Using Linguistic Data Summarization in the study of creep data for the design of new steels

Carlos Alberto Donis Díaz*, Rafael Bello Pérez, Eduardo Valencia Morales Department of Computer Science

Universidad Central “Marta Abreu” de Las Villas Santa Clara, Cuba

* cadonis@uclv.edu.cu

Abstract— A procedure for the design of new creep resistant ferritic steels that involves a large systematic search of combinations of parameters using a neural network model, was proposed in a paper published few years ago. In the present work we study the effectiveness of the Linguistic Data Summarization technique to be used as a tool to discover a credible and useful creep behavior in a way that it can be used as a guide in the mentioned search. Experiments are performed similar to those discussed in the paper mentioned in order to make an effective comparison of the behavior of the creep. We propose the use of an indicator that measures the degree of representativeness of the linguistic terms for the summarizer in our experiments context. As a result, the effectiveness of the Linguistic Data Summarization to discover hidden creep behavior stored in creep data and the usefulness of the representativeness indicator was confirmed.

Keywords: Linguistic Data Summarization; creep rupture stress

I. INTRODUCTION The creep rupture stress (creep) is one of the most

important mechanical properties considered in the design of new steels. It measures the stress level in which the steel structure fails when it is exposed to quite aggressive conditions over time periods as long as 30 years.

The basic principles of alloy design for creep resistance are well established and well founded on experience. It is nevertheless difficult to express the design process quantitatively given the large number of interacting variables.

Several modeling techniques have been developed to predict the behavior of creep in the design of new alloys [1]. Papers proposing models that use machine learning’s techniques like Artificial Neural Networks [2-4], Gaussian Processes [5] and Support Vector Machine for Regression [6] have shown the best results.

In these works [2-5] the used methodology involves a large systematic search using the specific model and based in the criteria of an expert. The aim is to search a combination of parameters for the steel-building process which corresponds with a desired creep behavior. We think that the efficiency of this process could be improved with a pre-processing of data that would reduce the search space to desired values of the parameters. In this context, the Linguistics Data Summarization (LDS) emerges as a possible effective technique.

LDS constitutes an interesting and promising approach mean to help decision makers make rational use of (vast) amounts of data that exist in their environment within which they operate. It was proposed by Yager [7] and further extended by Kacprzyk and Yager [8], and Kacprzyk, Yager and Zadro ny [9]. In a posterior work [10], Zadeh proposed the concept of protoform that organizes the mining of linguistics summaries since the user interface’s point of view. Recently, these approaches have been treated in several papers [11-13].

In the study of creep for the design of new steels, the linguistic summaries obtained (by LDS) with a high degree of validity and representing the sought creep characteristics, can be used to determine a desired subset of values in the parameters. This subset can then be refined using other techniques such as those proposed in [2-6].

To use the previous idea it is necessary to validate the hypothesis that LDS is able to adequately describe trends between creep behavior and the parameters used in the design of new steels. This is the main aim of this work. To this end, several experiments were developed to compare the behavior of LDS regarding trends discovered in [2] and reviewed in [4] using a neural network model.

II. THEORETICAL BACKGROUND

A. Linguistic Data Summarization via Fuzzy Logic with linguistic quantifiers

In this paper we consider a simple yet effective and efficient approach to the linguistic summarization of data sets proposed by Yager [7], that was presented in a more advanced, and implementable form by Kacprzyk and Yager [8], and Kacprzyk, Yager and Zadro ny [9]. We have: (1) Y={y1, . . . , yn} is a set of objects (records) in a database, e.g., the set of workers, and (2) A = {A1, . . .,Am} is a set of attributes characterizing objects from Y, e.g., salary, age, etc. in a database of workers, and Aj(yi) denotes a value of attribute Aj for object yi.

