Using Localized Features for Analyzing College Students ’ Imagination

The analysis of imagination has become popular in recent years because imagination is one of the key components of creativity and innovation. For extracting students’ implicit degrees and thought processes of imagination, we use frequent pattern mining and association rule extraction to localize the features and explain the deep meanings of imagination in the study. By our observations, these two methods may sometimes explore meaningless frequent patterns and rules on a global sparse dataset. In order to eliminate such phenomena when mining with these two methods, we use a localized feature approach called forecast with clustering and integration (FCI) to improve the drawbacks of two methods on a sparse dataset. The approach consists of two strategies. One is clustering and the other is the prediction based on integration from (1) frequent patterns, (2) association rule pruning with correlation, and (3) forecast with linear regression. The former strategy can reduce the number of samples and highlight the features of imagination and the latter strategy can prune meaningless information and predict the trend of scores from imagination input data. Experimental results show both proposed approaches can localize special features, thereby providing supervisors with meaningful information about students’ degrees and thought processes of imagination.


INTRODUCTION
Imagination analysis has become popular in recent decades (Safar and Alkhezzi, 2016;Huang et al, 2017) because imagination is a major driving force of human progress in stimulating creativity and innovation, and a critical factor promoted by education researchers (Eisner, 2001).For boosting imagination, the primary task is comprehending imagination cognition by measuring imagination with an imagination scale.By evaluating the degrees of students' imagination, the implicit information can be obtained, such as the exploration of inspired processes as they answer the scale.To comprehend students' imagination cognition, statistical analyses such as correlation, linear regression, may be used at the initial stage.However, compared to statistical analyses, mining technique may be an approach to support supervisors to obtain the deep meanings behind imagination input data.The objective of this paper thus is to adopt mining techniques with statistical analyses to analyze imagination for widely comprehending and evaluating students' imagination as well as predicting students' imagination degrees and students' imaginative thought processes.The derived results can assist supervisors in various aspects of exploration.
In mining technique, frequent pattern mining (FPM) and association rule exploration (ARE) (Ge & Xia, 2016) are two common mining approaches to show frequent items and frequent rules from a dataset.According to the properties of using frequent items and inferring rules, supervisors can easily reveal various imaginative phenomena from input datasets.However, in some situations, the size of datasets is usually small, and then results in spare datasets.Unfortunately, in the case of sparse input datasets, the results obtained using these two approaches may be usually imprecise and inaccurate because of low relative supports.To avoid the effect of low Liu & Liao / Localized Features for Analyzing Imagination 2 / 16 relative supports, we concentrate on localized feature strategies to evaluate frequent items and, thereby, infer the kernel association rules for enhancement.
The first localized feature strategy is top-K clustering.By top-K clustering, the data can be separated into several groups, and the representative features can be explored from the largest K groups individually.Generally, clustering, an unsupervised learning, can aggregate similar data into several groups and use in constructing the data model from a dataset with unknown categories.The clustered groups can be expressed as the localized features of all input data because clustering can separate global data into several local data.To localize students' imagination, in this study, we use the density-based clustering scheme, Density-Based Spatial Clustering of Applications with Noise (DBSCAN), for aggregating input data.Unlike centroid-based clustering approaches such as kmeans approaches, DBSCAN aggregates input data without restricting circle-shaped clusters because it considers the relationships such as Euclidean distance or cosine similarity between two data points.This feature of relationship results in grouping the similar input data into the same clusters.By contrast, the results obtained using a centroid-based clustering approach may be different due to the locations of the initial centroids.Hence, clustering using DBSCAN can yield more stable results than that with any centroid-based clustering approach.After clustering, researchers can choose the largest K clusters for analysis.These K clusters contain the features that most students have.Therefore, the results of frequent pattern mining and association rules by using top-K clustering reflect most of the degrees and relationships of imagination.
The second localized feature strategy is integrated prediction (IP).This approach combines three different types of information for prediction.The first type of information is the correlated association rules.Generally, the association rules imply frequent relationship among data with different variables.However, in terms of the relationships among imaginative thought processes, the derived association rules cannot offer any information about the variables.For considering the relationships among imaginative thought processes, we employ a correlated concept along with association rules to prune meaningless association rules.
The second type of information is regarding the forecast using linear regression.Generally, association rules cannot cover all possible worlds, which are the combinations of all questions.As a result, we can use linear regression for forecasting the score of a question by the previous highly correlated questions.The third type of information comprises the frequent scores obtained by DBSCAN.The frequent scores are representative guides for displaying imagination.If a question cannot be predicted from the previous two information entries, we use the frequent scores as a substitute.
In order to cultivate the imagination of college students and give the necessary training courses, we focus on measuring freshmen's and sophomores' imagination for analysis by using forecast with clustering and integration (FCI).The imaginative evaluations are performed with an imagination scale proposed by Lin and Tsau (2012).The scale has been validated with high reliability for college students, and contains four components, namely initiation, fluency, flexibility, and originality.Lin and Tsau (2012) considered that imagination belongs to divergent thinking and can be defined seriously as four portions derived from the theoretical basis of IERG's five understandings (Cant, 2012).Therefore, in accordance with the sequential processes of initiation, fluency, flexibility, and originality, the degrees, relationship, and thought processes can be easily revealed if students answer the questions on the scale sequentially.
Our contributions include the following.First, we adapted mining technologies to concentrate on the special features of imagination in a spare dataset.Second, we contributed strategies for co-analyzing correlations and association rules to reduce meaningless relationships.Third, we proposed a model to comprehend the imaginative processes of questions that students answered for sequential input data.Finally, we evaluated students' imagination and predicted the degree of imagination with the aforementioned evaluation of students' imagination by using FCI.
In this paper, we introduce localized feature concepts for analyzing imagination to determine the degrees and relationships of each component of imagination and predict the trend of the students' thought processes in a sparse dataset.The balance of this paper is organized as follows.We first briefly review the literature on imagination, approaches for evaluating imagination, and mining algorithms.Then, an overview of the proposed approach and the design of the scale used herein are presented.After that, the experiments and related results are discussed and

