International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm) Volume 1, Number 2, Pages 79–104, June 2003 Publisher Item Identifier S 1542-5908(03)10205-9/$20.00 Article electronically published on November 28, 2002 at http://www.YangSky.com/ijcc12.htm. Please cite this paper as: hCathy M. Helgason and Thomas H. Jobe, “Perception-Based Reasoning and Fuzzy Cardinality Provide Direct Measures of Causality Sensitive to Initial Conditions in the Individual Patient(Invited Paper)”, International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm), Volume 1, Number 2, Pages 79–104, June 2003i.
PERCEPTION-BASED REASONING AND FUZZY CARDINALITY PROVIDE DIRECT MEASURES OF CAUSALITY SENSITIVE TO INITIAL CONDITIONS IN THE INDIVIDUAL PATIENT(INVITED PAPER) CATHY M. HELGASON AND THOMAS H. JOBE
Abstract. Background : Clinical trials in medicine use probability -based statistics. Statistics separate the patient’s physiologic elements (specified as variables when given numeric form) from his or her body and define causal correlation for the group. Diagnostic and clinical decisions at the individual patient level are currently based on these definitions of causation. Because data is grouped and averaged, the relationship to initial conditions and their connection to the individual patient is lost. We develop an alternative method that directly measures “causality” in the individual patient that is sensitive to initial conditions. Methods: We define the measure of causal connection between elements in the individual patient. Necessary and Sufficient Causal Ground, Formal Causal Ground and Clinical Causal Effect are derived from the fuzzy subsethood theorem defined by Kosko. From these causal measures, we derive the clinical efficiency measure K from units of fuzzy cardinality. It is how much causal effect is present per unit of a specific patient’s initial conditions. Practically, as “sets as points” in a unit hypercube, each patient is represented as a fuzzy set of defined elements at different points in time. Efficiency K is defined as the extracubal causality measure for any process not represented in the cube acting on the patient between two points in the unit hypercube. For any process, 1/K gives us the dosage necessary of that agent needed to move that specific patient’s initial condition per unit of causal effect. Results: The measures of formal causal ground and clinical causal effect are in units of fuzzy cardinality. Thus the Received by the editors November 27, 2002 / final version received December 6, 2002. Key words and phrases. Causation, initial conditions, fuzzy hypercube, fuzzy cardinality, fuzzy cognitive map, fuzzy “sets as points”, fuzzy causality, fuzzy rules, fuzzy subsethood, cellular automata, perception-based science, deduction, stroke, evidence-based medicine. c °2002 Yang’s Scientific Research Institute, LLC. All rights reserved.
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clinical efficiency measure K (clinical causal effect/ formal causal ground) is unit less. Using data from real patients, we derived formal causal ground and clinical causal effect, which in each specific patient’s clinical fuzzy cognitive map represents the causal edge strength for nodes of therapeutic or other presence for that patient’s different measured states. The clinical efficiency measure K was calculated for any process that carries the patient from one state to another in the unit hypercube. Each patient has a unique measure for K, consistent with the physiologic elemental context for that individual and representative of the measure of interaction of those elements represented in the patient’s fuzzy set “causal nexus” at initial conditions. Conclusion: Our causal measures represent changes in fuzzy cardinality in a unit hypercube and can be used instead of probability- based statistics to directly measure the causal relation of medical therapies or conditions to the individual patient. K is the measure of causal efficiency with which a patient moves from one state to another as a result of intervention of treatment, pathology or other intervening force. These measures are connected to the individual patient unlike group based derived measures which cannot be extrapolated to the individual case. The efficiency measure K shows the criticality of initial conditions and captures numerically a simple rule that can be applied to the representation of actual causal forces within the patient’s body. Intuitively, these rules are captured by the expert physician when he adjusts the dosage of any medication or other therapy towards the individual patient, but are obscured when the “one dose for all” approach is applied in therapeutics. Our method lends numeracy to the physician’s intuition. The causal rules discovered for each patient can then be applied to other math models, for example, a fuzzy cognitive map or a multi-way causal c sequence. Copyright °2002 Yang’s Scientific Research Institute, LLC. All rights reserved. 1. Introduction In limiting itself to mathematical models, scientific medicine currently follows exclusively a probabilistic approach and has formally abandoned the perception- based approach of physician- based judgment out of which expertise traditionally develops. Both the more general mathematical and the specific probability –based models oversimplify the complexity of the unique context of the patient and put an artificial barrier between the patient and physician. Unlike public health matters, medical information relevant to the individual patient need not be probability-based because decisions in practice always involve the individual patient [1-4]. When group-based evidence is relied upon for individual patient care decisions, the result is a clinical
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decision paradox. This is because each individual patient is the unique contextual body of interaction for physiologic, environmental, therapeutic and diagnostic elements. The group-based-probability-based statistical evidence correlates these elements as variables at the collective level where they are removed from their natural proximity in the individual patient’s own bodily context. Causation can be represented only indirectly in terms of the collective as statistical correlation because it has lost the natural condition of connectedness in physiologic context. In order to solve the clinical decision paradox, we propose a two-fold approach to capture causation at the level of individual patient biology. First, the use of a fuzzy mathematical approach to model the interrelation of elements (variables) at the level of physiology of the patient and, second, a return to perception-based science. The two approaches are dependent on each other, because the adoption of fuzzy math allows for the use of powerful new perception based and math models such as the of development of rules for cellular automata and fuzzy cognitive causal maps that can simulate the patient context with a high degree of precision. This is because in our model, the individual patient remains intact as a dynamic physiologic system of interacting elements where causal relationships can be captured both qualitatively and quantitatively. By using a fuzzy mathematical approach, we make rules of cellular automata more complex, but thereby render the evolution of the patient system more regular and predictable. By moving from measurements to perceptions, we eliminate the barrier to interaction between the patient and his or her physician. Herein, we investigate real physiologic interactions of elements in the individual patient’s clinical context and develop a causal model based on initial conditions. We elucidate clinical “causal ground” in order to provide a useful framework for diagnosis and treatment of the individual patient. The same method will also provide a way to determine physiological interactions in a patient’s system in order to better understand the pathology of disease. Thus, we also develop the measure of clinical “causal effect”. Considered together, the “efficiency” of an intervening action from outside the elements represented and measured in the patient also can be measured. The physiological-biochemical interaction of elements in an individual patient is the basis of scientific medicine. If probability-based statistics cannot capture this, a new model must be sought and applied. Thus, the apparent ambiguity that results from large statistical studies as to the efficacy of treatments immediately resolves itself at the bedside when the individual patient either recovers or gets worse.