A linguistic summary of a data set D consists of: • a summarizer S, i.e. an attribute together with a

linguistic value (fuzzy predicate) defined on the domain of attribute Aj (e.g. ‘low salary’ for attribute ‘salary’);

• a quantity in agreement Q, i.e. a linguistic quantifier (e.g. most);

• truth (validity) T of the summary, i.e. a number from the interval [0, 1] assessing the truth (validity) of the

160978-1-4577-1676-8/11/$26.00 c©2011 IEEE

summary (e.g. 0.7); usually, only summaries with a high value of T are interesting;

• optionally, a qualifier R, i.e. another attribute together with a linguistic value (fuzzy predicate) defined on the domain of attribute Ak determining a (fuzzy subset) of Y (e.g. ‘young’ for attribute ‘age’).

Thus, linguistic summaries may be exemplified by

T (most of employees earn low salary) = 0.7 (1)

T (most of young employees earn low salary) = 0.7 (2)

and their foundation is Zadeh’s [14] linguistically quantified proposition corresponding to either, for (1) and (2):

Qy’s are S (3)

QRy’s are S (4)

The T, i.e., the truth value of (3) or (4) may be calculated by using either original Zadeh’s calculus of linguistically quantified statements [14], or other interpretations of linguistic quantifiers. Using the first, a (proportional, nondecreasing) linguistic quantifier Q is assumed to be a fuzzy set in [0, 1] and the values of T are calculated as

(5)

In 2002, Zadeh [10] introduced the concept of a protoform defined as a more or less abstract prototype (template) of a linguistically quantified proposition. The most abstract protoform correspond to (3) and (4), while (1) and (2) are examples of fully instantiated protoforms. Thus, protoforms form a hierarchy, where higher/lower levels correspond to more/less abstract protoforms. Going down this hierarchy one has to instantiate particular components of (3) and (4), i.e., Q, and S and R. A protoform may provide a guiding paradigm for a user interface for the mining of linguistic summaries. In Table I basic types of protoforms/linguistic summaries are shown, of a more and more abstract form. Each of fuzzy predicates S and R may be defined by listing its atomic fuzzy predicates (pairs of “attribute/linguistic value”) and structure, i.e., how these atomic predicates are combined.

TABLE I. CLASSIFICATION OF PROTOFORMS/LINGUISTIC SUMMARIES

Type Protoform Given Sought 0 QRy’s are S All Validity T 1 Qy’s are S S Q 2 QRy’s are S S and R Q 3 Qy’s are S Q and structure of S Linguistic values in S�4 QRy’s are S Q, R and structure of S Linguistic values in S�5 QRy’s are S Nothing S, R and Q

B. Creep prediction for the design of new alloys 1) Designing creep resistant ferritic steels As mentioned, there have been several studies in the

design of creep resistant ferritic steels using machine learning techniques. The present work will be based in the study proposed by F. Brun in [2] because their proposals have been tested recently in practice and the results reported in [4], so that we have a way of verifying our study.

F. Brun proposed the use of a neural network to predict the behavior of the creep rupture stress in relation to 37 input variables.

After several tests with different topologies of the network a committee of models (neural networks) was obtained. This model permits the estimation of the uncertainty of the prediction related to possible errors in the data or to little amount of it.

Then, the committee was used to propose some novel alloys with creep rupture stress properties which where predicted to be better than the existing steels creep properties. The procedure involved a systematic search of the input space focusing on directions which led to a maximization of certainty. The investigated trends where too voluminous and the calculations involved the modification of the standard 10CrMoW steel thus all the results where compared against that alloy. The attempts led to the design of two novel steels: steel A and steel B.

During the process changes in the concentration of various chemical components like cobalt (Co), chromium (Cr), molybdenum (Mo), wolfram (W), silicon (Si), aluminium (Al), nickel (Ni), boron (B) and manganese (Mn) were made. The normalizing temperature was modified too. Table II shows the input values used in the experiments for the parameters of the steels.

2) Database The database used in the present study is the same

referred in [2] and can be downloaded from [15]. The database compiled from the published literature consisted of 2066 combinations of creep rupture stress and, originally, 30 inputs including the time to rupture (creep rupture time), working temperature (temperature), chemical composition, and heat treatment.

