Investigation into hybrid data mining and soft

area, so they are also a source of weakness. Consequently, it is important to determine the failure strength and failure modes of these pinned connections in order to utilize the full potential of composite materials as structural elements. The assessment of stresses around fastener holes is critical for reliable strength evaluations and failure predictions.

COMPOSITES 2005Convention and Trade Show American Composites Manufacturers Association September 28-30, 2005 Columbus, Ohio USA

Investigation into hybrid data mining and soft computing techniques to aid to design of composite joints by S.Shirazi Kia, S.Noroozi. B.Carse, J.Vinney, M.Rabbani Faculty of Computation, Engineering and Mathematical Science (CEMS) University of the West of England (UWE), Bristol, United Kingdom.

Abstract The evaluation and prediction of the failure probability and safety levels of composite components and structures is of extreme importance in structural design and manufacturing. A new application of data mining techniques for predicting the behavior of pin-loaded composite joints is presented. The proposed system consists of combining different data mining and soft computing techniques such as classification and clustering with fuzzy logic. By using these techniques, the relationship between different parameters, such as edge distance and tensile strength of composite joints, is modeled. A classification approach based on fuzzy clustering yielded the best predictive results.

Keywords: C 4.5, classification, composite, fuzzy clustering, pin-loaded hole, tensile stress.

1 Introduction Composite materials have many advantages over conventional metal materials, due to their comparatively high strength to weight and stiffness to weight ratios. For this reason, in advanced engineering fields, composite materials have found extensive applications in structures. The mechanical fastener is a common technology for assembling these structural components. Pin joints (bolted joints) are often selected to join several parts of structures. The reasons are their low cost, simplicity of assembly and disassembly for repair and for their tolerance to environmental effects [16]. However, this type of joints in components incur a weight penalty and often cause a reduction of load capacity of the composite structure due to stress concentration created near the hole

In the literature, several approaches have been used to predict the stresses, failure strengths and failure modes of composite laminates containing pinloaded hole such as analytical [14,19,28], numerical [10,11,12,15,20,21], experimental [2,4,13] methods, starting with Knight [18,24]. During the last three decades, data mining and soft computing techniques have also provided an attractive alternative to accommodate the non-linearity and imprecise information found in the real world for modeling the complex systems. Soft computing is a set of methodologies (like fuzzy logic) that its aim is to exploit the tolerance for imprecision, uncertainty, approximate reasoning and partial truth in order to achieve robustness, low solution cost and close resemblance with human like decision-making [30]. Data mining is also concerned with algorithmic means by which patterns are enumerated from data. Soft computing tools are turning out to be strong candidate for performing data mining tasks efficiently. At present, integrating soft computing and data mining are being available in many papers in various fields. In this study, a novel application of classification and fuzzy clustering is introduced. The objective of this study is to evaluate the effectiveness of classification model to identify joint failure. Secondly, we compare the performance of classification model with the model initialized by fuzzy c-means clustering. Our analysis indicates that the performance of classification model initiated by fuzzy c-means clustering in evaluating potential behavior is slightly superior to the performance of classical classification. This paper is divided into five sections: Section 2 explains the design of C4.5 classification algorithm and fuzzy c-means clustering for prediction. Section 3 defines the problem and describes the data used for predicting the behavior of pin-loaded joints. Section 4 analyzes the results of applied methodologies and finally, Section 5 concludes and summarizes the study.

2 Data Mining and Soft Computing Techniques Data mining is the exploration and analysis of data in order to discover meaningful patterns [6]. It is an interdisciplinary field that combines artificial intelligence, computer science, machine learning, database management, data visualization, mathematics algorithms and statistics. This technology provides different methodologies for decision-making, problem solving, analysis, planning, diagnosis, detection, integration, prevention, learning and innovation [1,5,6,22,23,26]. The two primary goals of data min-

ing tend to be prediction and description. Prediction involves using some variables or fields in the database to predict unknown or future values of other variables of interest. The second goal which leads to descriptive model, describes patterns in existing data which may be used to guide decisions as opposed to making explicit predictions. In this paper we have considered classification as the main methodology for predicting the behavior of composite/aluminum joints. Classification is learning a function that maps (classifies) a data item into one of several predefined classes [29]. In general, it assumes that we have an object s, which may belong to one of a finite set of classes C = { C1 , C 2 ,...., C n }. It normally consists of a training phase and an application phase. In the training phase, the classifier is induced by learning the training set, which is a collection of objects of known observations from each type. In the application phase, the classifier is applied to the test set (a collection of unknown classes), to make a decision on what their classes are likely to be. Tree-based classifiers are one of the most popular predictive techniques for building understandable models because of their tree structure and ability to easily generate rules. Specifically each branch of the tree is a classification question and the leaves of the tree are partitions of the data set with their classification. Among all available classification methods, C4.5, a well known decision tree algorithm, has been utilized in this work.

