Evidence combination 
The purpose of blunder detection is to classify observations into those to be kept in the model 
(‘good’ observations) and to those to be rejected (‘bad’ observations). This classification is usually 
considered as a statistical decision process in which each residual is tested against the null hypothesis 
that it does not deviate significantly from zero. It is then critical to select the risk level a. Too 
large a risk level will result in the rejection of too many ‘good’ observations. On the other hand, too 
small a value will degrade the power of the test and too many ‘bad’ observations will be accepted. 
The selection is to be based on heuristics and experience. A typical choise is a = .001 or a = .01. 
The problem can also be approached from a Baysian-classification point of view. An observation is 
classified as ‘good’ or ‘bad’ based on probability criteria. Adapting some of the terminology from 
(Buchanan and Shortliffe, 1985), Bays' theorem about conditional probability can be written as 
P(d;)P(e|d.) 
Pldile) = g————"— 
3 i=1 Pld;) Pleld;) 
where d; are the disjoint diagnosis or sub-classes, in this application d; is the class of ‘good’ 
observations and d; is the class of ‘bad’ observations; P(d;) is the a priori probability for each 
class, (3° P(d;) = 1); P(e|d;) is the probability that an observation belonging to class d; has 
symptoms and signs represented by evidence e; and P(di|e) is the conditional probability that an 
observation belongs to sub-class d; if the evidence e is present. P(d;)’s and P(e|d;)'s are required 
as input. 
Let us investigate a specified observation k having an externally Studentized residual vector wy of 
length |w;| = az. Vector wy is used as an indicator (evidence) for the presence of a: blunder: the 
larger |w,|, the more likely is the presence of a blunder. 
The Baysian decision rule can also be written as (Fukunaga, 1972) 
P, (di) P. (e|d;) < Pk (dz) Pk(e|da) — obs., € fe 
P, (da) can be interpreted as an a-priori-known probability about the occurence of a blunder in the 
observation k, P,(d1) = 1 — Pa (d2). Pi(e|di) is given as o in conventional methods for blunder 
detection: 
co 
P,(e|d1) = Pr(|wk| > ax) - f Pr(x|d,)dz 
Gk 
where p,(x|d;) is the density function of |wi| assuming that the observation belongs to dy, to 
the class of ‘good’ observations. Because of the external Studentization, pk(z|d;) follows a T- 
distribution. The T-distribution is here defined such that if z follows a F-distribution then /z 
follows a T-distribution. The T-distribution is multi-dimensional, opposite to the t-distribution 
which is one-dimensional. Also, 
P,(e|d2) = Pr(|wx| < ak) = I" Pk(z|d2)dz 
where pi (z|d2) is the density function of |wk| assuming that the observation belongs to d5, the 
class of ‘bad’ observations. Unfortunately this distribution is generally not known. Because an 
empirically validated distribution is lacking, some hypothetical distribution must be used. Adapting 
the mean-shift model, the distribution would be a non-central T-distribution with an unknown 
non-centrality parameter. If the frequency of different magnitudes of the mean shifts among the 
blunder-observations can be estimated, even subjectively, a linear combination of the corresponding 
non-central T-distributions can be used as p(x|dz). The general appearances of p(x|d;) and p(z|d;) 
are show in Figure 1. For any realistic choice of p(z|dz) it must hold that 
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200 
  
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