Ch3.3.2 pg. 1

Environmental Modeling and Decision Support Systems

Prof. Dr. Struss, Dr. Dressler

Chapter 3.3.2 Qualitative Modeling

In order to motivate the need and objectives for qualitative modeling, we revisit some of the numerical models introduced earlier.

The Algal Bloom – A “Numerical Model”

Figure 3.3.2.1 Numerical Model

Figure 3.3.2.1 shows again the numerical model of Algal Bloom. Although it is stated as a real-valued function, no one would expect that it precisely mirrors the relevant interdependencies. On the one hand, the set of scenarios for which this model has been developed has a huge amount of variations, which would require numerical modifications – theoretically, because the required information and measurements would not be available. Obviously, the functions representing the extinction of light, length of daylight, or temperature dependence of the process are only approximate. Extinction of light is not linear and cannot be expressed by a location-independent function. The available daylight is not simply given by the period between sunrise and sunset, but would have to reflect the effect of clouds (which certainly cannot be represented exactly), etc. The functions have been chosen, because they qualitatively match the observations.

Intraspecific Competition

This numerical model, 1/N*dN/dt = r-(r/K)N (shown again in Figure 3.3.2.2), assumes that the reproduction rate r is linearly decreasing. (K is the maximal capacity of the species determined by the available resources). Certainly, in reality, r is not a linear function; it is only convenient to assume this. All we can probably state about it is that it is a monotonically decreasing curve. And, after all, it is not a function anyway, because it would have to reflect many influences that are not captured by the model, but produce “noise".

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Figure 3.3.2.2 Intraspecific competition, revisited

Qualitative modeling - Motivation

2 Knowledge about the current state of the environmental system is inevitably

incomplete, since in real scenarios neither spatial nor temporal coverage by measurements can be achieved and, hence, numeric values can only be approximate. Other observations are of an inherently qualitative nature, e. g. sighting of fish swarms, health conditions of particular fish, or changes in water color.

Even under such circumstances, offering little information, humans have the ability to characterize broad categories of outcomes to ascertain what might happen. For many tasks this is enough: Knowing that a valuable fragile object might be pushed off a table is sufficient reason to rearrange things so that it cannot happen. For other tasks, knowing the possible outcomes suggests further analysis, perhaps involving more detailed models. For example, an engineer designing a tea warmer must keep the tea at a drinkable temperature, while not allowing it to boil. Reasoning directly with qualitative models can capture important behavior patterns, automatically producing descriptions that are closer to the level of what people call insights about system behavior, making them useful for science, engineering, education and decision-support.

In qualitative models we capture partial knowledge and information. "Partial" means we do not know precisely how certain quantities are related to each other or that we do not have exact data that can be simply transfered into numerical values/formulae.

In summary (illustrated in Fig. 3.3.2.3, qualitative models are meant to reflect

What we know (or, rather, do not know): e.g. even for a particular species under particular circumstances, we have no exact knowledge about the function r(N)

What information is available: all empirical evidence is given by a finite set samples of data that are (for continuous quantities) never exact.

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What we are interested in: if we are, for instance, only interested in deciding whether or not and how the iron concentration in the drinking water can be reduced, a complex numerical model might be overkill, even if it could be constructed.

Figure 3.3.2.3 Motivation of qualitative models

Qualitative Modeling

In qualitative modeling, system models are built to express partial knowledge and process partial information. A good understanding of the investigated domain often proves crucial for systems that deal with large datasets of structurally complex objects. But we may have a rough understanding of dependencies, for example about the extinction of light, or replace imprecise or missing data. This allows us to model classes of systems and conditions, rather than only very special situations.

In pursuing qualitative modeling, we do not drop the requirement of a precise semantics, formal definitions, and a calculus for qualitative domains. Qualitative modeling is not heuristic reasoning.