Contribution of this paper to the literature
• We contribute strategies for analyzing correlations and association rules to reduce meaningless relationships when analyzing imagination.
• We evaluate students' imagination and predict the degree of imagination using FCI as well as provide representative features for supervisors to conclude what the students need to improve the imagination.
• Our proposed model can reveal the special features efficiently and be widely applied to explore the representative relationships from spare datasets.

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3 / 16 compared to those obtained without using localized features.Finally, we provide the conclusion.In the following, we use the same notations for the consistency in the study.The summary of the notations is shown in Table 1.

LITERATURE REVIEW
For clarity, we briefly review the literature on imagination and then review mining and statistical technologies sequentially.The definition of imagination is first discussed in this section.Before analysis, imagination shall be defined clearly because it can be misinterpreted easily as other concepts such as fantasy and creativity.Second, we recall the DBSCAN and FPM technology for extracting localized features.Finally, for localized features and predictions of students' imagination, a few statistical approaches-correlations and linear regressions-are presented.

Imagination
Imagination is a native ability of human beings, the basis of creative activities, and the result of cognitive and emotional processes.By the operation and exercise of imagination, Human beings not only construct scientific theories but also create new inventions to improve life through the process of constant thinking, trial and error.For example, in the 19th century, some people imaged eagerly they could fly in the sky.This imagination promoted artists, scientists and engineers to improve with various of past experiences and trial experiments to invent airplanes.Therefore, imagination is the key of innovation.
The definition of imagination is broad and vague.Generally, imagination is usually misinterpreted by the public because of its similarity to creativity.However, the meanings of imagination are not the same as the ones of creativity.The gyroscope theory proposed by Lin and Tsau (2012), who extended and redefined the activities related to the development of new ideas about physical objects or psychological feelings from the Vygosky theory (Lindqvist, 2003;Vygotsky, 2004), demonstrated the difference between creativity and imagination.Its concept is shown in Figure 1.The gyroscope figure shows the definition of creativity.It has eight elements, namely, stimuli from database, fluency, flexibility, originality, crystallized imagination, techniques, elaboration, and evaluation.The top four elements belong to convergent thinking, and the bottom four elements belong to divergent thinking.When human beings desire to create new concepts or objects, they first diverge to various lines of thinking and then arrive at the target concept by using their special insights.Such divergent thinking is called imagination.After a series of improvements and considerations of experience and intelligence, divergent thinking can converge into an extreme idea for new concepts or objects.The process of combining divergent and convergent thinking is creativity.For example, if students consider improvements and techniques, the thoughts are already limited and reduced with realistic considerations.The thoughts belong the creativity.
Nowadays, there exist popular imagination models for training and defining students' imagination such as IDeAL training models (Wang et al., 2010), IDV (Ho et al., 2013), and IFFO (Lin & Tsau, 2012).In the first model, the researchers considered the imagination can be formed in four components, including initiation, development, alternatives and links.The initiation component is the same as the stimuli from database that we mentioned in the The number of data in the dataset Sup(Pat) The frequency/count of pattern Pat RSup (Pat) The ratio of the frequency/count of pattern Pat to the number of data N Pat1⇒Pat2 The association rule regarding Pat1 implies Pat2 CPat1⇒Pat2 The confidence of the association rule regarding Pat1 implies Pat2 ηMinAsso The minimum threshold defined by the supervisors for association rules Xi A variable Xi in the dataset cov (X1, X2) The covariance between variables X1 and X2 σ(Xi) The standard deviation of variable Xi ρX1,X2, The Pearson's product moment coefficient from variables X1 and X2 RX3, X1X2 The multiple correlation coefficient of the independent variables X1 and X2, and the dependent variable X3 hθ (Xi) The dependent variable in linear regression θi The coefficient of the variable Xi Qi A question in the scale where 1 ≤  ≤ 20 Qi(score) A presentation of the score that students get in the question Qi (Qi, Q(i+1), …) The scores modeled in a vector form Alternatives implies reversed thoughts from the goal.Generally, the thoughts from the development may be suspended due to the size of students' databases.To stimulate more thoughts for students, the reversed thoughts are proposed.By considering the thoughts in the different way, the imagination can be inspired.The component of links is the thoughts that connect from initiative/developed thoughts and alternative thoughts.By these four components, imagination can be trained.
In the IDV model, the imagination can be described in the three stages, namely initiation stage, dynamic adjustment stage and virtual implementation stage.In initiation stage, the property of the thoughts is the same as the ones that we mentioned in IDeAL.The next stage is the dynamic adjustment stage.The key concept of the stage focuses on the association which can be briefly divided in two parts.One part is related to how to envision relationships among thoughts; that is, students should link related thoughts, extend the concepts behind thoughts, or identify the contradictions between thoughts and reorganize them accordingly (Ho et al, 2013); the other part is the quality which involved in conferring new meanings on a thought within an associative network to transform it into a novel thought.As for the virtual implementation stage, the key concept focuses on refining particular thoughts.
In the last model, IFFO, the imagination can be combined by four components.Those are initiation, fluency, flexibility, and originality.The first one component is the same as the one in IDeAL.Both of them emphasize the stimuli from database.As for the fluency, it is partially similar to the second component in IDeAL.In fluency, the authors stressed on the thoughts which belong to the same property (e.g., students inspired clocks from watches) or category (e.g., students inspired flowers from trees).As for the thoughts with different properties and categories, the authors think that such thoughts should be classified into the component, flexibility.By distinguishing the fluency and flexibility, the thoughts can easily be defined.The last component, called originality, is the special point in imagination because it is an ability to generate a product or thought with unique or unusual, unexpected, and first of its kind.In comparison with IDeAL and IDV, the last component, originality, emphasizes that students should combine all experience in initiation and think in association to create new thoughts which are different from others.With originality, the thoughts will be crystallized in imagination.
According to the above imagination models, for promoting students' imagination and precisely measuring imagination, we adopt the model, IFFO, proposed by Lin and Tsau because the model disassembles imagination more meticulous than the other models.We also consult some creativity scales such as creativity assessment packet (Williams, 1980).Unfortunately, questions in creativity scales should be verified with the imagination aspects and reliability tests and may not be suitable for IFFO.We thus use the scale proposed by Lin and Tsau who designed evaluation questions related to divergent thinking and roughly generalized the questions into IFFO.The scale concentrates on the degrees in divergent thinking and does not limit students' thoughts because the questions in the scale are designed in an open-ended manner.By analyzing the scores of these four components, the degree of imagination can be evaluated.