TABLE II. INPUT PARAMETERS FOR THE 10CRMOW STEEL AND DESIGN PARAMETERS FOR STEEL A

Parameter 10CrMoW Steel A Parameter 10CrMoW Steel A Normalizing temperature

1338 1473 Ni, wt % 0.32 0

Tempering temperature

1043 1073 Cu, wt % 0.86 0

Annealing temperature

1013 1013 V, wt % 0.21 0.21

C, wt % 0.12 0.12 Nb, wt % 0.01 0.01Si, wt % 0.05 0 N, wt % 0.064 0.064

Mn, wt % 0.64 0.48 Al, wt % 0.022 0P, wt % 0.016 0.0016 B, wt % 0.002 0.008S, wt % 0.001 0.0001 Co, wt % 0.015 1.25Cr, wt % 10.61 9 Ta, wt % 0.0003 0.0003Mo, wt % 0.44 0.75 O, wt % 0.01 0.01W, wt % 1.87 3 Re, wt % 0.0003 0.0003

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III. USING LDS IN THE CREEP PREDICTION

A. Configuration of the fuzzy logic features The input variables related to the chemical composition

were modeled as fuzzy variables with seven linguistic terms ("very low", "low", "medium-low", "medium", "medium-high", "high", "very high"). Trapezoidal membership functions uniformly distributed over the universe of discourse were used.

The variables temperature and creep rupture time (crt – expressed as log(crt)) were used as base line in the experiments. Due to this, they were configured with a uniform distribution of the membership functions that correspond, approximately, with the key values used in [2]. Then, for the temperature and crt, six and eight trapezoidal membership functions were used respectively to express its linguistics terms. These were defined as ("very low", "low", "medium-low ", "medium-high", "high", "very high") and ("very short", "short", " short-medium", "medium", "medium-long", "long", "very long", "extra-long") respectively.

For the creep (variable used in the summarizer) a special distribution of the membership functions was made. The values of creep in the dataset are between 18 and 530 but the observed values in the experiments made in [2] are concentrated in the range from 18 to 330 approximately. Taking into account this situation, the fuzzy variable creep was designed using a concentration of eight functions between the values 18 and 330 and another function to cover the rest of the universe (Fig. 1). The linguistics terms were interpreted as: "very low", "low", "medium-low", "medium", "medium-high", "high", "very high", "extra high" and “ideal”.

Figure 1. Creep membership functions.

A fuzzy variable with fifth linguistic terms represented by bell functions uniformly distributed over the interval 0 to 1 (Fig. 2) was used as linguistic quantifier. Its terms were interpreted as: few, some, half, much and most.

Figure 2. Membership functions used for the quantifier.

Finally, the operator and was used as the algebraic product t-norm t(a, b) = a . b

B. Characteristics of summaries and protoforms Due to the interest of analyzing the behavior of creep

against other variables, the used summaries have the following characteristics:

• have the form defined in (4), i.e., don’t have sense the use o summaries of the form established in (3),

• the summarizers only have the variable creep as component,

• the terms in the quantifier are aggregated using only the operator and.

For dealing with the linguistics summaries, a protoform similar (not equal) to the described in Table I as number 4 was used. For future references in this work it will be called creep-protoform. It is defined as: Type Protoform Given Sought

creep-protoform QRy’s are S R and structure of S

Q and linguistic values in S�

The truth value was calculated using the original Zadeh’s calculus of linguistically quantified statements represented in this work by (5).

C. The indicator of representativeness In our study it was necessary the use of an indicator that

helped to discover the most representative term of the fuzzy creep variable given the sets of summaries represented by the creep-protoform.

For example, we may suppose that we want to study the creep behavior in a specific alloy by means of the variable crt. In this case it is necessary: (step 1) to discover the linguistic term that better represent the variable creep (the summarizer) for each linguistic term of the variable crt, (step 2) to do the analysis between each “representative” linguistic term of the variable creep. In the first step, and considering the linguistic term “medium” for the variable crt, the summaries represented in Table III should be taken into account.

