries in the input file are inconsistent
with the Kalman-Filter algorithm, the program will respond with an error messages,
trying to identify the type of inconsistencies it encountered. In this case
the input file has to be examined and corrected.
RESULTS
After the successful run, results can be viewed. By selecting
"Plot" from the "Results" menu, one can see a plot of
successive estimates of the mean of unobservable state of nature Xt for the
case of general Kalman Filter. In case of reliability assessment parameters
a and b will be plotted against sequence of integer numbers each
representing an order of corresponding dose level in the input set. If the
parameter is a vector, the program will ask to specify a dimension, which
has to be plotted. By selecting "View Output" from the
"Results" menu, one invokes a display, containing means and
variances of state of nature for the case of general Kalman Filter and
parameters a and b for reliability assessment for each observation for
forward and backward recursion methods as well as an extrapolated estimate
of unknown value. These also could be saved in a specified file or printed
out.
OTHER FEATURES
The program has all the usual features of a Windows-based program,
including full file management facilities and the facility to print results
either in tabular or graphical form.
REFERENCES
Richard J. Meinhold and Singpurwalla N. D. 'Understanding the
Kalman-Filter.' In The American Statistician, May 1983, 37, 2,
123-127.
Richard J. Meinhold and Singpurwalla N. D. 'A Kalman-Filter Smoothing
Approach for Extrapolationsin Certain Dose-Response, Damage-Assessment, and
Accelerated-Life-Testing Studies.' In The American Statistician, May
1987, 41, 2, 101-106.
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