Dickson Owuor
email: doowuor@lirmm.fr

Anne Laurent
email: laurent@lirmm.fr

Joseph Orero
email: jorero@strathmore.edu

Fuzzy-Temporal Gradual Patterns

Keywords: fuzzy membership, gradual patterns, temporal tendencies

Gradual patterns allow for retrieval of correlations between attributes through rules such as "the more the exercise, the less the stress". However, it may be the case that there is a lag between changes in some attributes and their impact on others ones, current methods do not take this into account. In this paper, we extend existing methods to handle these situations in order to retrieve patterns such as: "the more the exercise increases, the more the stress decreases 1 month later". We also extend our gradual rules to include fuzzy temporal constraints such as "the more the exercise increases, the more the stress decreases almost 1 month later". For this kinds of patterns, we designed three algorithms that were implemented and tested on real data.

I. INTRODUCTION

Mining gradual patterns enables the testing of data crossings that can detect relevant correlations between the attributes of a data-set. One of the methods for extracting gradual patterns is to apply gradual rules in the form of "the more/less A1, ..., the more/less An" that correlates n attributes A1, A2, ..., An. For instance the "the greater the number of exercise activities, the lower the level of stress" [START_REF] Aryadinata | Mining Epidemiological Dengue Fever Data from Brazil: A Gradual Pattern Based Geographical Information System[END_REF], [START_REF] Ayouni | Fuzzy gradual patterns: What fuzzy modality for what result?[END_REF]. Example 1 is an instance applying gradual rules in the data-set shown in Table I.

Example 1. we consider a data-set containing the types of physical exercises that a person performed together with the stress levels reading for different dates. It is important to note that the values in Table I are arbitrary. Each tuple in the dataset correspond to a daily record of a person. A key point to realize is that deriving the support for a gradual pattern involves at least two or more records because the patterns are built on the increasing or decreasing nature of an attribute. In the case of pattern {(exercise, ↑), (stress, ↓ )} sup=3 , the support is 3 because we can order records <r1, r2, r5> successively to match the gradual pattern.

Because of the complexity deriving support, the efficiency of gradual patterns relies on the antimonocity property which states that no frequent pattern containing n attributes can be built over a pattern containing a subset of these n attributes [START_REF] Aryadinata | M2LGP : Mining Multiple Level Gradual Patterns[END_REF]. For example if the pattern "the greater the A, the greater B" is not relevant, then it is impossible for the pattern "the greater the A, the greater the B, the greater the C" to be relevant.

In comparison to association rules, extracting gradual patterns allows for discovery of more meaningful correlations between attributes of a data-set beyond finding frequent related item-sets. However, it may be the case that the value an attribute causes a ripple effect on other attribute with respect to time. For instance in Table I, it may be the case that exercise causes stress to reduce a few days (or weeks) later and not on the same day.

In order to for us to extract a pattern that correlates attributes with a time lag, we need to extend the existing methods for extracting gradual patterns in order to capture the temporal aspects of such data-sets.

II. PRELIMINARY DEFINITIONS

We recall below some definitions taken from literature that describe gradual pattern mining. Definition 1. Gradual Item. A gradual item is a pair (i, v) where i is an item and v is a variation v ∈ {↑, ↓}. ↑ stands for an increasing variation while ↓ stands for a decreasing variation.

Example 2. (exercise, ↑) is a gradual item that can be interpreted as "the more the exercise". Definition 2. Gradual Pattern (also known as Gradual itemset). A gradual pattern is a set of gradual items, denoted by GP = {(i 1 , v 1 ), ..., (i n , v n )}. The set of all gradual patterns that can be defined by GP. Example 3. {(jogging, ↑), (walking, ↑), (stress, ↓)} is a gradual item-set that can be interpreted as "the more the jogging, the more the walking, the less the stress".

It is important to note that gradual pattern mining aims at extracting the frequent patterns, in contrast to the classical data mining framework that aims to extract frequent item-sets through techniques such as association rules.

Definition 3. Given a threshold of a minimum support σ, a gradual pattern GP is said to be frequent if supp (GP ) ≥ σ.

There is a need to describe what frequent means in the context of gradual patterns. The principle idea that the support is based on, is that of counting the proportion of tuples in a data-set that respects the gradual pattern [START_REF] Aryadinata | Mining Epidemiological Dengue Fever Data from Brazil: A Gradual Pattern Based Geographical Information System[END_REF]. For instance in Table I, we see that for records r1 and r2 the number of exercise activities increase while the stress level decrease simultaneously, since 'swim' < 'jog, swim' and 4 > 2.

One support proposed in [START_REF] Di-Jorio | Mining frequent gradual itemsets from large databases[END_REF] is based on the length of the longest path of exercises that can be built on this pattern. While [START_REF] Laurent | Graank: Exploiting rank correlations for extracting gradual itemsets[END_REF] and [START_REF] Berzal | An alternative approach to discover gradual dependencies[END_REF] consider the number of tuples that are concordant by exploiting the Kendall's τ rank correlation. Definition 4. The support of gradual pattern GP is given by the following formula: supp (GP ) = max L∈l (|L|)

|R|

, where L is the set of rows that when ordered, match the gradual pattern GP and R is the set of all rows in the data-set D.

In order to determine the longest path, a precedence graph is built for the pattern considered as shown in Figure 1. The precedence graph shown can also be represented in a binary matrix, which allows to optimize the computations. Let us consider the graph for the pattern {(exercise, ↑), (stress, ↓)}, there is one long path in this instance: < r1, r2, r5 >. Therefore the support is equal to 3 5 . Similarly, the technique of gradual ranking based on Kendall's τ can be applied on the data-set in Table I, in order to compute concordant pairs for the pattern {(exercise, ↑ ), (stress, ↓)}. The possible number of ordering pairs is given by the formula: n(n-1)

2
where n is the number of tuples.

In our case, the possible ordering pairs are 10:

[r1,r2], [r1,r3], [r1,r4], [r1,r5], [r2,r3], [r2,r4], [r2,r5], [r3,r4], [r3,

III. RELATED WORK

According to [START_REF] Roddick | A bibliography of temporal, spatial and spatio-temporal data mining research[END_REF], temporal data mining concerns the analysis of events ordered by one or more dimensions of time. Further, they distinguished the field by two main areas: one involves discovering similar patterns within the same or among different time sequences; the other involves discovering causal relationships among temporally-oriented events.

The first area, also known as trend analysis has been a field of active research for a long time. Srikant and Agrawal are among the first contributors, they tried to solve the problem of 'absence of time constraint' in an algorithm that they had introduced earlier 'AprioriAll' [START_REF] Agrawal | Mining sequential patterns[END_REF], for discovering sequential patterns [START_REF] Srikant | Mining sequential patterns: Generalizations and performance improvements[END_REF], [START_REF] Chand | Sequential pattern mining : Survey and current research challenges[END_REF].

For instance, a shop does not care if someone bought 'bread', followed by 'bread and jam' three weeks later; they may want to specify that a customer should support a sequential pattern only if adjacent elements occur within a specified interval, say 3 days. (So for a customer to support this pattern, the customer should have bought 'bread and jam' within 3 days of buying 'bread').

So, [START_REF] Srikant | Mining sequential patterns: Generalizations and performance improvements[END_REF] proposed a new algorithm known as 'Generalized Sequential Pattern' (GSP) that allowed users to set a time gap that was used to generate candidates for the frequent pattern. GSP was 5 times faster than AprioriAll since it counted less candidates. However, [START_REF] Masseglia | Efficient mining of sequential patterns with time constraints: Reducing the combinations[END_REF] proposed a more efficient algorithm than GSP known as 'Graph for Time Constraint' (GTC). GTC handled time constraints prior to and separately from the counting step of the data sequence, thus making it perform faster.

The latter area, discovering causal relationships can easily be conceptualized by a gradual pattern that correlates the causal effect among gradual items. As an illustration, [START_REF] Fiot | Evolution patterns and gradual trends[END_REF] proposed algorithms that could mine evolution patterns and gradual trends such as: An increasing number of purchases of jam during a short period is frequently followed by a purchase of bream a few days later. The overall principle entailed converting a quantitative database into a variation database, which was converted into a membership degree database (also known as the trend database) which is mined for evolution patterns.