C4.5 is an inductive algorithm developed by Quinlan [24]. It represents acquired knowledge in the form of a decision tree. The essence of the algorithm is to construct a decision tree from the training data. Each internal node of a tree corresponds to a principal attribute, while each outgoing branch corresponding to a possible range of that attribute. Leaf nodes of the tree represent the class to be assigned to a sample. To build a decision tree from training data, C4.5 employs information theoretically measured based on “gain” and “gain ratio”. Given a training set T of pinloaded joint behavior, each sample has the same structure with 5 dimensions (i.e. the material type, the edge distance, load step, load value and the behavior). For the classification of aluminum joint behavior into levels of “Elastic”, “Plastic” and “Failure”, the training set T can be partitioned into three classes, i.e., E (Elastic level), P (Plastic level) and F (Failure level). Then the information (I) needed to identify the class of an element of T is given by

⎛| A |⎞ |U | ⎛|U |⎞ | A| ⎟⎟ − ⎟⎟ log 2 ⎜⎜ log 2 ⎜⎜ |T | ⎝|T |⎠ |T | ⎝|T |⎠

n

I(Load, T) =

| Ti |

∑ | T | I (T )

(1)

If the training set T was partitioned on the basis of the value of Load (the 4th attribute in data set) into sets T1 , T2 ,..., Tn , then, the information needed to identify

(2)

i

i =1

The information gained on a given attribute is the difference between the information needed to identify an element of T and the information needed to identify an element of T after the value of the attribute has been obtained. Therefore, the information gained on Load is:

Gain (Load, T) = I(T) –I(PC1, T).

(3)

The root of the decision tree is the attribute with the greatest gain. The process is repeated to build the decision tree where each node locates the attribute with the greatest gain among the attributes not yet considered in the path from the root. The gain measure has disadvantageous effects for the attributes with a large number of values. To cope with this problem, the gain ratio was introduced instead of the gain. For example, the gain ratio of Load is defined as: Gain ratio (Load, T) =

2.1 The C4.5 Classification Algorithm

I(T) = −

the class of an element of T can be calculated by the weighted average of I( Ti ) as follows:

n

Split (Load, T) =

Gain( PC1, T ) Split ( PC1, T )

(4)

⎛ | Ti | ⎞ ⎜⎜ ⎟⎟ ⎝ |T |⎠

(5)

| Ti |

∑ | T | log i =1

2

Split (Load, T), in equations 4 and 5, is the information due to the split of T on the basis of the value attribute Load. The decision tree obtained by recursively partitioning a training set, sometimes, may become quite complex with long and uneven paths. To deal with this shortcoming, the decision tree is pruned by replacing a whole sub-tree with a leaf node through an error-based strategy [24].

2.2 Clustering and Fuzzy Models Clustering is a common descriptive task. This technique is essentially the task of partitioning a set of patterns into a number of homogeneous classes (clusters) with respect to a suitable similarity measure. The patterns belong to any one of the clusters are similar and the patterns of different clusters are as dissimilar as possible. The classical clustering is described by a conventional crisp membership function. This function assigns each object to one and only one of the clusters, with a degree of membership equal to one. However, the boundaries between the clusters

are not often well-defined and this description does not reflect the reality. The fuzzy clustering, founded upon fuzzy set theory [30], is meant to deal with not welldefined boundaries between clusters. In fuzzy clustering, the membership function is represented by gradual variation between zero and one. Therefore, this membership indicates how the object is classified into each cluster. This can be advantageous in the boundary region which may not be precisely defined. The most widely used fuzzy clustering algorithm is the fuzzy c-means algorithm (FCM). FCM was originally introduced by Jim Bezdek in 1981 [9] as an improvement on earlier clustering methods. It is a data clustering technique wherein each data point belongs to a cluster to some degree that is specified by a membership grade. This technique provides a method that shows how to group data points that populate some multidimensional space into a specific number of different clusters. FCM is an objective function optimization approach to solve the following problem [8,9]: Minimize J m (U , V ) =

