This example shows how to use subtractive clustering to model traffic patterns in an area based on the demographics of the area.

In this example we attempt to understand the relationship between the number of automobile trips generated from an area and the area's demographics. Demographic and trip data were collected from traffic analysis zones in New Castle County, Delaware. Five demographic factors are considered: population, number of dwelling units, vehicle ownership, median household income and total employment.

Hereon, the demographic factors will be addressed as inputs and the trips generated will be addressed as output. Hence our problem has five input variables (five demographic factors) and one output variable (number of trips generated).

Load the input and output variables used for this example into the workspace.

tripdata

Two variables are loaded in the workspace, `datin`

and `datout`

. `datin`

has 5 columns representing the 5 input variables and `datout`

has 1 column representing the 1 output variable.

subplot(2,1,1) plot(datin) legend('population','num. of dwelling units','vehicle ownership',... 'median household income','total employment') title('Input Variables','fontsize',10) subplot(2,1,2) plot(datout) legend('num of trips') title('Output Variable','fontsize',10)

**Figure 1:** Input and Output variables

The number of rows in `datin`

and `datout`

, 75, represent the number of observations or samples or datapoints available. A row in `datin`

, say row 11, constitutes a set of observed values of the 5 input variables (population, number of dwelling units, vehicle ownership, median household income and total employment) and the corresponding row, row 11, in `datout`

represents the observed value for the number of trips generated given the observations made for the input variables.

We will model the relationship between the input variables (demographics) and the output variable (trips) by first clustering the data. The cluster centers will then be used as a basis to define a fuzzy inference system (FIS) which can then be used to explore and understand traffic patterns.

Clustering can be a very effective technique to identify natural groupings in data from a large data set, thereby allowing concise representation of relationships embedded in the data. In this example, clustering allows us to group traffic patterns into broad categories hence allowing for easier understandability.

Fuzzy logic is an effective paradigm to handle imprecision. It can be used to take fuzzy or imprecise observations for inputs and yet arrive at crisp and precise values for outputs. Also, a fuzzy inference system is a way to build systems without using complex analytical equations.

In this example, fuzzy logic is used to capture the broad categories identified during clustering into a Fuzzy Inference System (FIS). The FIS will then act as a model that will reflect the relationship between demographics and auto trips.

Clustering and fuzzy logic together provide a simple yet powerful means to model the traffic relationship that we want to study.

`subclust`

is the function that implements a clustering technique called subtractive clustering. Subtractive clustering, [1], is a fast, one-pass algorithm for estimating the number of clusters and the cluster centers in a dataset.

In this section, we will see how subtractive clustering is performed on a dataset and in the next section we will explore independently how clustering is used to build a Fuzzy Inference System(FIS).

[C,S] = subclust([datin datout],0.5);

The first argument to the `subclust`

function is the data to be clustered. The second argument to the function is the `radii`

which marks a cluster's radius of influence in the input space.

The variable `C`

now holds all the centers of the clusters that have been identified by `subclust`

. Each row of `C`

contains the position of a cluster.

C

C = 1.8770 0.7630 0.9170 18.7500 1.5650 2.1830 0.3980 0.1510 0.1320 8.1590 0.6250 0.6480 3.1160 1.1930 1.4870 19.7330 0.6030 2.3850

In this case, `C`

has 3 rows representing 3 clusters with 6 columns representing the positions of the clusters in each dimension.

`subclust`

has hence identified 3 natural groupings in the demographic-trip dataset being considered. The following plot shows how the clusters have been identified in the 'total employment' and 'trips' dimensions of the input space.

clf plot(datin(:,5),datout(:,1),'.',C(:,5),C(:,6),'r*') legend('Data points','Cluster centers','Location','SouthEast') xlabel('total employment','fontsize',10) ylabel('num of trips','fontsize',10) title('Data and Clusters in selected two dimensions of the input space','fontsize',10)

**Figure 2:** Cluster centers in the 'total employment' and 'trips' dimensions of the input space

The variable `S`

contains the sigma values that specify the range of influence of a cluster center in each of the data dimensions. All cluster centers share the same set of sigma values.

S

S = 1.1621 0.4117 0.6555 7.6139 2.8931 1.4395

`S`

in this case has 6 columns representing the influence of the cluster centers on each of the 6 dimensions.

`genfis`

is the function that creates a FIS using subtractive clustering. `genfis`

employs `subclust`

behind the scenes to cluster the data and uses the cluster centers and their range of influences to build a FIS which will then be used to explore and understand traffic patterns.

myfis=genfis(datin,datout, ... genfisOptions('SubtractiveClustering','ClusterInfluenceRange',0.5));

The first argument is the input variables matrix `datin`

, the second argument is the output variables matrix `datout`

and the third argument is the `radii`

that should be used while using `subclust`

.

`genfis`

assigns default names for inputs, outputs and membership functions. For our understanding it is beneficial to rename the inputs and outputs meaningfully.

Assign names to the inputs and outputs.

myfis.Inputs(1).Name = "population"; myfis.Inputs(2).Name = "dwelling units"; myfis.Inputs(3).Name = "num vehicles"; myfis.Inputs(4).Name = "income"; myfis.Inputs(5).Name = "employment"; myfis.Outputs(1).Name = "num of trips";

An 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. `genfis`

constructs the FIS in an attempt to capture the position and influence of each cluster in the input space.

`myfis`

is the FIS that `genfis`

has generated. Since the dataset has 5 input variables and 1 output variable, `genfis`

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 using the Fuzzy Logic Designer app.

fuzzyLogicDesigner(myfis)

**Figure 3:** The graphical editor for building Fuzzy Inference Systems (FIS)

As can be seen, the FIS has 5 inputs and 1 output with the inputs mapped to the outputs through a rule base (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 `fuzzyLogicDesigner`

.

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 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.

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 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 defuzzification.

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.

[1] - S. Chiu, "Fuzzy Model Identification Based on Cluster Estimation," *J. of Intelligent & Fuzzy Systems*, Vol. 2, No. 3, 1994.