Defining the Uncertainty Boundaries:

The next step in the classification is to convert the confidence factor representing each variable into a BOE using the Dempster-Shafer Theory. The BOE consists of three belief functions:

1. m(OA) - The degree of belief in OA

2. m(NP) - The degree of belief in NP (i.e. healthy) function 3. m(Θ) - The associated ignorance (uncertainty)

The relationship between the belief functions and the confidence factors followed the work of Safranek et al. (1990), where:

m(OA) = B 1 − 𝐴cf(v) − AB 1 − 𝐴 (3.52) m(𝑁𝑃) = B 1 − 𝐴cf(v) + B (3.53) m(Θ) = 1 − m(OA) − m(NP) =1 − 𝐴 − 𝐵 1 − 𝐴 (3.54)

Where A represents the dependence of the m(OA) on the confidence factor, B represents the maximal support which can be assigned to either m(OA) or m(NP). Jones (2004) states that the values of A and B should be assigned based on knowledge of the upper

ΘU, and lower ΘL boundaries of uncertainty. These were related to upper ΘU, and lower

ΘL boundaries of uncertainty as follows:

𝐴 = Θ𝑈− Θ𝐿 1 + Θ_𝑈− 2Θ_𝐿

(3.55)

𝐵 = 1 − Θ𝐿 (3.56)

It is, therefore, possible to assign different values for A and B depending on the variable. For example, the user might have previous knowledge of the level of ignorance within a particular variable. In the majority of the following research using the DST classifier, the same limits have been applied to each individual variable. This is likely because the classifier has been used as an objective tool, and the manual selection of A and B for each individual variable introduces a level of subjectivity. Furthermore, defining ΘL and

ΘU based on expert knowledge is not an instinctive process. For this reason, the same

values of A and B will also be applied to every variable.

Another way to understand the effect of the assignment of A and B is to first consider the situation where there is zero ignorance, as shown in Figure 3.21. In this instance, the belief m(OA) is equivalent to the confidence factor, and m(NP) is one minus the confidence factor. As previously discussed OA in a disease with a vast number of symptoms and is generally diagnosed through the collection of multiple pieces of evidence. Furthermore, all biomechanical variables are interdependent and each one considered alone only accounts for a small piece of a much larger picture. It is, therefore, unrealistic to expect that any one piece of evidence would result in a 100% belief of OA or NP.

Figure 3.22 introduces uncertainty, or ignorance. The maximal degree of belief any one piece of evidence can contribute is 70% and consequently, it would require further pieces of evidence to approach a belief of 100%. The upper ΘU, and lower ΘL boundaries of uncertainty have been set to 0.3, or 30% in this example; therefore, A=0 and B=0.7. Table 3.3 displays an example of two different input variables, gait velocity and peak flexion angle, we had two different items of evidence, gait velocity and peak flexion angle, for which the confidence values have already be found as 0.5 and 0.6 respectively.

Figure 3.21 The conversion between confidence factor and belief functions m(OA) and m(NP)

Notice how the belief attributed to the peak flexion angle (which by itself would not favour either a belief of NP or uncertainty) has both increased the belief m(OA) and the belief m(NP), while decreasing uncertainty. Also, notice that while both values have increased, the ratio m(OA):m(NP) has decreased. This result is challenging for the following reasons:

i) There has been a marked decrease in uncertainty: it doesn’t seem intuitive that a belief function which favours neither hypothesis would remarkably reduce the uncertainty of the classification. If there were multiple pieces of evidence which equally supported both OA and NP function, and these truly were the only two possibilities, in practice our ignorance or uncertainty would not decrease.

Figure 3.22 The conversion from confidence factor to a BOE when both upper and lower

boundaries of uncertainty are 0.3 (30%).

Table 3.3 Example of the combination of the bodies of evidence from two different variables.

cf(v) m(OA) m(NP) m(Θ) Gait velocity 0.6 0.42 0.28 0.3 Peak flexion angle 0.5 0.35 0.35 0.3 Combined NA 0.5 0.38 0.12

ii) The belief in m(NL) increased by a relatively larger amount than m(OA): if we were initially quite sure of OA function due to the gait velocity and we then find the peak knee flexion angle doesn’t support our hypothesis, either way, it might seem intuitive that this would then decrease our relative belief of OA. However, what if there was a high amount of error in the measurement of peak knee flexion angle? In this situation, the sigmoid curve might have a very small steepness coefficient k, and hence a large range of values of v would result in a confidence factor approaching 0.5. If multiple pieces of evidence, where errors are too large to discern differences were additionally combined, not only would uncertainty approach zero, but both m(OA) and m(NL) would approach 0.5. This is counter-intuitive when there is the possibility that the aforementioned variables might just have a very high level of error.

Figure 3.23 shows the same example as the previous figure, with the addition of an upper boundary of uncertainty ΘU, 0.5. Notice that the uncertainty increases as the confidence factor approaches 0.5, while both m(OA) and m(NL) change at an increased rate. Notice also that there is still a region where both m(OA) and m(NL) can be assigned simultaneously.

Figure 3.23 The conversion from confidence factor to a BOE when the upper and lower

As mentioned previously, in the absence of a clear objective way of selecting these boundaries, Jones (2004) investigated the effect that different uncertainty boundaries had on the final classification of a training body of NP and OA subjects. Jones argued that, for that specific dataset, the optimal boundaries were ΘU =1 and ΘL =0.8. These

boundaries and are plotted within Figure 3.24. It can be seen from the figures that the value of ΘU affects the rate of change of m(OA) and m(NL) as cf(v) approaches 0.5, and

the value of ΘL affects the maximum belief one piece of evidence can contribute.