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2. Methods: Definition, Derivation and Clinical Importance of Measures of Causation Some of the definitions and data presented in this paper have been published in part in the Proceedings of the 2002 Annual Meeting of the North America Fuzzy Information Processing Society [5]. In this paper they are expanded upon and new clinical data is presented and analyzed. The discussion of the relevance of the causal model presented in this paper sheds new light on the role it might play in scientific discovery in living systems. Definition of Causal Ground. We define “causal ground ” as the outer limit of measurable individual patient physiologic and contextual elemental (variable) interaction. Causal ground uses the measurement of elements, called individual –based variables, given as values in the unit interval [0,1], within the dynamic physiology and context of the individual patient. These are elements that are given their value [0,1] based on measurements or expert opinion. Measurements for the patient are typically done in the laboratory, and clinical expert opinion is numerically represented in the NIH Stroke scale, Glasgow coma scale, Barthel Index, various pain scales, to name a few examples. Causal ground is therefore never separable from the patient and his/her unique context. This context includes both the internal and the external environmental elements. Regardless of where they are, within the context, elements interact and the interaction itself is inseparable from and unique to the patient. A complex of elements contextual to the patient can be used to represent the patient as a fuzzy set of those elements (when each element takes on a value [0,1]) and this is called the “causal nexus” of that patient. Every living being can be represented as a causal nexus of interacting elements. Every living being may be represented as a unique conformation and interaction of these elements. The interaction of these elements in unique conformation is measurable. We call this measure “causal ground ” when the causal nexus represents initial conditions in a causal sequence. It is indirectly related to the measure of efficiency of any action or process termed “extra-cubal causality” responsible for moving one patient as set as point to another location or state of the patient as “set as point” in the unit hypercube. Another term for extra-cubal causality is K or causal efficiency. The measure of causal ground is the measure of initial conditions in this context. In this paper, from laboratory- based measurements of the patient, we take the patient’s elements as a “fuzzy set” and interpret that set as a point in a unit hypercube [6]. The explicit interpretation of a fuzzy “set as point” in a unit hypercube has been formulated by Bart Kosko [7]. We
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define causal ground in terms of the fuzzy cardinality or “Σ-count” of the elements of interest for an individual patient. Σ-count here is simply the sum of the values comprising a fuzzy set [7]. Our definition of causal ground builds on the concept of causality developed by Kosko where causality is fuzzy. Degree of fuzzy subsethood represents fuzzy causality and the fuzzy set membership of one concept in another concept’s power set defines the causal edge function for the fuzzy cognitive map (FCM) [8]. Derivation of Causal Ground: Necessary, Sufficient and Formal. Causal Ground. To visualize the patient as a graded union of unique elements with potential for interaction and to view the consequence of change in these elements, we define several measures of changes in cardinality that are founded in fuzzy subsethood [9]. Clinically, these changes might be the result of therapeutic or diagnostic interventions, or pathophysiology of disease: a) Necessary Causal Ground (NCG) is given by the formula: M (A ∩ B) µ ¶2 M (A ∩ B) M (A) N CG = = , M (A) M (A) M (A ∩ B) where A represents the patient (or object of interest) at an initial state, i.e., the start of a change or intervention, and B represents the same patient after the change or intervention. Necessary Causal Ground is a cubal causal measure. The ∩ symbol refers to fuzzy set intersection defined fitwise by pairwise minimum (picking the smaller of the two elements). NCG is 1 when M (A) = M (A ∩ B), and is strictly less than 1 otherwise. Here M is the “Σ-Count” or fuzzy cardinality. NCG is equal to the squared “subsethood” [9] of the antecedent A in the power set of the consequent B. It measures the degree to which the antecedent is present whenever the consequent is present. If A is completely necessary for B, there is “no B without A”. Here the notion of “subsethood” follows the view developed by Kosko [9]. (Kosko’s original formulation of subsethood measures the degree to which one fuzzy set is contained in another). Here, given two sets A and B, the amount of A outside of A ∩ B weakens the causal relation between A and B, as does the inverse, the amount of A ∩ B that is outside of pure A. To model this, we chose to square the fuzzy subsethood of A to B. b) Sufficient Causal Ground (SCG) is given by the formula:
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M (A ∩ B) µ ¶2 M (A ∩ B) M (B) SCG = = , M (B) M (B) M (A ∩ B) where A represents the patient at an initial state, i.e., the start of a change or intervention, and B represents the same patient in a final state, after the change or intervention. Sufficient causal ground is also a cubal causal measure. SCG is 1 when M (B) = M (A ∩ B), but is strictly less than 1 otherwise. As above, M is the Σ-Count. To derive SCG, we square the “subsethood” of the consequent B in the power set of the antecedent A. It measures the degree to which the consequent B is present whenever the antecedent A is present. If A is completely sufficient for B, there is “no A without B”. Again, the relationship is weakened if given two sets A and B, the greater the amount of B outside of A ∩ B, as does the inverse, the amount of A ∩ B outside of pure B. c) Formal Causal Ground (FCG) is the product of NCG and SCG: µ
¶2 µ ¶2 M (A ∩ B) M (A ∩ B) F CG = M (A) M (B) Formal causal ground FCG is a measure of the degree to which both NCG and SCG are simultaneously present. We call this measure Cubal Causal Ground for living systems. Since M (A ∩ B) is always dominated both by M (A) and M (B), both NCG and SCG always lie in the unit interval [0, 1]. This in turn implies that FCG is always in that same interval. Clinical Importance of Causal Ground. When there is a therapeutic, diagnostic, or other maneuver that displaces a patient (seen as a “set as point” in a unit hypercube of N dimensions) from location A to location B in that cube, and where the cube’s axes are defined by clinically measurable elements N in number, then we take the consequent of the maneuver as B and the antecedent of the maneuver as A. Necessary causal ground, NCG, is the squared degree to which A, the antecedent, is a subset of the power set of B, the consequent. Sufficient causal ground, SCG, is the squared degree to which B, the consequent, is a subset of the power set of A, the antecedent. Formal causal ground FCG measures the degree to which both NCG and SCG are simultaneously present. In the most rigorous case of complete formal causal ground, there will never be an A not followed by a B, nor a B not preceded by an A. In the maximal case A = B, and F CG will be 1. In other words, it becomes
