Learner to Learner Fuzzy Profiles Similarity Using a Hybrid Interaction Analysis Grid

In a socio-constructivist approach, interactions between learners play a dynamic role in individual learning [5]. Nonetheless, the analysis of remote discussions is not yet at the same level as the face-to-face ones. Evidently, the retrieved data tends to be more imperfect than those obtained from direct interaction. In this line, several studies have focused on the implementation of automatic systems for analyzing traces of interactions.

The Bales IPA has been used in some research in the area of Computer Sciences. Birnholtz et al. [6], for example, presented an experimental study of two-people groups (dyads) editing documents together. The focus of the study, was to analyze group maintenance, impression management and relationship-focused behavior. Savolainen [7] studied the extent to which blogs are used as interactive forums in which people can ask and share information by considering four reactions of the IPA categorization: showing positive and negative reactions, and asking and answering questions.

There are some related works that rely on IPA classifications but these ones are made manually by human experts. Kim et al. [8], for example, manually classified video recordings segments into IPA categories. Löfstrand and Zakrisson [9] followed a similar approach with the aim of identifying competitive and non-competitive behavior.

Approaches such as Academic Talk [10], Group Leader Tutor [11] and EPSILON [12] attempted to reduce the effort needed to encode group dynamics into IPA categories by introducing the concept of sentence openers. Sentence openers predefine a limited set of sentences that are allowed to start a communication act. Each sentence opener is directly matched to one IPA category.

Contrary to Bales, a limited number of researchers have relied on PLETY’s research on ethology. George [13] has proposed an environment called SPLASH that uses the PLETY grid to analyze mediated synchronous conversations.

The present review of the literature allows us to highlight the differences with our approach that will be presented in this article:

1. We use a hybrid grid (Bales and PLETY) which allows us to automatically classify learners according to behavioral and relational profiles.
2.  Instead of using sentence openers and limiting the IPA categories, we’ll use the intentions of the exchanged messages, not their contents, through positive and negative speech acts.
3. To improve the accuracy of the classification, we use a heuristic method to calculate the coefficients of the profiles.
4.  The use fuzzy logic for the analysis of that is closet to human reasoning.
5. Introduction of data mining methods for the measurement of similarity between learners and grouping learners with the same profiles.

4.1 Behavior fuzzy analysis

The fuzzy logic is an extension of Boolean logic, introduced with the 1965 proposal of « fuzzy set theory » by Lotfi Zadeh. It gives very appreciable flexibility to the reasoning that use it, which makes it possible to take into account imprecisions and uncertainties [17].

One of the interests of fuzzy logic to formalize human reasoning, is that the rules are expressed and stated in natural language. This is appropriate with our desire to draw up a learner profile in the group according to the discussions made during a collaboration session, although it is always difficult to try to model human behavior.

The fuzzy system [18] is divided into Five steps: (1) the input, (2) the Fuzzification, (3) the Inference engine, (4) the Defuzzification (5) the Output.

For each of the behavioral profile pattern (Organizer, Verifier, Seeker and Independent) and for the relational profile (Collaboration) we will do a fuzzy analysis what will generate 5 fuzzy systems (as shown in Figure 2).

2.png

Figure 2. Synoptic view of the fuzzy systems

Step 1: Profiles coefficients

To analyze automatically the conversations between learners, we have to calculate some heuristic formulas using the indications given above and the messages exchanged between learners. In reality we’ll use the intentions of the exchanged messages and not their contents, by means of positive and negative speech acts [19].

• For the relationship profile, and for each learner, we’ll calculate the orientation and the decision indices given by the formulas: Eq. (1) and Eq. (2).

The Orientation Index (Ind_ort_p (L)):

Ind_ort $(L)=\frac{\sum A c t_{-} \text {pos_ort }(L)}{\sum \text { Act_neg_ } \operatorname{ort}(L)}$     (1)

where:

• Act_pos_ort is the number of orientation positive acts,
• Act_neg_ort is the number of orientation negative acts.

