# The modelling of business rules for dashboard reporting using mutual information

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4 Our aim is to examine ow tese two business rules function in conveying information to te next level of te ierarcy, under different levels of organisational uncertainty. Drawing from te examples of Section 2, organisations tat face certain environments ave low entropy wilst ig uncertainty implies rapid canges over time in te value of te key performance indicators, tat is, ig entropy. To calculate te mutual information for te two rules, a model of te key performance indicator values (te value of te nodes) must be formed at level 0 of te ierarcy (te base nodes)-te interface between te organisation and te environment. Our approac is to define a set of distributions parameterised by entropy value in te following way. Assume tere are m subordinate nodes tat report to te coordinator or fuser. Suppose X 0 = ( X 0, X 02,, X 0m ) ten in order to tune te distribution over X0 to ave entropy te first m nodes are set or frozen to te green state, X 0 = X02 = = X0 ( m ) = G, wit probability. Te remaining nodes are set to ave eiter te green or red state wit probability / 2. Tus Pr ( X ) = Pr( X ) Pr( X ) Pr( X ), Pr( X = G) = / 2, i m +,, m. o m + m +2 m i It is easy to sow tat te total entropy of te subordinate nodes defined by tis distribution as entropy bits. Te base nodes tat can switc from red or green states wit probability / 2 supply te uncertainty into te ierarcy. Tus te entropy, is bot te measure of te uncertainty of te organisation and te number of subordinate nodes tat can switc between te red and green states. Wit tis explicit model distribution, we are able to calculate te mutual information precisely for te two business rules. Te examination of te effectiveness of eac rule can ten be ascertained by varying te entropy, tat is canging te number of nodes tat can switc between te red and green states. To calculate te mutual information te conditional entropy between te fusion node states and te subordinate node states H( X Y) Pr( Y) Pr( X Y)log Pr( X Y ) = Y=R, G X is evaluated. We omit te details of te derivation albeit to say tat te conditional distributions are evaluated troug te application of Baye s Teorem, Pr( X Y) = (Pr( Y X) Pr( X)) / Pr( Y). Upon viewing te conditional probabilities, te two business rules can now be expressed in general form. For bot rules te conditional probability were for te report by exception rule 2 for X S Pr( Y = R X) = R 0,oterwise, EXCEPTION { X } S R,exception = : te number of red subordinate reports and for te report by te majority rule wit m subordinates reporting to te coordinator { X m } SR,majority = : te number of red subordinate reports / 2. MAJORITY Figure 2. Illustration of te exception and minority reporting rules. On te left and side a red state is seen so under te exception rules a red is reported for te fusion. On te rigt and side tere are more green reports tan red, so under te majority rule a green is reported. Put in tis form, te reporting rule can be generalised to any tresold rule as we will discuss in te following section 4. Te mutual information between te subordinate nodes and te coordinator node for te exception rule, given information entropy level of te subordinate nodes and te distribution specified above can be sown to be 597

5 ( ) I exception ( X, Y ) = log2 2, >0. 2 Te report by majority rule can be sown to ave mutual information K 2 log2( 2 K) + log2 I majority ( ) X, Y = 2 K for > m/2, 0oterwise, were te number of combined subordinate states wit total entropy in wic te majority report is red (tat is in S ) R,majority K = m/2 m/2 + Te following figure plots te mutual information for te exception and te majority rules, as a function of te information entropy, assuming tat tere are eleven suc subordinate nodes. Mutual Information exception majority Total Subordinate Entropy (Nodes tat can switc states) Figure 3: Mutual information between subordinate and fusion nodes of te exception and majority reporting rules as a function of te subordinate nodes total information entropy, wic is te number of nodes tat can switc between red and green states, given eleven subordinate nodes. It is clear from inspection of Figure 3 tat te exception reporting rule is most effective wen te information entropy is low. Te interpretation for te preparedness reporting system is tat te exception rule applies wen most subordinate reports are consistently good (green) over time, reporting few inadequate key performance indicators. However, as te uncertainty increases te majority reporting rule ten becomes more informative as to te state of te subordinate nodes below. Analogously, tis reflects an organisation were te base key performance indicators are in flux and cange over time. Referring to Figure 3, in te left limit as te information entropy approaces one, any canges in te subordinate states are precisely reflected in te report of te lowest common ancestor, as tere is one subordinate state canging. Hence te mutual information is maximised for te exception rule. Conversely, in te rigt limit, wen all subordinate states can cange randomly and equiprobably, te majority rule will reflect tese canges, toug not precisely as tere is information loss. Te mutual information approaces te maximum of one (tis will be te maximum for te coordinator node as it can only be in two states, red and green, one bit). Te mutual information reaces tis maximum value only because our assumption tat all subordinate nodes can cange in an equiprobable way. If nodes can cange from across red and green states wit different probabilities ten te mutual information for te majority rule will not reac one. We discuss tis issue in te following section. 598