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impossible to distinguish between A and B and causal ground is maximal and equal to 1. In this instance, the set as point where A = B is the timeless instantiated causal nexus of that set. This is where the full causal force of the action of A into B has taken place; this is the full causal force of one set into another. This is a measure of involutionary causation of initial conditions at A [10]. A “involves” B into it. In other words, the degree that something other than A causes the change, to that degree B will “differ” from A after the change or intervention is instituted that separates the patient or object of interest from the initial state as set as point A to final state as set as point B. Our definition of cubal causal ground is consistent with the causal measure of subsethood, but the product of the squared functions of N CG and SCG represents the non-linearity characteristic of living systems. Since the subsethood measure is always [0, 1], for those cases where A 6= B, the causal function of initial conditions at A is strictly decreasing as A and B are more distinct. This is consistent with the causal concept of how much does A “cause” B. The more distinct A and B are as sets as points in the unit hypercube, the less the causal flow from A into B and the more something else besides A is causing B. Definition of Causal Effect. Causal Effect. For a clinical move that distinguishes A from B, the inverse function would define the clinical causal effect CCE of that move. This is consistent with an intuitive increasing causal effect of the clinical maneuver the more distinct A is from B in the hypercube context. The measure of such a clinical maneuver is clinical causal effect and is represented in terms of the square root of subsethood (A, B) and subsethood (B, A) and will be thought of as an evolutionary causation of the final state [10]: B evolves from A. Derivation of Causal Effect: Clinical Causal Effect. Clinical is given by the formula: p Causal Effect (CCE) p CCE = M (A ∩ B)/M (A) × M (A ∩ B)/M (B) Clinical Importance of Causal Effect. Clinical causal effect defines a new measure of interaction of variable elements of the patient at a final state. It is the evolutionary causal measure of the final state at B. The greater CCE, the more A and B are separated as sets as points and the less A is instantiated in B, i.e., the less A causes B. The Application of F CG and CCE to Fuzzy Cognitive Maps. It is important to note that the clinical F CG and CCE are causally directionally dependent on the starting source state equal to the initial conditions of the
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patient. Thus if one moves the patient from state A to state B, then F CG is the cause at A and CCE the final state B. However, moving from B to A, F CG starts at B and CCE is at A. Initial conditions are paramount. The degree of connectedness between elements at initial conditions can either facilitate or hinder the change in cardinality of the patient. If the degree of connectedness is great, then the move to separate A and B is difficult, if it is tenuous, it is easy. If the degree of connectedness is high, and yet A and B are separated, then the extra-cubal process that separated A and B was able to overcome the high degree of connectedness and was very efficient. If the degree of connectedness is low, and A and B are separated, the efficiency of the process separating A and B is low. If the degree of connectedness at initial conditions is high and the extra-cubal process does not separate A and B, it is inefficient. If the connectedness is low and they are not much separated, the efficiency is very very low.
CCE
Node A
FCG
Node B
Figure 1. Fuzzy Cognitive Map (FCM) of Causal Edge Strengths F CG and CCE Between Initial Conditions at A and Final Conditions at B, where Conditions are Changeable Elements in the Patient’s Physiology. Fuzzy Cognitive maps model causal reasoning. The causal edge function of a fuzzy cognitive map is defined by the fuzzy set membership of concepts and the degree of subsethood of concepts [8]. In our causal model, concepts are the patient as set as points at A and B. Sets as points at A and B are conditions of the patient. The conditions of the elements are present to degree and interact to degree. Our measure of F CG can be viewed as the value of the fuzzy causal edge strength that connects the nodes of a clinical F CM where an extra-cubal maneuver moves the patient from set as point A to set as point B in the unit hypercube. The measure of F CG gives us the involutionary causal flow from A into B and CCE that evolutionary flow
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of B out of A. It will be noted that involutional and evolutional changes parallel to the central axis of the cube going from {0,0,0. . . } to {1,1,1. . . } always are greater than comparable changes parallel to the dimension of probability, the n − 1 dimension in the fuzzy hypercube, where all fuzzy sets are of the same cardinality. Causal Efficiency. Definition of Causal Efficiency. We derive a measure of the effect of fuzzy cardinality [7], that is, the “cardinality” responsible for change of a set as point in a unit hypercube. Because cardinality represents the number of elements that are contextually co-localized to the patient, any addition or subtraction of cardinality must make room for the new elements or removal of the old ones. That means that the interaction of the patient elements at initial conditions must be redefined depending on the addition or subtraction of new or old elements. Any process that changes the cardinality of the patient must account for disruption of old interactions. Another word for the interaction between elements of a patient is connectedness. Derivation of Causal Efficiency. Referring to formal causal ground (F CG) as the involutional cause of initial conditions and clinical causal effect (CCE) as an evolutionary cause of the final state, we define Ki as the amount of dimensionality per unit of F CG, in the cube of N dimensions, where each fuzzy set as point is made up of N elements. We define Ke as the amount of dimensionality per unit of CCE in the cube of N dimensions, where each fuzzy set as point is made of N elements. Our measure K, combines the causal power of F CG and CCE in terms of units of cardinality (dimensionality), and is given by the formulae: Ki = N/F CG and Ke = N/CCE. The smaller F CG, that is the closer to 0, the more dimensions are needed per unit F CG to represent it, the same with CCE. If F CG and CCE are close to 1 in any given hypercube, then that cube is sufficient to describe the movement from set as point A set to set as point B in that unit hypercube. The weaker these involutionary and evolutionary factors are, the more dimensions are required to represent the change from set as point A to set as point B. (F CG and CCE can never be 0 ontologically subsethood-wise because in our model everything is causally connected to some degree, and because F CG and CCE are derived from subsethood [9]).