Data Analysis
In terms of data analysis, data mining analyses and statistical research are two popular techniques.In general, statistical research uses the cores/algorithms of data mining.The only difference is the behavior of data analysis.Data mining is the process of automatically discovering implicit information from masses of data, whereas statistical research focuses on the verification of hypotheses defined by researchers.For exploring data, data mining techniques may serve as a guide for supervisors to provide interesting information and situations before creating a hypothesis.
Generally, there exist many mining approaches, such as classification, clustering, FPM and ARE (Aher & Lobo, 2011;Slater et al., 2017).However, using only one single approach for data analysis cannot provide accurate results.For obtaining realistic results, the improvement with the combination from two or more approaches is necessary.For example, Hahsler and Karpienko (2017) extracted association rules with hierarchical clustering.For revealing accurate students' imagination in the study, we adopt some mining approaches and statistical approaches, and thereby combine these approaches to analyze and predict students' imagination.To accurately predict students' imagination, we review some mining and statistical approaches which can assist with concentrating on localized features.
The clustering approaches fall under the category of mining analysis, and they are usually adopted to extract preliminary difference of data information for supervisors.The main purpose of clustering is to differentiate the degrees of data from unknown categories by aggregating them into several groups.Supervisors can then focus on clusters of interest.
DBSCAN (Khan et al., 2014) is a well-known clustering approach, and it uses the density model to aggregate data.Unlike the centroid-based clustering model, DBSCAN provides stable clustering results based on two factors because it does not consider the initial locations of centroids.The two factors are called neighborhood radius (eps) and minimum number of data required to form a dense region (minpts), where eps represents the degree of data similarity in a region and minpts denotes the lower bound of cluster size.By setting the values of these two factors, clustering results can be generated steadily.
In addition to DBSCAN, FPM is used to look for interesting information in data analysis.To filter meaningless information, supervisors can extract interesting information from patterns/itemsets by comparing the support of patterns (Sup(.))with a user-defined threshold, called the minimum support (MinSup).These patterns, which are very frequent in a dataset, are called frequent patterns, and they can be described as follows.Assume a pattern Pat is generated from any combinations of all variables, X where X={Xi | i = 1, …, m, and m is the number of variables}.As a result, Pat ⊆ X and ∃Xi ∈ Pat.Pat will become a frequent pattern if the support of Pat, (Sup(Pat)), that is the frequency(/count) of Pat in the dataset, is equal to or larger than MinSup.Many approaches have been proposed for obtaining frequent patterns by using possible worlds such as the Apriori approach (Inokuchi et al., 2000) or by using tree-based approaches such as the frequent pattern tree approach (Han et al., 2004).These approaches can yield the same frequent patterns.
ARE is a rule discovery technique to extract frequent interesting relationships of appearances from frequent patterns.Based on the confidence value, rules that appear very frequently can be treated as association rules.The association rules can be calculated using relative support RSup(.), which is the proportion of the number of data in the dataset that contains the item, and can be expressed as follows.Consider patterns Pat1, and Pat2 in a dataset and the number of data is N, and Pat1≠Pat2.The rule of Pat1 ⇒ Pat2 implies that if Pat1 appears, Pat2 appears as well.To verify whether the rule Pat1 ⇒ Pat2 is an association rule, the confidence of the rule (CPat1⇒Pat2) should be calculated with relative support (RSup(.)) by the using following formula: Liu & Liao / Localized Features for Analyzing Imagination 6 / 16 If CPat1⇒Pat2 is higher than the minimum threshold (ηMinAsso) defined by the supervisors, the rule is an association rule.
However, in the real world, the process of ARE may be complicated for the variable-oriented data.As mentioned previously, all patterns only combine from variables.If a variable is a random variable, patterns will be difficult to be obtained.To simplify the process of ARE, researchers used a concept of the quantitative association rule (Salleb-Aouissi, Vrain, & Nortet, 2007) for categorical and quantitative data.The kernel of the concept is that all values in a variable are discrete and countable.Therefore, a pattern can be organized from any variables with values and variable Xi will be yield in a pattern at most once.
Example 2: Consider the situation from Example 1, and supervisors set the values of the MinSup and the ηMinAsso are 3 and 0.6 respectively.According to the description of the frequent patterns, the frequent patterns in Pat1 which is yield by the variable A are scores 2 and 4, and the ones in Pat2 which is yield by the variable B are scores 2 and 3.After calculating the frequent patterns, the association rules can be derived from these frequent patterns.According to Formulas (1), ( 2) and the concept of the quantitative association rule, the rules are XA(4) 4), and XB(2) ⇒ XA(2) where the confidences of the rules are 1, 0.8, 1 and 1, respectively, and all of the confidences are larger than 0.6.
In addition to mining analysis, statistical analysis is also widely used in data analysis.Correlation analysis is used to quantify the relationships between two variables.It has two differences compared to association rule analysis.Causality is the main difference between these two analysis methods.Generally, correlation analysis is concerned only with the relationships among variables.That is, if variable X1 and variable X2 have high correlation, the relationship only objectively expresses the degree between X1 and X2.By contrast, association rule analysis strengthens causality in variables such as X1 ⇒ X2, which implies that if X1 appears, X2 appears as well.Besides causality, purpose is the other difference between the two analysis methods.Association rules refer to more generalized terms from a dataset, whereas correlation can be considered as a special case of association that implies a linear degree of dependence between the variables.
Correlations between/among variables are evaluated by determining Pearson's product moment coefficient or the correlation coefficient R, which are common approaches for providing correlation coefficients, ρ, to measure the relationship between two variables and among multiple variables, respectively.According to the values of ρ and R, the relationship can be revealed easily.The definitions of ρ for two variables and multiple correlation coefficient of R for multiple variables are given in Definitions 1 and Definitions 2, respectively.Definition 1 (Pearson's Product Moment Coefficient) Given variables X1 (independent variable), and X2 (dependent variable), the Pearson's product moment coefficient ρX2,X1 is equal to where ( 1 ,  2 ) is the covariance between  1 and  2 , and (.) is the standard deviation.
Regression analysis is a statistical process for estimating the relationships among dependent variables and independent variables.It is the best function to summarize associations among data and then easily predict the value of the dependent variable from the model.In simple linear regression, the model can be denoted as follows: where ℎ  (  ) is the dependent variable,   is the coefficient of the variable   , and  0 is a constant.By calculating the linear regression and then resolving the coefficients   , the predicted function can be obtained easily, and thus, supervisors can predict the dependent value by using independent variables.