Similarly, [START_REF] Phan | Moving Objects: Combining Gradual Rules and Spatio-Temporal Patterns[END_REF] proposed an a novel algorithm called, GSTD that combined two concepts: gradual patterns and spatiotemporal pattern to extract gradual-spatio-temporal patterns. This algorithm can be used to mine frequent moving objects such as: the more time is going on, the more objects are moving from east to west. They achieved this by defining a gradual-spatio-temporal rule that provided notations for direction (positive or negative) and time duration and object set variation.

IV. PROPOSITION

In our work, we aim at extending gradual patterns in such a way that they can include the temporal correlations between attributes. For instance "the more the exercise increases, the more the stress decreases 2 weeks later" denoted as: {(exercise, ↑), (stress, ↓) +=2weeks }. We also intend to extend our gradual rules to include fuzzy constraints such as "the more the exercise increases, the more the stress decreases almost 2 weeks later" denoted as: {(exercise, ↑), (stress, ↓ ) + 2weeks }. Definition 5. Time Lag. A time lag is the amount of time that elapses before or after the changes in a one gradual item affects the changes in another gradual item. Time lag is denoted as αβt where α is an operator α ∈ {+, -} and '+' implies after/later and '-' implies before/earlier; β is an operator β ∈ {=, } and '=' implies equal to and ' ' implies almost; t is the value of time lag and is given by the formula: t = M edial M ∈m , where M is a medial of sequence m, and m = (c i r 1 -c i r 1+k ), ..., (c i r n -c i r n+k ) where c is the column for time/date, r is a single tuple/row, i = 1, 2, ..., I, n = 1, 2, ..., N , and k = 1, 2, ..., K.

Remark 1. In definition 5, we consider the medial value of the sequence as the approximation for time lag when the largest proportion of members are split around it. Definition 6. Temporal Gradual Item. A temporal gradual item is made up of two parts: a gradual item and a time lag, denoted by (i, v) αβt where (i, v) is a gradual item and αβt is a time lag where β ∈ {=} so that '+ = t' implies a time lag of t later and, '-= t' implies a time lag of t earlier.

Definition 7. Fuzzy-Temporal Gradual Item. A fuzzytemporal gradual item is a temporal gradual item with a fuzzy time lag αβt where β ∈ { } so that '+ t' implies a time lag of almost t later and, '-t' implies a time lag of almost t earlier.

T GP = {(i 1 , v 1 ), (i 2 , v 2 ) αβt2 , ..., (i n , v n ) αβtn }.
Definition 9. Fuzzy-Temporal Gradual Pattern. A fuzzytemporal gradual pattern consists of one reference gradual item and a set of fuzzy-temporal gradual items, denoted by

T GP f = {(i 1 , v 1 ), (i 2 , v 2 ) αβt2 , ..., (i n , v n ) αβtn }.
Remark 2. In order for definitions 7 and 8 to be relevant, there must be one reference gradual item. A reference gradual item is the anchor gradual item selected by a user and, from which other temporal gradual items in the item-set are varied with respect to time. Example 6. {(jogging, ↑), (walking, ↑) -=1week , (stress, ↓ ) + 2weeks } is a fuzzy temporal gradual item-set that can be interpreted as "the more the jogging, the more the walking 1 week earlier, the less the stress almost 2 weeks later".

In the first place, our goal is to transform data-sets into a temporal format that allows for extraction of gradual patterns with the corresponding time information. We intend to achieve this by modifying an existing gradual pattern mining. However, it is very critical to highlight here that temporal correlations between attributes introduce more possible combinations of gradual patterns within a given data-set. For instance, the notation {(A, ↑) +/-=time , (B, ↓) +/-=time } introduces 4 possible pattern combinations if attributes A and B are used as reference item-sets interchangeably and time is held to a constant value.

V. DATA TRANSFORMATION

In this section, we will demonstrate how a typical data-set can be transformed into a temporal format in order to allow for extraction of temporal gradual patterns. The raw (or nontransformed) data in the data-set should be chronologically ordered with respect to time.

Definition 10. The representativity of a temporal gradual pattern T GP is given by the formula:

rep (T GP ) = |N |
|R| , where N is the set of all rows in the transformed data-set D and R is the set of all rows in the original data-set D.

Definition 11. Given a threshold of minimum representativity δ, a temporal gradual pattern T GP is said to be relevant if rep (T GP ) ≥ δ.

Remark 3. Definition 10 and 11 also hold for fuzzy-temporal gradual patterns which is denoted as T GP f . In Section V-A, we propose an algorithm for transforming data based on the representativity threshold set by the user.

Example 7. we consider a data-set containing the number of hours a person spent performing physical exercises together with the stress levels after irregular number of days. It is important to note that the values in Table II In the first place, we determine the longest path that match gradual pattern {(exercise, ↑), (stress, ↓)} for transformation r n+1 . Using the path <t2, t3>, the support is 2 4 . It is important to note that the rows that do not have values for stress level are removed from the computation.

We observe that transformation r n+1 represents 4 out of 5 tuples. On the negative side, there is a decrease in the representativity of the data as we progress our transformations to larger time gaps. However, representativity has a less significant effect on large data-sets because of their great number of tuples. Next, we determine the time lag between transformation r n and transformation r n+1 and at this point we observe that the 'day lags' in Table III vary. In order to approximate the most relevant time lag, we apply fuzzy logic which is described in the section that follows.

VI. BUILDING THE FUZZY MODALITY

First thing to remember is that there exists a great number of membership modalities that one can build functions from (for instance triangular, trapezoidal, Gaussian among others), and it is very difficult to determine which one will fit the data-set perfectly. However, it is enough to pick modalities that span the whole universe and remain scalable [START_REF] Ayouni | Fuzzy gradual patterns: What fuzzy modality for what result?[END_REF], [START_REF] Nath Mandal | In Search of Suitable Fuzzy Membership Function in Prediction of Time Series Data[END_REF], [START_REF] Zadeh | Fuzzy sets[END_REF], [START_REF] Schockaert | Fuzzifying allen's temporal interval relations[END_REF].

We recommend a triangular membership function (described in Figure 2), because we are interested in approximating the value medial time lag. Such that when this value is taken as the center membership function, the function should include a majority of the members without extending its boundaries. The TRUE center of a distribution is established when the largest proportion of members are closely spaced around it [START_REF] Montgomery | Applied Statistics and Probability for Engineers[END_REF]. In light of this, we initially can take the median as the center and slide the membership function left or right until we find the value that represents the TRUE center of the distribution, see proposed Algorithm 2.

       x-x1 x2-x1 if x1 ≤ x < x2 x-x3 x2-x3 if x2 ≤ x < x2 0
Algorithm 2: Slide, re-calculate membership function Input : selT sselected time-lags, allT sall time-lags, minSupminimum support Output: q2approximated medial value, supsupport In our modality, the triangular membership function will initially have the median time lag as the center and minimum and maximum time lags as the extremes so that it spans the entire universe of data-set. Figure 3 shows the membership function for the transformed data-set D 3 in Table III. When the membership function in Figure 3 is applied to the data-set in Table III, we generate Table IV. We observe that the membership degree support of ' 2.5' for path <t2, t3> is 0 2 , in this case the support is less than half.

1 q1 ← quartile (1, allT s), q2 ← quartile (2,
id 2.5days exercise stress (∈) (rn) (rn+1) t1 1 1 2 t2 0 2 3 t3 0 3 2 t4 .5 1 3 t5 - - - Table IV: Transformed data-set D 1 3
As can be seen, the problem may be that the membership function in Figure 3, is either be too narrow or is pivoted on a wrong median value. We shy away from widening the function since increases the size of the universe. We recommend sliding the median then re-calculating the membership degrees.

