myfis = setfis(myfis, 'output',1,'name','num of trips');%% Understanding the clusters-FIS relationship% A FIS is composed of inputs, outputs and rules. Each input and% output can have any number of membership functions. The rules dictate the% behavior of the fuzzy system based on inputs, outputs and membership% functions. |genfis2| constructs the FIS in an attempt to capture the the% position and influence of each cluster in the input space. % % |myfis| is the FIS that |genfis2| has generated. Since the dataset has 5% input variables and 1 output variable, |genfis2| constructs a FIS with% 5 inputs and 1 output. Each input and output has as many membership% functions as the number of clusters that |subclust| has identified. As% seen previously, for the current dataset |subclust| identified 3% clusters. Therefore each input and output will be characterized by 3% membership functions. Also, the number of rules equals the number of% clusters and hence 3 rules are created. % % We can now probe the FIS to understand how the clusters got converted% internally into membership functions and rules.fuzzy(myfis)%%% *Figure 3:* The graphical editor for building Fuzzy Inference Systems% (FIS)%%% |fuzzy| is the function that launches the graphical editor for building% fuzzy systems. |fuzzy(myfis)| launches the editor set up to edit% |myfis|, the FIS that we just generated. As can be seen, the FIS has 5% inputs and 1 output with the inputs mapped to the outputs through a% rulebase (white box in the figure).%% Let's now try to analyze how the cluster centers and the membership% functions are related.mfedit(myfis)%%% *Figure 4:* The graphical membership function editor%%% |mfedit(myfis)| launches the graphical membership function editor. It% can also be launched by clicking on the inputs or the outputs in the% FIS editor launched by |fuzzy|.% % Notice that all the inputs and outputs have exactly 3 membership% functions. The 3 membership functions represent the 3 clusters that were % identified by |subclust|.% % Each input in the FIS represents an input variable in the input dataset% |datin| and each output in the FIS represents an output variable in the% output dataset |datout|.% % By default, the first membership function, |in1cluster1|, of the first% input |population| would be selected in the membership function editor.% Notice that the membership function type is "gaussmf" (gaussian type% membership function) and the parameters of the membership function are% |[1.162 1.877]|, where |1.162| represents the spread coefficient of the% gaussian curve and |1.877| represents the center of the gaussian curve.% |in1cluster1| captures the position and influence of the first cluster% for the input variable |population|. |(C(1,1)=1.877, S(1)=1.1621 )|%% Similarly, the position and influence of the other 2 clusters for the% input variable |population| are captured by the other two membership% functions |in1cluster2| and |in1cluster3|. %% The rest of the 4 inputs follow the exact pattern mimicking the position% and influence of the 3 clusters along their respective dimensions in the% dataset.%% Now, let's explore how the fuzzy rules are constructed.ruleedit(myfis)%%% *Figure 5:* The graphical rule editor%%% |ruleedit| is the graphical fuzzy rule editor. As you can notice, there% are exactly three rules. Each rule attempts to map a cluster in the input% space to a cluster in the output space.% % The first rule can be explained simply as follows. If the inputs to the% FIS, |population|, |dwelling units|, |num vehicles|, |income|, and% |employment|, strongly belong to their respective |cluster1| membership% functions then the output, |num of trips|, must strongly belong to its% |cluster1| membership function. The (1) at the end of the rule is to% indicate that the rule has  a weight or an importance of "1". Weights can% take any value between 0 and 1. Rules with lesser weights will count for% less in the final output. %% The significance of the rule is that it succinctly maps cluster 1 in% the input space to cluster 1 in the output space. Similarly the other% two rules map cluster 2 and cluster 3 in the input space to cluster 2% and cluster 3 in the output space.%% If a datapoint closer to the first cluster, or in other words% having strong membership to the first cluster, is fed as input to |myfis|% then rule1 will fire with more <#28 firing strength> than the other two% rules. Similarly, an input with strong membership to the second cluster% will fire the second rule will with more firing strength than the other% two rules and so on.%% The output of the rules (firing strengths) are then used to generate the% output of the FIS through the output membership functions.%% The one output of the FIS, |num of trips|, has 3 linear membership% functions representing the 3 clusters identified by |subclust|. The% coefficients of the linear membership functions though are not taken% directly from the cluster centers. Instead, they are estimated from the% dataset using least squares estimation technique. %% All 3 membership functions in this case will be of the form |a*population% + b*dwelling units + c*num vehicles + d*income + e*employment + f|, where% |a|, |b|, |c|, |d|, |e| and |f| represent the coefficients of the linear% membership function. Click on any of the |num of trips| membership% functions in the membership function editor to observe the parameters of% these linear membership functions.%%% Using the FIS for data exploration% You can now use the FIS that has been constructed to understand the% underlying dynamics of relationship being modeled. surfview(myfis)%%% *Figure 6:* Input-Output Surface viewer%%% |surfview| is the surface viewer that helps view the input-output% surface of the fuzzy system. In other words, this tool simulates the% response of the fuzzy system for the entire range of inputs that the% system is configured to work for. Thereafter, the output or the response% of the FIS to the inputs are plotted against the inputs as a surface.% This visualization is very helpful to understand how the system is going% to behave for the entire range of values in the input space.%% In the plot above the surface viewer shows the output surface for two% inputs |population| and |num of dwelling units|. As you can see the% number of auto trips increases with increase in population and dwelling% units, which sounds very rational. You can change the inputs in the X and% Y drop-down boxes to observe the output surface with respect to the inputs% you choose.ruleview(myfis)%%% *Figure 7:* Rule viewer that simulates the entire fuzzy inference process%%% |ruleview| is the graphical simulator for simulating the FIS response for% specific values of the input variables. Now, having built the fuzzy system,% if we want to understand how many trips will occur for a particular% demographic setup, say an area with a particular population, a certain% number of dwelling units and so on, this tool will help you simulate the% FIS response for the input of your choice.%% Another feature of this GUI tool is, it gives you a snapshot of the% entire fuzzy inference process, right from how the membership functions% are being satisfied in every rule to how the final output is being% generated through <#28 defuzzification>. %%% Conclusion% This example has attempted to convey how clustering and fuzzy logic can% be employed as effective techniques for data modeling and analysis. %% Fuzzy logic has also found various% applications in other areas of technology like non-linear control,% automatic control, signal processing, system identification, pattern% recognition, time series prediction, data mining, financial applications% etc.,%% Explore other demos and the documentation for more insight into fuzzy% logic and its applications.%%% Glossary%% *input space* - it is a term used to define the range of all possible% values in the dataset. When using |subclust| the input space refers to% the entire range of values between the maximum and minimum in each% dimension (column) of the dataset.%% *defuzzification* - the process of transforming a fuzzy output of a fuzzy% inference system into a crisp output.%% *firing strength* - The degree to which the antecedent part of a fuzzy% rule is satisfied. Also known as degree of fulfillment.%% *fuzzy inference system (FIS)* - The overall name for a system that uses% fuzzy reasoning to map an input space to an output space%% *Reference:*%% [Chi94] - S. Chiu, "Fuzzy Model Identification Based on Cluster% Estimation," J. of Intelligent & Fuzzy Systems, Vol. 2, No. 3, 1994.displayEndOfDemoMessage(mfilename)##### SOURCE END #####-->   </body></html>