A Study to Generate a Weak Order from a Partially Ordered Set, Taken. . .
73
The regression equation based on 6 pairs (the pair (1,0) is realized three times)
has the striking structure:
epsav = a ∗ (1 − Dom (X 1 , X 2 ) ) for 0.8 < Dom (X 1 , X 2 ) ≤ 1
(15)
with the coefficient of determination, R 2 = 0.88 and the coefficient a around
1.35. This statistical result indicates that the relevant quantity is the deviation Δ:
:= 1 − Dom (X 1 , X 2 )
(16)
Hence, Eq. 15 expresses proportionality between the error epsav and Δ. The
crucial value 0.8 for separating relevant Dom(X 1 ,X 2 )-values from irrelevant ones,
may vary from case to case and is open for future research. Furthermore, Fig.
2 shows that the average error related to single objects is less than 0.25 and the
deviations from the regression line will be larger the smaller the value Dom(X 1 , X 2 )
is.
3.2 Application to Real Data Set
The estimation method needs the following steps:
1. Defining X 1 , X 2 and the Hasse diagram for the full set X
2. Calculation of Dom(X 1 , X 2 )
3. Providing the data for X 1 and X 2
4. Application of the lattice theoretical method: (a) for X, (b) for X 1 , (c) for X 2
5. Performing the calculations due to Eqs. 8 and 9
6. Inspecting epsav to check the quality of the results
Up to now there is no program performing all 6 steps. However, for steps (1)–
(3) the program package PyHasse (see for details Bruggemann et al. 2014) was
extended by the new module DomRkav. Its graphical user interface is shown in Fig.
3. The data is found in Table 1.
The corresponding Hasse diagram is shown in Fig. 4.
Some remarks concerning Fig. 4 may be useful here:
• Hg and As are minimal elements, they cause the least deviation from a natural
state
• Fe and Zn are maximal elements, they are most problematic because the deviation
of the natural state is very high.
• Incomparabilities, such as for Zn and Fe show that the loading of the lichens in
general is high, however with some geographical differentiation.
73
The regression equation based on 6 pairs (the pair (1,0) is realized three times)
has the striking structure:
epsav = a ∗ (1 − Dom (X 1 , X 2 ) ) for 0.8 < Dom (X 1 , X 2 ) ≤ 1
(15)
with the coefficient of determination, R 2 = 0.88 and the coefficient a around
1.35. This statistical result indicates that the relevant quantity is the deviation Δ:
:= 1 − Dom (X 1 , X 2 )
(16)
Hence, Eq. 15 expresses proportionality between the error epsav and Δ. The
crucial value 0.8 for separating relevant Dom(X 1 ,X 2 )-values from irrelevant ones,
may vary from case to case and is open for future research. Furthermore, Fig.
2 shows that the average error related to single objects is less than 0.25 and the
deviations from the regression line will be larger the smaller the value Dom(X 1 , X 2 )
is.
3.2 Application to Real Data Set
The estimation method needs the following steps:
1. Defining X 1 , X 2 and the Hasse diagram for the full set X
2. Calculation of Dom(X 1 , X 2 )
3. Providing the data for X 1 and X 2
4. Application of the lattice theoretical method: (a) for X, (b) for X 1 , (c) for X 2
5. Performing the calculations due to Eqs. 8 and 9
6. Inspecting epsav to check the quality of the results
Up to now there is no program performing all 6 steps. However, for steps (1)–
(3) the program package PyHasse (see for details Bruggemann et al. 2014) was
extended by the new module DomRkav. Its graphical user interface is shown in Fig.
3. The data is found in Table 1.
The corresponding Hasse diagram is shown in Fig. 4.
Some remarks concerning Fig. 4 may be useful here:
• Hg and As are minimal elements, they cause the least deviation from a natural
state
• Fe and Zn are maximal elements, they are most problematic because the deviation
of the natural state is very high.
• Incomparabilities, such as for Zn and Fe show that the loading of the lichens in
general is high, however with some geographical differentiation.