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This can be reformulated as: K = Ki/Ke = CCE/F CG, where K is derived from units of fuzzy cardinality, the Σ-count of elements of the patient as fuzzy set as point in the unit hypercube [7], but is unit less. As F CG moves towards 1, Ki involutional approaches N and of course becomes N when F CG is 1. This means no dimensions need to be added to distinguish A and B and A = B. Now, as F CG move s towards 0, K becomes large. As CCE goes towards 1, Ke approaches N , but as CCE is smaller, Ke approaches a huge number. Clinical Importance of Clinical Efficiency. The measure K is unique for each patient given each initial condition of elements and the therapeutic intervention. Any extra-cubal process is not represented as an element in the patient as set as point in the unit hypercube. The extra-cubal process can come from the patient’s environment or inward physiology. The measure K is a way of representing its presence and effect on the patient as sets as points in the cube. It gives the clinical efficiency of the therapeutic, diagnostic, pathologic intervention in terms of cardinality. The reciprocal of K, that is 1/K, is the relative dosage needed of the drug or process needed to move one unit of CCE and tells us how many F CGs or initial conditions are overcome by one unit dose of the drug. If this is many, the drug is efficient and the dose is small. If this is few, the drug is inefficient and the dose is large. Formal Causal Ground is the measure of initial conditions. Initial conditions and the final state of the patient are cubal representations of the patient in terms of elements measured as variables. The number of elements chosen to represent the patient can vary to a large degree, but visualization of all elements is impossible due to the perceptual limitation of the human brain. However, K is a way to capture the presence of un-represented elements as variables. This is because the un represented elements have presence in the interaction of measured elements in the causal nexus in any set as point. Because initial conditions are unique for each patient, 1/K is a universally applicable factor for understanding pathophysiology or the adjustment of dosage based on the unique context of the individual patient. If all patients were exactly identical in every aspect, and all elements were represented to their exact degree, then 1/K would be the same for everyone and initial conditions would not make a difference. To overcome the curse of dimensionality, we can avoid simply adding variable dimensional element upon variable dimensional element to make the hypercube more complex by simply adding or eliminating one dimension at
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a time. The goal would be to substitute dimensions that are more connected with the causal nexus. What is meant here is that some physiologic elements are more connected with one another than others. For example, the distance between A and B in the unit hypercube as sets as points would be less as the substituted dimensions capture more of the causal nexus. Of especial importance to note here in our model is that causality is not “correlation” but is “connection” between elements that are physically contiguous with one another and the patient. Our method does not address the question of whether something is a solitary or independent cause. Rather we measure the clinical causal efficiency K of a therapeutic, diagnostic, or pathologic process that results in the change in value of measurable elements of a patient. Clinical causal efficiency is based upon how much the clinical causal ground and causal effect of the patient as set as point A (at state A) and B (at state B) are instantiated in moving a patient from one location-state to another in a defined hypercube (elemental) context. Clinical causal ground and effect are direct measures of “cubal causality”. This method is applicable to the single patient whose unique clinical causal course is instantiated in the unit hypercube. Ultimately, this causal course will define the unique rule base for that patient’s system. Intuitively, the rule base is captured by the expert physician at the patient’s bedside who adjusts a dosage to that patient’s needs. The causal measures here of K and 1/K give numeracy to the expert’s perception. 3. Results In order to test the application of our causal measures in the real world, we first found causality measures for the vertices of a 3-dimensional hypercube, followed by those for anti-platelet therapy and a vitamin therapy given as one dose for all and finally for conditions where the dosage of an anti thrombotic agent was adjusted by the expert physician between conditional measures at sequential moments of testing. Results for A and B as Vertices of the Unit Hypercube. The calculations hold for A and B at any vertices of the hypercube except that of the vertice {0, 0, 0}. This is because in arithmetic division by 0 is undefined. In the following Table of Vertex Causality Edges, The K = CCE/F CG values are represented. In our model there is no such thing as nothing (see discussion). The true full significance of zero and undefined efficiency measures is not appreciated. For certain, the size of the set has not changed when going from one vertex to another where the sum of the elements adds to 1, or
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Table of Vertex Causality Edge K Efficiency Values where CCE/F CG = K, and division by 0 is undefined (undef). Vertice 000 010 011 101 110 111 100 001 000 undef undef undef undef undef undef undef undef 010 undef 1.0 2.84 0 2.84 6.11 0 0 011 undef 2.84 1.0 8.33 2.84 1.34 0 2.84 101 undef 0 8.33 1.0 8.33 1.34 2.84 2.84 110 undef 2.84 8.33 8.33 1.0 1.55 2.84 0 111 undef 6.11 1.34 1.34 1.55 1.0 0.93 0.93 100 undef 0 0 2.84 2.84 0.93 1.0 0 001 undef 0 2.84 2.84 0 0.93 0 1.0
anywhere where A intersect B is equal to 0, but clearly the quality of the set is different. Practical example 2: The Effect of Adding Fixed Dose Clopidogrel to Aspirin-Therapy. This example was previously presented in the Nafips Proceedings for 2002. Practical results show the application of our methods to clinical problems. In order to use data derived from a true clinical situation, we determined the clinical causal ground and effect of adding the drug clopidogrel to aspirin in patients already taking aspirin for prevention of recurrent ischemic stroke (stroke due to vascular occlusion). [Local IRB #2001-0046.] Briefly, in order to interpret the results, the usual effect of aspirin and clopidogrel is described. The effect of aspirin and clopidogrel is to inhibit the formation of thrombus in the arterial system by prevention of platelet aggregation. Platelets are blood cells that participate in thrombus formation by clumping and sticking together with fibrin protein strands in the flowing blood stream. The degree of platelet aggregation can be measured ex-vivo by drawing blood from the living patient. The effect of aspirin or clopidogrel on platelet aggregation can thus be measured directly. Aspirin affects cycloxygenase-dependent platelet aggregation. It usually directly inhibits epinephrine-induced platelet aggregation to some degree. It also directly inhibits collagen-induced platelet aggregation, and indirectly inhibits ADP-induced aggregation. Clopidogrel, on the other hand, tends primarily to inhibit ADP-induced platelet aggregation directly, while indirectly inhibiting epinephrine-induced and collagen-induced aggregation. The effects on platelet aggregation of these two drugs can be measured ex-vivo. Given together, an enhanced inhibitive effect on platelet aggregation is expected, but the precise levels of inhibition may differ among patients, the specific agonist, and the degree of interaction of the drugs. The interaction may
