Jerry Mendel - Academia.edu (original) (raw)
Papers by Jerry Mendel
arXiv (Cornell University), Jul 2, 2019
Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They h... more Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They have demonstrated superior performance in many applications. However, the operation of an IT2 fuzzy system is more complex than that of its type-1 counterpart. There are many questions to be answered in designing an IT2 fuzzy system: Should singleton or non-singleton fuzzifier be used? How many membership functions (MFs) should be used for each input? Should Gaussian or piecewise linear MFs be used? Should Mamdani or Takagi-Sugeno-Kang (TSK) inference be used? Should minimum or product t-norm be used? Should type-reduction be used or not? How to optimize the IT2 fuzzy system? These questions may look overwhelming and confusing to IT2 beginners. In this paper we recommend some representative starting choices for an IT2 fuzzy system design, which hopefully will make IT2 fuzzy systems more accessible to IT2 fuzzy system designers.
arXiv (Cornell University), Jun 1, 2019
There have been different strategies to improve the performance of a machine learning model, e.g.... more There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This paper proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: 1) train an initial global model using all training data; 2) identify from the initial global model the patches which contribute the most to the learning error, and train a (local) patch model for each such patch; and, 3) update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on 1D/2D/3D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.
Granular computing, Dec 18, 2015
This article compares three methods [Interval Approach (IA), Enhanced Interval Approach (EIA) and... more This article compares three methods [Interval Approach (IA), Enhanced Interval Approach (EIA) and Hao-Mendel Approach (HMA)] for estimating (synthesizing) an interval type-2 fuzzy set (IT2 FS) model for a word, beginning with data that are collected from a group of subjects, or from a single subject. It summarizes the stages for each of the methods in tables so it is possible to compare the steps of each stage side-by-side. It also demonstrates, by means of an example of three words, that using more information contained in the collected data intervals is equivalent to reducing the uncertainty in the IT2 FS model. It recommends the HMA because it uses more information contained in the collected data intervals than does the IA or the EIA, and because it is the only method to-date that leads to normal IT2 FSs. Such fuzzy sets are easier to compute with than are non-normal IT2 FSs.
IEEE Transactions on Acoustics, Speech, and Signal Processing, Jul 1, 1990
Several algorithms are developed to estimate the parameters of a causal nonminimum phase ARMA(p, ... more Several algorithms are developed to estimate the parameters of a causal nonminimum phase ARMA(p, q) system which is excited by an unobservable independent identically distributed (i.i.d.) non-Gaussian process; the output is contaminated by additive colored Gaussian noise of unknown power spectral density. First we present a fiindamental result pertaining to the identifiability of AR parameters, based on the Yule-Walker type equations drawn from a (specific) set of (p + 1) 1-D slices of the kth (k > 2) order output cumulant. Next, we develop several MA parameter estimation algorithms: one method uses q 1-D slices of the output cumulant; a second method uses only two 1-D cumulant slices. These methods do not involve computation of the residual (i.e., AR compensated) time series or polynomial factorization. Multidimensional versions of the various algorithms are also presented. A simulation study demonstrating the effectiveness of our algorithms is included.
IEEE Transactions on Automatic Control, 1972
Copyright© 1970, by Academic Press, Inc. all rights reserved. no part of this book may be reprodu... more Copyright© 1970, by Academic Press, Inc. all rights reserved. no part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without written permission from the publishers. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York ...
IEEE Transactions on Fuzzy Systems, Sep 1, 2020
There have been different strategies to improve the performance of a machine learning model, e.g.... more There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This article proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: first, train an initial global model using all training data; second, identify from the initial global model the patches that contribute the most to the learning error, and train a (local) patch model for each such patch; and, third, update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on one-dimensional (1-D)/2-D/3-D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.
SpringerReference, Aug 29, 2011
ABSTRACT
IEEE Transactions on Geoscience and Remote Sensing, 1983
ABSTRACT
This chapter focuses first on what exactly “design of a type-1 fuzzy system” means, and then prov... more This chapter focuses first on what exactly “design of a type-1 fuzzy system” means, and then provides a tabular way for making the choices that are needed in order to fully specify a type-1 fuzzy system, and introduces two approaches to design, the partially dependent approach and the totally independent approach. It then describes six design methods for designing a type-1 fuzzy system, namely: one-pass, least squares, derivative-based, SVD-QR, derivative-free and iterative. It then introduces and covers three case studies (forecasting of time series, knowledge mining using surveys, and fuzzy logic control, all of which are reexamined in Chap. 10), as well as the applications of forecasting of compressed video traffic, and rule-based classification of video traffic. Twelve examples are used to illustrate the chapter’s important concepts.