The Decision Index (Ind_dec_p (L)):

$\operatorname{Ind}_{-} \operatorname{dec}(L)=\frac{\sum A c t_{-} p o s_{-} \operatorname{dec}(L)}{\sum A c t_{-} n g_{-} \operatorname{dec}(L)}$     (2)

where:

• Act_pos_dec is the number of decision positive acts,
• Act_neg_dec is the number of decision negative acts.

• From the description of each behavioral profile p (where p $\in$ {o, v, s, i} for respectively organizer, verifier, seeker, independent) we’ll calculate the intervention and reaction coefficients and the intervention ratio given by the formulas: Eq. (3), Eq. (4) and Eq. (5).

The Intervention Coefficient (Coef_int_p(L)): We see that the intervention coefficient of each profile is a ratio between the number of participations that characterize this profile and the total number of participations of the learner during a session.

Coef_int_p $p(L)=\frac{\sum A c t_{-} \text {int } p(L)}{\sum \text { Act }_{-} \operatorname{int}(L)}$     (3)

where:

• Act_int_p is the number of positive and negative intervention acts according to the profile p,
• Act_int is the total number of intervention acts.

The Reaction Coefficient (Coef_reac_p(L)): We see that the reaction coefficient of each learner is a ratio between the reactions of his peers to the interventions during a session.

Coef_reac $_{-} p(L)=\frac{\sum \text { Act_reac }_{-} p(L)}{\sum \text { Act_int } p(L)}$     (4)

where:

• Act_reac_p is the number of positive and negative reaction acts according to the profile p,
• Act_int_p is the number of positive and negative intervention acts according to the profile p.

The Intervention Ratio (IR(L)): It is the learner's participation or interventions (Act_int(L)) rate compared to the group's average participation during a session (Avr_int_grp).

$\operatorname{IR}(L)=\frac{\sum A c t_{-i n t}\quad(L)\quad}{\sum A v^{*}_{-} \text {int }_{-} g r p}$    (5)  

Step 2: The fuzzification

The second step involves construction of membership

functions. It corresponds to the identification of the linguistic variables. Two profiles’ analysis are to be implemented:

Step 2- Stage 1: Behavioral PLETY analysis:

We have three inputs for each of the profiles "Organizer, Verifier, Seeker and Independent” with as variable, the interaction coefficient for input1, the reaction coefficient for input2 and the intervention or participation ratio for input 3.

In light of the profiles’ characteristics shown in the hybrid grid the following fuzzy sets (linguistic variables) are proposed:

• The organizer input1 = {Passive, Active}
• The organizer input2 = {Negative, Positive}
• The verifier input1 = {Indifferent, Interested}
• The verifier input2 = {Little, Enough}
• The seeker input1 = {Incurious, Curious}
• The seeker input2 = {rejected, Accepted}
• The independent input1 = {Present, Absent}
• The independent input2 = {Heard, Disregarded}
• The behavioral profiles input3= {Low, Average, Important}

The membership function of input1 as shown in Figure 3, for all the behavioral profiles, is a sigmoid function defined on the interval [0 .. 1] and given by the formula Eq. (6):

$f 1(x)=1 /\left(1+e^{\pm 14(x-0.5)\quad}\right)$     (6)

The membership function f2 of input 2 (Eq. (7)), for the behavioral profiles «Organizer, Independent and Seeker», is a sigmoid function too (Figure 3), defined on the interval [0 . 1]:

$f 2(x)=1 /\left(1+e^{\pm 14(x-0.5)\quad}\right)$     (7)

3.png

Figure 3. Linguistic variables of the organizer

The membership function f_v2 of input 2, for the Verifier profile is a gaussian function in the interval [0..0.5] and a sigmoid function in the interval [0.25..1] (Eq.(8)) as shown in Figure 4:

$f_{v} 2(x)=\left\{\begin{array}{l}1 / e^{-(x-0.25) / 0.02} \\ 1 /\left(1+e^{\pm 11(x-0.625)\quad}\right)\end{array}\right.$      (8)