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not be enhancement, but additive or inhibitory and depend on the individual patient and his initial conditions. The effect of this drug interaction has previously been described as additive, synergistic or incompatible using alterative fuzzy definitions interaction: synergy, additivity and incompatibility [11]. Typically, platelets in the ex-vivo platelet aggregation test are stimulated with agonists to make them aggregate. These agonists are adenosine diphosphate (ADP), epinephrine (EPI), and collagen (COLL) in certain concentrations. The degree of inhibition of platelet aggregation is recorded with a spectrophotometer and read as light transmission over time [12]. This response then can be easily expertly graded within the interval 0 to 1, i.e., [0, 1], with the following operational definitions: 1.0 absent, 0.8 markedly decreased, 0.6 moderately decreased, 0.4 decreased, 0.2 mildly decreased, 0.0 normal or none. These definitions have been described in our previous work [11]. This method is similar to the defining the grading scale for the NIH stroke scale, scales of Pain, Hunt and Hess score for subarachnoid hemorrhage or the Glascow coma scale. The grading judgment in each individual case where the scale is applied rests solely on the perception of the expert examiner. The formulation of the scales themselves depends on the perception of experts in consensus. The original fit value data for inhibition of platelet aggregation for the 16 patients is given in Table 1 . Again, each fit value is a value [0,1] and means ‘fuzzy information value’. Given data from actual patients, we calculated the effect that adding clopidogrel to aspirin has on the inhibition of platelet aggregation by aspirin alone and give the clinical causal ground and causal effect measures for this clinical move. These measures determine the clinical causal edge weights for causal arrows between the effects of giving aspirin alone (F CG) into that of clopidogrel on inhibition of platelet aggregation on to that of aspirin alone (CCE) in a given single patient. Data was obtained from measurements on sixteen actual patients (Table 1). FCG
→
clopidogrel
aspirin CCE
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Epi asp 0.4 0.4 0.4 0.8 0.1 0.2 0.8 0.8 0.8 0.2 0.0 0.4 0.4 1.0 0.8 0.4
Coll asp 0.8 1.0 0.8 1.0 0.2 0.4 0.8 0.8 0.8 0.4 0.2 0.8 0.8 0.8 0.8 0.8
Adp asp+clop Epi 0.2 0.2 0.4 1.0 0.4 0.2 0.8 0.2 0.8 0.8 0.4 0.8 0.8 0.2 0.8 0.6
asp+clop 0.8 0.8 0.4 0.8 0.8 0.0 1.0 0.8 0.8 0.4 0.4 0.4 0.4 0.8 0.8 0.6
Coll asp+clop 0.8 0.8 0.4 1.0 0.8 0.1 1.0 0.8 1.0 0.8 0.8 0.4 1.0 1.0 1.0 0.8
Table 1. Original Data with Fit Values (fuzzy information values) for Inhibition of Platelet Aggregation for patients 1 through 16 taking first Aspirin alone and then combination Aspirin plus Clopidogrel therapy. These are determined by direct visualization of the platelet aggregation by the expert. Patient Adp asp 1 0.2 2 0.2 3 0.2 4 0.8 5 0.2 6 0.1 7 0.2 8 0.4 9 0.2 10 0.1 11 0.1 12 0.2 13 0.2 14 0.4 15 0.4 16 0.2
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To explain the meaning of Table 2 , the data for patient 1 is detailed below; the other patient computations are similar. Patient 1 on aspirin alone has A = {ADP, Epi, Coll} = {0.2, 0.4, 0.8}, and on aspirin with clopidogrel has B = {ADP, Epi, Coll} = {0.2,0.8,0.8}. Thus M (A) = 1.4, M (B) = 1.8, and M (A ∩ B) = M {0.2, 0.4, 0.8} = 1.4. µ ¶2 M (A ∩ B) N CG for Patient 1 = = 1. M (A) SCG for Patient 1 = µ ¶2 µ ¶2 M (A ∩ B) 1.4 = = 0.6049. M (B) 1.8 F CG for Patient 1 = (1)(0.6049) = 0.6049. Table 2 below displays the computations for each patient where N = 3. In this study, each patient produced a unique causal number for FCG in the range [0,1] that represented the degree to which A “causes” B. Clinically, for these 16 patients this is the degree to which the initial inhibition of platelet aggregation by aspirin involutes that of the addition of clopidogrel into it, and is represented by the squared function of subsethood. The inverse, CCE, the degree to which the inhibition of platelet aggregation by clopidogrel evolves that by aspirin alone is expressed in terms of the square root function of subsethood and is the clinical causal effect of the addition of that drug. These results indicate the clinical efficiency of clopidogrel over aspirin for each individual patient. Each patient has a different measure of K indicating his/her uniqueness with regard to connectedness of elements at the initial conditions. The extra-cubal therapeutic intervention of adding clopidogrel to aspirin, overcame K in order to move the patient to B. The dose of clopidogrel necessary to move the patient one unit of CCE initial conditions is 1/K, or one unit of the the causal force of B. Practical Example 3: The Effect of fixed dose Foltxr on lowering plasma homocysteine. In addition to looking at platelet aggregation in patients with stroke we studied the effect of the combination of vitamin B12 1000 mcg plus vitamin B6 25 mg and Folicin 2.5 mg as one tablet a day of Foltxr in lowering plasma homocysteine levels over a 6 to 12 month period of time. Again we used the representation of patients as “sets as points” and cardinality measure M of each patient before and after treatment in order to judge the efficiency of the drug in effecting plasma homocysteine lowering in each patient. When each patient was considered as a set of elements, we defined the patient in terms of compliance with treatment medication and normalized homocysteine values. The homocysteine value that is normalized is the
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M (B) 1.8 1.8 1.2 2.8 2.0 0.3 2.8 1.8 2.6 2.0 1.2 1.6 2.2 2.0 2.6 2.0
M (A ∩ B) 1.4 1.4 1.0 2.6 0.5 0.2 1.8 1.8 1.8 0.7 0.3 1.0 1.4 1.8 2.0 1.4
N CG 1 0.7656 0.5100 1 1 0.0816 1 0.81 1 1 1 0.5101 1 0.6693 1 1.4
SCG 0.6049 0.6048 0.6943 0.8621 0.6250 0.4443 0.4132 1 0.4792 0.1225 0.0625 0.3906 0.4049 0.8100 0.59169 0.4900
F CG 0.6049 0.4630 0.3540 0.8621 0.6250 0.03626 0.4132 0.8100 0.4792 0.1225 0.0625 0.1995 0.4049 0.5421 0.59169 0.4900
CCE 0.8819 0.8249 0.7714 0.4225 0.5000 0.4364 0.8017 0.9486 0.8320 0.5916 0.5000 0.5996 0.7977 0.8580 0.8770 0.8366
K = CCE/F CG 1.457 1.782 2.179 0.499 0.800 12.03 1.940 1.171 1.736 4.829 8.000 3.005 1.970 1.583 1.483 1.707
1/K 0.69 0.56 0.46 2.00 1.25 0.08 0.52 0.85 0.58 0.21 0.13 0.33 0.50 0.63 0.67 0.59
Table 2. Causal Measurements per Patient. Here A is the patient on aspirin alone, B the patient on aspirin plus clopidogrel and the numerical measures the value of inhibition of platelet aggregation for adp, epi and collagen induced platelet aggregation. Patient M (A) 1 1.4 2 1.6 3 1.4 4 2.6 5 0.5 6 0.7 7 1.8 8 2.0 9 1.8 10 0.7 11 0.3 12 1.4 13 1.4 14 2.2 15 2.0 16 1.4
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laboratory value obtained in the patient before and after fixed dose Foltxr therapy. Compliance was gauged by expert opinion of the physician caring for the patient and homocysteine values were normalized from real lab test values to a value [0,1]. Each patient was represented as a set as point before A and after B treatment in the unit hypercube (Table 3). Again, to get the patient to move at all from A to B in the unit hypercube one has to overcome the resistance of the interactive power of the original causal nexus of A. The force needed to add on any fractal or full dimension that was added on in the individual instance is K. The dose needed to overcome the causal power of initial conditions at A is 1/K. Efficiency here means that the causal agent was able to cause the move at all. K is the measure of the dimensionality of the nexus in that particular patient. If K is huge, the nexus has so many dimensions that the drug had great effect. Clinically, a decision about dosing should be able to be made on the basis of this data. Persons in whom the K measure is high need smaller doses of the agent in order to get any where for change in homocysteine. So for patient 1, the Foltxr had to overcome 13 dimensions in order to move each homocysteine units he did and 1/K is the dosage needed to move one CCE unit. The addition of another dimension does change the results of our measures. When the genetic mutation for Methyltetrahydrofolate reductase C677T mutation is accounted for and represented as another measured element as variable in each patient, the K changes for the same patient. Patient 2 and 9 are heterozygous and patient 10 homozygous for the mutation. The fit value given for the heterozygous mutation is {0.5} and the homozygous {1.0}. Absence of the mutation is {0}. The other patients did not carry the mutation, except for patient 5 for whom no result is available. The addition of another element in each patient’s set as point had a dramatic effect on the K value even though the elemental value did not change at patient as set A and B. Therefore, there does not seem to be a linear relation between this mutation and the K value and the additional of another element at initial conditions changes their causal measure (Table 3 final column CCE/F CG = K). Practical example 3: Effect of Coumadinr dosing for Goal INR (International Normalized Ratio). Comparison of Efficiency Measures over time in the same patient. Finally, a third set of data was reviewed for 7 patients taking Coumadinr (warfarin) for prevention of recurrent stroke. Each patient was represented in the hypercube context as a fuzzy set of elements {INR, Normalized total Coumadinr dosage}. Three points in the cube were measured first for each patient. Table 4 shows the causality and efficiency measures for the patient