Type-reduction in a type-2 fuzzy logic system (FLS) is an “extended” version of the defuzzificati... more Type-reduction in a type-2 fuzzy logic system (FLS) is an “extended” version of the defuzzification operation in a type-1 FLS. In this paper, we briefly review the structure of a type-2 FLS and describe type-reduction in detail. We focus on a center-of-sets type-reducer, and provide some examples to illustrate it. We also provide some practical approximations to type-reduction computations for certain type-2 membership functions.
IEEE Transactions on Automatic Control, 1967
IEEE Transactions on Automatic Control, 1977
: This paper develops time-domain state space models for lossless layered media which are describ... more : This paper develops time-domain state space models for lossless layered media which are described by the wave equation and boundary conditions. Our models are for non-equal one-way travel times; hence, they are more general than existing models of layered media which are usually for layers of equal one-way travel times. Full state models, which involve 2K states for a K-layer media system, as well as half-state models, which involve only K states are developed and related. Certain transfer functions, which appear in the geophysics literature in connection with models of layered media with equal travel times, are generalized to the situation of non equal travel times. Our state space models represent a new class of equations, causal functional equations, some of whose properties and approaches to simulation are discussed.
IEEE Transactions on Fuzzy Systems, 2021
This article is about explainable artificial intelligence (XAI) for rule-based fuzzy systems [tha... more This article is about explainable artificial intelligence (XAI) for rule-based fuzzy systems [that can be expressed generically, as <inline-formula><tex-math notation="LaTeX">$y({{\bf x}}) = f({{\bf x}})$</tex-math></inline-formula>]. It explains why it is <italic>not valid</italic> to explain the output of Mamdani or Takagi–Sugeno–Kang rule-based fuzzy systems using IF-THEN rules, and why it <italic>is valid</italic> to explain the output of such rule-based fuzzy systems as an <italic>association</italic> of the compound antecedents of a small subset of the original larger set of rules, using a phrase such as “these linguistic antecedents are <italic>symptomatic</italic> of this output.” Importantly, it provides a novel multi-step approach to obtain such a small subset of rules for three kinds of fuzzy systems, and illustrates it by means of a very comprehensive example. It also explains why the choice for antecedent membership function shapes may be more critical for XAI than before XAI, why linguistic approximation and similarity are essential for XAI, and, it provides a way to estimate the quality of the explanations.
2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2016
This paper provides a mathematical analysis that shows how the crisp output of an IT2 FLS that is... more This paper provides a mathematical analysis that shows how the crisp output of an IT2 FLS that is obtained by using the Begian-Melek-Mendel (BMM) formula compares to the one obtained by using center-of-sets type-reduction followed by defuzzification (COS TR + D). This is made possible by reformulating the structural solutions of the two optimization problems that are associated with COS TR, and then expanding each of them using a Maclaurin series expansion. As a result of doing this, we show that BMM is the zero-order approximation to COS TR + D. Additionally, by retaining the zero-order and first-order terms from the Maclaurin series expansions, we provide a new Enhanced BMM, one that is non-iterative, has a closed form and is much faster than using the EKM algorithms for COS TR. Although the Enhanced BMM formula is slower than BMM, we demonstrate, by means of extensive simulations, that it is from 5% to 50% more accurate than is BMM for achieving the same numerical solution that is obtained from COS TR + D; and, it is at least 94% faster than when EKM is used for COS TR +D, which makes the Extended BMM a very strong candidate for use in real time applications of IT2 FLSs.
International Journal of Intelligent Systems, 2018
Atanassov's intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembe... more Atanassov's intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembership function, and a hesitancy function, is a generalization of a fuzzy set. Various aggregation operators are defined for AIFSs to deal with multicriteria decision-making problems in which there exists a prioritization of criteria. However, these existing intuitionistic fuzzy prioritized aggregation operators are not monotone with respect to the total order on Atanassov's intuitionistic fuzzy values (AIFVs), which is undesirable. We propose an intuitionistic fuzzy prioritized arithmetic mean based on the Łukasiewicz triangular norm, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a consumer selects a car to illustrate the validity and applicability of the proposed method aggregation operator.