Another task (not discussed in much detail in this lecture) is a formal analysis of relationships among models of different granularity. Especially, it is important to understand how the results of a qualitative model that has been obtained from a numerical one relate to the results obtained from the original model (discussed later in this chapter).

Qualitative modeling is not only more intuitive than numerical modeling. There are more benefits expected. We obtain a finite representation, which in turn often results in more efficient processing. For instance, rather than running a large number of simulations (for different combinations of parameter values and initial conditions) the qualitative model should predict the possible system behavior(s) with the same coverage in one step.

We introduce the key ideas and principles using an extremely simple example, modeling of a resistor.

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A (Very General) Representation of Behavior Models

In section 3.2, we presented a model capturing the reproduction of a species under conditions of intra-specific competition (see figure 3.3.2.4):

dN/dt = N*r = N*r0*[1 – (N/K)]

Figure 3.3.2.4 Interdependency between reproduction rate and population size N

Although one can use this equation directly to calculate the derivative of the population size from the other quantities, the equation seen as a model (rather than as a calculation step), has a different meaning. It constrains the possible tuples of values, which do occur in reality, if this “law” holds. For instance, when fixing r0 = 2 and K = 1000, the tuple (r, N) = (1, 500) is possible, while (r, N) = (1, 100) and (r, N) = (-1/2, *) are excluded from the behavior. Hence, what really matters is not the equation, but the relation it specifies. Rr,N represents this behavior relation.

Representation of Qualitative Behavior Models

Qualitative models should only make the essential and possible distinctions. But what is essential depends on the task. For consistency checking, those distinctions are essential, that can contribute to the detection of inconsistencies.

Assume we are interested in expressing only the qualitative knowledge that N is never greater than K (and not negative), and that r lies between 0 and r0. This results in the relation

Rqr,N = {[r0, r0 ] [0, 0]} {[0, 0 ] [K, K]} (0 , r0) (0 , K)

depicted in fig. 3.3.2.5

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Figure 3.3.2.5 A – very rough – qualitative model of intraspecific competition

For this model, (r, N) = (1, 500) Rqr,N : i.e. the tuple is consistent, as for the numerical

model, but also

(r, N) = (1, 100) Rqr,N is now consistent, according to the purely qualitative model,

while (r, N) = (-1/2, *) Rqr,N and remains inconsistent.

Extended Qualitative Model

We may want to also express the hypothesis that r decreases with increasing N, as indicated by Fig. 3.3.2.5, and we can do so by including dr/dN as a variable.

Fig. 3.3.2.6 r decreases if N increases.

The extended Relation is then

Rqr,N,dr DOM(r, N, dr/dN):

Rqr,N,dr =

{[r0, r0 ] [0, 0] [0, 0] } {[0, 0 ] [K, K] [0, 0] } (0 , r0) (0 , K) (- , 0)

Refined Qualitative Model

We may refine the model by expressing

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• “If N is close to 0, r is close to r0“ • “If N is close to K, r is close to 0” • “If N is in between, r is in between”

which results in the relation

Rq’r,N,dr DOM’(r, N, dr/dN):

Rq’r,N,dr = {[r, r0 ] [0, K ] [dr , 0] } {[0, r ] [K, K] [dr ,0] }

(r , r) (K , K) (- , 0)

using appropriate thresholds for r and N (see fig. 3.3.2.7).

Fig. 3.3.2.7 A refined qualitative model of intraspecific competition

If we associate symbolic names with the intervals between adjacent thresholds (called “landmarks”), the relation can be stated in very intuitive terms by three tuples:

Rq’r,N,dr ={ (small, small, neg), (large, large, neg), (medium, medium, neg)}

Generalization: Relational Behavior Models

For every model, a representational space is (v, DOM(v)) has to be chosen, where v is a vector of local variables and parameters (“ local” is meant with respect to the model fragment (process) or system and Dom(v) is the domain of v, i.e. a Cartesian product of the domains of the variables in this vector. A particular behavior can then be (statically)

described as a relationR DOM(v).