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(A)∩(B) 0.30 0.26 0.16 0.20 0.28 0.20 0.10 0.30 0.28 0.16
N CG 0.42 1.0 0.45 0.50 0.20 0.83 1.0 0.88 0.49 0.32
SCG 0.06 0.21 0.02 0.06 0.06 0.11 0.06 0.09 0.08 0.03
F CG 0.03 0.21 0.01 0.03 0.01 0.09 0.06 0.08 0.04 0.01
CCE 0.41 0.68 0.32 0.42 0.33 0.54 0.50 0.53 0.45 0.31
K∗ 1/K∗ 13.67 0.07 3.24 0.31 32.00 0.03 14.00 0.07 33.00 0.03 6.00 0.17 8.33 0.12 6.63 0.15 11.25 0.09 31.00 0.03
CCE/F CG = K ∗ ∗ 1/K** 0.41/0.03=13.67 0.07 0.85/0.52=1.63 0.61 0.32/0.01=32.00 0.03 0.42/0.03=14.00 0.07 No gene test 0.54/0.09=6.00 0.17 0.50/0.06=8.30 0.12 0.53/0.08=6.62 0.15 0.68/0.21=3.24 0.31 0.73/0.29=2.52 0.40
Table 3. Causal Measures per Patient including the Measure of Efficiency K of Foltxr in lowering plasma homocysteine values in stroke patients before and after fixed dose therapy. Here K∗ is the value when two elements, compliance and normalized plasma homocysteine are considered and K ∗ ∗ when three elements, MTHFR (methylenetetrahydrofolate reductase C677T) mutation (Nucleotide C to T substitution at position 677), compliance, and normalized plasma homocystiene are considered. Note the change in K as knowledge of and initial conditions change. Patient M (A) M (B) M 1 0.46 1.2 2 0.26 0.56 3 0.24 1.06 4 0.28 0.80 5 0.62 1.18 6 0.22 0.60 7 0.10 0.40 8 0.32 1.0 9 0.40 0.98 10 0.28 0.96
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at three different INR checks A, B and C. Note that though the initial conditions change, the K remains the same for any given patient. This is because the dose is adjusted to fit the patient and is consistent with the idea that no drug can be given at one fixed dose and expect to have the same effect on all persons at that dosage. The “K” is intuitively found by the physician who changes the dosage of warfarin accordingly. Had one dosage of warfarin been given to all, then the causal methodology would find the 1/K dosage of warfarin appropriate for each patient. In this instance the K shows the efficiency of the physician in dosing the drug and the minor changes in K capture the contextual differences of the patient at conditions for A, B and C that are not represented in the elements comprising the patient as fuzzy set. So now we have initial conditions so the dosage can be differentially applied according to the formula 1/K. For every unit of F CG, how much CCE has taken place, and this is K. The less causality of A the more evolution pushes A and B apart, and the less A causes B. 4. Discussion This paper discusses a new approach to the problem of the direct measure of causality which transforms the meaning of connectedness and explanation. It uses fuzzy mathematics to solve its equations. It abandons probability –based statistics and by doing so, no physical rupture from the individual patient (or object of interest) and any causal measure exists in this model. Our measures of causality, F CG, CCE, are in units of fuzzy cardinality. The definition of causal ground rests on the concept of necessity and sufficiency in the medical setting. It is founded in the subsethood measure where the degree of fuzzy causality is represented, but is stricter in the sense that by squaring the subsethood measure we account for both necessity and sufficiency in the framework of assuming that A and B will be impacted by their interaction A ∩ B. In this sense there is an involutionary concept of subsethood and there is a bi-directional causal flow at the level of both necessity and sufficiency. This makes the system responsive to dynamic factors of the individual rather than static factors of the group. In the clinical setting, A is antecedent and B consequent to some therapeutic move on the patient as set as point A which results in B. Initial conditions at A draw B into A. The inverse causal measure, called causal effect, separates A and B and is represented by taking the squared root of the “stricter sense” subsethood measure. Causal effect represents an evolutionary concept of subsethood. For definition purposes, in the instance where A = B at the outset, time is absent or no change occurs and we are speaking of what we term the
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M (B) 0.42 0.83 0.83 0.53 0.52 0.68 0.34
M (C) 0.43 0.82 NA 0.54 0.64 0.70 0.45
FCG A-B 0.74 0.94 0.86 0.88 0.90 0.52 0.44
FCG B-C 0.94 0.96 NA 1.00 0.66 0.94 0.58
CCE A-B 0.92 0.98 0.96 0.97 0.97 0.85 0.81
CCE B-C 0.98 0.98 NA 1.00 0.90 0.98 0.87
K A-B 1.24 1.04 1.12 1.10 1.08 1.63 1.84
K B-C K CD 1.04 1.02 NA 1.00 1.00 1.36 1.04 1.14
Table 4. K values INR check at initial conditions A, B and C and in one patient D. INR is normalized by 10 and the patient’s weekly Coumadinr dosage by 150. Data is presented from 7 patients. Pt M (A) 1 0.49 2 0.85 3 0.89 4 0.56 5 0.55 6 0.94 7 0.37
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causal nexus. The causal nexus is the patient or object as a “set as point” of any number of elements in the unit hypercube context. The cardinality measure itself is the outer-limit of measure of causal interaction within the timeless causal nexus. It is the sum of the values of the elements of the set , not the sum of the number of the elements in the set. It thus represents the “size” of the fuzzy set. The causal nexus persists through initial and final states. Our argument does not guarantee all measures within the causal nexus itself, but rather provides the groundwork for the measurement of clinical maneuvers on a patient as causal nexus. By this measure, one can gage the totality of effect on many variables within one patient at a time of any clinical maneuver. This is readily visualized by using the fuzzy hypercube context. The concepts of causal ground , causal effect and causal efficiency can be generalized to clinical settings other than stroke. Kosko has used the medical example of smoking and lung cancer [8]. Furthermore, these concepts may be more generalized to other living systems. The basic assumption of formal causal ground and cubal causality in the living system is complete determinism in the physical setting. Ontologically speaking everything is causally connected to everything else to some degree. This may be zero, but it is measurable. This means that when the causal connection between A and B is being represented, a bi-direction flow is occurring as well resulting from their interaction. This is most often a non-linear relation as is reflected by the formula for formal causal ground similar to most catalysis chemical reactions. The key to these results is that they represent degrees of causal ground, not probability, and degrees in terms of the amount of causal connection between A and B and not conditional inference. Previous efforts to find an exact measure of the amount of causality between two states or events has been hampered by thinking of group processes only. This in turn was framed by the assumption that probability based statistics provides the only path towards truth and certainty. The uniqueness of our approach is derived from the individual patient because a direct measure of causality must take into account the role of what is unknown in each context. It must be an emergent measure rather than an externally imposed measure. Our measures of F CG, CCE and K allow us to understand the interactive force of connectedness between elements of any patient as sets as point and the causal efficiency necessary to disrupt that interaction. All the unknown factors that enter into a causal path need not be determined, measured or articulated, in order to obtain a measure of causality in the individual causal nexus. Outside the individual one needs to determine every variable that is involved in the system correlation. Within, one remains in the individual
