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weibull

Derives the shape and scale parameters for the Weibull distribution via maximum likelihood estimates.

Available in version 6.3.0 and later.

Prototype

	function weibull (
		x        : numeric,  
		opt  [1] : logical,  
		dims [*] : integer   
	)

	return_val  :  float or double, see return value description below

Arguments

x

Array of any dimensionality. Missing values (x@_FillValue) are allowed.

opt

If opt=True, two options are available:

  • opt@nmin: An integer specifying the minimum number of non-missing values that must be present before computations will be performed. If fewer than nmin are present, missing values will be returned for the shape and scale parameters. Defaults to 75% of the number of points if not set.

  • opt@confidence: A float or double scalar specifying the desired confidence limits on the returned coefficients: 0.0 <= confidence < 1.0. Actually, the (1-confidence) confidence limits will be calculated. If confidence=0.95, the 2.5% and 97.5% limits will be returned.

dims

The dimension(s) of x on which to calculate the Weibull distribution parameters.

Return value

For each series specified by dims, the shape and scale will be returned.


       (0) = shape 
       (1) = scale 
If opt@confidence is specified an additional 4 estimates are returned:

       (2) = shape low  confidence limit 
       (3) = shape high confidence limit
       (4) = scale low  confidence limit 
       (5) = scale high confidence limit

The return type will be double if x is double, and float otherwise.

Description

This function computes maximum likelihood estimates of the two parameter (shape and scale) Weibull distribution. This distribution is also known as the Extreme Value Type III distribution.

Reference

   Weibull Maximum Likelihood Parameter Estimates with Censored Data
   J. Bert Keats, Frederick P. Lawrence, and F. K. Wang 
   Journal of Quality Technology, Volume 29, Number 1, pp. 105--110, January 1997

See Also

ftest, rtest, betainc, kolsm2_n

Examples

As always when fitting a distribution, it is best if the sample size is large.

Example 1

           q   = (/0.01,0.08,0.13,0.16,0.55,0.90,1.11,1.62,1.79,1.59,1.83,1.68,2.09,2.17,2.66 \
                  ,2.08,2.26,1.65,1.70,2.39,2.08,2.02,1.65,1.96,1.91,1.30,1.62,1.57,1.32,1.56 \
                  ,1.36,1.05,1.29,1.32,1.20,1.10,0.88,0.63,0.69,0.69,0.49,0.53,0.42,0.48,0.41 \
                  ,0.27,0.36,0.33,0.17,0.20/)

           wF = weibull(q, False, 0)
           print(wF)

           opt = True
           opt@confidence = 0.95
           wT = weibull(q, opt, 0)
           print(wT)

will yield
           wF(0) = 1.509     ; shape
           wF(1) = 1.299     ; scale 
and
           wT(0) = 1.509     ; shape 
           wT(1) = 1.299     ; scale 
           wT(2) = 1.152     ; shape low  limit ( 2.5%) 
           wT(3) = 1.867     ; shape high limit (97.5%)
           wT(4) = 1.051     ; scale low  limit ( 2.5%) 
           wT(5) = 1.548     ; scale high limit (97.5%) 

An estimate of the Weibull location parameter may be obtained from:

           qavg = avg(q)                       ; data average
           wavg = wF(0)*gamma(1 + 1/wF(1))   ; theoretical Weibull avg
           wloc = qavg - wavg                  ; location parameter

           print("qavg="+qavg+"  wavg="+wavg+"  wloc="+wloc)
yields
           qavg=1.1862  wavg=1.3939  wloc=-0.207703

           ------------------------ R ---------------------------
For comparison, the R software package [Note: 'library(MASS)' is required] returns

           library(MASS)
           qWeib <- fitdistr(q, "weibull")    
           qWeib

                   shape       scale  
                 1.5090848   1.2993179 
                (0.1824047) (0.1268983)

           qWeibCon <- confint(qWeib)
           qWeibCon 

                       2.5 %   97.5 %
                shape 1.151578 1.866591
                scale 1.050602 1.548034
           ------------------------ R ---------------------------
Example 2 Same as Example 1 but
           q = q+10
yields
           wF(0) = 17.37123   ; shape
           wF(1) = 11.52268   ; scale
and
           wT(0) = 17.37123   ; shape
           wT(1) = 11.52268   ; scale
           wT(2) = 13.66275
           wT(3) = 21.07971
           wT(4) = 11.32812
           wT(5) = 11.71724

           qavg=11.1862  wavg=16.621  wloc=-5.43483

           ------------------------ R ---------------------------
The R software package returns

          shape         scale  
        17.37123526   11.52268040
       ( 1.89211608) ( 0.09926876)
           ------------------------ R ---------------------------

Example 3 Let p(ntim,nlat,mlon). Determine Weibull parameters at each grid location.

           wF  = weibull(p, False, 0)   ; wF(2,nlat,mlon)

           opt = True
           opt@confidence = 0.90
           wT = weibull(p, opt, 0)      ; wT(6,nlat,mlon)
would return
           wF(0,:,:) ; shape 
           wF(1,:,:) ; scale 
and
           wT(0,:,:) ; shape 
           wT(1,:,:) ; scale 
           wT(3,:,:) ; shape low  limit ( 5.0%) 
           wT(4,:,:) ; shape high limit (95.0%
           wT(4,:,:) ; scale low  limit ( 5.0%) 
           wT(5,:,:) ; scale high limit (95.0%) 

Example 4 Let wind(ntim,nlat,mlon). Determine Weibull parameters at each grid location. Let's require that at least 48 values must be present.

           opt      = True
           opt@nmin = 48
           wndp = weibull(wind, opt, 0)   ; wndp(2,nlat,mlon)