Valid behavior models

The relational notion of a behavior model is (deliberately) very general and independent of a particular syntactical form, in which the behavior may have been specified in the first place. Its origin could be an equation, a logical expression, a set of data, a verbal description. It should answer the question: “what set of states are allowed by a

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particular behavior?” When would such a relation be called a valid model of a behavior that appropriately describe the behavior of the system? Intuitively stated: when for every situation that might occur in reality a taken measurement (i.e. an instantiated tuple vs,0 of the vector v) is covered by this relational model. This definition means: Whenever we take a measurement of the reality and obtain a tuple, which is not an element of the relation, we will assume that the modeled behavior is not the one that currently occurs in reality.

More formally, if we denote the set of all real-world senarios under this behavior by SIT, then, for every situation, every possible measurement outcome vs,0 has to be in the relation:

" sSIT Val(vS , vS,0, s) vS,0 RS

As indicated by Fig. 3.3.2.8, this definition requires only that the model relation is a superset of the measurements taken in real-world situations and not identical to this set of measurement. This reflects that, at least for systems modeled over a continuous infinite domain, we are unable to exactly characterize this set. For Ohm’s Law, we should not use the linear function, but some envelope around it, because the resistance will vary and has some tolerance. However, we cannot describe the boundaries of this envelope. On the negative side, this notion of a valid model seems to be quite weak, because the trivial model given by R=DOM(v) is all a valid model because it certainly covers all possible measurements. However, it fulfills the purpose that a tuple not contained in the behavior relation allows the safe conclusion that the modeled behavior is not present. In other words: if an observation is found to be inconsistent with the behavior model is detected, then it is guaranteed to be inconsistent with the modeled behavior, which can, hence, be safely refuted. The model may fail to detect some inconsistency, but this will only lead to a larger set of consistent behaviors, which still contains the one that is really present. This is crucial to consistency-based problem solving which is the basis for our approach to decision support.

Figure 3.3.2.8 Valid behavior models

Types of qualitative abstraction

There are different types of qualitative abstraction. It may be related to

The functional dependencies. Instead of providing a real-valued function (and pretending that it precisely reflects the dependency), we may just state that it is a

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monotonic dependency. For example: the increase of Diclofenac carcasses has a negative impact on the vulture population size.

Orders of magnitude: we frequently ignore certain factors, because their contribution to a certain phenomenon is negligible compared to other influences (e.g. because it is smaller than the noise associated with the decisive factors or because it is below the granularity required for answering certain questions. For instance, the variation in cloud coverage may be considered irrelevant to algae biomass in trout streams).

The granularity of the domain of variables. For instance, if we are only interested in whether or not a population size is below a critical value. This domain abstraction means aggregating values that lead to the same class of behaviors, as in the illustrative example in Fig. 3.3.2.9.

Figure 3.3.2.9 Intervals between landmarks

Interval Arithmetic

The sign algebra is only a special case of applying arithmetic to intervals. Interval mathematics (independently of qualitative modeling) has investigated this. In interval arithmetic, values are represented as closed intervals whose end points are specified numerically. To express tolerances, values are described as a numerical value plus a numerical tolerance, essentially a small interval around the given value within which the real value can be found. Arithmetic operations on intervals can be defined as follows:

Addition of Intervals: (α1, ω1) (α2, ω2) = (α1+α2, ω1+ω2)

Subtraction of intervals: (α1, ω1) (α2, ω2) = (α1- ω2, ω1- α2)

Multiplication: (α1, ω1) (α2, ω2) = (min(α1*α2, α1*ω2, α2*ω1, ω2*ω1), max(α1*α2, α1*ω2, α2*ω1, ω2* ω1))

Division: (α1, ω1) Ø (α2, ω2) = (min(α1/α2, α1/ω2, ω1/α2, ω1/ω2), max(α1/α2, α1/ω2, ω1/α2, ω1/ω2)),

for 0(α2,ω2).