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system; all factors are present and can have effect but do not necessarily require measurement. To illustrate using the example we studied, the molecular mechanism by which clopidogrel interacts with the aspirin-platelet system need not be defined nor articulated to get an actual measure of causal effect because we are here dealing with the individual patient. At the group level, the model of inference implies exact knowledge of every detail in the system, so that only correlations –and not connectedness– can be measured. In the actual molecular system for one patient there will be the presence of peptides that inhibit or facilitate the action of clopidogrel, whereas in another patient these same peptides may be diminished or absent. Our measure of causal ground respects the presence of all of these detailed molecular effects, but does not require their determination or measurement in order to reach an efficiency measure K of clopidogrel’s extra-cubal causal action on that same molecular system. Our extrapolation of FCG and CCE to the Fuzzy Cognitive Map model may be an important new observation because derivations of edges in FCM’s have for the most part relied upon expert opinion as to the degree of causal connection between two conceptual nodes. The implicit assumption in the expert opinion process is that the expert is thinking in terms of groups or aggregate processes. Dickerson et al have obtained fuzzy causal edge weights from three sources: expert opinion, peer reviewed literature and micro- array data [13, 14] An attempt to objectify expert opinion has been made by combining many experts’ opinions to strengthen the validity of the causal linkage [15]. But the move to numeric values derived from natural biologic data as was done in our study appears to be possible only because individual patients, the natural objects of interest, are considered relevant. Even with this method, relying on numeric values from lab data, the expert interpretation of these values is still required. In this paper we carry our analysis of fuzzy causality further, but again use an involutionary and evolutionary approach to derive a measure of clinical efficiency or the efficiency of pathogenesis. In our representation of cubal causal ground and effect, we build on the subsethood theorem of Kosko [7]. The present attempt at causal representation deals with the curse of dimensionality. This latter refers to the fact that each new dimension adds more knowledge in terms of actual measures of the causal nexus of the patient represented at set as point of elements in a unit hypercube. Each element of the set as point is a cubal dimension present to degree. Causal nexus is defined in the patient as a fuzzy set of elements representing his/her state at any given instant. We are handling the curse of dimensionality by not actually measuring all the dimensions that are interacting in the nexus, but
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whose effect can be assumed to be present when a set as point moves from one location to another in a unit hypercube context. This is made possible because the Fuzzy Subsethood theorem carries the concept of fuzzy membership to the level of set’s membership in each other. We present a technique which permits the unknown elements of a causal nexus to have a measured role without being explicitly measured or formulated itself [3]: the effect of the unknown on the known without actually measuring the unknown. With our new measure of clinical efficiency, K, physician presence at the bedside of the patient becomes numerically productive in terms of our causal measures. Typically one thinks of the physician’s domain of knowledge as consisting of results of statistical studies on the one hand, and a complex of past experience on the other. The contact between the physician and the patient activates the part of the expertise related to statistical studies and leads to the selection of various lab tests related to those studies. This will then yield medical inferences similar to, but, fundamentally different from those of the Bayesian type. However, the past experience of the physician, with so many individual patients, each of whom is unique in his/her variable complex, plays little or no role in the standard statistical numerical exchanges. Our method now permits a numerical exchange to occur based on perception between the physician’s past experience and the patient’s contextual physiological state. Therefore, we can numerically leverage the vast potential of this physician perceptual experience that has been ignored or thwarted by the more classical statistical approach. Bayes Theorem to individualize the patient experience through the use of statistics is flawed because it is based in group-based derived data, which ignore the specific initial conditions of the individual patient at hand. Intuitively our approach to the use of fuzzy subsethood in medicine is based on the notion that each patient as fuzzy set can be viewed as an individual such that numbers can be attached to elements without losing that individuality. This concept we have defined as the “incarnation/discarnation” or “separation from body of interest” problem and distinguishes our medical application of fuzzy logic from that of others [6]. Our use of normalization and assignment of fit values is perception based because we have used direct visual assessment of platelet aggregation and expert opinion has carried us from perception to measures and back again to perception. This is because our measures of K and 1/K allow us to measure and to perceive the relative difference between individual patients that are otherwise intuitively perceived by the expert physician at the bedside. In this model, all the dimensions but one can be zero for any set as point. This is because ontologically nothing does not exist as per Kosko’s proof.
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[16] This makes sense because causality must involve something. So, this is consistent with the idea of conservation of mass and energy where nothing can be created or destroyed. The advantage of using the exponential function is that the rate of change of subsethood is measured. As we are interested in causality of extra-cubal processes, it is not only necessary to capture the subsethood relation of all objects of interest (everything exists according to subsethood) in the cube instantiated at any one point in time, but the force of any process brought to bear on the objects in the cube. That we have done here. Causality is the rate of change of subsethood in both directions between and A and B. This is analogous to velocity and acceleration in mechanics in physics. Something is happening and subsethood is changing. WE are asking how much is the antecedent A influential in changing into the subsequent B. That is cubal-inertia. We ask how much B extends A, and that is cubal causal causal effect. We then ask, how much was the rate of dimensional change by which this was achieved. This rate depends on the effect of all unseen and unmeasured dimensions in each object of interest and we call this K. It is a measure of how easily or difficult the move was made from A to B by an outside extra-cubal process, such as a drug, pathology, or other means. 5. Conclusion Stephan Wolfram, who devised Mathematica, a software program, has discovered a new kind of science of complexity [16]. How does fuzzy causality relate to Stephan Wolfram’s Principle of Computational Equivalence (PCE), since both concepts relate to complexity? The PCE states that most phenomena that are not obviously simple will be Turing “universally” computable and that any complex system can therefore simulate any other complex system. The key terms “obviously simple” mean that repetition and computational reduction can be detected by either human perception or by mathematical analysis. Both perception and mathematics work only on computations that are reducible, where a short hand symbolism, for example, can be used to represent long stretches of a repeated pattern. Wolfram’s discovery that simple algorithms or simple programs can produce complexity that rapidly outstrips both perception and mathematics to look, for all intent and purposes, perfectly random, is paradoxically matched by the observation that starting with the complex pattern, it appears to be impossible to discover the actual program that lawfully accounted for it other than by trial and error. This is a paradox in that there is no way to logically infer the algorithm from the pattern. In many respects, we can view the simple program as the” cause” of the complex pattern, as in cellular automata. Fuzzy causality refers to fuzzy rules that generate complex patterns.