PROPOSED MODEL AND SCALE DESIGN
For analyzing imagination, we measure imagination by exploring frequency scores and implicit generalized association rules of imagination components, namely, initiation, fluency, flexibility, and originality.To weaken the effect of noise on frequent patterns and association rules, we proposed localized feature strategies to highlight common features and to display some interesting relationships among different variables.Clustering is the first EURASIA J Math Sci and Tech Ed 7 / 16 strategy in this study because it prevents noise data from having any effect on frequent patterns and association rules.In addition to clustering, we also pruned association rules by using correlations to explore meaningful rules for thought processes.In the following, we explain the proposed model in detail.
For the sake of clarity in the proposed model, we analyze imagination in three phases, namely, the test and score phase, feature localization phase, and prediction phase, according to the flowchart is shown in Figure 2. In the test and score phase, the purpose is to collect students' scores by having them answer the questions on the imagination scale.For comprehending the degrees and thought processes of students' imagination, students were asked to answer the questions one by one and within a short period (e.g., 150 s).The questions, Qis, where i ∈ ℤ + | 1 ≤ i ≤ 20, on the scale are shown in Table 2.In the scale, questions in initiation are represented in literature, such as "What do you think when you see a starfish?".As for the remainder questions in the other three components, the questions are illustrated in geometric graphics.After the students finished answering all questions, all scores were determined using the guide given in Table 3, which contains the scoring criteria for each question defined by referring to experts' opinions.
To comprehend students' imaginative qualities for development in the future, we first used two concepts of localized features to analyze scores that students obtained in the second phase.The first concept is regarding the use of the clustering technique to classify the data into several degrees.Generally, frequent patterns and association rules show the frequency of patterns and rules in a dataset.However, without the use of localized features, this information may confuse supervisors, leading them to misjudge.For strengthening the features of various groups and avoiding the effect of noise, the clustering technique shall be executed before analysis.
Clustering approaches can rapidly aggregate students with similar degrees of imagination into clusters.For decreasing the drawback in determining the number and locations of clusters in advance, we adopted the densitybased approach, DBSCAN, for clustering.As explained previously, the DBSCAN approach uses eps, and minpts to aggregate data into clusters.To determine the optimal value of eps, we selected a "knee" for eps by using a k nearest neighbor distance histogram (Kurumalla & Rao, 2016a), where a knee corresponds to a threshold at which a sharp change occurs along the k-distance curve.
The other concept in the feature localization phase pertains to the integration of association rules and variable correlations for preparations in the prediction phase.The association rules mainly highlight frequent relationships among variables by using conditional probability, called confidence, to reveal the implicit common information in a dataset.However, in a sparse dataset, plenty of association rules are occasionally confusing supervisors determining policies in the real world.For decreasing the number of association rules, we use correlation analysis to indirectly determine meaningful association rules.
For increasing the number of meaningful localized features of imagination, we use the strength of correlation to prune association rules by co-analyzing correlations and association rules.That is, if variables in an association rule have a strong correlation, the association rule will become a confidence rule for predictions in the future.By contrast, if two variables have non-correlation or weak correlation, the association rules containing these two variables describe only the generalized features with frequency and cannot be confidence rules for predictions.Therefore, these association rules can be ignored, even though there exists a generalized relationship between the two variables.
For calculating the correlations between two variables from association rules such as X1 ⇒ X2 , the correlation coefficient can be calculated easily by Pearson's ρ.As for complex association rules that contain variables in the leftmost of the right arrow, such as X1X2 ⇒ X3 and X1X2X3 ⇒ X4, the correlation coefficient can be calculated by multiple correlation analysis.For simplifying the processes of multiple correlation, we evaluate the R value as correlation coefficient from the value of R 2 in multiple linear regression.
According to the strength of correlation, the association rules can be distinguished into two types, namely, (1) non-correlation or weak correlation (0 ≤ R ≤ 40%), and (2) moderate or strong correlation (R > 40%).If the variables in an association rule fall under type (1), the implicit information displays that the association rule is meaningless because the variables in the rule may be "independent" or there might be "similar amounts of negative and positive data."As for type (2), these associations will be meaningful and confident association rules because the variables have strong correlations.Therefore, the meaningless association rules with the correlation coefficient R ≤ 40% can be pruned.
In the last phase, we predict the degrees and thought processes of imagination by integrating previous association rules, frequent scores, and results of linear regression analysis.Prediction is the kernel phase that occurs after the feature localization phase to forecast imagination-related information.However, this prediction requires activation by the most principal starting point.Here, we used the question Q1, which belongs to the initiation, as the starting point.The reasons for this selection are given below.First, owning to extension in thinking (i.e., thinking would become rich in the same component when students answered more of the same type of questions.), most degrees and thought processes were determined when the students answered the first Q1.By contrast, obtaining a frequent score on Q1 would be representative of imagination.Second, given that the students answered the questions sequentially, Q1 is the first question that the students came across.Finally, Q1 belongs to the initiation, which is the base of the three components.Therefore, all predictions will start from Q1.