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Note that the result of division by an interval, containing zero does not yield an interval, but the union of two intervals. For instance, given Z = X/Y, with X = [1,2] and Y = [-1,1], then Z = {[-∞, - 1], [1,∞]}.

The difference to qualitative modeling is usually, that this uses a given set of landmarks to define the considered intervals. Since in this case, the results of the arithmetic operations on interval boundaries in general do not coincide with one of these given landmarks, one has to choose the smallest of the admissible intervals that contain the exact result, which adds a “rounding error”.

Properties of interval arithmetic

While interval operations as defined above are associative and commutative, distributivity holds only in a weaker form:

i1 (i2i3) (i1i2) (i1i3)

Defining what a solution to a set of interval “equations” is, follows the intuition that a set of specific intervals satisfies a set of equations if there is a value that lies in both the interval resulting from the right-hand side and the one resulting from the left-hand side.

x1=i1,x2=i2,…

satisfies fl(x1,x2,…,xn)fr(x1,x2,…,xn) iff fl(i1,i2,…,in)∩fr(i1,i2,…,in)≠Ø

One may think that this is equivalent to the statement that a tuple of intervals solves an interval equation if it is an abstraction of a solution to the underlying numerical equation. Unfortunately, this is not true. Subdistributivity already indicates this, because multiplying out does not modify the set of real-valued solutions, but changes the interval result. We get back to this below.

Arithmetic of Signs

Ohm’s Law is formulated as an equation, and the behavior relation is the graph of the respective function. The equation can be combined with other equations (e.g. characterizing the behavior of other components), and the resulting set of equations can be symbolically manipulated and solved. We can attempt to do the same with “equations” over qualitative domains. The simplest version is the sign algebra used in qualitative economics some decades ago, where parameters are characterized as positive (+), negative, (-), or zero (0). The sign algebra is surprisingly powerful. It is the simplest system that enables continuity constraints to be applied: A qualitative value cannot jump directly from + to -, or from – to +, without first going through 0. When applied to derivatives, it provides a natural formal expression for the intuitive idea of a parameter either being increasing, decreasing, or remaining constant.

But reasoning only with signs obviously has rather strong limitations, as indicated by the fact that adding a positive and a negative number does not yield a restriction on

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the result. However, if we really do not know the relative magnitude of the quantities, then this ambiguity is simply realistic and honest. One can introduce a new value, often labeled ?, to explicitly represent this ambiguity. From a relational point of view, it simply means that all three tuples (+, -, -), (+, -, 0), and (+, -, +) do occur.

Real numbers are mapped to qualitative values by means of the operator [ ]. [x] indicates the qualitative values of variable x with the set of possible values {+,-,0}, where [ ]: R+ +, R- - and {0} 0. Supplementary to the information about its sign, a variable is further described by the qualitative value of its derivative, written as

x. x denotes [dx/dt] and may also take any value from the set {+,-,0}, indicating

whether variable x is increasing, decreasing or not changing at all. Thus describes the dynamics of the system, its change in time and the direction of change.

The operation and , the qualitative counterparts to addition and multiplication defined on R, are defined on this limited value set. The table for addition is not completely filled, the symbol ? denotes an open result. Thus, the sign algebra of the device-centered approach is not closed with respect to addition. Since + indicates (0,

∞) and - (- ∞,0), the operation + - has an ambiguous result.

Figure 3.3.2.10 Arithmetic on Signs

Domain abstraction – Formally

In general, such a domain expection is transformation from one domain to another one:

τi : DOM0(vi) → DOM1(vi) .

More specifically, the aggregation of values that lead to the same qualitative behavior maps each original value to an equivalence class of values. Hence, it can be formalized as a mapping from one domain to a different one, which is the subset of the power set of the original domain:

τi : DOM0(vi) → DOM1(vi) P(DOM0(vi))

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For continuous models, such equivalence classes will mainly be intervals (or unions of intervals), including, perhaps, their boundaries (“landmarks”) as degenerate intervals; i,e, τi is a function which maps real values to a domain DOM1(vi) which is subset of the

set of intervals τi : IR∞ → DOM1(vi) I(IR∞)).