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It is precisely because causality is perception-based that mathematical analysis can decipher repetitions in the data. To apply numbers to causality is to bring mathematics closer to perception. It is to apply perception to mathematics and vice versa. Fuzzy logic is unique in combining math and perception because only a human brain can determine by perception the numbers to give partial membership values to fuzzy membership or fuzzy subsethood. The act of assignment of a number is a perceptual act. Fuzzy logic permits a creative fusion of math and perception to perform computational reductions. The act of “ positioning” a fuzzy hypercube is also an act of perception. The perceiver can assign a hypercube and add or subtract dimensions at will. The field of perception is always broader, “ sees more” than the hypercube and we refer to this as extra-cubal causality. The perceiver has a view as to how much of a phenomenon is being included in the fuzzy representation and how much is being left out. Cubal causality applies to what is in the cube. The hypercube is like a magnifying glass. The field of vision must be large enough to allow the positioning of the magnifying glass directly over the area to be more carefully observed. The strength of the magnification is analogous to how many dimensions (elements) are included in the hypercube. In conclusion, we can say that in traditional clinical science, at the group level, using probability based reasoning, two objects or “events” A and B, are assumed not to be causally related until proven otherwise, whereas at the level of the fuzzy hypercube, two objects “events” A and B are assumed always to be causally related until proven otherwise. Therefore, our use of a fuzzy hypercube-based causal measure puts deduction at the center of perception-based reasoning and forms the basis of a perception-based science.
References [1] C.M. Helgason, D.S. Malik, S-C. Cheng, T.H. Jobe and J.N. Mordeson, “ Statistical versus fuzzy measures of interaction in patients with stroke”, Neuroepidemiology, vol 20, 2001, pp. 77-84. [2] D. Kirschner, “Reconstructing microbial pathogenesis. Mathematic models can simulate the complex dynamics between host and pathogen during various phases of infection”, ASM News, vol 67, 2001,pp. 566-573. [3] D.L. Sackett, W.S. Richardson, W. Rosenberg and R.B. Haynes, Evidence-Based Medicine, Churchhill Livingstone, Edinburgh, 1998. [4] G. Guyatt, H Schunemann, D. Cook, R. Jaeschke, S. Pauker and H Bucher, “Grades of recommendation for anti-thrombotic agents”, Sixth ACCP Consensus Conference on Anti-thrombotic Therapy, Chest vol 119 (suppl), 2001, pp. 3S-8S. [5] Fuzzy Information Processing Society, 2002. Proceedings Nafips, 2002. Annual Meeting North America, 2002 pp 117-123 and 434-439.
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[6] C.M. Helgason and T.H. Jobe, “ The fuzzy cube and causal efficacy: Representation of concomitant mechanisms in stroke”, Neural Networks, vol 11, 1998, pp. 549-555. [7] B. Kosko, “Fuzziness versus Probability: The Subsethood Theorem”, Neural Networks and Fuzzy Systems. A Dynamical Approach to Machine Intelligence, Prentice Hall, Inc., Englewood Cliffs, NJ 07632, 1992. Chapter 7, pp. 269 and 274. Also: B. Kosko, “Fuzziness versus Probability”, Int J General Systems, vol 17, 1990, pp.216. [8] B. Kosko, “Fuzzy Cognitive Maps”, Int. J Man-Machine Studies vol 24, 1986, pp. 65-75. [9] B. Kosko, “Fuzziness versus Probability: The Subsethood Theorem”, Neural Networks and Fuzzy Systems. A Dynamical Approach to Machine Intelligence, Prentice Hall, Inc., Englewood Cliffs, NJ 07632, 1992, Chapter 7, pp. 278-289. [10] A.A. Klaff, Arithmetic Refresher, Dover Publications Inc., New York, 1964, pp. 218 and 227. [11] C.M. Helgason, T.H. Jobe, L.D. Brace and J.N. Mordeson, “ The fuzzification of platelet aggregation response for interpretation of interactive effect of aspirin – ticlopidine therapy in patients with stroke”, Proceedings 18 th International Conference of the North American Fuzzy Information Processing Society-NAFIPS. New York June 10-12, 1999, IEEE Catalogue Number 99TH8397, pp 283-288. [12] C.M. Helgason, K.L.Tortorice, S.R. Winkler, D.W.Penney, J.J. Schuler, T.J.Mc Cleeland and L.D. Brace, “Aspirin dosage and failure in cerebral infarction”, Stroke, vol 24, 1993, pp. 345-350. [13] J.A. Dickerson, Z. Berleant, Z. Cox, A.W. Fulmer and E. Wurtele, “Creating and modeling metabolic and regulatory networks using text mining and fuzzy expert systems”, Computational Biology and Genome Informatics, eds. Cathy H. Wu, Paul Wang and Jason T.L. Wang, World Scientific Press, 2002; Chapter 9, In Press. [14] J.A. Dickerson, Z. Cox, E.S. Wurtele and A.W. Fulmer, “Creating metabolic and regulatory metabolic pathways using fuzzy cognitive maps”, Nafips Proceedings from the Joint 9 th IFSA World Congress and 20 th NAFIPS International Conference, July 25-28, 2001, Vancouver, British Columbia, Canada. IEEE Catalogue number 01TH8569,pp. 2171-2175. [15] R. Taber, R.R.Yager and C.M.Helgason, “Small-sample quantization effects on the equilibrium behavior of combined fuzzy cognitive maps”, IEEE Fuzz 2001, Melbourne, Australia. December, 2001. [16] B. Kosko. Fuzzy Thinking. Hyperion Press New York, New York 1993. pp 273. [17] S. Wolfram. A New Kind of Science. Wolfram Media, Inc. 2002: 547-636. a Cathy M. Helgason and b Thomas H. Jobe, Departments of Neurologya and Psychiatryb , University of Illinois College of Medicine at Chicago, Chicago, Illinois. E-mail address:
[email protected](C. M. Helgason)