Fluency Geometric graphics
The fluency score is arrived at based on the number of answers drawn by a student.Akin to initiation, the fluency score is proportional to the number of answers.

Flexibility Geometric graphics
The degree of flexibility score relies on the number of categories drawn by a student.The greater the number of categories (i.e., different macro taxonomies) drawn by a student, the higher is her/his score.(e.g., If a student drew a flower and a car, both should be grouped into different categories.If a student drew a flower and a tree, both should be grouped into the same category-plant.)

Originality Geometric graphics
For measuring originality, we pre-investigated the answers of 800 other students with ages similar to the students in the present study and then used the answers as an originality guide.The degree of originality score depends on the uniqueness of the answers of a student, which is decided based on whether they are in the guide.Therefore, students would get high scores if they drew the items that are not in the guide.
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According to remainder of the association rules after correlated pruning, the degrees and thought processes of imagination can be predicted from Q1 with the following priorities: 1. Predict the score of Qi by checking the association rules from the component to which Qi belongs 2. Predict the score of Qi by checking the association rules from two components, where one of the components contains Qi 3. Predict the score of Qi by using linear regression analysis with Qj(s), where j < i and Qi and Qj belong to the same component 4. Set the score from the frequent scores.
For example, consider four questions Q1-Q4, where Q1-Q3 belong to a component, and Q4 belongs to another component.For analyzing the imaginative thought processes, students answered these questions sequentially, and the score of each question is denoted as Qi(score).After the analysis of localized features, information about frequent scores, association rules, and correlation is shown in Table 4.The number of predictions of scores from Q1-Q4 is three because three association rules are involved in Q1.These three predictions can be calculated as follows.In the first prediction, the score of Q1 is set to three based on priority because there exists an association rule-Q1(3) ⇒ Q2( 4) with strong correlation.According to the previous association rule, the scores of Q1 and Q2 are three and four, respectively.Q3 is the special case in the prediction because it cannot be predicted by the third priority according to the rule Q1(4) ⇒ Q3( 6), where the score of the Q1 is four.As a result, we will use the third priority to calculate the score of Q3 by multiple linear regression with Q1 and Q2.If R corresponds to weak or no correlation, the score of Q3 would be equal to four based on the fourth priority; otherwise, the score of Q3 would be equal to m, which is the value predicted by using multiple linear regression with the scores of Q1 and Q2.As for the last question, Q4, the score cannot be obtained from the association rule Q2(4) ⇒ Q4(5) because Q2 and Q4 have weak correlation.Therefore, the score of Q4 will be determined using the fourth priority, and the prediction of Q1-Q4 in vector form can be denoted as (3, 4, 4 or m, 3).
In the second prediction, the scores of Q1 and Q3 can be predicted and are equal to four and six, respectively by using the association rule Q1(4) ⇒ Q3( 6).The score of Q2 can be determined using the third priority because (1) Q2 cannot be supposed from any association rule and (2) Q1 and Q2 are strongly correlated.As for Q4, its score can be obtained by using the fourth priority.As a result, the prediction of the Q1-Q4 in vector form can be denoted as (4, m, 6, 3).
In the third prediction, the score of Q1 is set to four according to the fourth priority because the score of Q1 cannot be obtained from the first three priorities.As for the score of Q2, even though Q1(3) ⇒ Q2(4) and Q1 and Q2 are strongly correlated, it cannot be predicted from the association rule Q1(3) ⇒ Q2(4) because there exists no appropriate score of Q1 in the association rules to predict the score of Q2.However, the score of Q2 can still be predicted from Q1 by linear regression because Q1 and Q2 are strongly correlated.The score of Q3 can be obtained easily and is equal to six from the association rule, Q1(4) ⇒ Q3( 6) because the score of Q1 equals to four, and Q1 and Q3 are strongly correlated.As for the score of Q4, it follows the fourth priority and is equal to three which is the frequent score of Q4.The first reason is Q4 is the first question in another component; therefore, the first priority is not suitable for Q4.The second reason is that Q2 and Q4 are weakly correlated, and therefore, the score of Q4 cannot be predicted from the association rule Q2(4) ⇒ Q4(5).The last reason is Q4 and Q1, Q2, and Q3 belong to different components.In the light the last reason, the score of Q4 cannot be predicted by multiple linear regression from Q1, Q2, and Q3.Therefore, the prediction of Q1 to Q4 in vector form is denoted as (4, m, 6, 3).
In the prediction phase, there may be plenty of trends connected to the scores of the questions because many correlated association rules that contain Q1 may generate different trends.For providing more information for analysis, we illustrate each trend with a probability as a weight.Unfortunately, the probabilities of the scores from frequent patterns and linear regression are difficult to measure.For simplicity, we are concerned only with the probabilities of confidence that arise from the association rules.The probability of each trend can be modeled from the product confidence.By using these probabilities, supervisors can make any decision using different trends.

EXPERIMENTS
For revealing students' imagination and reducing misinterpretations owing to the various degrees in the data, we used localized features, clustering, and co-analysis between correlations and association rules for analyzing imagination components before extracting frequent scores and association rules.Owing to the use of clustering, the imagination scores will show the frequencies of similar degrees, and thereby supervisors can easily obtain less impure frequent scores and association rules.In addition to clustering, we used the mechanism of supporting the frequent scores with predictions.By using association rules pruned by correlation analysis of components, linear regression analysis, and frequent scores, the trends of students' imagination can be obtained easily.For simplifying the tasks, we experimented with students' imagination input data in MATLAB 6.2 on the Windows 7 platform.

Datasets
We performed a proficiency assessment of imagination on freshmen and sophomores at two universities in Taiwan.One consists of the engineering institute and the college of design; the other consists of the college of arts.The number of data points was 332, and the backgrounds of the students included computer science, civil engineering, electrical engineering, fine arts, business management, industrial design & creative design, architecture, and electronics engineering.In these data, the gender ratio was unbalanced; therefore, an analysis of the gender aspect was beyond the scope of our study.

Experimental Setup
DBSCAN requires two parameters, namely, minpts and eps.We set minpts=5 as the lower bound for a cluster for less computing cost and avoiding the cases of generating lot of noise data (In our experiments where K=1, when minpts<229, the results are the same); clusters having fewer than minpts data items would be viewed as noise.With respect to eps, we used the kNN distance histogram (Kurumalla & Rao, 2016b) which is a popular approach determining the optimal values of eps in DBSCAN.The different values of k do not affect the eps because the value of eps determined this way depends on k, but it does not change dramatically as k changes.For example, the values of eps in the initiation component are shown in Figure 3.According to the kNN distance histogram, the optimal values of eps for the different components listed in Table 5 can be obtained.For simplifying the imagination analysis, we only listed results from top-K clustering, where K=1.

