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The network that we obtain explicitly depicts the causal model of system behaviour where the nodes represent variables of interest (it is useful to think of them as discrete variables, which values represent intervals of the actual domain) and the links represent informational or causal dependencies among the variables. To translate the causal network in a Bayesian Causal Network we must express the conditional probabilities which link any variable value with any value combination of its parent nodes. The solving algorithms provide to spread the effect of an assertion (made by updating the likelihood of a variable), to any variable of the net. In this way the causal framework represented in the net, highlights the system states compatibles with observed data, by increasing the likelihood of specific values of the nodes. As written, this feature allows the investigator to answer a variety of queries, including: abductive queries, such as ‘What is the most plausible explanation for a given set of data?’; and control queries; such as ‘What will happen if we intervene and enlarge the inlet?’. Answers to these queries depend on the causal knowledge embedded in the network. The probabilistic basis of Bayesian networks offers a coherent semantics for co-ordinating top-down and bottom-up inferences, thus bridging information from high-level concepts and low-level percepts. This capability is important for achieving selective attention that is, selecting the most informative data in a specific step of the design. In other words the student is guided not only in diagnosing the reason of a behaviour but also in focusing its attention to weigh the importance of each variable in determining that response or in confirming the diagnosis. The capability of integrating in a unique graphical model, directly accessible to the user, many abstraction levels of the domain is a novel feature which enhance the tutorial role of the expert system, and modify the interaction with the user that directly interact with the graphical knowledge representation envisioning the effect of data he provides directly on the whole knowledge. This allows a sort of sensivity analysis about variables and parameters relevance, which enhances the user skill in perceiving the integrity of design problem normally hidden in other expert systems. 5. APPLICATION In this section we’ll demonstrate the approach of causal network modelling through a simple case of stack ventilation of a single zone building under steady state conditions of heat transfer. 5.1 STACK VENTILATION OF A SINGLE ZONE BUILDING Consider a simple single-zone model of a building utilizing a stack ventilation strategy2. A steady wind, characterized by a stagnation pressure p0, approaches the building from the left as air passes through a window ‘a’ of a cross sectional area Aa and exits at a higher opening ‘c’ of a cross sectional area Ac, located a distance Dz above the lower opening. These three variables (Aa, Ac, Dz) will be taken as the key design parameters that will be adjusted to achieve thermal comfort. Wind pressure coefficients at the windows Cpa and Cpc, building conductances SUA, internal gains qgain, outdoor air temperature T0 and heat capacity of air p c ˆ , are assumed known a priori. Two unknown state variables are associated with this simple idealization, the zone air temperature Ti and the zone air pressure pi defined relative to a specific elevation, which will be taken along the horizontal centreline through the lower window. With the problem thus defined, we can form the heat transfer system equations by demanding conservation of thermal energy: We’ll model the airflow through the windows using the familiar orifice equation as it has proven to be a reliable model. In the absence of wind-driven pressures, both indoor and outdoor air pressures will be assumed to vary hydrostatically in proportion to the indoor and outdoor air densities ri and r0 respectively. Then: ( ) i a d a p p A C m - = 0ˆ 2r and ( ) ( ) ( ) z g p z g p A C m i i c d cD - - D - = 0 0ˆ 2 r r r The airflow system equations may be formed by demanding conservation of airflow, namely by using the equation ma+mc=0. Finally we’ll use the ideal gas law to estimate air densities indoor and out. In this case the heat transfer and airflow equations are coupled through both the airflow rate and buoyancy terms and the resulting nonlinearity is pathological. Consequently, it was not possible to explicitly express the resulting equations for the indoor air temperature Ti in terms of the design variables Aa, Ac and Dz. Nevertheless, we can numerically solve the equation and plot the indoor air temperature as depicted in figure 5. In addition the inlet air velocity is also plotted as it varies with the inlet window opening Aa: To specify the probability distribution of the network, one must give the prior p robabilities of all root nodes (nodes with no predecessors) and the conditional probabilities of all non-root nodes given all possible combinations of their direct predecessors. Considering the node v, we have to specify the values: P(v=0.2¸0.5|Aa=0.3¸1,Dz=2.5¸5); P(v=0.5¸2|Aa=0.3¸1,Dz=2.5¸5); P(v=2¸4|Aa=0.3¸1,Dz=2.5¸5); and so forth with all combinations of Dz and Aa. To express such values, say P(v=0.2¸0.5|Aa=0.3¸1,Dz=2.5¸5) we had to evaluate the probability that using design parameters included within the intervals (Aa=0.3¸1,Dz=2.5¸5) the inlet air velocity will be included between the values [0.2¸0.5 m/s]. Many approaches are available to solve and automate this problem, which is typical in the fuzzy set theory3. Once the Bayesian network has been assembled, it allows effectively investigating the stack ventilation design problem both assessing the consequences of a decision on the comfort variables and going back up the causes of a given behaviour. For instance we may set the variable Ti to a given value and the variable Comfort to the value ‘Good’, getting the most likely design parameter values, which achieve the selected state. Afterwards, by setting one or more design parameters we can address the decision about the remaining design parameters by selecting those values with higher related probability. In this manner the design could achieve a good result in terms of internal comfort also overlooking the investigation about the inlet air velocity, which value is constrained by the ‘Good Comfort’ and the air temperature selections. Moreover the graph we have assembled can be clearly expanded, introducing further causal dependencies between design parameters (i.e. parameters as T0 that we have fixed in this example, or expressing a design parameter by means of more detailed features) and by adding external variables, which affect the microclimate conditions. Suppose for instance, we want to introduce the relationship between the direction of airflow through the inlet opening and the pattern of air circulation inside the room.
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