TABLE III. SUMMARIES ANALYZED TO SELECT THE MOST REPRESENTATIVE TERM FOR THE VARIABLE CREEP

Qualifier (given)

Summarizer (sought)

Quantifier (sought) T No.

… and crt is medium

and …

creep is high

few 1.000 1 some 0.062 2 half 0.000 3

much 0.000 4 most 0.000 5

creep is medium-high

few 0.000 6 some 0.000 7 half 0.000 8

much 0.062 9 most 1.000 10

creep is medium

few 1.000 11 some 0.062 12 half 0.000 13

much 0.000 14 most 0.000 15

To select the most representative linguistic term for the

variable creep it is necessary to analyze: (1) the value of truth and (2) the linguistic strength of the quantifier (e.g. “most” is linguistically stronger than “much”, “half”, “some”

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or “few”). In the example shown in Table III it is easy to select the term “medium-high” as the most representative one for the variable creep because between all the summaries related to this term (from summary 6 to 10), the summary with the quantifier “most” (summary 10) has the greater truth value (1.0 –the maximum) while for the others terms, the summaries with quantifier “few” are those that have the greater truth value between the respectively related summaries (1 to 5 and 11 to 15). This selection is not so obvious when there are various summaries with similar truth values between the related ones (for each term). In this case it is very useful an indicator that measures the representativeness degree.

In the present work, it is proposed the use of an indicator called representativeness (E). It is calculated as a weighted sum of the summaries truth values

(6)

where n is the amount of linguistics terms of the quantifier, Ti is the truth value of the summary that uses the i-th linguistic term of the quantifier and wi is the weight assigned to the i-th linguistic term of the quantifier. The values used for the weights were 1.26, 1.68, 2.25, 3 and 4 for the linguistics terms “few”, “some”, “half”, “much” and “most” respectively. In our experiments, the minimum value registered for this indicator was 1.364 which was obtained every time that the truth value of 1.0 belonged to the summary with quantifier “few”. When the truth value of 1.0 belonged to the summary with quantifier “most”, the maximum value of 4.186 was obtained.

In the example, for the linguistic term “high” of the variable creep, the value of E can be calculated as: E = 1.26(1.0) + 1.68(0.062) + 2.25(0) + 3(0) + 4(0) = 1.364

The other two calculations of E result in: • E(“medium-high”) = 4.186 and • E(“medium”) = 1.364

As the greater value of E was for the linguistic term “medium-high”, we consider it like the most representative one for the variable creep in relation to the qualifier that includes the linguistic term of “medium” for the variable crt. Using this reasoning for the above mentioned step 2, it is said that for the steel described by the mentioned qualifier, when the crt is “medium” then the creep is “medium-high”.

IV. EXPERIMENTS, RESULTS AND ANALYSIS In the present study the experiments were made to

evaluate the use of LDS as a way to guide the systematic search for combinations of parameters used in the design of novel steels with a desired level of creep by using a committee of neural networks.

The experiments were developed based on the results presented in [2] and reviewed in [4] to check if the trends shown by the neural networks model coincide with those expressed by LDS.

The first experiment is related to the behavior of creep in relation with the temperature and the crt. A total of 450

summaries that comply with the following characterization of the creep-protoform were built:

• Qualifier (given): crt is x and temperature is y and Mn percent is medium and Cr percent is high and W percent is medium and Ni percent is low and Cu percent is very high and Al percent is medium-low and B percent is very low and Tempering temperature is high

• Values used for x: from short-medium to very long • Values used for y: medium-high, high The linguistic terms used in the qualifier for the different

variables are the better matches by the values of the steel 10CrMoW used in [2] (Table II).

Table IV shows the calculations of E for the linguistic terms of creep. Some conclusions arise from its analysis.