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2019
Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a non-memb... more Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a non-membership function, and a hesitancy function, is a generalization of a fuzzy set. There are various intuitionistic fuzzy hybrid weighted aggregation operators to deal with multi-attribute decision making problems which consider the importance degrees of the arguments and their ordered positions simultaneously. However, these existing hybrid weighed aggregation operators are not monotone with respect to the total order on intuitionistic fuzzy values (AIFVs), which is undesirable. Based on the Łukasiewicz triangular norm, we propose an intuitionistic fuzzy hybrid weighted arithmetic mean, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a company intends to select a project manager to illustrate the validity and applicability of the proposed aggregation operator. Moreover, we extend this kind of hybr...
IFAC Proceedings Volumes, 1981
Thi s paper surve ys appl ications of state variable techn o l ogy i n the area s of mo del i ng ... more Thi s paper surve ys appl ications of state variable techn o l ogy i n the area s of mo del i ng and signal processing for problems in reflecti o n se i smo logy. Forward and inver s e problems are described, and, wherever po ss i b le, new theoret i cal pr o blems wh ich ha ve been spawned by our quest to appl y s tat e vari ab l e method s t o reflection s ei smology , are pointed out.
SPE Western Regional Meeting, 2015
This paper presents a new method for forecasting post-fracturing responses in a tight oil reservo... more This paper presents a new method for forecasting post-fracturing responses in a tight oil reservoir using historical hydraulic fracturing data. The methodology is based on a nonlinear regression method called Variable Structure Regression (VSR). Data for 80 frac jobs were used to calibrate and predict responses. Information per fracturing job included feet of perforation, number of perforations per stage, number of stages, pad volume, slurry volume and sand volume. Information for a set of hydraulically fractured vertical wells was used to test the proposed technique. The VSR method establishes the optimal nonlinear dependencies of variables to predict the cumulative liquid production after fracturing. This method determines the relationships between fracturing parameters and post-fracturing productions in the form of linguistic terms, thereby providing a physical understanding of the process. The exact mathematical structure of these linguistic terms and the number of terms are est...
arXiv (Cornell University), Jul 2, 2019
Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They h... more Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They have demonstrated superior performance in many applications. However, the operation of an IT2 fuzzy system is more complex than that of its type-1 counterpart. There are many questions to be answered in designing an IT2 fuzzy system: Should singleton or non-singleton fuzzifier be used? How many membership functions (MFs) should be used for each input? Should Gaussian or piecewise linear MFs be used? Should Mamdani or Takagi-Sugeno-Kang (TSK) inference be used? Should minimum or product t-norm be used? Should type-reduction be used or not? How to optimize the IT2 fuzzy system? These questions may look overwhelming and confusing to IT2 beginners. In this paper we recommend some representative starting choices for an IT2 fuzzy system design, which hopefully will make IT2 fuzzy systems more accessible to IT2 fuzzy system designers.
arXiv (Cornell University), Jun 1, 2019
There have been different strategies to improve the performance of a machine learning model, e.g.... more There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This paper proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: 1) train an initial global model using all training data; 2) identify from the initial global model the patches which contribute the most to the learning error, and train a (local) patch model for each such patch; and, 3) update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on 1D/2D/3D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.
Granular computing, Dec 18, 2015
This article compares three methods [Interval Approach (IA), Enhanced Interval Approach (EIA) and... more This article compares three methods [Interval Approach (IA), Enhanced Interval Approach (EIA) and Hao-Mendel Approach (HMA)] for estimating (synthesizing) an interval type-2 fuzzy set (IT2 FS) model for a word, beginning with data that are collected from a group of subjects, or from a single subject. It summarizes the stages for each of the methods in tables so it is possible to compare the steps of each stage side-by-side. It also demonstrates, by means of an example of three words, that using more information contained in the collected data intervals is equivalent to reducing the uncertainty in the IT2 FS model. It recommends the HMA because it uses more information contained in the collected data intervals than does the IA or the EIA, and because it is the only method to-date that leads to normal IT2 FSs. Such fuzzy sets are easier to compute with than are non-normal IT2 FSs.