For instance, we slide the membership function for transformation r n+1 to the left as shown in Figure 4. We observe that the support for ' 1.5' for path <t2, t3> is 1 2 . The fuzzy-temporal gradual pattern {(exercise, ↑ ), (stress, ↓) + 1.5days } has a support of 2 4 , a representativity of 4 5 and the time lag: ' 1.5' has a support of 1 2 .

VII. EXPERIMENTS

This section seeks to show that our proposed approach allows to discover new temporal knowledge in gradual patterns that previously could not be discovered using existing gradual pattern mining techniques.

A. The Proposed T-GRAANK Approach

The proposed algorithm known as T-GRAANK (denotes Temporal GRAANK) modifies the GRAANK algorithm proposed by [START_REF] Laurent | Graank: Exploiting rank correlations for extracting gradual itemsets[END_REF] in order to extend its functionality to mining gradual patterns with temporal tendencies. More precisely, the algorithm works as illustrated in Algorithm 3:

It is important to mention that since the proposed algorithm is based on GRAANK, it inherits all the good that comes with it. For instance, the algorithm benefits from the computational efficiency and low computational complexity since it is also based on binary matrices.

On one hand, the proposed algorithm seems to be more computationally intensive than the original GRAANK algorithm proposed by [START_REF] Laurent | Graank: Exploiting rank correlations for extracting gradual itemsets[END_REF] because it executes 4 additional functions. On the other hand, the increase in computations can be justified by the fact that new knowledge about time lag is extracted which was not possible previously.

B. Short Performance Analysis

The runtime performances of our algorithm for temporal gradual pattern mining shown in Figure 5 were obtained from the execution of dummy data containing 50 tuples and 2 gradual items. The runtime values were generated by a python code that recorded the start-time and stop-time. In Figure 5 (a), as the representativity threshold is decreased, the number of data-set transformations (or number transformed data-sets) to be mined increase hence the increase in runtime. In Figure 5 (b), as minimum support threshold is decreased, the number of possible gradual patterns increase which in turn increases the number of scans in the data-set.

C. Results for Temporal Gradual Patterns

In order to test the efficacy of the T -GRAAN K, we performed two separate tasks related to weather and compared their results. The aim was to confirm the conclusions of [START_REF] Davenport | On the relation between rainfall and the normalized difference vegetation index for diverse vegetation types in east africa[END_REF], that the NDVI (Normalized Difference Vegetation Index) is a sensitive indicator of the inter-annual variability of rainfall in the East African region.

In the first task, we retrieved the historical rainfall distribution amounts for 4 towns in Kenya (October-December 2013 and 2015) from Kenya Meteorological Service weather report in [START_REF]Review of weather in kenya during the octoberdecember 2013[END_REF], [START_REF]Review of weather in kenya during the octoberdecember 2015[END_REF], shown in Table V In the second task, we first generated NDVI data (for year 2013 and 2015) from LANDSAT 7 satellite images over Kenya through a novel tool known as data-cube. The data-cube is a great tool for the expanded use of satellite data in an Open Source framework, see also https://www.opendatacube.org.

Lastly in the second task, we applied our approach on the NDVI data and we obtained the results shown in Table VI We note here that it is difficult to get clear satellite images after short intervals due to cloud coverage; therefore, the datacube tool runs an algorithm that re-creates the image based on previous images. It may be for this reason that the time approximation for pattern {W AJ+, M AK-} is slightly less than 2 years.

VIII. CONCLUSION

In this paper, we propose an approach for extending the existing GRAANK algorithm in order to extract fuzzy temporal patterns. This approach integrates two main areas: fuzzy logic and mining gradual patterns (with temporal tendencies). We provide formal definitions for temporal and fuzzy temporal patterns based on existing formal definitions for gradual patterns. Further, we demonstrate a step-wise transformation of the data-set and recommend a technique for approximating the medial time lag using a fuzzy membership function.

Apart from extensive experimentation, including both computation efficiency and semantics (relevance of the extracted patterns), further works include scaling and optimizing the technique in order to allow for multi-level temporal gradual pattern extraction. Moreover, we aim at improving the fuzzy modality in order to increase its accuracy in perfectly fitting the distribution of time lags.

Figure 1 :

 1 Figure 1: Precedence graph for {(exercise, ↑), (stress, ↓)}

Example 4 .

 4 (stress, ↓) +=2weeks is a temporal gradual item interpreted as the "the less the stress 2 weeks later". Example 5. (exercise, ↑) -1week is a fuzzy-temporal gradual item that can be interpreted as the "the more the exercise almost 1 week earlier". Definition 8. Temporal Gradual Pattern. A temporal gradual pattern consists of one reference gradual item-set together with a set of temporal gradual items, denoted by

  Otherwisethe values must satisfy the condition: x1 < x2 < x3

Figure 2 :

 2 Figure 2: A triangular membership function

Figure 3 :

 3 Figure 3: Membership function for r n+1

Figure 4 :

 4 Figure 4: Modified membership function for r n+1

Algorithm 3 :

 3 The T -GRAAN K algorithm Input : Dtransformed data-set, T dtime differences, minSupminimum support Result: Fset of Frequent gradual patterns, T lcorresponding approximated time lags

Figure 5 :

 5 Figure 5: (a) Temporal gradual pattern mining runtime according to minRep, (b) temporal gradual pattern mining runtime according to minSup

Table I :

 I Sample data-set D 1

	Correlation between exercise and stress levels
	id	date	activity	stress
		(day/month)	(exercise)	levels
	r1	01/06	swim	4
	r2	02/06	jog,swim	2
	r3	03/06	walk,jog	3
	r4	04/06	walk	5
	r5	05/06	walk,jog,swim	1

Table III :

 III are arbitrary. The main goals of the algorithm are: to calculate the time lags for each transformation, and to generate a new data-set table that includes columns that have been restructured (stepwise) while excluding the Time column. The time lags in the Time column are separately processed by a fuzzy modality, see Section VI. Mining temporal gradual patterns Input : Ddata set, ref Columnreference column, minSupminimum support, minRepminimum representativity Output: Fset of Frequent Gradual Patterns, T lcorresponding approximated time lag 1 sM ax ←maximum number of steps w.r.t minRep; Let us transform Table II (using Algorithm 1) so that we compare the hours of exercise in r n with the corresponding stress level in r n+1 , as illustrated in Table III. Transformed data-set D

	Algorithm 1: 2 rM ax ←totalRows (D);
	3 for s ← 1 to sM ax do
	4	for i ← 0 to (rM ax -s) do
	5	d ← Cell [i+s] -Cell [i] ;	/* time is in the 1st column */
	6	tempRow.append (ref Column);
	7	cM ax ← totalCols (Row [i] );
	8	for j ← 1 to cM ax do
		;		/* excluding 1st column */
	9	if Column [j] is same as ref Column then
	10	skip;
	11	else
	12	tempRow.append (Cell [j][i+s] ); /* Cell [col][row] in D
	13	end if	*/
	14	end for
	15	D .append (tempRow);
	16 17	T d .append (d); end for
	18 19 20 end for F, T l ← T-GRAANK(D , T d , minSup); display F, T l ;
	21 return D , T d
		id	days lag	exercise	stress
		(rn -rn+1)	(rn)	(rn+1)
		t1	3	1	2
		t2	1	2	3
		t3	5	3	2
		t4	2	1	3
		t5	-	-	-
					id	date	exercise stress
					(day/month)	(hours)	levels
					r1	01/06	1	4
					r2	04/06	2	2
					r3	05/06	3	3
					r4	10/06	1	2
					r5	12/06	3	3
					Table II: A sample data-set D 3
					A. The Data Transformation Algorithm
					In this section, we propose an algorithm calculates the num-
					ber of possible transformations based on the representativity
					threshold, and extracts the temporal gradual patterns for each
					transformation. T -GRAAN K is the proposed algorithm for
					mining temporal gradual patterns, see Section VII-A.