The membership function f3 of input3 (Figure 3 and Figure 4), for every behavioral profile is defined by Eq. (9) as follow:

$f 3(x)=\left\{\begin{array}{l}1 / e^{-(x-1) / 02} \\ 1 /\left(1+e^{-10(x-1.5)\quad}\right) \\ 1 /\left(1+e^{10(x-0.5)\quad}\right)\end{array}\right.$     (9)

4.png

Figure 4. Linguistic variables of the verifier

For the Output, the following fuzzy sets are proposed:

• The organizer, the verifier and the seeker output = {Weak_p, Insufficient_p, Medium_p, Satisfactory, Good_p} where p$\epsilon${o, v, s}
• The independent output = {Integrated, Upper_accepted, Accepted, Less-accepted, Isolated}

The membership function (Figure 5), for all the behavioral profiles is defined as a gaussian function (Eq. (10)) on the interval [0 .. 12]:

$F(x)=1 / e^{-(x-\mu)^{2} / 0.8}$     (10)

where, μ is the expected value.

Step 2- Stage 2: Relational analysis

It concerns the Collaborator profile. The latter has three variables: orientation index for input1, decision index for input 2 and collaboration index for output.

The fuzzy sets proposed for this profile are:

•  The collaborator input1 = {Weak, Good}
• The collaborator input2 = {Weak, Good}
• The collaborator output= {Weak_c, Average_c, Good_c /c = collaborator}.

The Input1 and input2 membership functions (Figure 6) are both sigmoid defined on the interval [0 . 2] (Eq. (11)):

$f_{c} 1(x)=f c 2(x)=1 /\left(1+e^{\pm 7(x-1)}\right)$     (11)

5.png

Figure 5. Output Linguistic variables of the verifier

6.png

Figure 6. Linguistic variables (Orientation, Decision and Collaboration)

For the Output, the membership function as shown in Figure 6, is a gaussian function (Eq. (12)) defined in the universe of discourse X = [0.12]:

$F c(x)=1 / e^{-(x-\mu)^{2} / 0.8}$      (12)

Step 3: The inference engine

The third step involves developing a set of rules for the knowledge base.

Given below are the set of rules using IF-THEN logic.

The fuzzy knowledge base is made up of all the fuzzy rules according to each profile. Let’s take as an example the two profiles: the organizer and the collaborator.

Step 3- Stage 1: Definition of fuzzy rules

The organizer inference rules:

1. If Coef_int_o is Active and Coef_reac_o is Positive and IR is Important then Coef_organization is Good_o
2.  If Coef_int_o is Active and Coef_reac_o is Positive and IR is Average then Coef_organization is Satisfactory_o
3.  If Coef_int_o is Active and Coef_reac_o is Negative and IR is Important then Coef_organization is Satisfactory_o
4. If Coef_int_o is Passive and Coef_reac_o is Positive and IR is Important then Coef_organization is Satisfactory_o
5.  If Coef_int_o is Active and Coef_reac_o is Positive and IR is Low then Coef_organization is Medium_o
6.  If Coef_int_o is Passive and Coef_reac_o is Positive and IR is Average then Coef_organization is Medium_o
7. If Coef_int_o is Active and Coef_reac_o is Negative and IR is Average then Coef_organization is Medium_o
8.  If Coef_int_o is Active and Coef_reac_o is Negative and IR is Low then Coef_organization is Insufficient_o
9. If Coef_int_o is Passive and Coef_reac_o is Positive and IR is Low then Coef_organization is Insufficient_o
10. If Coef_int_o is Passive and Coef_reac_o is Negative and IR is Important then Coef_organization is Insufficient_o
11. If Coef_int_o is Passive and Coef_reac_o is Negative and IR is Average then Coef_organization is insufficient_o
12. If Coef_int_o is Passive and Coef_reac_o is Negative and IR is Low then Coef_organization is Weak_o The collaborator inference rules:
13. If Ind_ort is Good and Ind_dec is Good then Ind_collab is Good_c
14. If Ind_ort is Good and Ind_dec is Weak then Ind_collab is Average_c
15. If Ind_ort is Weak and Ind_dec is Good then Ind_collab is Average_c
16. If Ind_ort is Weak and Ind_dec is Weak then Ind_collab is Weak_c

Step 3- Stage 2: Selection of fuzzy operators

The choice of the fuzzy operators allows to determine the inference engine, which is generated in our case by using the Min/Max Zadeh operators [12].

