Forecasting for time series has always been of interest to statisticians and data scientists because it offers a lot of benefits in reality. This study proposes the fuzzy time series model which can both interpolate historical data, and forecast effectively for the future with the important contributions. First, we build the universal set based on the percentage of the original data variation, and divide it to clusters with the suitable number by the developed automatic algorithm. Next, the new fuzzy relationship between each element in series and the obtained clusters is established. The bigger the variation is, the more the clusters are divided. Finally, combining the two above improvements, we propose the new principle to forecast for the future. The experiments on many well-known data sets, including 3003 series of M3-competition data show that the proposed model has shown the outstanding advantage in comparing to the existing ones. Because the proposed model is established by the Matlab procedure, it can apply effectively for real series.

This article builds the fuzzy clustering algorithm for interval data (FCAI). In the proposed algorithm, we use the overlap distance as a criterion to cluster for interval data. The FCAI can determine not only the suitable number of clusters, the elements in each cluster but also the probability of assigning the elements to the established clusters at the same time. In addition, we also consider the convergence of the proposed algorithm by the theory and illustrated it by the numerical examples. The FCAI is applied well in image recognition, a problem with many challenges nowadays. Using the Grey Level Co-occurrence matrices (GLCMs), we propose a novel texture extraction approach to generate featured intervals. The complex computations of the FCAI can be performed conveniently and efficiently by the established Matlab program. We utilize the corrected rand indexes (CR) to find the suitable number of clusters while a partition entropy (PE) and partition coefficients (PC) are applied to argue the quality of fuzzy clusters. As a result, the experiments on the data sets having different characteristics and elements show the reasonableness of the proposed algorithm and its advantages in comparison to the existing ones. Regarding our best knowledge, it has also shown potential in the real application of this study.

This study establishes the new results for Cluster Width of probability Density functions (CWD). There are the upper and lower bounds of CWD and the relationships of CWD to other measures in statistical discriminant. The CWD for two and more two probability density functions is determined in the di fferent cases. Based on CWD, we propose a measure called similar coefficient to evaluate the quality of the established clusters. Furthermore, CWD is also used as a criterion to build two algorithms: to determine the suitable number of clusters and to analyse the fuzzy clusters. The numerical examples are given to illustrate the proposed algorithms and to prove their advantages over existing methods.