3 : transformation: r n+1

  allT s), and q3 ← quartile (3, allT s); 2 boundaries ← append (q1, q2, q3); 3 lef t, right ← F alse, slice ← (0.1 * q2), and sup ← 0;

	4 while sup < minpSup do
	5	memberships ← fuzzTrimf (selT s, boundaries);
	6	sup ← countAverage (memberships);
	7	if sup >= minSup then
	8	return q2, sup;
	9	else
	10	if lef t == F alse then
	11	center ← minimum (selT s);
	12	if center <= q2 then
	13	q1 ← (q1 -slice), q2 ← (q2 -slice), q3 ← (q3 -slice);
	14	boundaries ← append (q1, q2, q3);
	15	else
	16	lef t ← T rue;
	17	end if
	18	else if right == F alse then
	19	center ← maximum (selT s);
	20	if center >= q2 then
	21	q1 ← (q1 + slice), q2 ← (q2 + slice), q3 ← (q3 + slice);
	22	boundaries ← append (q1, q2, q3);
	23	else
	24	right ← T rue;
	25	end if
	26	else
	27	return F alse, F alse;
	28	end if
	29	end if
	30 end while

Table V :

 V . Here, we selected two observable patterns: {(M AK, ↓), (W AJ, ↑)} +=2years and {(ELD, ↑), (N RB, ↑)} +=2years , see also http://www. meteo.go.ke/index.php?q=archive. Rainfall distribution in Kenya

	town	amount
		2013 2015
	MAK	104	75
	WAJ	49	69
	ELD	174	200
	NRB	44	223

  . It can be seen that the patterns built by our algorithm match the selected patterns in TableV; except, the time lag is slightly less for pattern {W AJ+, M AK-}.

	Ref. Item	Pattern : Sup	Time : Sup	Rep
	NRB	{ELD+, N RB+} : 0.666	≈ +1.999yrs : 1.0	50%
		{W AJ+, N RB+, M AK+} : 0.666	≈ +1.999yrs : 1.0	50%
	WAJ	{ELD+, W AJ+} : 0.600	≈ +1.223yrs : 0.5	62.5%
		{W AJ+, M AK-} : 0.600	≈ +1.747yrs : 0.5	62.5%

Table VI :

 VI NDVI Temporal Pattern Results

Acknowledgment. The authors would like to thank the members of staff at the Co-operation and Cultural Service, Embassy of France in Kenya and Campus France (Montpellier) for their involvement in creating the opportunity for this work to be produced.


Linguistic Modifiers with Unbalanced Term Sets in Multi-valued Logic

Keywords: Imperfect knowledge, Multi-valued logic, Unbalanced terms, Linguistic modifiers

Modeling human knowledge by machines should be as faithful as possible to reality. Therefore, it is imperative to take account of inaccuracies and uncertainties in this knowledge. This problem has been dealt with through different approaches. The most two common approaches are fuzzy logic and multi-valued logic. These two logics propose a linguistic term modeling. Generally, problems modeling qualitative aspect use linguistic variables assessed in linguistic terms that are uniformly distributed on the scale. However, in many cases, linguistic information needs to be defined by unbalanced term sets whose terms are not uniformly and/or not symmetrically distributed. In the literature, many researchers deal with these term sets in the context of fuzzy logic. Thereby, in our work, we introduce a new approach to represent and treat such term sets in the context of multi-valued logic. First, we propose an algorithm that allows representing terms within an unbalanced set. Then, we describe a second algorithm that permits the use of linguistic modifiers within unbalanced multi-sets.

INTRODUCTION

Knowledge handled by humans is often imperfect. These imperfections may be due to ambiguity, incompleteness, imprecision, uncertainty, inconsistency, etc. Several approaches were suggested in the literature for such knowledge representation and treatment. The most known are fuzzy logic [START_REF] Zadeh | Fuzzy sets[END_REF] and multi-valued logic [START_REF] De Glas | Knowledge representation in a fuzzy setting[END_REF][START_REF] Akdag | A qualitative theory of uncertainty[END_REF].

We notice that humans are able to perform reasoning without any exact measurements. They mostly use abstract terms of natural language (young, old, mature, etc.) and symbolic data rather than numerical values or qualitative ones. Terms can also be composed by using adverbs, such as little, more or less and slightly. Both fuzzy logic and multi-valued logic propose a linguistic term modeling to allow using words in reasoning process. They use linguistic variables that take values in a set of linguistic terms [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning -I[END_REF]. These latter express the various nuances of the processed information using words.

Generally in Knowledge-Based Systems, experts use linguistic terms that are uniformly and symmetrically distributed on a scale. However, in some cases, we need to assess qualitative aspects by means of variables using linguistic term sets which are not uniformly distributed. For example, in the evaluation process, we often consider a single negative term, i.g. Fail, and many positive terms such as Medium, Good, Excellent, etc. The gap between these terms is unequal.

In this paper, we focus on multi-valued logic. It allows to symbolically represent imprecise knowledge using ordered adverbial expressions of natural language [START_REF] De Glas | Knowledge representation in a fuzzy setting[END_REF]. We have noticed that in the context of this logic, few studies have treated unbalanced linguistic term sets (Abchir, 2013,Chaoued andBorgi, 2015). The aim of this paper is to establish a methodology to represent and manage this kind of data. It is based on the work of [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] that express unbalanced terms using a uniform multi-set. In the present work, we apply our proposal to an Information Retrieval System (IRS). The latter aims to retrieve a set of documents that satisfies a user query. This system is composed of three units [START_REF] Herrera-Viedma | A model of an information retrieval system with unbalanced fuzzy linguistic information[END_REF]): 1. A documentary archive or a database including a set of documents. They are represented by means of index terms describing their subject content. 2. A query subsystem presenting user needs by means of weighted queries. It indicates the topics that he/she is asking for. 3. An evaluation subsystem allowing the evaluation of documents according to their relevance compared to the user query. Usually, users are interested by documents whose contents are the most relevant to their queries. This implies the use of more precise labels in the left (positive) interval than in the right (negative) one [START_REF] Herrera-Viedma | A model of an information retrieval system with unbalanced fuzzy linguistic information[END_REF]. Thus, in IRS the use of non-uniformly distributed term sets is recommended. Indeed, it is more appropriate to use such kind of multi-set to represent the relevance degrees of the documents or to express the weights of index terms in the queries. To achieve this, we will use the unbalanced set S un = {None, Low, Medium, High, QuiteHigh,VeryHigh, Total} = {N, L, M, H, QH,V H, T } (Herrera-Viedma and López-Herrera, 2007)(Fig. 1). The first concerns the representation of terms within unbalanced multi-set. The second deals with the management of such kind of knowledge. Fig. 2 illustrates the different steps of this process. In the first one, we apply the single scale algorithm [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] to express a term (L in the figure) using a uniform multi-set (L is represented by τ 2 ). Afterwards, an existing tool, as aggregation operator or modifiers, is applied to the obtained term (τ 2 is modified into τ 3 ). The last step aims to express the result of the computational phase with a term from the initial multi-set (Approximately L in the figure). This article is organized as follows. We start, in section 2, with an explanation of what is unbalanced term sets. In section 3, we introduce the basic concepts of multi-valued logic and of linguistic modifiers. Existing works that concern the representation of unbalanced multi-sets are presented in section 4. Then, section 5 introduces our approach to express terms within an unbalanced set (Fig. 2 Step 3). Finally, in section 6, we propose a new way to use Generalized Symbolic Modifiers with unbalanced linguistic terms (Fig. 2 Step 2).

PRELIMINARIES

Most Knowledge-Based Systems use linguistic sets with terms that are uniformly and symmetrically distributed. However, there are other cases that need to assess qualitative aspect by means of variables using linguistic terms which are not uniformly and/or symmetrically distributed, named unbalanced linguistic sets. In many real-life situations, these latter are used as in project investment, negotiation process, evaluation process, etc. Asymmetric linguistic information can be a consequence of the nature of the linguistic variables involved in the problem such as personal examination or evaluation system (Fig. 3) [START_REF] Martínez | An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges[END_REF]. Terms are not equidistant, e.g. the distance between Poor and Average is greater than between Average and Good. This difference indicates the expert's interest in having more precise definition of a part of the domain, that leads to the use of more labels in this interval. Moreover, in many problems, several decision makers are involved. In fact, it is more reliable to obtain a decision based on the opinion of several experts than on a single one. In a multi-experts decision making process, each expert may assess his knowledge with a particular scale having a specific granularity (Fig. 4) [START_REF] Martínez | An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges[END_REF]. It indicates their different knowledge backgrounds or judging abilities. In each used linguistic set, terms are uniformly distributed.