∑∑ u i

m ik

d 2 ( x k ,Vi )

(6)

k

with respect to U = [ u ik ] ∈ R , a fuzzy c-partition of n cn

unlabelled data set X = { x1 , x 2 ,...., x n } ∈ R

pn

and to

V, a set of c fuzzy cluster centers V=( V1 ,.....,Vc )

∈ R pc . The parameter m >1 is used as the fuzziness index. If m=1, then the algorithm is reduced to the hard c-means algorithm [8, 9]. The process starts with an initial guess for the cluster centers ( V0 = {V10 ,...,Vc 0 } ), which are intended to mark the mean location of each cluster. Additionally, every data point is assigned a membership grade for each cluster ( U = [u ik ] ). The initial guess for these cluster t

t

centers is most likely incorrect. By iteratively updating the cluster centers and the membership grades for each data point, the cluster centers are moved to the “right” location within a data set. These iterations are based on minimizing the function defined in equation 6 that represents the distance from any given data point to a cluster centre weighted by that data point’s membership grade. The output of this procedure is a list of cluster centers (V) and a number of membership grades (U) for each data point.

3 Problem statement This study investigates the classification accuracy of C4.5 algorithm together with fuzzy c-means clustering system to predict the behavior of pin-loaded joints. It is desired to find the maximum failure load P that can be applied before the joint fails for each geometry. To construct the model, the study used data sets obtained from experimental tests on pin-loaded composite as well as aluminum joints. A laminate composite (carbon fiber re-

inforced plastic) plate of length L, thickness t and width W with a hole of diameter D with a pin is used. Pin strength is very high compared with that of the composite plate. For this reason the failure of the pin has been neglected. The parametric dimensions of composite plate are shown in Fig. 1. The hole is located along the centerline of the plate at a distance E from one end of the plate. A uniform tensile load P is applied to the plate. The load is parallel to the plate and is symmetric with respect to the centerline. Similar configuration is also applied for aluminum plates. It is desired to find: • •

The load-displacement tensile response for each specimen The ultimate failure load

In the tensile test, different edge distance/diameter (E/D) ratios were considered. Each plate was cut and prepared with a circular hole, centrally placed with respect to the width and at a predetermined distance from the end. The specimens were tested in Instron-8033 testing machine of 75 KN load capacity. All specimens were mounted in the testing machine using a set of test fixture specially designed for this experiment and loaded at a constant crosshead rate (0.5 mm/min) to minimize any catastrophic failure and to allow the time for observation of damage and its progression. For each test, applied load and pin-displacement were continuously recorded from the chart recorder attached to the testing machine. The magnitude of the applied load was measured by using a load cell mounted on the testing machine. The displacement of bottom grip was also recorded during the test. Specimens were tested to final failure. It was observed how failure is affected by geometry, especially the variation of plate edge distance. Similar experiment was repeated with aluminum plates, as well. However, the failure mechanisms for composite materials are generally not similar to those observed in metals, in which the metal exhibits considerable yielding prior to fracture. Yielding does not occur in carbon fiber composites. Therefore, for applying classification method we have considered three levels for the behavior of aluminum, i.e. Elastic, Plastic and Failure, while considering only two levels for composites, Non-failure and Failure levels. A classification algorithm, C4.5 has been applied to model these behaviors of composite and aluminum joints. This model can be used to predict the behavior of a joint under specific amount of load. In order to improve the performance of the above algorithm, fuzzy clustering has been applied for determination of the boundaries between different levels of the behavior. The system is a four input-one output system. The input variables are Material type, Edge distance, Load step and Load value and the Joint behavior at that specific step is taken as the output variable. As mentioned above, there are three possible behaviors for aluminum joints while for composite there are only Non-Failure and Failure levels.

In order to take the material properties into account, two models have been generated one for each material type. In both cases, four different edge distances (i.e. 0.7, 1, 1.5 and 3 cm) have been used for training the systems while they were tested with edge distance 1.25 cm. For each edge distance, three sets of data have been applied. The models usually have high accuracy when describing their own case. However, it might not be appropriate for a different case, even though the cases share similar steps of loading. The last step after developing the models is generating the rules belonging to the model and their evaluation in terms of accuracy and error rate. The models have been implemented using Java under Windows 2000 on a Compaq Workstation for the C4.5 algorithm [25] and MATLAB (6.5.1 release 13, 1997) for fuzzy c-means clustering. These two methodologies have been reviewed in the next section.