Localized Features and Predictions
For highlighting the frequent scores and association rules of students that have similar degrees of imagination, we first considered two types of strategies-using DBSCAN (localized aspect) and non-localized features (global aspect) to analyze each component and the relationship between two components.In the frequent score analysis, we defined various levels for displaying the degrees of the components according to the scores.The levels include zero (score =0), low (1≤score≤3), moderate (4≤score≤7), high (8≤score≤12), and rich (>12).By revealing the levels of frequent scores and association rules, supervisors can obtain (1) the degrees of the imagination component and the processes of students' imaginative thinking along with the order of questions to describe the students' imagination.After the DBSCAN strategy, we integrated (1) the frequent scores, (2) association rules obtained from correlation pruning, and (3) conjecture by linear regression of each component as the degrees and the processes of students' imagination.
To localize the features, we first use the DBSCAN to fetch similar data.Table 6 shows the information of the number of clusters and number of data in the largest cluster (i.e., displaying the number of data where K=1) in various components after DBSCAN.According to the clustering, some data will be pruned from the data set.Table 7 shows the moderate levels of frequent scores and the most popular association rules in initiation from the data in Table 6.Both strategies yielded the same frequent scores and similar association rules, and the association rules Q1(4),Q3(2) ⇒ Q4(3) and Q1(4),Q3(3) ⇒ Q4(4) in the two strategies can be thought to represent the students' initiation.This is because the scores of the questions in the itemsets related to both strategies are close to the frequent scores in initiation.However, the association rules related to the two strategies are slightly different.In terms of the localized aspect, the association rules reflect some additional information such as Q2(1) ⇒ Q3(1) and Q3(2) ⇒ Q4(3) that indirectly reveals the students' initiation strengthened gradually when answering the questions.On the contrary, in terms of the global aspect, the association rules show the students had constant initiation because the additional information was hidden by other students having different degrees.In terms of the localized aspect, our strategy can provide more information to supervisors.
In the fluency analysis, the frequent scores and association rules are similar.In Q5, the two strategies reveal that a student has a high level when starting to answer the first question related to fluency.The reasons for this include: (1) the thinking accumulated in initiation explodes when using it for derivation, and (2) the two parallel lines are easy to answer.After the students answered Q5, the frequent scores reflected that most students had moderate fluency levels, except on Q7.
Although both frequent scores on Q7 are low after accumulated thinking exploded, the declines of the scores on Q7 have different meanings.According to the score pertaining to the localized aspect, we infer that most students showed a short burnout period for repaying the thinking because they asked for an advance on their thinking in Q5.However, the information about frequent scores may imply students' thought processes were unstable.On the contrary, the scores pertaining to the global aspect show that thinking becomes steady after Q5, with the change being unclear because the value of the differences from Q6 to Q7, and Q7 to Q8 are equal to 1. Therefore, compared to the scores obtained using non-localized features, those obtained using DBSCAN can represent students' thought status more accurately and precisely.
With respect to the association rules in fluency, both strategies provided two common facts, which are the same to those obtained using non-localized features.One is that the students achieved high levels in any question from Q6 to Q8, and they achieved the high levels in the remainder of the questions pertaining to fluency; the other is that the students achieved low levels on Q6 and Q8 or Q7 and Q8, and they achieved low levels in Q7 and Q6, respectively.These two facts indicate that the students' partial thought processes diverge into two levels after answering Q5.However, these association rules cannot explain the moderate levels on Q6 and Q8.Therefore, we can infer that most of the students' thought processes remained unsteady after answering Q5.The aforementioned inference from the association rules corresponds to the one from the strategy using DBSCAN.Therefore, compared to the strategy that uses non-localized features, the strategy that uses DBSCAN can explain the students' thought processes in the real world.All frequent scores and association rules in fluency are explored from data in Table 6 and shown in Table 8.
Table 9 shows the same frequent scores and similar association rules for students' flexibility in both strategies from data in Table 6.After the unsteady thinking process in fluency, the frequent scores converged toward stable status, where the scores are approximately equal to 3. We infer the changes from different types of questions are the main reason for this because different types of questions would stop students from thinking in the same way and break the model of thinking from an infinite thinking cycle.4),Q3(3) ⇒ Q4(4) (75%) The association rules in the two strategies also support the aforementioned main reason obtained from the frequent scores because the scores on the itemsets in the association rules are approximately equal to the ones in the frequent scores.They show the students had similar frequent scores and also imply that the students answered questions with steady thought processes.Although both strategies reveal the same information about the students' flexibility, the strategy that uses DBSCAN provides richer information than the one using non-localized features.