Let S be the set of ordered pairs (fuzzy variables, crisp variables).

S = {(Coef_int_o(Active), Xo1), (Coef_int_o(Passive), Xo2), (Coef_reac_o(Positive), X’o1), (Coef_reac_o(Negative), X’o2), (IR(Important), I1), (IR(Average), I2), (IR(Low), I3), (Coef_organization(Good_o), Yo1), Coef_organization (Satisfactory_o), Yo2), (Coef_organization(Medium_o), Yo3), (Coef_organization(Insufficient_o), Yo4), (Coef_organization (Weak_o), Yo5), (Ind_orient(Good), X1), (Ind_orient(Weak), X2), (Ind_dec(Good), X’1), (Ind_dec(Weak), X’2), (Ind_collab(Good_c), Y1), (Ind_collab(Average_c), Y2), (Ind_collab(Weak_c), Y3)}.

Step 3- Stage 3: Apply fuzzy rules

We apply the fuzzy rules, quoted above, with the MIN/MAX operators and obtain:

The organizer:

Yo1 = Min (Xo1, X’o1, I1)

Yo2 = Max (Min (Xo1, X’o1, I2), Min (Xo1, X’o2, I1),

   Min (Xo2, X’o1, I1))

Yo3 = Max (Min (Xo1, X’o1, I3), Min (Xo2, X’o1, I2),

   Min (Xo1, X’o2, I2))

Yo4 = Max (Min (Xo1, X’o2, I3), Min (Xo2, X’o2, I1),

   Min (Xo2, X’o2, I2), Min (Xo2, X’o1, I3))

Yo5 = Min (Xo2, X’o2, I3)

The collaborator:

  Y1 = Min (X2, X’2)

Y2 = Max (Min (X1, X’1), Min (X2, X’2))

Y3 = Min (X1, X’1)

Step 4: The defuzzification

The Mean of Maxima (MOM) method is applied here [20]. In this method, the defuzzied value is taken as the element with the highest membership values. When there are more than one element having maximum membership values, the mean value of the maxima is taken.

Let p a behavioral profile, p$\in$ {o, v, s, i}.

Yp = Max (Yp1, Yp2, Yp3, Yp4, Yp5).

Y = Max(Y1, Y2, Y3).

We calculate the inputs (or abscissas) named Ap, A corresponding to the outputs Y_p, y respectively:

$Y=e^{(A-\mu)^{2} / 0.8}$      (13)

$Y_{p}=e^{\left(-A_{p}-\mu_{p}\right)^{2} / 0.8}$       (14)

where, μ $\in$ {2, 6, 10} and μp $\in$ {2, 4, 6, 8, 10}

Let A be a fuzzy set with membership function Y(x) defined over x $\in$ X, where X is the universe of discourse.

The defuzzied value is let say x^∗ of a fuzzy set is defined by the formula Eq. (15):

$x^{*}=\left(\sum_{x_{i} \in M} x_{i}\right) /|M|$     (15)

where:

• M = { xi| Y ( xi ) is equal to the height of the fuzzy set A},
• |M| is the cardinality of the set M.

Step 5: The output decision

It corresponds to the definitive decision:

1. Giving the relational profile of the learner according to Bales with the percentages corresponding to the values Good_c, Average_c and Weak_c and the final collaboration index A, corresponding to the mean of MOM method.
2. Giving the behavioral profile of the learner according to PLETY with the percentages corresponding to the values Weak_p, Insufficient_p, Medium_p, Satisfactory_p and Good_p for each of the profiles Organizer, Verifier and Seeker, as well as the final indices: Animation, Check and Quest; calculated by the mean of MOM method.
3. Giving the behavioral profile of the learner according to PLETY with the percentages corresponding to the values Integrated, Upper_accepted, Accepted, Less-accepted and Isolated for the Independent, as well as the final index Independence; calculated by the mean of MOM method.

