Harimino Andriamalala Rajaonarisoa, Irrish Parker Ramahazosoa, Hery Zojaona Tantely Stefana Zafimarina Reziky, Adolphe Andriamanga Ratiarison, University of Antananarivo, Madagascar

#### Abstract

The objective of this research is to find the best conventional high order fuzzy time series model for annual precipitation series in southern Madagascar. This work consists on finding the hyper parameters (number of partition of the universe of discourse and model order) to obtain the best conventional high order fuzzy time series model for our experimental data. In previous works, entitled spatial and temporal variability of precipitation in southern Madagascar, we subdivided the study area between 22 ° S to 30 ° S latitude and 43 ° Eto 48 ° E longitude into four zones of homogeneous precipitation. In this article, we seek to model annual precipitation data representative of one of these four areas. These data were taken between 1979 and 2017. Our approach consists on subdividing the data: data obtained from 1979 to 2001 (60%) for the training and data from 2002 to 2017 (40%) to test the model. To determine the number of partitions and model order, we fix first the number of partitions to 10 and then to 15, 20, 25,30, 35, 40, 45 and 50.For each of these values, we vary the model order from 1 to 10.Thenwe locate the model order which corresponds to the minimum of the average curve between the Mean Absolute Errors (MAE) between the training data and the test data. Thus, the orders of the candidate model are 2, 3, 5, and 6.The next step is to fix the model order with the previous values and vary the number of partitions from 3 to 50.For each couple of hyper parameter of the model (number of partitions, model order), we locate the value of number of partitions corresponding to the minimum of the average curve between the absolute mean of the errors or MAE (Mean Absolute Error) between the train and test data. We obtain the hyper-parameter pairs (37, 2), (20, 3), (35, 5) and (35, 6).The first pair gives the lowest Mean Absolute Error. As a final result, we obtain the best high order fuzzy time series model with hyperparameters number of partition equals thirty seven and of order equals two for annual precipitation in Southern of Madagascar.

#### Keywords

High Order Fuzzy Time Series, Mean Absolute Error, Number of Partition, Model order, Precipitation, Southern Madagascar.