Figure 4 -Fuzzy multi-granular scales used by experts [START_REF] Herrera | A fuzzy linguistic methodology to deal with unbalanced linguistic term sets[END_REF] considered that an unbalanced fuzzy set is a set with a minimum term, a maximum term and a central term s c called also midpoint. The remaining terms are not uniformly distributed and asymmetrically on both side of central term :

S = S L ∪ S C ∪ S R (1)
With :

• S L : the subset including terms on the left of the central term s c and #(S L ) its cardinality;

• S C : the singleton containing the term s c ;

• S R : the subset including terms on the right of the central term s c and #(S R ) its cardinality.

Hence, S is described by:

S = {(#(S L ), density L ), 1, (#(S R ), density R )} (2)
where 1 is the midpoint. The density corresponds to a high granularity around the central term or the end of fuzzy partition. Therefore, the density value can be middle or extreme. For example, the set represented by Fig. 3 will be described by S = {(1, extreme), 1, (3, extreme)}. In fact, the central term is {Average}. The left subset includes only the term {Poor} which is the minimum term. While the right subset includes {Good, Very Good, Excellent}. These terms are closer to the maximum term Excellent. So the density corresponds to extreme for the two subset.

Xu [START_REF] Xu | An interactive approach to multiple attribute group decision making with multigranular uncertain linguistic information[END_REF] proposed the unbalanced set as a set of (2t-1) labels, with t a positive integer, defined by:

S (t) = {s (t) β |β = (1 -t), 2 3 (2 -t), 2 4 (3 -t), ..., 0, ..., 2 4 (t -3), 2 3 (t -2), (t -1)} (3)
Thus, the central term has index 0 and other terms have positive or negative indices. The main idea of this definition is that the absolute value of the deviation between the indices of two successive terms increase regularly proceeding from the center term to the end of the scale. Fig. 5 [START_REF] Xu | An interactive approach to multiple attribute group decision making with multigranular uncertain linguistic information[END_REF] illustrates an unbalanced set with t = 4. We notice that terms are symmetrically distributed around the central term. In this work, we propose to define an unbalanced linguistic term set S un as a set of degrees that includes a minimum degree, False, a maximum degree, True, and the remaining ones are not necessarily uniformly distributed on the scale: the gaps between adjacent terms may be unequal. Set granularity, i.e. its cardinality, can be odd or even. Each term is defined by its position on the scale. They are supplied directly or by specifying the gap between each successive terms.

To illustrate our proposal, let us use the set represented in Fig. 3. This set will be described using the distance between terms :

• Distance Poor -Average:

d 1 = 1 2 • Distance Average -Good: d 2 = 1 4 • Distance Good -Very Good: d 3 = 1 8 • Distance Very Good -Excellent: d 4 = 1 8
There are two important subjects to treat when dealing with imperfect knowledge: (1) their representation and (2) their management. Many researches had dealt with these term sets in the fuzzy logic context [START_REF] Herrera-Viedma | A model of an information retrieval system with unbalanced fuzzy linguistic information[END_REF][START_REF] Martínez | An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges[END_REF][START_REF] Herrera | A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decisionmaking. Systems, Man, and Cybernetics, Part B: Cybernetics[END_REF][START_REF] Wang | A normalized numerical scaling method for the unbalanced multi-granular linguistic sets[END_REF][START_REF] Xu | An interactive approach to multiple attribute group decision making with multigranular uncertain linguistic information[END_REF][START_REF] Marin | Induced unbalanced linguistic ordered weighted average and its application in multiperson decision making[END_REF][START_REF] Bartczuk | A new method for dealing with unbalanced linguistic term set[END_REF][START_REF] Jiang | The power average operator for unbalanced linguistic term sets[END_REF]. Researchers have proposed approaches for their representation [START_REF] Martínez | An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges[END_REF][START_REF] Herrera | A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decisionmaking. Systems, Man, and Cybernetics, Part B: Cybernetics[END_REF][START_REF] Bartczuk | A new method for dealing with unbalanced linguistic term set[END_REF] and their management, such as operators for unbalanced aggregation [START_REF] Marin | Induced unbalanced linguistic ordered weighted average and its application in multiperson decision making[END_REF][START_REF] Jiang | The power average operator for unbalanced linguistic term sets[END_REF]. In the context of multi-valued logic, only Abchir [START_REF] Abchir | Towards fuzzy semantics for geolocation applications[END_REF] and [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] proposed algorithms to represent unbalanced term sets within uniform set. Our work, developed in the multi-valued context, has two targets: the first one is to propose a new way to represent terms within unbalanced multi-sets; the second intends to define an approach to use linguistic modifiers with such knowledge.

MULTI-VALUED LOGIC AND GENERALIZED SYMBOLIC MODIFIERS

Multi-valued logic is based on De Glas's multi-set theory [START_REF] De Glas | Knowledge representation in a fuzzy setting[END_REF]. In this approach, each linguistic term is represented by a multi-set. Knowledge is expressed thanks to an ordered and finite scale of M symbols denoted by (De Glas, 1989, Akdag and[START_REF] Akdag | Incertitude et logique multivalente, première partie : Etude théorique[END_REF]:

L M = {τ 0 , τ 1 , ..., τ (M-1) }; M ≥ 2 (4) τ i is the membership degree to the multi-set (i ∈ [0, M-1]).
It should be noted that symbolic degrees are connected only by the total order relation ≤, defined by [START_REF] Adkag | Une approche logique du raisonnement incertain[END_REF]

τ α ≤ τ β ⇐⇒ α ≤ β; ∀α and β ∈ [0, M -1] (5)
In most existing works on multi-valued logic, these degrees are assumed to be uniformly distributed on the scale. The membership relation in multi-valued logic is partial:

x ∈ α A ⇐⇒ x belongs to A at a degree α (6)
To express the imprecision of a predicate, a qualifier ϑ α is associated with each degree:

x is ϑ α A ⇐⇒ (x is ϑ α A) is true ⇐⇒ "x is A" is τ α true
The work of Zadeh [START_REF] Zadeh | A fuzzy-set-theoretic interpretation of linguistic hedges[END_REF], in fuzzy logic, and more recently those of Akdag et al. [START_REF] Akdag | A symbolic approach of linguistic modifiers. Information Processing and Management of Uncertainty in Knowledge-Based Systems[END_REF][START_REF] Akdag | Linguistic modifiers in a symbolic framework[END_REF] and Truck [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF], in multi-valued logic context, consider that any fuzzy subset or multi-valued symbol can be considered as a modifier of another fuzzy subset or multi-valued symbol respectively. A modifier allows modeling knowledge and gradual reasoning to build new terms from the initial ones, or to compare two values by finding the modifier that allows the transformation from one to another.

In the multi-valued logic, a membership in a multi-set is characterized by a symbolic membership degree τ i defined on a scale of ordered degrees L M . Thus, the data modification is a transformation of a degree and/or the scale of the multi-set. Indeed, some linguistic modifiers preserve the same multi-set, but modify the membership degree. Others transform a multi-set towards another one. Hence, it leads to an expansion or erosion of the original scale.

Symbolic linguistic modifiers have been proposed by Akdag et al. [START_REF] Akdag | A symbolic approach of linguistic modifiers. Information Processing and Management of Uncertainty in Knowledge-Based Systems[END_REF][START_REF] Akdag | Linguistic modifiers in a symbolic framework[END_REF] and they were generalized and formalized by Truck [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF], Truck et al., 2002). They were named Generalized Symbolic Modifiers (GSM). According to Truck [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF], a GSM (Definition 1) is a triplet of parameters: radius, nature (i.e. dilated, eroded or preserved) and mode (i.e. reinforcing, weakening or central). The radius is denoted by ρ with ρ ∈ N * . The higher ρ, the more powerful the modifier.