4.2 Learner-to-learner behavior similarity

The analysis of interactions between learners is based on data obtained by the participants' own activity during collaborative sessions. The learner profiles calculated for each of these sessions allow us to obtain a multivariate time series (MTSLP = Multivariate Time Series for Learner Profiles). Each MTS [21] concerns a learner. It is a series of final indices xi (t); [i = 1, .., n; t= 1, … , m], calculated sequentially through discrete time (collaborative sessions) where i indexes the calculation made at each time point t, thereby n = 5, since the columns of the matrix represent the collaboration, animation, check, quest and independence indices, calculated above (Figure 7).

7.png

Figure 7. Synoptic of an MTSLP

An MTSLP:

• should be treated as a whole, since there are usually important correlations among the variables in MTS data,
• may not be transformed into one long univariate time series.

Step 6: Learner to learner similarity

In this section, we aim to calculate the learner-to-learner similarity which provides a way of quantifying the degree of agreement between two learners behaviors.

The multidimensional similarity measure aims to indicate the level of similarity between several databases or groups of data simultaneously.

To measure the similarity between two learners L_A and L_B, we consider their MTSLP A, B of respective dimension d_A × 5 and d_B × 5.

These matrices (MTSLPs) have the same number of columns (5), but not necessarily the same number of rows, because:

• a learner within a group can be absent during a collaborative session,
• the number of work sessions may differ from one group to another.

We will use two methods dealing with Multivariate Time Series MTS:

• Principle Component Analysis Similarity factor (PCAS).
• The Eros method (Extended Frobenius norm).

Both methods apply to principal components of matrices. Hence, we need to pre-process our data. Algorithm1 computes the principal component of two MTSLPs:

1. Apply the SVD (Singular Value Decomposition) on the M_A and M_B covariance matrices, and we obtain the decomposition of each into (U Σ VT). V is called the right eigenvector matrix, and U the left eigenvector matrix.
2. Consider V_A and V_B, the two right eigenvector matrices resulting from the SVD.V_A and V_B are expressed in the following form: V_A = [a1, · · · , an] and V_B = [b1, · · · , bn], with ai and bi column orthonormal vectors of size n=5.

Because SVD is applied to covariance matrices, the eigenvectors and the principal components are used interchangeably [22].

Step 6.1: Learner to learner similarity using PCA factor

PCAS computes the similarity between the first k principal components [23].

1. PCAS firstly obtains the k principal components for each matrix A and B.
2. Intuitively, PCAS measures the similarity between two matrices by computing the squared cosine values between all the combinations of the n principal components from two matrices using Eq. (16).

$\operatorname{PCAS}(A, B)=\sum_{i=1}^{k} \sum_{j=1}^{k} \cos ^{2} \theta_{i j}$      (16) 

where: θ_ij is the angle between the i^th principal component of A and the j^th principal component of B.

3. The range of PCAS is between 0 and n, to obtain a range between 0 and 1 we divide PCAS by n.

The result of the calculations carried out during this step between each pair of learners, will make it possible to obtain the learner-to-learner similarity matrix X as shown in Figure 8.

Step 6.2: Learner to learner similarity using EROS method

The intuition behind this method lies in its ability to process a different set of observations for each group of data with the same number n of variables (5 profiles).

8.png

Figure 8. Learner to Learner PCA similarity matrix X

9.png

Figure 9. Learner to Learner Eros similarity matrix Y

This method is freed from the dimension problem by considering not the group of data but the eigenvalues and eigenvectors of the covariance matrix which, themselves, are of identical sizes [24].

Another advantage provided by this method concerns the dimension reduction of the data.