4 Evaluations of the Different Methods The evaluation of mentioned methods was performed to check the efficiency of the proposed methods with respect to three issues: (a) prediction performance (b) robustness and (c) initialization capabilities as far as the fuzzy c-means clustering is concerned. To study the first issue, the model’s prediction performance is compared with a new case (edge distance 1.25 cm), for which an actual test has been carried out and its real performance is compared with the predicted performance. The edge distances considered for training were: 0.7 cm, 1 cm, 1.5 cm and 3 cm, while for testing the edge distance 1.25 cm was used. Table 1 and 2 show the distribution of training sets for both aluminum and composite joints and table 3 indicates the prediction accuracy for edge distance 1.25 for both cases. It can be seen that decreasing the size of training set so that it includes only one set of data for each edge distance instead of three, has diverse effects on aluminum and composite joints. Although it slightly improves the prediction performance for composite joints, it causes abrupt decrease in aluminum prediction performance. Fig. 2 and 3, also, show the prediction performance of initial tree classifier for each edge distance, while it was trained with the rest of data sets (called Dataset 1 and 2). These two graphs indicate that the initial tree classifier may lead to about 95% accuracy in prediction regarding the composite joints while for aluminum joints it is just above 80%. The robustness is taken into account using the holdout cross-validation method, according to which the data set is separated into two sets called training data set and test data set. The training data set is issued to build the tree classifier model and the test data set to validate its performance. Then, the robustness is checked by the model prediction performance on the test data set. Regarding the robustness issue, table 5 and 6 summarize the root mean square errors (RMSE) as well as the predicted accuracy for both material types. As it was expected, the accuracy of the system is better when it is

trained and tested against the same data rather than being tested with another data set. Finally, the initialization capabilities, as far as the fuzzy c-means clustering is concerned, are checked by comparing the model predictions before and after the c-clustering tuning procedure is applied. The model obtained before the implementation of the cmeans clustering is called initial tree classifier, while the model obtained after the application, is called final fuzzy model. In Fig. 4 and 5, initialization capability of c-means clustering has been depicted. It is shown that the final fuzzy model performs much better on aluminum data rather than composites.

5 Discussions and Conclusion This work has explained the application of classification and fuzzy logic in predicting the effect of geometric factors and material parameters on failure loads of a pin-loaded carbon fiber composite as well as aluminum joints. The distance from the upper edge of the plate to the hole canter was taken as the geometric factor. The pin-loaded plates used in this study were made of two different materials: carbon fiber composite and aluminum. The main scope was to explore the benefits that could be earned from using a fuzzy c-means based classifier instead of using the classical classifier. The performance of the prediction accuracy of the initial tree classifier, obtained from composite joints, verified that the classical classification tree is able to model the mentioned problem with around 95% accuracy. However, for aluminum joints, the comparison of the initial tree classifier with the final fuzzy model indicated that the application of the fuzzy cmeans provided much better results in terms of modeling and accuracy. The performances of two methods for classification of the behavior of pin-loaded joint were compared using five different edge distances. For aluminum joints, it was found that the accuracy of the classical tree is around 80 % which is improved by almost 2 % in final fuzzy model. There was an exception for edge distance 1 cm, in which the two models performed similarly. For the composite joint, the initial tree classifier created a model with average 95% accuracy, while some inaccuracy could be observed in boundary edge distances, which are 0.7 cm and 3 cm. As it was expected, fuzzy clustering could not improve much the composite model. The reason is due to the behavior of composite that does not yield. This kind of behavior makes the data not-overlapping and therefore the boundaries of data are actually crisp rather than fuzzy.

