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incremental structure

in an incremental structure, input values are incorporated in multiple stages to refine the output values in several levels. for example, the previous figure shows a three-level incremental fuzzy tree having fuzzy inference systems fisin, where i indicates the index of a fis in the nth level. in an incremental fuzzy tree, i = 1, meaning that each level has only one fuzzy inference system. in the previous figure, the jth input of the ith fis in the nth level is shown as input xijn, whereas the kth output of the ith fis in the nth level is shown as input yikn. in the figure, n = 3, j = 1 or 2, and k = 1. if each input has m membership functions (mfs), each fis has a complete set of m² rules. hence, the total number of rules is nm² = 3 × 3² = 27.

the following figure shows a monolithic (n = 1) fis with four inputs (j=1, 2, 3, 4) and three mfs (m = 3).

in the fis of this figure, the total number of rules is nm⁴ = 1 × 3⁴ = 81. hence, the total number of rules in an incremental fuzzy tree is linear with the number of input pairs.

input selection at different levels in an incremental fuzzy tree uses input rankings based on their contributions to the final output values. the input values that contribute the most are generally used at the lowest level, while the least influential ones are used at the highest level. in other words, low-rank input values are dependent on high-rank input values.

in an incremental fuzzy tree, each input value usually contributes to the inference process to a certain extent, without being significantly correlated with the other inputs. for example, a fuzzy system forecasts the possibility of buying an automobile using four inputs: color, number of doors, horse power, and autopilot. the inputs are four distinct automobile features, which can independently influence a buyer’s decision. hence, the inputs can be ranked using the existing data to construct a fuzzy tree, as shown in the following figure.

for an example that illustrates creating an incremental fuzzy tree in matlab^®, see the "create incremental fis tree" example on the reference page.

aggregated structure

in an aggregated structure, input values are incorporated as groups at the lowest level, where each input group is fed into a fis. the outputs of the lower level fuzzy systems are combined (aggregated) using the higher level fuzzy systems. for example, the following shows a two-level aggregated fuzzy tree having fuzzy inference systems fisinn, where i_n indicates the index of a fis in the nth level.

in this aggregated fuzzy tree, i₁ = 1,2 and i₂ = 1. hence, each level includes a different number of fis. the jth input of the i_nth fis is shown in the figure as input xinj, and the kth output of the i_nth fis is shown as output yink. in the figure, j = 1,2 and k = 1. in other words, each fis has two inputs and one output. if each input has m mfs, then each fis has a complete set of m² rules. hence, the total number of rules for the three fuzzy systems is 3 m² = 3 × 32 = 27, which is the same as an incremental fis for a similar configuration.

in an aggregated fuzzy tree, input values are naturally grouped together for specific decision-making. for example, an autonomous robot navigation task combines obstacle avoidance and target reaching subtasks for collision-free navigation. to achieve the navigation task, the fuzzy tree can use four inputs: distance to the closest obstacle, angle of the closest obstacle, distance to the target, and angle of the target. distances and angles are measured with respect to the current position and heading direction of the robot. in this case, at the lowest level, the inputs naturally group as shown in the following figure: obstacle distance and obstacle angle (group 1) and target distance and target angle (group 2). two fuzzy systems separately process individual group inputs and then another fuzzy system combines their outputs to produce a collision-free heading for the robot.

for an example that illustrates creating an aggregated fuzzy tree in matlab, see the example create aggregated fis tree on the reference page.

variation on aggregated structure

in a variation of the aggregated structure known as parallel structure [1], the outputs of the lowest-level fuzzy systems are directly summed to generate the final output value. the following figure shows an example of a parallel fuzzy tree, where outputs of fis1 and fis2 are summed to produce the final output.

the fistree object does not provide the summing node σ. therefore, you must add a custom aggregation method to evaluate a parallel fuzzy tree. for an example, see the "create and evaluate parallel fis tree" example on the reference page.

cascaded or combined structure

a cascaded structure, also known as combined structure, combines both incremental and aggregated structures to construct a fuzzy tree. this structure is suitable for a system that includes both correlated and uncorrelated inputs. the tree groups the correlated inputs in an aggregated structure, and adds uncorrelated inputs in an incremental structure. the following figure shows an example of a cascaded tree structure, where the first four inputs are grouped pairwise in an aggregated structure and the fifth input is added in an incremental structure.

for example, consider the robot navigation task discussed in aggregated structure. suppose that task includes another input, the previous heading of the robot, taken into account to prevent large changes in the robot heading. you can add this input using the incremental structure of the following diagram.

for an example that illustrates creating an aggregated fuzzy tree in matlab, see the "create cascaded fis tree" example on the reference page.