NORMAL KERNEL DENSITY MIXTURE

NORMAL KERNEL DENSITY MIXTURE (LET)

Let Subcommand

Given a set of means and associated uncertainties, compute the mixture of their normal kernels.

Data for reference materials, interlaboratory studies, and key comparisons is often given as a mean and an associated standard deviation. It is often assumed that the data for each laboratory is from a normal distribution.

This command does the following:

1. For each individual laboratory, compute the \( \bar{y} \pm 4s \) points where \( \bar{y} \) and s denote the mean and standard deviation of the laboratory.

2. Find the minimum and maximum value of these points over all the laboratories. Define a grid of 1,000 equally spaced points between the minimum and maximum point.

3. At each grid point, compute the normal probability density function (PDF) for each laboratory (each laboratory will use its own mean and standard deviation for the parameters of the normal distribution). The PDF values for all laboratories are summed and then the value at the grid point is normalized by dividing by the number of laboratories.

LET <y> <x> = NORMAL KERNEL DENSITY MIXTURE <ymean> <ysd>
                        <SUBSET/EXCEPT/FOR qualification>
where <ymean> is a variable containing laboratory means;
            <ysd> is a variable containing the standard uncertainties associated with the <ymean> variable;
            <x> is a variable containing the values where the normal kernel density mixture values are computed;
            <y> is a variable containing the normal kernel density mixture values;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET YKERN XKERN = NORMAL KERNEL DENSITY MIXTURE YMEAN YSD
LET YKERN XKERN = NORMAL KERNEL DENSITY MIXTURE YMEAN YSD SUBSET TAG = 1

None

None

CONSENSUS MEAN	=	Compute the consensus mean and associated uncertainty.
NORMAL KERNEL DENSITY MIXTURE PLOT	=	Generate a normal kernel density mixture plot.
KERNEL DENSITY PLOT	=	Generate a kernel density plot.

Duewer (2008), "A Comparison of Location Estimators for Interlaboratory Data Contaminated with Value and Uncertainty Outliers", Accredited Quality Assurance, Vol. 13, pp. 193-216.

Consensus means, interlaboratory studies, key comparisons

2017/02

 
. Step 1:   Define the data
.           (from David Duewer paper)
.
read ymean ysd
102.02   0.16
102.47   0.84
102.98   0.90
103.69   0.45
104.08   1.16
105.04   0.39
105.36   0.32
107.26   1.50
108.26   0.77
end of data
.
. Step 2:   Generate the normal kernel density mixture
.
let y x = normal kernel density mixture  ymean ysd
.
. Step 3:   Plot the result
.
label case asis
title case asis
title offset 2
x1label Measurement Value
y1label Density
title Normal Kernel Density Mixture for Duewer Data
xlimits 95 115
.
plot y x