Correlation Pattern between Temperatur, Humidity and Precipitaion by using Functional Canonical Correlation

 Understanding dependence structure and relationship between two sets of variables is of main interest in statistics. When encountering two large sets of variables, a researcher can express the relationship between the two sets by extracting only finite linear combinations of the original variables that produce the largest correlations with the second set of variables. When data are continuous functions of another variable (generally time), the methods of multivariate analysis can not be applied to the data. Therefore, some theoretical justification is needed to provide the required definitions and concepts regarding the essential nature of the data. This leads to difining canonical correlation for pairs of random functions called functional canonical correlation (FCCA). If the data related to functional phenomena, are observed discretely, the first task is to convert these observations to appropriate curves. This is due to functional nature of the phenomena that the data related to. On the other hand, the functional quantity of interest may be measured with error. In such cases, we should first remove the observational error bay taking a smoothing procedure to account. In this paper, Iranian weather data, collected in 2006, are treated by using FCCA. The data set contains discret measurments of three phenomena: temperature,humidity and precipitation and was collected from 102 weather stations. we have fitted continuous curves to the original data, and then extrated the correlation patterns between each pair of the three phenomena.