Definition 1. [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] Let τ i be a symbolic degree, such that i ∈ N, in a scale L M of M terms (M ∈ N * \{1}) with i < M. Let m be a GSM with radius ρ denoted m ρ . The modifier m ρ is a function that performs a linear transformation of τ i to a new degree τ i ∈ L M (where L M is the linear transformation of L M ) according to a radius ρ :

m ρ (i) = i ; m ρ (M) = M
For each linguistic degree, Akdag et al. [START_REF] Akdag | A symbolic approach of linguistic modifiers. Information Processing and Management of Uncertainty in Knowledge-Based Systems[END_REF] associate a numerical rate, i.e. an intensity level. In fact, an item from the multi-set can be considered as a precision degree of a proposition. This latter is the quotient prop(τ i ) = p(τ i )

M-1 associated with each τ i such that p(τ i ) is its position in the scale. Thus, Prop(τ i ) is the weight of the degree τ i relatively 1 . is the flour function.

to the linguistic set granularity, i.e. its intensity compared to the truth degree τ (M-1) (True).

Symbolic modifiers are classified as: weakening when the proportion decreases, i.e. prop(τ i ) < prop(τ i ) (the four weakening modifiers defined in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] are EW ρ , DW ρ , DW ρ and CW ρ ); reinforcing when it increases, i.e. prop(τ i ) > prop(τ i ) (the four reinforcing modifiers proposed in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] are ER ρ , DR ρ , DR ρ and CR ρ ) or central if prop(τ i ) does not change, such modifiers may act as a zoom on the base (the four central modifiers presented in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] are EC ρ , EC ρ , DC ρ and DC ρ ). Some examples of GSM are presented in Table 1.

REPRESENTING UNBALANCED MULTI-SETS

In the context of multi-valued logic, few works have treated the non-uniformly distributed knowledge. Abchir proposed in the discussion of his thesis [START_REF] Abchir | Towards fuzzy semantics for geolocation applications[END_REF] a first approach based on the use of Generalized Symbolic Modifiers (GSM). Chaoued and Borgi also proposed a modified version of the Abchir's algorithm to represent unbalanced degrees on a single uniform scale [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF].

Abchir's Algorithm

This approach aims to represent numerical input values v 1 , v 2 , ..., representing the position of terms on a scale, as symbolic degrees of a uniform multi-set L M . The algorithm starts by calculating a γ coefficient multiplier to transform input values (v i ) to integer ones, denoted val i , if they are not. Then, each obtained integer value is transformed into a couple (τ i , L M ) according to the following rule:

M = max(2, val i+1 ); τ i = τ val i (7)
The procedure is iterative. It stops when all input values have been treated. The author denotes the first M by M 1 . If the next value to treat cannot be integrated in this set, i.e., val 2 ≥ M 1 , a new value M, denoted by M 2 , is calculated such that:

M 2 = val 2 + 1.
Thus, the couple (val 2 , M 2 ) leads to (τ val 2 , L M 2 ). When M changes, previously calculated couples must be represented within the new scale L M 2 . The weakening expanding modifier DW(M 2 -M 1 ) (Table 1) is used to do this.

The process continues for each value, ensuring that the multiplier coefficient γ always generates integer values. If this is not the case, the multiplier γ should be changed in a new γ that allows transforming [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] Mode Nature Modifier Effect

Weakening

Erosion EW ρ m(i) = max (0, i -ρ ) m(M) = max (2, M -ρ ) Dilatation DW ρ m(i) = i m(M) = M + ρ DW ρ m(i) = max (0, i -ρ ) m(M) = M + ρ Conservation CW ρ m(i) = max (0, i -ρ ) m(M) = M Reinforcing Erosion ER ρ m(i) = i m(M) = max (i + 1, M -ρ ) ER ρ m(i) = min (i + ρ , M -ρ -1) m(M) = max (1, M -ρ ) Dilatation DR ρ m(i) = i + ρ m(M) = M -ρ Conservation CR ρ m(i) = min (i + ρ , M -1) m(M) = M Central Erosion EC ρ m(i) = max ( i ρ , 1) 1 m(M) = max ( M ρ + 1, 2) 1 Dilatation DC ρ m(i) = i ρ m(M) = M ρ -ρ + 1
all considered values into integers, its old value will be denoted by γ old . Then, all previous values must be recalculated using the central expanding modifier DC(c) (Table 1). The multiplier c allows to transform γ old into γ. It is obtained as follows:

γ = c * γ old ( 8
)
For this algorithm, neither considered values to treat nor their number is known in advance. Partitioning will be done as the value is specified. The final result remains the same if a change is made in input terms order. The main criticism made to this approach is its iterative part for recalculating pairs representing terms already treated. This is done each time that the used multi-set or the value of the multiplier γ is modified.

Single Scale Algorithm

The difference between Abchir's algorithm [START_REF] Abchir | Towards fuzzy semantics for geolocation applications[END_REF] and Chaoued and Borgi approach [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] is in the input data as well as in their treatment. In this latter, input data can be the positions of the values to partition or the distances between them. This last case, not allowed with Abchir's approach, offers more flexibility to the users.

The gap between each pair of successive terms reflect a difference in their meaning. In this proposal, inputs are expressed by numerical values. These latter are supplied directly (d i ) (Fig. 6-a) or specified by the position of each term on the scale (v i ) (Fig. 6-b). In the latter case, the distance between the terms (more precisely their positions) v i+1 and v i are calculated as follows: This proposal is based on Espinilla et al. work [START_REF] Espinilla | An extended hierarchical linguistic model for decision-making problems[END_REF]. The authors proposed a new definition of linguistic hierarchies: extended linguistic hierarchies (ELH), within fuzzy logic. To build an ELH, they use a finite number of levels l(t, n(t)) with:

d i = v i+1 -v i (9)
• t: the number indicating the hierarchy level with t=1,..,m; such m is the number of experts ;

• n(t): the granularity of the linguistic term sets corresponding to the level t. Each linguistic term has a triangular membership function, uniformly and symmetrically distributed on [0,1]. The granularity of each level is always odd.

To express his knowledge, each expert can use a specific level, i.e. a particular granularity [START_REF] Espinilla | An extended hierarchical linguistic model for decision-making problems[END_REF]. Espinilla et al. proposed to add a new level l(t * , n(t * )) (Definition 2) to the hierarchy with t * = m + 1. This level retains all the modal points of all the previous levels. The modal points have a membership degree equal to one. Definition 2. [START_REF] Espinilla | An extended hierarchical linguistic model for decision-making problems[END_REF] Let {S n(1) , ..., S n(m) } be the set of linguistic scales with any odd value of granularity. A new level, l(t * , n(t * )) with t * = m + 1, that keeps the former modal points of the previous m levels can have the following granularity:

n(t * ) = 1 + LCM[(n(1) -1), (n(2) -1), ..., (n(m + 1) -1)]
With m the number of experts In a similar way, Chaoued and Borgi [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] proposed to express unbalanced terms within a new uniform multi-set. In fact, the granularity of the uniform multi-set equals the LCM of the sets representing initial terms granularity. The input data are unbalanced linguistic terms set S un (L M , M the number of terms) and the gap between terms. These are provided directly or by specifying the position of each term.

First, the granularity M k of each uniformly distributed linguistic set L M K used to represent each input term v k is calculated. In this set, the distance between each two successive terms is denoted by d k .

M k = (1 + d k ) d k ; k = 1, ..., (M -1) (10)
Once the sets L M K including all the terms of the initial set are determined, the granularity of the uniform set L M is deduced as:

M = 1 + LCM M-1 k=1 (M k -1) (11)
Each value will be expressed on the new uniform scale L M as a couple (τ γ k , L M ) according to the following rule:

d k = γ k * d ; k = 1, ..., (M -1) ( 12
)
Where d is the distance between any pair of successive degrees in L M . For a uniform linguistic set, this distance is:

d = 1 (M -1) (13)
To express the distances d k according to the distance d , the multiplier γ k is calculated using the rule (12).