The frequent scores and association rules in terms of originality, as given in Table 10, are more different than the ones in the three components because the input data are approximately equal to zero.Clearly, in terms of originality, with both strategies, the frequent scores are zero and the association rules are similarly zero.We infer that the lack of originality is the main factor leading to the zero scores.According to the arousing mechanisms of imagination and creative activity, originality is the last component of imagination.It relies very heavily on the other three components, and a long time is required to crystalize and converge thoughts into unique ideas.Therefore, based on (1) the decreasing trend from the scores of three components and (2) the zero scores in originality, we infer that these students lacked originality.
In addition to discussions on the degrees and thought processes of each component, we discuss the effects of interactions of two components from among initiation, fluency and flexibility.Owing to zero scores in originality, we ignore the interactions regarding originality.Table 11 displays the interactions of two components in different strategies.In the DBSCAN strategy, the scores of the association rules in Table 11 are close to the frequent scores in aforementioned three components in Tables 7, 8, and 9.By contrast, these association rules honestly reflect the relationship between two components.According to the number of association rules, we infer that initiation is strongly related to fluency, and fluency is related to flexibility; otherwise, the relationship between initiation and flexibility is weak.
Generally, between the two strategies, there is a difference in these relationships between initiation and fluency, initiation and flexibility, and fluency and flexibility.The difference is regarding the scores on association rules.In the DBSCAN strategy, the association rules in three relationships show similar scores to the frequent scores in the three aforementioned components.As a result, these association rules reflect the relationships that are strongly related to most of the scores of a student.By contrast, when using non-localized features, the association rules  (Q13-Q20) (0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0) Association rules (confidence) EURASIA J Math Sci and Tech Ed 13 / 16 display the relationship regarding Q5(8).In addition, there exist no rules in the relationship between initiation and flexibility.Therefore, the strategy using DBSCAN contributes more meaningful information than the one using non-localized features.
For obtaining precise association rules, we filter the association rules based on the second localized featurecorrelations.To increase the association rules, we prune the association rules in which the correlation of the questions with the association rules are equal to or greater than 0.4.Therefore, the association rules in DBSCAN can be pruned and the results are shown in Table 12.These association rules provide the prediction for the next phase.
The degrees of imagination are listed in Figure 4 for comparison with two different strategies, namely nonlocalized features, DBSCAN, and FCI.The degrees of the imagination in non-localized features, DBSCAN, and FCI  The trends of imagination that originate from the scores of Q1-Q20 in two strategies-non-localized features and DBSCAN-display similar evolutions of components.The students under these two strategies had moderate initiation and fluency, low flexibility, and zero originality.As for the FCI approach, there exist three predictions denoted by square marks of different sizes in the experiments.The red one (FCI (1)) used the fourth priority to Q1 for the prediction; and the purple and cyan ones (FCI (3) and FCI ( 2)) used the first priority of initiation to Q1 for prediction.All three FCI predictions show decreasing trends in terms of thought processes.
The three predictions made using FCI are different from the ones made using the two strategies.In FCI (1), one of the predictions has higher weight (31.1%) than the other two predictions and most of the scores obtained with FCI (1) are similar to the ones with the two strategies.The differences between FCI (1) and the two strategies are the scores on Q5, Q7, and Q12, and the scores are five, four, and three points, respectively.The scores of Q5, and Q7 were obtained by using second priority, and Q12 was inferred with linear regression using the variables Q9, Q10, and Q11.These three scores imply (1) a slight increasing trend when students started to answer the first questions in fluency and (2) steady thought processes in fluency and flexibility.
In addition to FCI (2), the prediction has medium weight among the three predictions and shows the worst case of localized features in terms of imagination because the scores of initiation, fluency, and flexibility are unsteady.This prediction is considerably different from those made using the two strategies.We inferred that these students may not have concentrated when answering the questions.Although they were unsteady, the scores of initiation, fluency, and flexibility were within the upper bound of the lower levels.As for FCI (3), the prediction is very similar to the one in FCI (1) and has the lowest weight in FCI predictions.Moreover, like FCI (1), it has slight differences with the prediction made using the other two strategies.We inferred these students are similar to the ones predicted in FCI (1).The only difference is they might have a slight weakness on some questions.