If we have a finite set of landmarks, then domain abstraction maps every value onto the interval between two related landmarks it is contained in:

Let L IR∞ be a finite set of landmarks of real values.

τi : IR∞ → DOM1(vi) IL(IR∞), T is function which maps every real value to DOM1, which is a subset of Intervals between Landmarks L of real values.

Model abstraction induced by domain abstraction

A domain abstraction τ(DOM0(vs) → DOM1(vs)) induces a model abstraction because

the image of a relation is a relation in the new domain: Rs DOM0(vs) →τ(Rs)

DOM1(vs). It is an important question which properties the transformed relation model has. It is easy to see that validity of a model is preserved under the abstraction transformation: If the original relation specifies a valid model of a behavior, then the same is true for its abstraction.

Again, this is important for consistency checking. If the abstraction of an observation is not contained in the abstract relation, and, hence, inconsistent with the behavior model, then respective behavior cannot be present. Figure 3.3.2.11 illustrates this. Of course, the qualitative model may fail to detect some inconsistencies that the original model could reveal. It is possible however, to first use the abstract model for pruning the solution space and then apply the more fine-grained (and probably more costly) model to the reduced space.

Figure 3.3.2.11 Domain abstraction

Lotka-Volterra Predator-Prey Model – A Qualitative Analysis

In 3.2, we have introduced the Lotka-Volterra Predator-Prey Model and it's two functions:

dN/dt = (r-a*P)*N, dP/dt =(f*a*N-q)*P.

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Actually, the importance of this model did not lie in the exact calculation of population sizes, but in the interpretation at a qualitative level, namely the statement about the resulting oscillating behavior, Figure 3.3.2.12.

Figure 3.3.2.12 Evolution of prey and predator population sizes over time.

In the following, we investigate whether we can obtain the same result following a different path: rather than using a numerical equation and then analyzing the solution space qualitatively, we would like to use qualitative abstraction.

Qualitative Lotka-Volterra Predator-Prey Model

Before we apply the qualitative abstraction, we perform a translation for reasons of convenience. Lets take the differential equations:

dN/dt = (r-a*P)*N dP/dt =(f*a*N-q)*P

and transfer N and P by shifting equations and variables:

N’=N+q/(f*a) P’=P+r/a dN’/dt = -a*P’*(N’-q/(f*a)) dP’/dt =f*a*N(P’-r/a) Then we apply the qualitative abstraction: [x]:=sign(x)

x :=[dx/dt]

and obtain: δN’ [P’] [N’-q/(f*a)]=0

P’ =[N’] [P’-r/a]=0 using N,P>0, because population size should be positive.

Finally, we get the set of qualitative equations:

N’ [P’]=0

P’=[N']

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Qualitative Lotka-Volterra – Relational Model

The behavior relation tells which ones are the valid tuples of this model, i.e. satisfy both confluences. This set is the Cartesian product of the solutions of the individual confluences, since they do not share variables.

N’ [P’]=0, P’=[N’],

RLVPP Dom(p’,P’, N’,N’): {(-,-),(0.0),(+,+)}×{(-,+),(0,0),(+,-)}

This describes all possible states of the system.In the general case, it can be obtained by applying constraint satisfaction algorithms as described in Ch. 3.4.

Qualitative Lotka-Volterra – Qualitative States

Figure 3.3.2.13 shows the possible states of the system.

Figure 3.3.2.13 Qualitative States of the Lotka-Volterra prey-predator model obtained from its qualitative

version

Qualitative Lotka-Volterra – Transition between states

In order to characterize the evolution of the system, we have to ask which transitions between states (and then paths following these transitions) are possible. This analysis exploits

Continuity: e.g. a state with [N’]=+ cannot be directly followed by one with [N’]=-

Qualitative derivatives: if [N’]=+ and N’=+, then [N’]=+ must hold also in the next state.