Algorithm 1: Single Scale algorithm

Input: M unbalanced linguistic terms Normalized distances d k with k=1, ..., (M-1) begin Calculate granularities M k ; M k ∈ N * \{1} with k=1, ..., (M-1) Calculate granularity:

M = 1 + LCM M-1 k=1 (M k -1) Deduce distance d = 1 (M -1) s 0 ← (τ 0 , L M ) Calculate γ k = d k d with k=1, ..., (M-1) s k ← (τ γ k , L M ) Output: S = {(τ 0 , L M ),(τ γ 1 , L M ),..., (τ γ p-1 , L M )}
The minimum term is represented by (τ 0 , L M ) and denoted by s 0 . The process, presented in Algorithm 1, continues for other values. The obtained couples (τ γ k , L M ) are denoted by s k . To ensure the succession of terms, the calculated multiplier γ k is added to the one calculated in the previous iteration named γ old . Thus, the comparison is always done regarding the first term of the set L M , i.e., τ 0 .

This algorithm allows to bring us back to the uniform case which will give us the capacity to use different existing tools as linguistic modifiers [START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF], aggregation operators, etc.

Considering the complexity, this proposal is less complex (O(p)) than Abchir's (O(p 2 )), with p the number of input terms [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF]. Indeed, in this approach the treatment is done after introducing all data. Hence, the couples representing terms are calculated once.

To illustrate this algorithm, let us consider an Information Retrieval System (IRS) and the linguistic term set S un = L 7 = {N, L, M, H, QH, VH, T} [START_REF] Herrera-Viedma | A model of an information retrieval system with unbalanced fuzzy linguistic information[END_REF]) (Fig. 1). This latter is used to express the relevance of documents in the retrieval process. We aim to represent each term of L 7 using a term from a uniform linguistic set. As input, we consider the normalized distances in L 7 as:

• Distance N -L: d 1 = 1 4 = 0.25 • Distance L -M: d 2 = 1 4 = 0.25 • Distance M -H: d 3 = 1 8 = 0.125 • Distance H -QH: d 4 = 1 8 = 0.125 • Distance QH -VH: d 5 = 1 8 = 0.125 • Distance VH -T: d 6 = 1 8 = 0.125 First, we calculate the values M k : M 1 = 1+d 1 d 1 = 5 = M 2 ; M 3 = 1+d 3 d 3 = 9 = M 4 = M 5 = M 6
Then, we determine the granularity M of the uniform set and the distance between any of its successive terms d :

M = 1 + LCM(5 -1, 9 -1) = 9; d = 1 M -1 = 0.
125 We associate the couple (τ 0 , L 9 ) to the term N, and we treat the other values:

• For the term L (k = 1)

γ old ← 0; γ 1 ← γ old + d 1 d = 2; L ← (τ 2 , L 9 ) • For the term M (k = 2) γ old ← 2; γ 2 ← γ old + d 2 d = 4; M ← (τ 4 , L 9 ) • For the term H (k = 3) γ old ← 4; γ 3 ← γ old + d 3 d = 5; H ← (τ 5 , L 9 )
• For the term QH (k = 4)

γ old ← 5; γ 4 ← γ old + d 4 d = 6; QH ← (τ 6 , L 9 )
• For the term VH (k = 5)

γ old ← 6; γ 5 ← γ old + d 5 d = 7;V H ← (τ 7 , L 9 )
• For the term T (k = 6)

γ old ← 7; γ 6 ← γ old + d 6 d = 8; T ← (τ 8 , L 9 )
The algorithm output is illustrated in Fig. 8.

REPRESENTATION OF TERMS WITHIN UNBALANCED SET

In the previous section, we have presented Chaoued and Borgi's work [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] to represent unbalanced term set S un within a uniform term set L M . Thereby, existing linguistic modifiers or aggregation operators can be applied to its terms. The obtained values are also expressed using the uniform set L M . However, it is more understandable for the user to represent these results using terms from the initial set S un .

In this section, we focus on the representation of a term τ i from the uniform set L M with a term from S un denoted by L M . We suppose that the used distances on both sets are normalized. For more clarity, we denote the degrees of the uniform set L M by τ 0 , ..., τ i ... and those of S un by τ 0 , ..., τ i ... (knowing that τ 0 and τ 0 correspond to the same position 0 and τ M and τ M to 1). We propose a way to determine the nearest position, in S un , to an initial term τ i , represented by its position v i in L M . For that purpose, first, we need to define the proportion of an unbalanced degree from S un . Definition 3. Let τ i be a symbolic degree of an unbalanced multi-set S un (L M ). Its proportion is defined as:

Prop(τ i ) = i ∑ j=1 d j
With d j the normalized distance between each pair of successive degrees in the unbalanced multi-set S un .

We aim to identify the term τ pos in S un which distance between it and the first term (τ 0 ) of L M is the closest to v i , the position of τ i on L M . Thus, we will compare v i with the sum l k of the distances separating successive terms in L M :

l k = k ∑ j=1 d j .
The process will stop when the value of l k is higher or equal to v i . If the values v i and l k are equal, the position of τ i in L M is that of τ k-1 . If v i < l k , we check the closest term to v i between τ k-1 and τ k . In this case, a proportion error, denoted by α, exists and its value is the difference between the position of a term τ i and that of the closest term, i.e. τ k-1 or τ k . If these terms are equidistant to τ i , we choose the term with the smallest index. This approach is described in Algorithm 2.

The inputs for this function are the position v i of the term τ i , the M terms of the unbalanced multiset S un and the normalized distances d k between its successive terms. As output, we obtain the couple (τ pos , α).

To illustrate our approach, let us continue with the IRS already described. Let us suppose that a document D is described in the database with a set of 5 index t i (i ∈ {1, ..., 5}): D = 0.7/t 1 + 0.5/t 2 + 0.9/t 3 + 0.6/t 4 + 0.4/t 5 .

In fact, the system indicates for each index term a numeric value corresponding to its importance in describing the subject discussed in the document. If the index term t i is not linked to the document subject content, its value is equal to 0. However, index value equals to 1 means that t i is too important in the document subject. We aim to represent each index value, i.e. v i , by means of a couple (τ pos , α); such τ pos is a membership degree from L 7 (S un ).

The counter k is initialized to 1 and the distance l k to 0. For the first index value v 1 :

• k = 1: l 1 = 0.25 < 0.7 = v 1 • k = 2: l 2 = 0.25 + 0.25 = 0.5 < 0.7 • k = 3: l 3 = 0.5+ 0.125 = 0.625 < 0.7 • k = 4: l 4 =0.625 + 0.125 = 0.75 > 0.7 In this case v 1 is lower than l 4 (= 0.75). We can say that the index value is between τ 3 , High (H), and τ 4 , Quite High (QH). So, we check the closest term between τ 3 and τ 4 by comparing values of (l 4 -v 1 ) and (v 1 -(l 4 -d 4 )): (0.75 -0.7) < (0.7 -(0.75 -0.125)).

Algorithm 2: Representation of terms within unbalanced multi-set Input:

The position v i of the term τ i , τ i ∈ L M M linguistic terms of L M (S un ) Normalized distances d k in L M ; k=1, ..., (M-1) begin k = 1 l 1 = 0 while l k < v i do

l k = k ∑ j=1 d j k ← k + 1 if l k = v i then pos ← k-1 α ← 0 else if l k -v i < v i -(l k -d k ) then pos ← k α ← -(l k -v i ) else pos ← k-1 α ← v i -(l k -d k )
Output: The couple (τ pos , α)

Hence, τ 4 is closer than τ 3 and the proportion error is α = -0.05. Thus, v 1 is represented by (QH, -0.05) (Fig. 9). We can say that the degree τ 4 is weakened of 0.05. We proceed using the same approach for the

LINGUISTIC MODIFIERS WITH UNBALANCED MULTI-SETS

In multi-valued logic, any symbol can be considered as a modifier of another one. Thereby, modifiers allow to represent small variations of imprecise characterizations of a linguistic variables. Symbolic modifiers may transform simultaneously the membership degrees and the scale of the multi-set. We propose in this section to illustrate how using linguistic modifiers with unbalanced imperfect knowledge. We present an approach to apply Generalized Symbolic Modifiers [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] (Table 1), initially designed for balanced sets, to unbalanced term sets.

