Consider a Takagi - Sugeno - Kanga (TSK) Model consisting of rules of the form : If x 1 is A i1 and ... and x r is A ir THEN y = f i (x 1, x 2, ...., x r) = b i0 + b i1 x1 + b ir xr assume, α i is the matching degree of rule i, then the total output of the model is given by :
Understanding Takagi-Sugeno-Kang (TSK) Fuzzy Model Output
The question asks about the formula for the total output of a Takagi-Sugeno-Kang (TSK) fuzzy model. TSK models are a type of fuzzy inference system known for their clarity and effectiveness in modeling complex systems.
A TSK fuzzy model consists of rules that have a specific structure. As described in the question, each rule \(i\) follows the form:
IF \(x_1\) is \(A_{i1}\) AND ... AND \(x_r\) is \(A_{ir}\)
THEN \(y = f_i(x_1, x_2, \ldots, x_r)\)
Here:
- \(x_1, x_2, \ldots, x_r\) are the input variables.
- \(A_{i1}, \ldots, A_{ir}\) are fuzzy sets defined on the input variables in the antecedent (the "IF" part) of rule \(i\).
- \(f_i(x_1, x_2, \ldots, x_r)\) is the consequent (the "THEN" part) of rule \(i\). In a typical TSK model, this consequent is a linear function of the input variables, given as \(f_i(x_1, x_2, \ldots, x_r) = b_{i0} + b_{i1}x_1 + \ldots + b_{ir}x_r\).
- \(\alpha_i\) is the matching degree (also known as firing strength or weight) of rule \(i\). This value represents how well the input values match the antecedent of rule \(i\). It is typically calculated by applying a T-norm operator (like minimum or product) to the membership degrees of the inputs in the corresponding fuzzy sets \(A_{i1}, \ldots, A_{ir}\).
Calculating the Total Output in a TSK Model
Unlike other fuzzy models (like Mamdani) which use defuzzification methods to convert a fuzzy output set into a single crisp value, the TSK model's output is directly calculated as a weighted average of the consequent functions \(f_i\).
The process involves two main steps after computing the matching degree \(\alpha_i\) for each rule \(i\):
- Calculate the contribution of each rule: This is done by multiplying the matching degree \(\alpha_i\) by the output of the consequent function \(f_i(x_1, \ldots, x_r)\). So, for rule \(i\), the contribution is \(\alpha_i \cdot f_i(x_1, \ldots, x_r)\).
- Calculate the total output: The total output \(y\) is the weighted average of the contributions of all \(L\) rules. The weights are the matching degrees \(\alpha_i\), normalized by the sum of all matching degrees.
The formula for the total output \(y\) is given by:
\[ y = \frac{\sum_{i=1}^L \alpha_i f_i(x_1, x_2, \ldots, x_r)}{\sum_{i=1}^L \alpha_i} \]
This formula represents the weighted sum of the individual rule outputs \(f_i\), where each \(f_i\) is weighted by its corresponding matching degree \(\alpha_i\), and the sum is normalized by the sum of all matching degrees to ensure it's a proper average.
Analyzing the Given Options
Let's compare this formula to the provided options:
Option 1: \(y = \mathop \sum \limits_{i = 1}^L {\alpha _i}{f_i}\left( {{x_1},\;{x_2}, \ldots .,\;{x_r}} \right)\)
This formula represents the weighted sum of the rule outputs but lacks the normalization step (division by the sum of matching degrees). While \(\alpha_i f_i\) gives the contribution of each rule, summing them up without normalization is not the standard TSK output calculation.
Option 2: \(y = \frac{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}{f_i}\left( {{x_1},\;{x_2}, \ldots .,\;{x_r}} \right)}}{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}}}\)
This formula exactly matches the standard TSK output calculation formula: the sum of weighted rule outputs divided by the sum of the weights (matching degrees). This provides the weighted average.
Option 3: \(y = \frac{{\mathop \sum \nolimits_{i = 1}^L {f_i}\left( {{x_1},\;{x_2}, \ldots .,\;{x_r}} \right)}}{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}}}\)
This formula sums up the rule consequent functions \(f_i\) directly, without weighting them by the matching degrees \(\alpha_i\), and then divides by the sum of matching degrees. This is incorrect as it doesn't account for the degree to which each rule fires.
Option 4: \(y = \max[\alpha_i f_i (x_1, x_2,....x_r)]\)
This formula suggests taking the maximum contribution among all rules. This is not how the total output is calculated in a TSK model, which uses an aggregation (summation and averaging) rather than a selection of the maximum.
Based on the standard definition and calculation method for the Takagi-Sugeno-Kang fuzzy model output, Option 2 provides the correct formula.
Summary of TSK Model Output Calculation
| Step | Description | Formula/Concept |
|---|---|---|
| 1 | Fuzzify Inputs | Determine membership degrees of inputs in fuzzy sets. |
| 2 | Evaluate Antecedents | Calculate the matching degree \(\alpha_i\) for each rule \(i\) using T-norm. |
| 3 | Calculate Rule Output | Evaluate the consequent function \(f_i\) for each rule \(i\) using current inputs. |
| 4 | Aggregate Outputs | Calculate the total output \(y\) as a weighted average. |
The aggregation step (Step 4) is specifically where the formula from Option 2 is applied to get the final crisp output \(y\).
Revision Table: Key TSK Concepts
| Concept | Description |
|---|---|
| Rule Structure | IF-THEN rules with fuzzy antecedents and crisp consequent functions (often linear). |
| Antecedent | The "IF" part, combining fuzzy propositions with AND/OR operators. |
| Consequent | The "THEN" part, specifying a function (e.g., \(y = b_{i0} + b_{i1}x_1 + \dots\)). |
| Matching Degree (\(\alpha_i\)) | Strength of activation of rule \(i\), calculated from antecedent evaluation. |
| Total Output | Weighted average of rule consequent functions, weighted by matching degrees. |
Additional Information on TSK Models
TSK fuzzy models are widely used in control systems, system identification, and prediction because of several advantages:
- Interpretability: Although the consequent is a function, the rules still provide some level of interpretability regarding the system's behavior.
- Computational Efficiency: The calculation of the crisp output is often faster than the defuzzification required in Mamdani models.
- Mathematical Tractability: The linear consequent functions make TSK models amenable to analysis and integration with conventional control techniques. They can also be easily trained using methods like least squares estimation or gradient descent.
- Universal Approximators: TSK models are known to be universal approximators, meaning they can approximate any continuous function to any desired degree of accuracy given enough rules.
The choice between a Mamdani model and a TSK model often depends on the application and the desired balance between interpretability and mathematical tractability.
Important Questions from Fuzzy Sets - Teaching
Let A α0 denotes the α-cut of a fuzzy set A at α 0. If α 1 < α 2, then
Consider the following models:
M 1: Mamdani model
M 2: Takagi – Sugeno-Kang model
M 3: Kosko’s additive model (SAM)
Which of the following option contains examples of additive rule model?Consider the following methods:
M 1: mean of maximum
M 2: Centre of area
M 3: Height method
Which of the following is/are defuzzification method(s)?A fuzzy conjunction operator denoted as t(x,y) and fuzzy disjunction operator denoted as s(x,Y) form dual pair if they satisfy the condition:
A fuzzy conjunction operators, t(x, y), and a fuzzy disjunction operator, s(x, y), form a pair if they satisfy:
t(x, y) = 1 – s(1 – x, 1 - y).
If \(t\left( {x,\;y} \right) = \frac{{xy}}{{\left( {x + y - xy} \right)}}\) then s(x, y) is given by
