Qualitative Modelling of Time Series Using Self-Organizing Maps: Application to Animal Science

Transcription

1 Proceedings of the 6th WSEAS International Conference on Applied Coputer Science, Tenerife, Canary Islands, Spain, Deceber 16-18, Qualitative Modelling of Tie Series Using Self-Organizing Maps: Application to Anial Science EMILIO SORIA, JOSE D. MARTIN, CARLOS FERNANDEZ, RAFAEL MAGDALENA, ANTONIO J. SERRANO, ÁLVARO MORENO Departaento de Ingeniería Electrónica, E.T.S.E, University of Valencia C/ Dr. Moliner 50, Burjassot (Valencia). Departaento Producción Anial, Universidad Cardenal Herrera CEU, Edificio Seinario s/n Moncada, Valencia, Spain Abstract: - In this wor, a new ethodology for odeling qualitative teporal processes, is proposed. In this developent, two stages are considered. First, odels are obtained and adapted fitting their free paraeters according to the existing tie series. Second, Self- Organizing Maps (SOM) are used to establish a clustering of the data using the paraeters obtained in the first stage. This ethodology is successfully applied to a proble of il yield prediction in goat herds. Key-Words: - Tie Series, Neural Networs, Matheatical Models. 1 Introduction There is an extensive literature about tie odeling of processes, including both linear and nonlinear odels [1-3]. In all these odels, there are a nuber of free paraeters that should be fit. Once these paraeters are fit, it should be analyzed the goodness of the odel as well as its capacity of generalization. There are any probles in which the processes to odel do not present a unique behavior, that is, considering the sae atheatical odel base for all the processes, the value of the different paraeters differs according to the process. As an exaple, we can thin about a odel that is a daped sine; the value of decay of the exponential and the frequency of the sine ay vary for the different processes. In this situation, the developent of local odels for each process can be carried out after a classic analysis of clustering [4]. Therefore, this approach will need a first segentation and characterization of the tie series in order to carry out the odeling. We propose a different approach in this counication. First, a basic odel is taen into account and then, a clustering of processes is carried out using the paraeters of the developed atheatical odel. The present counication shows the advantages of the approach proposed, copleenting this description with an exaple of tie series in Anial Sciences. The reainder of the paper is organized as follows: in Section 2, the self-organizing aps are explained due to its essential role in this wor. Section 3 presents the proposed procedure and, Section 4 the obtained results, ending up the paper with the conclusion of the wor in Section 5. 2 Self-Organizing Maps The Self-Organizing Map (SOM) is a neural odel that is inspired on the fact that siilar tass tae place in

2 Proceedings of the 6th WSEAS International Conference on Applied Coputer Science, Tenerife, Canary Islands, Spain, Deceber 16-18, siilar areas of the brain. In this odel, neurons are arranged in two layers. The first layer is called the input or sensorial layer, and it consists of n neurons, being n the nuber of input variables. The processing is carried out in the second layer or copetition layer, that fors the ap of characteristics [3-4]: Fig. 1. SOM architecture. In this structure, a vector of weights is assigned to each neuron of the output layer; this vector should have the sae diension as the input vectors, that is, for the neuron i,j (first index row, second colun): w = [ w... w ] 1. The processing of this networ aps siilar input patterns into close areas of the output layer of SOM [4]. To achieve this goal, the following learning algorith for the synaptic coefficients is used [3-4]: 1) Weight initialization. 2) An input pattern is selected: x = x 1... x. [ ] 3) Calculation of the siilarity between the weights associated with all the neurons and the corresponding input vector. If we consider the Euclidean distance as a coparison easure, then: d ( w, x ) = ( s s 2 w x ) (1) s= 1 4) Deterination of the closest neuron to the input pattern, the socalled Best Matching Unit (BMU). 5) Update of the synaptic weights; if the quadratic function is used as siilarity function, then: w ( NG w ) ( x w ) = w + α h, (2) where α is the learning rate and h(x) is nown as neighborhood function. This function is a function whose value depends on the distance between the BMU and the updated neuron. This function is thus responsible of aintaining the topological relationships aong patterns when the input space is apped into the two-diensional space defined by the networ architecture [4]. 6) If the axiu nuber of iterations is reached, end; otherwise, go to step 2. At the end of the learning stage, a two-diensional ap is obtained. This ap provides qualitative, valuable inforation about how the input variables are related aong theselves. With the SOM, a data clustering is obtained, deterining siilar groups within our data. It should be ephasized the role that SOM plays as a visual tool in data ining; it is possible to use a colorbased representation of the coponents of the vectors corresponding to every neuron in the ap, and therefore, it is straightforward to figure out the relationships aongst the input variables. 3 Proposed ethodology There are two ain approaches to odel a tie series: either a paraetric or a nonparaetric odel. There is still a third option, the so-called seiparaetric odeling, but it is not necessary to study in depth this third option for