IEEE Transactions on Acoustics, Speech, and Signal Processing, Jul 1, 1990
Several algorithms are developed to estimate the parameters of a causal nonminimum phase ARMA(p, ... more Several algorithms are developed to estimate the parameters of a causal nonminimum phase ARMA(p, q) system which is excited by an unobservable independent identically distributed (i.i.d.) non-Gaussian process; the output is contaminated by additive colored Gaussian noise of unknown power spectral density. First we present a fiindamental result pertaining to the identifiability of AR parameters, based on the Yule-Walker type equations drawn from a (specific) set of (p + 1) 1-D slices of the kth (k > 2) order output cumulant. Next, we develop several MA parameter estimation algorithms: one method uses q 1-D slices of the output cumulant; a second method uses only two 1-D cumulant slices. These methods do not involve computation of the residual (i.e., AR compensated) time series or polynomial factorization. Multidimensional versions of the various algorithms are also presented. A simulation study demonstrating the effectiveness of our algorithms is included.
IEEE Transactions on Automatic Control, 1972
Copyright© 1970, by Academic Press, Inc. all rights reserved. no part of this book may be reprodu... more Copyright© 1970, by Academic Press, Inc. all rights reserved. no part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without written permission from the publishers. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York ...
IEEE Transactions on Fuzzy Systems, Sep 1, 2020
There have been different strategies to improve the performance of a machine learning model, e.g.... more There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This article proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: first, train an initial global model using all training data; second, identify from the initial global model the patches that contribute the most to the learning error, and train a (local) patch model for each such patch; and, third, update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on one-dimensional (1-D)/2-D/3-D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.
SpringerReference, Aug 29, 2011
ABSTRACT
IEEE Transactions on Geoscience and Remote Sensing, 1983
ABSTRACT
This chapter focuses first on what exactly “design of a type-1 fuzzy system” means, and then prov... more This chapter focuses first on what exactly “design of a type-1 fuzzy system” means, and then provides a tabular way for making the choices that are needed in order to fully specify a type-1 fuzzy system, and introduces two approaches to design, the partially dependent approach and the totally independent approach. It then describes six design methods for designing a type-1 fuzzy system, namely: one-pass, least squares, derivative-based, SVD-QR, derivative-free and iterative. It then introduces and covers three case studies (forecasting of time series, knowledge mining using surveys, and fuzzy logic control, all of which are reexamined in Chap. 10), as well as the applications of forecasting of compressed video traffic, and rule-based classification of video traffic. Twelve examples are used to illustrate the chapter’s important concepts.
Type-reduction in a type-2 fuzzy logic system (FLS) is an “extended” version of the defuzzificati... more Type-reduction in a type-2 fuzzy logic system (FLS) is an “extended” version of the defuzzification operation in a type-1 FLS. In this paper, we briefly review the structure of a type-2 FLS and describe type-reduction in detail. We focus on a center-of-sets type-reducer, and provide some examples to illustrate it. We also provide some practical approximations to type-reduction computations for certain type-2 membership functions.
IEEE Transactions on Automatic Control, 1967
IEEE Transactions on Automatic Control, 1977
: This paper develops time-domain state space models for lossless layered media which are describ... more : This paper develops time-domain state space models for lossless layered media which are described by the wave equation and boundary conditions. Our models are for non-equal one-way travel times; hence, they are more general than existing models of layered media which are usually for layers of equal one-way travel times. Full state models, which involve 2K states for a K-layer media system, as well as half-state models, which involve only K states are developed and related. Certain transfer functions, which appear in the geophysics literature in connection with models of layered media with equal travel times, are generalized to the situation of non equal travel times. Our state space models represent a new class of equations, causal functional equations, some of whose properties and approaches to simulation are discussed.
IEEE Transactions on Fuzzy Systems, 2021
This article is about explainable artificial intelligence (XAI) for rule-based fuzzy systems [tha... more This article is about explainable artificial intelligence (XAI) for rule-based fuzzy systems [that can be expressed generically, as <inline-formula><tex-math notation="LaTeX">$y({{\bf x}}) = f({{\bf x}})$</tex-math></inline-formula>]. It explains why it is <italic>not valid</italic> to explain the output of Mamdani or Takagi–Sugeno–Kang rule-based fuzzy systems using IF-THEN rules, and why it <italic>is valid</italic> to explain the output of such rule-based fuzzy systems as an <italic>association</italic> of the compound antecedents of a small subset of the original larger set of rules, using a phrase such as “these linguistic antecedents are <italic>symptomatic</italic> of this output.” Importantly, it provides a novel multi-step approach to obtain such a small subset of rules for three kinds of fuzzy systems, and illustrates it by means of a very comprehensive example. It also explains why the choice for antecedent membership function shapes may be more critical for XAI than before XAI, why linguistic approximation and similarity are essential for XAI, and, it provides a way to estimate the quality of the explanations.