References [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

S.S.R. Abidi, “Knowledge management in healthcare: towards knowledge-driven decisionsupport services”, International Journal of Medical Informatics, 63, 5-18, 2001. S.F.M. Abd-El-Naby, L. Hollaway, “The experimental behavior of bolted joints in pultruded glass/polyester material. Part 1: single-bolt joints”, Journal of Composite Materials, 24, 531-538, 1993. R. Agrawal, T. Imielinski, A. Swami, “Database mining: a performance perspective”, IEEE Transactions on Knowledge and Data Engineering, 5 (6), 914-925, 1993. A. Aktas, M.H. Dirikolu, “The effect of stacking sequence of carbon epoxy composite laminates on pinned-joint strength”, Composite Structures, 62, 107-111, 2003. S.S. Anand, D.A. Bell, J.G. Hughes, “EDM; a general framework for data mining based on evidence theory”, Data and Knowledge Engineering, 18, 189-223, 1996. S.S. Anand, A.R. Patrick, J.G. Hughes, “A data mining methodology for cross-sale”, Knowledge-based Systems, 10, 449-461, 1998. M. Berry, G. Linoff, Data Mining Techniques: for Marketing, Sales, and Customer Support, John Wiley & Sons, New York, 1997. J.C.Bezdek, Fuzzy mathematics in pattern classification, Ph.D.Dissertation, Cornell Univ.,Ithaca,NY, 1973. J.C.Bezdek, “A convergence theorem for the fuzzy ISODATA clustering algorithms”, IEEE Trans.Pattern Anal..Machine Intell.PAMI, 2 (1), 1 –8, 1981. P.P. Camenho, F.L. Matthews, “A progressive damage model for mechanically fastened joints in composite laminates”, Journal of Composite Materials, 33, 2248-2280, 1999. F.K. Chang, R.A. Scott, G.S. Springer, “Strength of mechanically fastened composite joints”, Journal of Composite Materials, 16, 470-494, 1982. F.K. Chang, “The effect of pin load distribution on the strength of pin loaded holes in laminated composites”, Journal of Composite Materials, 20, 401-408, 1986. T.A. Collings, “The strength of bolted joints in multi-directional CFRP laminates”, Journal of Composites, 43-45, 1997. T. DeJong, “Stresses around pin-loaded holes in elastically orthotropic or isotropic plates”, Journal of Composite Materials, 11, 313-331, 1977. F. Hassani, M.M. Shokrieh, L.B. Lessard, A fully non-linear 3D constitutive relationship for the stress analysis of a pin-loaded composite laminate”, Composites Science and Technology, 62, 429-439, 2002. B.M. Icten, R. Karakuzu, “Progressive failure analysis of pin-loaded carbon-epoxy woven composite plates”, Composites Science and Technology, 62, 1259-1271, 2002.

[17]

[18]

[19] [20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

A.K. Jain, R.C. Dubes, Algorithms for Clustering Data, Englewood Cliffs, N.J, Prentice-Hall, 1988. R.C. Knight, “Action of a rivet in a plate of finite breadth”, Philosophical Magazine Series, 19, 517-540, 1935. S.G. Lekhnitskii, “Anisotropic plates”, Cordon and Breach, 1998 L.B. Lessard, M.M. Shokrieh, “Two dimensional modelling of composite pinned-joint failure”, Journal of Composite Materials, 29, 671-697, 1995. C.C. Lin, C.H. Lin, “Stresses around pinloaded hole in composite laminates using direct boundary element method”, International Journal of Solid Structures, 36, 763783, 1999. F.Y. Lin, S. McClean, “A data mining approach to the prediction of corporate failure”, Knowledge-based Systems, 14, 189195, 2001. H.R. Nemati, D.M. Steiger, L.S. Iyer, R.T. Herschel, “Knowledge warehouse: an architectural integration of knowledge management, decision support, artificial intelligence and data warehousing”, Decision Support Systems, 33, 143-161, 2002. E. Persson, E. Madenci, “Composite laminates with elliptical pin-loaded holes”, Engineering Fracture Mechanics, 61, 279-295, 1998. J.R. Quinlan, .5: Programs for Machine Learning. Morgan Kaufman Publishers, San Mateo, California, 1993. M.J. Shaw, C. Subramaniam, G.W. Tan, M.E. Welge, “Knowledge management and data mining for marketing”, Decision Support Systems, 31, 127-137, 2001. B. Thuraisingham, Data Mining: technologies, techniques, tools and trends, CRC Press, 1999. J.T. Wang, C.G. Lotts, D.D. Davis, “Analysis of bolt-loaded elliptical holes in laminated composite joints”, Journal of reinforced plastics and composites, 12, 128-138, 1993. S.I. Weiss, C. Kulilowski, Computer Systems That Learn: Classification and Prediction Methods from Statistics, Neural Networks, Machine Learning and Expert Systems, San Francisco, Calif., Morgan Kaufman Publisher, 1991. L.A. Zadeh, “Fuzzy sets”, Inform.Control, 8, 338 –353, 1965.