The Eros similarity between two MTSLP A and B; to obtain the learner-to-learner similarity matrix Y as shown in Figure 9; is given by:

$\operatorname{Eros}(A, B, w)=\sum_{i=1}^{n} w_{i}\left|\cos \theta_{i}\right|$     (17)

where:

• cos θi is the angle between ai and bi,
• w is a weight vector with:

$\sum_{i=1}^{n} w_{i}=1$ and $w_{i} \geq 0$     (18)

w vector is obtained by:

1. Taking the eigenvalues obtained from all the MTSLP items in the dataset.
2. Aggregating the eigenvalues obtained into one vector using the mean function.
3. Normalizing the weights ($\sum \mathrm{w}_{\mathrm{i}}=1$ for $\mathrm{i} \in[1 . \mathrm{n}]$) by applying the Eq. (19).

$w_{i} \leftarrow w_{i} / \sum_{j=1}^{n} w_{j}$      (19)

The range of Eros is between 0 and 1, with 1 being the most similar.

4.3 Learners’ profiles clustering

We’ll use two clustering approaches: soft clustering and hard clustering, and we’ll focus on three main methods:

Step 7.1: Hierarchical Ascendant Classification (HAC)

It is a technique whereby an indexed hierarchy is constructed among the elements of the data set [25]. Elements are successively ''merged,'' that is, subsumed into entities or clusters comprising two or more elements, based on their distance in factor space, and on application of a merging rule. As distance, the generalized Euclidean distance in factor space is normally used (Eq. (20)). In accordance with the merging rule, each merger yields an index of similarity, which can be used to construct a tree that depicts the hierarchical relationships. Such a tree is called a dendrogram [26].

$d\left(x_{i}, y_{i}\right)=\sqrt{\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2}}$     (20)

Step 7.2: K-means

Unlike HAC methods, which are difficult to implement on large datasets, K-means [27] is a clustering algorithm (algorithm 2) which is widely used in such situations. It is an iterative algorithm that proceeds as follows:

The Euclidean distance, given by Eq. (20), is generally considered to determine the distance between each data object and the centers of the cluster.

The Euclidean distance (d) between two vectors X = {x₁, x₂,…, x_n} and Y = {y₁, y₂,…, y_n} is given by:

When all data objects are included in some clusters, the first step is complete and early grouping is performed. This iterative process continues until the criterion function is minimized.

Step 7.3: Fuzzy c-means

The application of fuzzy logic for various scientific and technical goals has been commented on for decades [28]. This approach differs from the classical hard clustering where each object of the data set finds its own cluster. Thus, an object either belongs to a defined cluster or is out of it. The application of Fuzzy theory to the problem of finding similarity between objects of interest leads to the conclusion that a particular object can belong simultaneously to more than one cluster, but with different degrees of membership (DOMs) between 0 and 1 [29].

To calculate the weights and determine the clusters of a set of data points X = {x1, x2, x3 ..., xn} we perform the steps of Algorithm 3. Firstly, we randomly select a set of centroids V = {v1, v2, v3 ..., vc}, then, for each data point and for each cluster, we calculate the fuzzy membership of a data i to the cluster j using Equ. 21. Next, we calculate the new fuzzy centers v_j using Equ. 22. We repeat the calculation of the fuzzy memberships and the fuzzy new centers until the change in the center of the clusters between two iteration steps k and k+1 is less than the termination criterion β (||Z^(k+1) – Z^(k)|| < β) or I iterations value is achieved.

$z_{i j}=\frac{1}{\sum_{k=1}^{c}\left(\frac{\left\|x_{i}-v_{j}\right\|}{\left\|x_{i}-v_{k}\right\|}\right)^{2 /(m-1)}}$     (21)

$v_{j}=\frac{\sum_{i=1}^{n} z_{i j}^{m} \cdot x_{i}}{\sum_{i=1}^{n} z_{i j}^{m}}$     (22)

where:

• m is the fuzziness index $m \in\lceil 1 \ldots \infty]$,
• c represents the number of cluster center,
• z_ij represents the membership of i^th data to j^th cluster center,
• n is the number of data points,
• v_ij represents the j^th cluster center and c represents the number of cluster center.