EXPLANATION FILE OF PROGRAM PARAFIT
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  Parametric Least Squares
  ------------------------

    Sa Far in this chapter, the discussion has been limited ta polynomial least- 
  squares fits in .which the coefficients appeared in linear form. ln fact, this
  linearity property was key to the iteration procedure discussed in the pre-
  vious section. 

    Many problems lend themselves to this form of analysis. For example, the 
  Weibull cumulative-distribution function is often used to analyze the failure
  modes in manufactured products: 

                        P(t) = 1 - exp(-(t/to)^w) 
                        t0 > 0                                  (1.8.1) 
                        w > O 
                        t > 0 

  (See References 1 and 13 for further discussion). P(t) is the probability that
  a failure occur by time t. Often, the procedure is ta record the failure times
  and to fit a Weibull curve graphically in order to de termine the characteris-
  tic time t0 and the parameter w (which has implications with respect to the
  failure mode). This function can be linearized for least-squares fitting as
  follows: 

                        1 - P(t) = exp(-(t/t0)^w) 
                        ln [1 - P(t)] = -(t/t0)^w               (1.8.2)
                        ln {ln [1 - P(t)J} = -w ln t + w ln t0 

  We define the following: 

                        x = ln t 
                        y = ln {ln [1 - P(t)]} 
                        A = -w 
                        B = w ln t0 

  Equation (1.8.2) then becomes 

                        y = AX + B 

  This last equation has the standard linear form that can easily be treated
  with the routine given in program LSTSQRl. The resulting calculated coef-
  ficients, A and B, can be employed to estimate t0 and w:
   
                        w  = -A 
                        t0 = exp(-B/A)                          (1.8.3)
                         
    The linearization method is interesting, but it should not be relied upon
  for dealing with nonlinear coefficient situations. One problem with the
  approach is that not aIl equations can be linearized. Another problem is
  that the least-squares fit is not with respect to the original equation, but
  rather with respect to the linearized form. ln the case of the Weibull fit,
  if P(t) is near unity, 1 - P(t) is near zero and Y can become large. Thus, it
  is possible for errors in P(t) in this region to unduly influence the fitted
  parameters. 

    An alternative procedure is to iteratively seek the values of the parameters
  that best satisfy the original equation. We will calI this method parametric
  least-squares fitting. 

    In file lstsqr.txt, we found that the values of the parameters that minimi-
  zed the variance (and thereby the standard deviation) were unbiased estimates.
  Thus, the minimization of the standard deviation is a good statistical crite-
  rion on which to base an iterative procedure.
   
    We use a heuristic method for finding the value of a parameter that minimi-
  zes the standard deviation. We will calI this method the shuffle. The basic 
  assumptions are that the sign of the optimum value of the parame ter is known,
  and that there is only one minimum. The basic concept behind the method is
  that if the estimates are getting better, then you take bigger steps; and
  when the optimum is passed, you turn around and halve the step size.

    This approach can be generalized to several parameters by taking a step in
  reducing the standard deviation one parameter at a time, and repeatedly pas-
  sing through the parameter set. 

    Assuming that the equation to be fitted can be written in the Form y(x) = 
  f(x;A1;A2;A3; ... ), this algorithm has been implemented in program Parafit. 

    The inputs to PARAFIT are the number of data points (N), the number of coef- 
  ficients (L), the data pairs [X(I), Y(I)], the initial step size (0 < El < 1),
  and a convergence factor (E). The returned results are the estimated coeffi-
  cients [A (I)], the standard deviation of the fit using the se coefficients
  (D), and the number of iterations employed (M). It is assumed that the func-
  tion Y(X) = f[X;A(I); ... ] is available in the functions subroutine. 

    El and E must be specified very carefully. El is related to the step size,
  and must be in the range 0 < El < 1. If El = 0, no progress will be made and
  the algorithm will fail. If El = 1, the procedure may become unstable. A good
  compromise choice is El = 0.5, and preferably less then 0.8. 

    E determines when the iteration is terminated. If the relative change in the
  standard deviation between two complete successive passes through the parame-
  ters is less th an E, then the iteration stops. 

    An example of the use of PARAFIT is shown in the program. ln this case, the 
  demonstration program generated a set of ten data pairs using the equation
  y(x) = 2exp(-(x-4.5)^2/3). This equation has the form of a gaussian, which is
  a function of considerable importance in statistics, communications, optics,
  physics, and other sciences. The values of the parameters calculated by PARA-
  FIT are in very good agreement with those used to generate the data set. 

    PARAFIT is a powerful optimization subroutine. lts chief disadvantage is its 
  slow execution speed. Also, some care is required in the selection of input
  parameters. PARAFIT does not check on the validity of the inputs. If L > N,
  the subroutine may never terminate. Also, PARAFlT is not always guaranteed to
  converge. However, it will usually do weIl if E == 0.1, El == 0.5, and if the
  initial guesses for the A(I) are at least correct in sign, and are reasonable
  overestimates. 

    FinaIly, as an exercise, try to apply PARAFlT to the Weibull function dis-
  cussed earlier in this section.

  From [BIBLI 01].
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End of file Parafit.txt