TABLE IV. CORE VALUES OF E TO ANALYZE THE BEHAVIOR OF CREEP IN RELATION WITH THE VARIABLES TEMPERATURE AND CRT

Qualifier Summarizer E Temperature is medium-high and

crt is short-medium and … any Naa

Temperature is medium-high and crt is medium and … creep is medium-high 4.186

Temperature is medium-high and crt is medium-long and … creep is medium-high 4.093

Temperature is medium-high and crt is long and … creep is medium 2.555

Temperature is medium-high and crt is very-long and … creep is medium 4.186

Temperature is high and crt is short-medium and … creep is medium-high 4.186

Temperature is high and crt is medium and … creep is medium 3.456

Temperature is high and crt is medium-long and … creep is medium-low 4.079

Temperature is high and crt is long and … creep is medium-low 3.621

Temperature is high and crt is very-long and … any Naa

a. Value not available because the absence of cases in the dataset that correspond with the qualifier

The first conclusion: when the temperature is “medium-high” (first half of Table IV) and the crt moves from “short-medium” to “very long” there is a movement in the creep from “medium-high” to “medium”, i.e. the creep decreases when the crt increases. If we analyze the base of the trapezoidal function (Table V) used to represent the linguistics terms of the variable creep, it can be observed that this linguistic variation in the creep coincides with a variation of the crisp values approximately from 215 to 123 (red line in Fig. 3(a)). This is very similar to the equivalent results shown in [2] (dashed line in Fig. 3a).

TABLE V. PARAMETERS OF THE TRAPEZOIDAL MEMBERSHIP FUNCTION USED FOR TWO LINGUISTICS TERMS OF THE VARIBLE CREEP

Linguistics terms of creep

Parameters of the trapezoidal function Base (left)

Center (left)

Center (right)

Base (right)

Medium 123.043 136.174 162.435 175.565 medium-high 162.435 175.565 201.826 214.956

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The second conclusion: when the temperature moves from “medium-high” to “high” it can be observed a deterioration in the behavior of creep through a movement of crt from “medium” to “long”, i.e., the creep moves from “medium-high” to “medium” (as observed in the previous analysis) and then it moves from “medium” to “medium-low” when the temperature is “high” (as can be observed in the second half of Table IV). These results match with those obtained in [2]. Comparing Fig. 3(a) with Fig. 3(b) it can be observed the mentioned difference in the trend of creep when the terms “medium-high” and “high” are used to refer the temperature. The red line shows, in both cases, the behavior interpreted from the values of the respective trapezoidal membership functions. These lines deal practically between the uncertainties boundaries of values predicted by the neural network model.

Figure 3. Behavior of creep vs crt as shown in [2] (consider only the

10.6Cr type of steel): (a) at temperature=600oC (“medium-high”), (b) at temperature=650oC (“high”).

The following experiments are related to the study of specifics variables in the behavior of creep for the steel 10CrMoW. Like in [2] it will be analyzed the following variables: Cobalt (Co) percent, Nickel (Ni) percent, Aluminium (Al) percent and Normalizing Temperature (NT).

For the experiments, summaries that comply with the following characterization of the creep-protoform were built:

• For all the experiments the first part of the qualifier (fpq) was: crt is long and temperature is medium-high and Cr percent is high

• The complete qualifier was defined as: Qualifiers for the analysis of variable

Co Ni Al NT fpq and Co is x

x: from very low to medium-

low

fpq and Ni is x x: from very

low to medium

fpq and Al is x x: from very low to high

fpq and NT is x x: from very

low to very high

As noted, the qualifiers don’t contain all the variables that describe the steel 10CrMoW. LDS works using only the cases in the dataset and not a potential estimated behavior as does the neural network model. This means that unlike the neural network model where you can use any value as input for the variables, in the LDS model the use of some values for the linguistic terms of the qualifier (a filter) can lead to the recovery of no case to create the summary. That’s why in the present study it was necessary to reduce the “filtering” of the cases needed to build some summaries.

The calculations of E for the analysis of the behavior of creep versus the analyzed variable are presented in Table VI.