To perform this, first, we express the unbalanced term to modify with a term from a balanced multi-set. Afterwards, we apply the GSM modifier on the obtained term within a balanced set. Then, we should find the closest matching term back in the original unbalanced multi-set. Our proposal has as input M linguistic terms of the unbalanced multi-set S un (represented on a scale L M ), a term τ i from this set, the normalized distances d k between its successive terms and the GSM m to apply. The treatment will be as follows (Algorithm 3):

1. Express the term τ i within a uniform multi-set L M using the single uniform scale approach (Subsection 4.2). The new term is denoted by τ i .

2. Apply the GSM m to the term τ i . Thus, we get a new uniform linguistic set L M , such that M = m(M ), and a new term τ i = m(τ i ).

3. Represent the term τ i within the initial unbalanced linguistic set L M (S un ) using the approximation function presented in Algorithm 2.

Thus, for this last step, we have to determine the nearest position to τ i in S un (L M ). First, we calculate the position v i , i.e. the distance between the term τ i and τ 0 . Afterwards, we determine the term τ pos by comparing v i to the sum l k of the distances between each successive terms in L M (d k ).

τ i , L M ) corresponding to (τ i , L M ) in L M (v i , M ) ← m (τ i , M ) v i ← i (M -1)
Calculate the closest term τ pos to v i in S un and the proportion error α (Algo.2) Output: The couple (τ pos , α)

To illustrate our approach, we use the same example of IRS previously presented. We consider the unbalanced set L 7 of relevance degrees (Fig. 10). We apply the reinforcing modifier CR(1) to the relevance degree Medium, i.e. τ 2 . The process is represented in Fig. 11. The granularity of the uniform set, previously calculated in Subsection 4.2, is 9 and the term τ 2 from S un is represented by τ 4 in L 9 . We notice that the proportion is preserved Prop(τ 4 ) = 4 8 = 0.5. For this example, we apply the GSM CR (1) to the couple (τ 4 , L 9 ) (Table 1): i = m(4) = min(4 + 1, 9 -1) = min(5, 8) = 5; M = m(9) = 9

The obtained couple is (τ 5 , L 9 ). We remind that the modifier CR is a reinforcing one, i.e. the proportion is increased: Prop (τ 5 ) = 5 8 > Prop(τ 4 ) = 4 8 . Finally, we must express the term τ 5 within the initial set L 7 . We first calculate the distance: v 5 = i (M -1) = 5 8 = 0.625. The counter k is initialized to 1 and the distance l 1 to 0.

• For k = 1; l 1 ← 0.25 < v 5

• For k = 2 ; l 2 ← 0.25 + 0.25 = 0.5 < v 5

• For k = 3 ; l 3 ← 0.5 + 0.125 = 0.625 = v 5 In this case, the value of l 3 is equal to v 5 . Thus, τ 3 is the correspondent term in the unbalanced set L 7 . The proportion of this term is Prop (τ 3 ) = 0.625. It corresponds to the proportion of the term τ 5 in the balanced multi-set L 9 .

We can say that the result of applying CR (1) to the relevance degree Medium corresponds to the relevance degree High.

CONCLUSIONS

This work deals with imperfect knowledge management in the multi-valued logic context. Specifically, it deals with the representation and management of unbalanced multi-sets. In fact, the available literature that involves unbalanced term sets are only concerned with the fuzzy logic. They also do not propose a formal definition of unbalanced term sets, which could be an important improvement in imperfect knowledge management.

This paper proposes an algorithm that allows to represent uniformly distributed terms within an unbalanced term set. This is performed by an approximation function that provides the closest term in the unbalanced multi-set to the desired value. In some cases, a proportion error may exist. Besides that, based on Chaoued and Borgi's work [START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] that allows the representation of terms on a single uniform scale, we introduce a second algorithm that applies linguistic modifiers, initially designed for balanced terms, to unbalanced ones.

As future work, we notice that, in both Abchir's and Nouha and Borgi's algorithms, the input terms for the representation are numerical. It would be interesting to propose a way to treat symbolic cases where the distance between terms are defined with words instead of numbers. It would also be more understandable by humans to express the proportion error using adverbs like little, more or less, slightly,... Another aspect in the management of imperfect knowledge that should be treated is the approximate reasoning [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning -I[END_REF]. It is based on the Generalized Modus Ponens. Its principle is to deduce a fact similar to the conclusion rule from an observation approximately equal to the premise rule. It would be important to propose a new Generalized Modus Ponens rules dealing with unbalanced multi-sets.

Figure 1 -

 1 Figure 1 -Unbalanced linguistic term set Two aspects are discussed in the present work.The first concerns the representation of terms within unbalanced multi-set. The second deals with the management of such kind of knowledge. Fig.2illustrates the different steps of this process. In the first one, we apply the single scale algorithm[START_REF] Chaoued | Representation of unbalanced terms in multi-valued logic[END_REF] to express a term (L in the figure) using a uniform multi-set (L is represented by τ 2 ). Afterwards, an existing tool, as aggregation operator or modifiers, is applied to the obtained term (τ 2 is modified into τ 3 ). The last step aims to express the result of the computational phase with a term from the initial multi-set (Approximately L in the figure).

Figure 2 -

 2 Figure 2 -Management of unbalanced linguistic term set

Figure 3 -

 3 Figure 3 -Set of 5 linguistic terms not uniformly distributed

Figure 5 -

 5 Figure 5 -A set of seven linguistic labels S (4)

Fig. 7 (

 7 Fig. 7[START_REF] Espinilla | An extended hierarchical linguistic model for decision-making problems[END_REF] shows an ELH with three levels of 3, 5 and 7 terms. The fourth level (t * ) has as granularity the LCM of the previous ones: 1 + LCM(3 -1, 5 -1, 7 -1) = 13.

Figure 7 -

 7 Figure 7 -Extended Linguistic Hierarchy with 3, 5, 7 and 13 terms

Figure 8 -

 8 Figure 8 -Result of the single scale algorithm

Figure 9 -

 9 Figure 9 -Representation of the index v 1 within an unbalanced set other index values. Thus, we obtain: D = (QH, -0.05)/t 1 + (M, 0)/t 2 + (VH, 0.025)/t 3 + (H, -0.025)/t 4 + (M, -0.1)/t 5 .

Algorithm 3 :

 3 Using GSM with unbalanced multi-set Input: M unbalanced linguistic terms of L M (S un ) The term τ i ∈ L M Normalized distances d k of L M ; k=1, ..., (M-1) The GSM m begin L M ← Single scale approach (M unbalanced terms, d k ) (Algo.1) Identify the couple (

Figure 10 -

 10 Figure 10 -Unbalanced set of relevance degrees As mentioned before, the distances d k are: d 1 = d 2 = 0.25; d 3 = d 4 = d 5 = d 6 = 0.125. The proportion of the term τ 2 in the unbalanced term set L 7 (Definition 3) is: Prop (τ 2 ) = d 1 + d 2 = 0.25 + 0.25 = 0.5The granularity of the uniform set, previously calculated in Subsection 4.2, is 9 and the term τ 2 from S un is represented by τ 4 in L 9 . We notice that the proportion is preserved Prop(τ 4 ) = 4 8 = 0.5. For this example, we apply the GSM CR (1) to the couple (τ 4 , L 9 ) (Table1):i = m(4) = min(4 + 1, 9 -1) = min(5, 8) = 5; M = m(9) = 9The obtained couple is (τ 5 , L 9 ). We remind that the modifier CR is a reinforcing one, i.e. the proportion is increased: Prop (τ 5 ) = 5 8 > Prop(τ 4 ) = 4 8 . Finally, we must express the term τ 5 within the initial set L 7 . We first calculate the distance: v 5 =

Figure 11 -

 11 Figure 11 -The process of applying CR (1) to relevance degree M

Table 1 -

 1 Some examples of Generalized Symbolic Modifiers