3 Proceedings of the 6th WSEAS International Conference on Applied Coputer Science, Tenerife, Canary Islands, Spain, Deceber 16-18, the goals of this wor [2]. Our ethodology is based on the use of paraetric odels. The paraetric odels assue a certain probability distribution, or a certain atheatical expression, as basis of the process that is wanted to describe. In these atheatical expressions, there are soe ters that ust be fit [2]. Therefore, the way to wor with these odels is first to define a basis odel in order to adjust the paraeters of this odel and then, to chec its perforance. However, one of the probles of odeling tie processes is the existence of groups that, for the sae basis odel, have different values of the paraeters. In these case, a classic approach to the proble is to carry out a clustering on the different processes depending on the tie characteristics; after this clustering, the paraeters can be fit to these groups. An extree approach in this sense is often ade in Anial Science [7-8]. In this approach, an average is carried out, and the paraeters are adjusted according to this average. Nonetheless, the use of this approach involves losing uch inforation about individual processes. Our approach is based on fitting the paraeters for each one of the tie processes, and then, obtaining a SOM, which is characterized by these paraeters. The ain advantages of our approach are the following: 1. It is fully generic and can be applied to any paraetric odel (linear or non-linear). 2. The clustering is applied to paraeters that depend on the odeling of the process, not on interediate results The fact of using a SOM allows a qualitative analysis for each group. Next section shows results achieved in a odeling exaple, using the proposed ethodology in a real proble, which presents serious difficulties in its odeling and ulterior inforation extraction: the prediction of Mil Yield (MY) in dairy goats [5]. It is a strong nonlinear proble, in which different goats have different tie characteristics. 4 Results Mil yield (MY) is a ey factor in the anageent of goat livestoc since it ay becoe the ain farer incoe. MY is iportant to identify which goats are the best il producers, also to analyze the current state of a certain livestoc. Therefore, MY prediction is very helpful as it enables the farer to ae decisions about livestoc anageent before the end of lactation, when decisions are usually ade. A hoogenous group of dairy goats within a coercial far (Excaur, S.L., Spain) was selected in this trial. This far is a eber of ACRIMUR (Murciano-Granadina Goat Breeder Association Asociación Española de Criadores de la Cabra Murciano-Granadina). The selected far is a representative saple of these inds of fars in South-Eastern Spain. The data set used in this wor was fored by 34 goats. A weely follow-up was carried out for every goat, being the study period of 22 wees. The following figure shows the il production for three different goats: Fig. 2. Weely il yield for three goats.

4 Proceedings of the 6th WSEAS International Conference on Applied Coputer Science, Tenerife, Canary Islands, Spain, Deceber 16-18, In order to odel MY of the different goats, a second order odel was taen into account, although the followed ethodology could be used for any type of odel. The solution of a second order differential equation can give the following odels: train the neural networ. Finally, the ap of coponents for the experiental data set was obtained, as it is shown in Fig. 4. Model_1: = + t e + 5 t e (3) 1 2 Model_2: 3 t = e cos( 4 t) (4) Model_3: 3 t = e sin( 4 t) (5) 4 The following step was to deterine the paraeters that appear in the equations (3)-(5) for each of the goats. A gradient procedure was followed using the least square function as cost function [2]. The third odel was the ost accurate one, and hence, this odel was analyzed using our ethodology. Figure 3 shows the achieved odeling. Fig. 3. Weely il production and odeling using the third odel. The different goats are arranged in the x-axis. As it is shown in Fig. 3, the odeling offered by the third odel was accurate enough. The following step was to obtain a SOM which was trained by using the values of the paraeters that appear in (5): 1, 2, 3, 4. An architecture with hexagonal neighbors was used, being the size of the ap equal to 65, according to the guidelines proposed in [4]. A batch algorith was used to Fig. 4. Map of coponents. Once the ap was obtained and using (5), it was possible to analyze the different tie behaviors that appeared in the data set, and it was also possible to copare the variation of the different paraeters altogether. As an exaple, the upper-left corner of the ap shows the lowest values of the paraeter 1 which correspond with the constant level throughout the wees; it also corresponds with the highest values of the paraeter 2 (factor that ultiplies the exponential). Moreover, 3 has a value close to zero, hence, this exponential becoes as a ultiplying factor of 1. Finally, the factor with lowest variation (or frequency) corresponds to paraeter 4. It eans that these goats, whose production is ore or less constant throughout the tie, are clustered in the sae group. A second exaple of inforation extraction can be obtained fro the lower-right area of paraeter 2 in Fig. 4; as it is shown, the tonality is rather dar, i.e., this paraeter is very close to zero. Therefore, the group of goats allocated in this area, present a ore or less constant MY

5 Proceedings of the 6th WSEAS International Conference on Applied Coputer Science, Tenerife, Canary Islands, Spain, Deceber 16-18, throughout the lactation and this production depends on the constant value of paraeter 1. In this area, there are 6 BMUs, which stands for a total of 17 goats (50% of the studied goats). Neuron 28 is the one that includes the highest nuber of goats (6 goats). It should be ephasized the relevance of the qualitative analysis that can be carried out on any area of the ap and, for any paraetric odel. 5 Conclusion In this counication, a new ethodology for the qualitative analysis of paraetric odels has been proposed. It is copletely general and can provide a descriptive nowledge of the processes involved in the odeling. Its effectiveness has been deonstrated on a nonlinear process, obtaining relevant qualitative inforation. References [1] G. Box, G.M Jenins, G. Reinsel, Tie Series Analysis: Forecasting & Control. Prentice [2] O. Nelles, Nonlinear Syste Identification: Fro Classical Approaches to Neural Networs and Fuzzy Models. Springer, [3] S. Hayin, Neural Networs: A Coprehensive Foundation, Prentice Hall, [4] T. Kohonen, Self Organizing Maps, Springer, [5] C. Fernández, E. Soria, J.D. Martín, A.J. Serrano, Neural Networs for Anial Science: Two case studies. Expert Systes with Applications, vol 31, pp , 2006.