TABLE VI. CALCULATIONS OF E FOR VARIABLES CO, NI, AL AND NT

Qualifier Summarizer E For the variable Co fpq and Co is very low creep is medium-high 1.911 fpq and Co is low No summaries availables fpq and Co is medium-low creep is high 2.518 For the variable Ni fpq and Ni is low creep is very-high 2.635 fpq and Ni is medium-low creep is very-high 4.203 fpq and Ni is medium creep is high 2.903 For the variable Al fpq and Al is very low creep is medium 1.905 fpq and Al is low creep is medium-low 4.186 fpq and Al is medium-low creep is medium 1.411 fpq and Al is medium creep is medium 1.405 fpq and Al is medium-high creep is medium 1.913 fpq and Al is high creep is low 1.911 For the variable NT fpq and NT is low creep is medium-low 3.825 fpq and NT is medium-high creep is medium-high 1.857 fpq and NT is very-high creep is high 2.558

For the variable Co, the analysis of values of E shows that the behavior of creep match with the trend reported and explained in [2], i.e., the creep increases from “medium-high” to “high” when the Co increases from “very-low” to “medium-low” values. A comparison between the trend estimated by the neural networks model and that obtained by the analysis of the parameters of the membership function used in LDS, can be observed in Fig. 4(a).

Figure 4. Behavior of creep versus: (a) Co, (b) Ni, (c) Al. The red line

shows the behavior analyzed in the present work using LDS.

Similar analysis for variables Ni, Al and NT, using the values of Table VI and figures Fig. 4(b), Fig. 4(c) and Fig. 5 respectively shown results that matches with those reported in [2].

The last experiment is related to the novel steel A proposed in [2]. This steel was designed through the modification of the parameters of the steel 10CrMoW (see (Table II) and it is expected with a better creep behavior. The quantifier used in the creep-protoform is: crt is x and temperature is medium-high and Mn percent is low and Cr percent is medium-high and W percent is high and Ni percent is very low and Cu percent is very low and Al percent is very low and B percent is low. Where x takes values from “medium” to “very long”.

log(crt, h)

(a) log(crt, h)

(b)

(a)

(b) (c)

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Figure 5. Behavior of creep versus NT. The red line shows the behavior

analyzed in the present work using LDS.

From Table VII it can be concluded that once again, the trend of creep obtained using LDS matches with that estimated by the neural network model.

TABLE VII. CORE CALCULATIONS OF E FOR THE ANALYSIS OF THE BEHAVIOR OF CREEP IN THE NOVEL STEEL A

Qualifier Summarizer E Temperature is medium-high and

crt is medium and … creep is high 4.186

Temperature is medium-high and crt is medium-long and … creep is high 2.539

Temperature is medium-high and crt is long and … creep is medium-high 4.186

Temperature is medium-high and crt is very-long and … creep is medium-high 4.186

Figure 6. Comparison of results related to the trend of creep. Consider

only the values for the steel N (10CrMoW) and the steel A.

Fig. 6 shows the same graph exhibited in [4] and modified here with the addition of the red line. As can be observed the measured trend of creep for steel A lies under the values predicted by the neural network model in [2]. The red line shows a reference to the trend obtained using LDS. This trend lies between the measured values and those estimated by the neural network model. This last observation together with all other results in this paper leads to the conclusion that LDS can discover credible descriptions of the behavior of creep.

V. CONCLUSIONS The following conclusions can arise from the study: (1)

the degree of representativeness is a useful indicator that

allows a proper selection of the most representative linguistic term within the context studied, (2) the parameters of membership functions of the linguistic terms of the summarizer can be used to form a line that serves as a reference for the trend discovered by the LDS technique and (3) the LDS can be an effective technique to discover hidden creep behavior stored in creep data; this is the reason why it can be used to guide the systematic search for combinations of creep variables used in the design of new steels by using the model of neural network proposed in [2].

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[15] http://www.msm.cam.ac.uk/map/data/materials/creeprupt-b.html, last review: 10/06/2011.

log(crt, h)

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