CONCLUSIONS AND FUTURE WORKS
In the study, we proposed two strategies using FPM and ARE for analyzing imagination data and provided degrees and thought processes for supervisors.We first used DBSCAN to objectively distinguish imagination input data into several groups to localize the most similar input data.By discussing the differences between the frequent scores and association rules by using non-localized features and the ones using DBSCAN, the degrees and thought processes of imagination can be revealed easily.Besides clustering, we performed a second imagination analysis to provide predictions for supervisors.By means of integration from association rules and the frequent scores generated after DBSCAN, we filtered the association rules with correlations to decrease the number of association rules.According to association rules, frequent scores, and linear regression, supervisors can obtain the most representative degrees and processes of imagination.
By analyzing and observing the trends from representative degrees, supervisors can conclude what the students need to improve the imagination.Based on the conclusion, supervisors can redesign the course.We use Figure 4 as an example.The trend of FCI (1) shows that students have moderate levels in initiation and fluency, low levels in flexibility, and zero levels in originality.According to those levels, supervisors can elegantly organize imaginative courses to strengthen students' initiation, fluency, flexibility and originality with the weights, 20%, 20%, 25% and 35%, respectively if the total percentage is equal to 100%.
In the future, we would like to explore the improvements on analyzing students' imagination and enhancing their imagination.For the improvement on analysis, we will consider correlation analysis.In the current study, we adopted linear correlation for pruning association rules.However, linear correlation sometimes cannot be satisfied with the current situation.To accurate the results, we will focus on the analysis of nonlinear analysis.For enhancing the students' imagination, some courses of the college of engineering can be redesigned with the feedbacks of students' imagination analysis.This work thus focuses on the course design and imagination evaluation.
paragraph.Development implies the association from any initiative thoughts and then expands thoughts.

Figure 4 .
Figure 4. Comparisons with three strategies for predicting students' imagination

Table 1 .
Summary of the use of notations

Table 3 .
Scoring criteria for imaginationFormatScoring criteriaInitiation LiteratureThe initiation score depends on the number of answers filled in by the student.If a student filled in many answers without duplicates, s/he would have a high initiation score.

Table 4 .
Example: frequent scores, association rules and correlations after feature localization

Table 6 .
Number of clusters and data in various components where K=1

Table 10 .
Comparisons of frequent scores and association rules in terms of originality(MinSup=6, and ηMinAsso=0.6)

Table 11 .
Comparisons of association rules between two components in different strategies(MinSup=6, and ηMinAsso=0.6)

Table 12 .
Association rules with correlation pruning Using