2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2016
This paper provides a mathematical analysis that shows how the crisp output of an IT2 FLS that is... more This paper provides a mathematical analysis that shows how the crisp output of an IT2 FLS that is obtained by using the Begian-Melek-Mendel (BMM) formula compares to the one obtained by using center-of-sets type-reduction followed by defuzzification (COS TR + D). This is made possible by reformulating the structural solutions of the two optimization problems that are associated with COS TR, and then expanding each of them using a Maclaurin series expansion. As a result of doing this, we show that BMM is the zero-order approximation to COS TR + D. Additionally, by retaining the zero-order and first-order terms from the Maclaurin series expansions, we provide a new Enhanced BMM, one that is non-iterative, has a closed form and is much faster than using the EKM algorithms for COS TR. Although the Enhanced BMM formula is slower than BMM, we demonstrate, by means of extensive simulations, that it is from 5% to 50% more accurate than is BMM for achieving the same numerical solution that is obtained from COS TR + D; and, it is at least 94% faster than when EKM is used for COS TR +D, which makes the Extended BMM a very strong candidate for use in real time applications of IT2 FLSs.
International Journal of Intelligent Systems, 2018
Atanassov's intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembe... more Atanassov's intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembership function, and a hesitancy function, is a generalization of a fuzzy set. Various aggregation operators are defined for AIFSs to deal with multicriteria decision-making problems in which there exists a prioritization of criteria. However, these existing intuitionistic fuzzy prioritized aggregation operators are not monotone with respect to the total order on Atanassov's intuitionistic fuzzy values (AIFVs), which is undesirable. We propose an intuitionistic fuzzy prioritized arithmetic mean based on the Łukasiewicz triangular norm, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a consumer selects a car to illustrate the validity and applicability of the proposed method aggregation operator.
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2019
Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a non-memb... more Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a non-membership function, and a hesitancy function, is a generalization of a fuzzy set. There are various intuitionistic fuzzy hybrid weighted aggregation operators to deal with multi-attribute decision making problems which consider the importance degrees of the arguments and their ordered positions simultaneously. However, these existing hybrid weighed aggregation operators are not monotone with respect to the total order on intuitionistic fuzzy values (AIFVs), which is undesirable. Based on the Łukasiewicz triangular norm, we propose an intuitionistic fuzzy hybrid weighted arithmetic mean, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a company intends to select a project manager to illustrate the validity and applicability of the proposed aggregation operator. Moreover, we extend this kind of hybr...
IFAC Proceedings Volumes, 1981
Thi s paper surve ys appl ications of state variable techn o l ogy i n the area s of mo del i ng ... more Thi s paper surve ys appl ications of state variable techn o l ogy i n the area s of mo del i ng and signal processing for problems in reflecti o n se i smo logy. Forward and inver s e problems are described, and, wherever po ss i b le, new theoret i cal pr o blems wh ich ha ve been spawned by our quest to appl y s tat e vari ab l e method s t o reflection s ei smology , are pointed out.
SPE Western Regional Meeting, 2015
This paper presents a new method for forecasting post-fracturing responses in a tight oil reservo... more This paper presents a new method for forecasting post-fracturing responses in a tight oil reservoir using historical hydraulic fracturing data. The methodology is based on a nonlinear regression method called Variable Structure Regression (VSR). Data for 80 frac jobs were used to calibrate and predict responses. Information per fracturing job included feet of perforation, number of perforations per stage, number of stages, pad volume, slurry volume and sand volume. Information for a set of hydraulically fractured vertical wells was used to test the proposed technique. The VSR method establishes the optimal nonlinear dependencies of variables to predict the cumulative liquid production after fracturing. This method determines the relationships between fracturing parameters and post-fracturing productions in the form of linguistic terms, thereby providing a physical understanding of the process. The exact mathematical structure of these linguistic terms and the number of terms are est...