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These mathematical rules can also be formulated as a set of constraints (on pairs of states) and solved by the methods discussed in Ch. 3.4. Figure 3.3.2.14 shows the result.

Figure 3.3.2.14 Qualitative behaviors allowed by the Lotka-Volterra prey-predator model

Qualitative Lotka-Volterra – Possible terminal states

However, if there is a transition from one state to another one, this does not mean that it must happen. Without more detailed information, it is not possible to conclude, for example, that the state

[P’] = +

[N’] = P’ = -

N’ = - will eventually be left, since P’ may decrease, but stay positive and never reach zero. Fig. 3.3.2.15 marks such potential terminal states by a transition to the same state.

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Figure 3.3.2.15 Possible terminal states of the system.

Qualitative Lotka-Volterra – Interpretation

Indeed does the last Figure capture the qualitative interpretation we derived from the numerical model: the oscillating behavior. However, there are more possible evolutions of the system; namely the ones that terminate in some state rather than oscillation forever. Actually, when we revisit the isocline analysis we performed intuitively in Ch. 3.2, we could not rule out these possibilities without adding further detail.

Qualitative Modeling with deviations

A very intuitive and powerful way to use qualitative models is by applying them not to the magnitudes of the variables, but to the deviations from some reference, usually given by the nominal behavior. This kind of analysis is often performed by humans, especially in diagnostic tasks: if pressure is higher than normal, this may be caused by an increased inflow, which in turn may result from a valve which is wider than normal.

This is captured in a mathematical and rigorous way by deviations which are defined as the difference between an actual and some reference value: Δx:= xact - xref

This definition yields equations that capture how deviations of certain quantities cause others to deviate. For instance, consider Kirchhoffs Law: Q1+Q2 =0

Combining the equation with the definition of deviations and perform qualitative

abstraction, we get the qualitative model fragment [ΔQ1] [ ΔQ2]=[0], which expresses “The inflow is higher/lower/unchanged, then there is more/less/the same flowing out” (note that we need to express the direction of flow by its sign).

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For a certain set of algebraic equations, we obtain algebraic equations stating interdependencies of deviations, as shown in Fig. 3.3.2.16 for instance, the deviation of the sum is the sum of deviation (for both the numerical and the qualitative model fragment). A little more complex are product and division. Even for functions, we can use this approach: if y is a monotonic function of x, f(x), then the deviations of x and y will have the same sign. The important observation is that these model fragments do not contain the reference values. So did our argument about the possible cause of an increased pressure.

Figure 3.3.2.16 Qualitative Modeling with deviations

Spurious Solutions in Qualitative Modeling

We already pointed out that there may exist solutions to an interval equation, that do not contain a real-valued solution to the real-valued equation from which it was derived. The following tiny example (Figure 3.3.2.17) shows this and also helps to understand the reasons.

Figure 3.3.2.17

It is derived from the equation x+y=y+z. The interval equation xy yz is solved by x=(1,2),y=(0,1),z=(0,1), because the intervals(1,3) and (0,2) have a non-empty

intersection, but it contains no real-valued solution: because x+y=y+zx=z, and x=(1,2) is disjoint to z=(0,1).

The deeper reason for this problem lies in the multiple occurrence of the variable y: one can pick values for x and y in the respective intervals, such that the sum is, say 1.5. We can also pick values for z and y in the respective intervals, such that the sum is also 1.5.

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However, we cannot pick the same value for y. But at the real-valued level, y has to be unambiguous.

Quatlitative Models – Implementation

Qualitative abstraction aims at a finite set of qualitative values, which allows stating it in terms of propositional logic and as a finite constraint satisfaction problem. More details will be shown in Chapter 3.4

Types of qualitative abstraction

Finally, we emphasize that we focused on domain abstraction, which is but one type of qualitative abstraction.