100

Accuracy (%)

80 60

Initial Tree Classifier Final Fuzzy Model

40 20 0 0.7

1

1.5

3

Edge Distance

Figure -1 Geometry of Specimen with Pin-loaded Hole

Figure -4 Performance comparison of tree classifier and final fuzzy -model for different edge distances of composite joint

120

Accuracy (%)

100 80 Dataset 1

60

Dataset 2

40 20

100

0 0.5

1

1.5

2

2.5

3

3.5

80

Edge Distance (cm)

Figure -2 Prediction performance for different edge distances of aluminum joint

Accuracy (%)

0

60

Initial Tree Classifier Final Fuzzy Model

40 20 0 0.7

1

1.5

3

Edge Distance (cm) 120

Accuracy (%)

100 80 Dataset 1

60

Dataset 2

40 20 0 0

0.5

1

1.5

2

2.5

3

3.5

Edge Distance

Figure -3 Prediction performance for different edge distances of composite joint

Figure- 5 Performance comparison of tree classifier and final fuzzy -model for different edge distances of aluminum joint

Table -1 Training sets distribution for aluminum samples Training Set Total samples Elastic Plastic Failure

1 (0.71) 128

2 (0.72) 128

3 (0.73) 256

4 (1-1)

5 (1-2)

7 (1.52) 256

8 (1.53) 512

9 (3-1)

10 (3-2)

11 (3-3)

512

6 (1.51) 256

256

1024

1024

512

32 69 27

33 74 21

35 92 129

41 111 104

44 118 350

51 41 164

54 180 22

53 169 290

58 206 760

58 351 615

57 343 112

Table -2 Training sets distribution for composite samples Training Set

1 (0.7-1)

2 (0.7-2)

3 (0.7-3)

4 (1-1)

5 (1-2)

Total samples

256

128

256

61

27 229

31 97

31 225

47 14

NoFailure Failure

7 (1.52) 256

8 (1.53) 512

9 (3-1)

10 (3-2)

11 (3-3)

48

6 (1.51) 256

356

590

512

47 1

67 189

64 192

78 434

355 1

363 227

99 413

Table -3 Prediction accuracy for Aluminum and Composite Joints Testing Set No of Training set

1.25 Composite Joint 3231

1419

4864

2304

No of Testing set

256

256

256

256

Accuracy (%)

98.04

92.18

43.35

99.60

1.25 Aluminum Joint

Table -4 Prediction performance for testing sets Testing Set Composite Joint Aluminum Joint

1 (0.71) 93.75

2 (0.72) 90.62

3 (0.73) 62.5

4 (1-1)

5 (1-2)

7 (1.52) 93.75

8 (1.53) 96.87

9 (3-1)

10 (3-2)

11 (3-3)

97.91

6 (1.51) 94.92

95.50

27.80

55.08

98.43

78.12

74.21

80.07

75.39

85.15

84.37

89.84

93.16

94.43

80.17

62.30

Table - 5 Cross-validation performance for aluminum joints Training Set

1 (0.71)

2 (0.72)

3 (0.73)

4 (1-1)

5 (1-2)

6 (1.51)

7 (1.52)

8 (1.53)

9 (3-1)

10 (3-2)

11 (3-3)

RMSE

0.102

0.102

0.072

0.072

0.051

0.072

0.088

0.051

0.036

0.044

0.051

Accuracy (%)

98.43

98.43

99.21

99.21

99.60

99.21

98.82

99.60

99.80

99.70

99.60

Table - 6 Cross-validation performance for composite joints Training Set

1 (0.71)

2 (0.72)

3 (0.73)

4 (1-1)

5 (1-2)

6 (1.51)

7 (1.52)

8 (1.53)

9 (3-1)

10 (3-2)

11 (3-3)

RMSE

0.063

0.088

0.063

0.128

0.146

0.063

0.088

0.044

0.053

0.041

0.044

Accuracy (%)

99.60

99.21

99.60

98.36

97.91

99.60

99.21

99.80

99.71

99.83

99.80