6 4. A GENERALISED THRESHOLD BUSINESS RULE Troug te preceding section we ave illustrated tat te exception business rule may not necessarily be effective in propagating te canging states of te subordinate nodes below. We modeled te majority business rule and found tat under te circumstances of dynamic state cange of subordinate nodes, tis rule better reflects suc a cange in states. Bot euristics are a version of te general tresold business rule as determined by te value of te conditional probability distribution Pr( Y = R X) wic is one (tat is te business rule reports te red state) under our distribution Pr ( X 0) for te set { X a} SRa, = : te number of red subordinate reports. For te exception rule a = and for te majority rule a = m / 2 were m is te number of subordinate nodes. It is natural to ask if tere is an generalised tresold business rule. Couced in communications teory terms we seek a rule maximises tat capacity between te subordinate nodes and te state of te fusion node. Under our distribution Pr ( X0) defined earlier, we can calculate te mutual information for a general value of a. Te resulting equation is identical to tat of te mutual information for te majority rule except tat te value of K is replaced by a term dependent on a, tis being te number of elements in SRa, wit entropy bits K( a) = a a+ One need only cycle over te values following Figure 4. 0 a H to find te optimum value of a. Tese values are sown in te a, te number of subordinate red states before reporting red Total Subordinate Entropy (Number of nodes at can switc states) Figure 4: Grap of te optimal number of red states a, given eleven subordinate nodes, before te business rule reports an overall state of red, as a function of te subordinate entropy. Referring to Figure 4, as per te observation in Figure 3, as te uncertainty in te organisation increases, te business rule tresold by wic te subordinate reports te red state also increases. As an approximation, a = /2. te tresold number of red reports [ ] We are able to calculate a generalised business rule under te assumption tat all te nodes tat can switc between red and green states do so wit equal probability / 2 over time. If tis assumption fails, ten inevitably, te difficulty of calculating exact expressions for te mutual information will increase substantially. To overcome tis problem, one need only employ Monte Carlo simulation for te given set of state switcing probabilities to determine wat te tresold will be. For tat set of state switcing probabilities Pr( = G, R), i =, m te information entropy may be exactly determined. Tis gives us an X i 599

7 upper bound on a as 0 a. It is only te conditional probabilities tat need to be estimated from simulation. 5. ESTIMATES OF BIAS IN THE REPORTING SYSTEM From te discussion above, we are able to establis criteria by wic reporting witin te preparedness system is biased. Tis measure of bias will determine te overall system effectiveness. Longitudinal data on te state canges of eac node in te current defence system is recorded and terefore te entropy of all subordinate nodes can be estimated. Tere are two approaces to estimating te internal preparedness system bias, given longitudinal data. One is in reference to te judgments of oter reports in te ierarcy and te oter is in reference to te deviation from te generalised business rule or te mandated business rule. To measure te internal bias in reporting, one can estimate, given a particular level of te ierarcy te probability of a fused state Pr( Y X). Differences in te fused state probabilities can be calculated and summed to form te bias estimate over te wole ierarcy. Assuming tat te subordinate states are te same, tat is X = X for two fused states Yi and Yj, a measure of te preparedness system bias is Yi Yj b = i Yi j Yj Hierarcylevels i, j Pr( Y X ) Pr( Y X ). An alternative is to look at te deviation from te result of te general business rule. For a fusion node, an indicator function may be defined * (Y) tat is one wen te fused state matces tat of te state found Y from te business rule or zero oterwise. Ten one measure of te preparedness system bias, given te set of fusion nodes wic we label A (tat is all nodes except te base nodes of te ierarcy) is 6. DISCUSSION AND CONCLUSION ( ) b= A *. Y Y i i i A We ave igligted tat te use of te report by exception rule, wilst appropriate for a preparedness system tat as few problems, may not be te appropriate rule wen te values of te key performance indicators sow dynamic canges over time. Te majority business rule or te general tresold rule as discussed in Section 4 is more appropriate. In Defence, decision makers know te problems wit te exception rule. At te moment tere is considerable subjectivity and variability in fusion decisions made, wit limited guidance as to wat rules sould be applied across different circumstances. Decision makers ave te option to fuse decisions based on exception or averaged results from te subordinates or fuse subjectively. Data regarding te longitudinal cange in states over time is recorded. As suc, te necessary infrastructure is in place at present to apply te best tresold business rules as derived in tis paper. Tis work sould improve te guidance on wat rules to apply to best relay preparedness information to senior leadersip. REFERENCES Butte, A. J. and Koane, I.S. (2000), Mutual Information Relevance Networks: Functional Genomic Clustering Using Pairwise Entropy Estimates. Pacific Symposium in Biocomputing, Irving, W. W. and Tsitsiklis, J. N., (994), Some Properties of Optimal Tresolds in Decentralized Detection. IEEE Transactions in Automatic Control 39(4). Luque, B. and Ferrera, A., (997), Measuring Mutual Information in Random Boolean Networks. Complex Systems,. Sannon, C.E. (948), A Matematical Teory of Communication. Te Bell System Tecnical Journal and Tay, W. P., Tsitsiklis, J. N., and Win, M. Z., (2008), Data Fusion Trees for Detection: Does Arcitecture Matter? IEEE Transactions in Information Teory 54(9),

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