Extended Abstract. When data are in the form of continuous functions, they may challenge classical methods of data analysis based on arguments in finite dimensional spaces, and therefore need theoretical justification. Infinite dimensionality of spaces that data belong to, leads to major statistical methodologies and new insights for analyzing them, which is called functional data analysis (FDA).

Dimension reduction in FDA is mandatory, and is partly done by using principal components analysis (PCA). Similar to classical PCA, functional principal components analysis (FPCA) produces a small number of constructed variables from the original data that are uncorrelated and account for most of the variation in the original data set. Therefore, it helps us to understand the underlying structure of the data.

Temperature and amount of precipitation are functions of time, so they can be analyzed by FDA. In this paper, we have treated Iranian temperature and precipitation in 2005, extract patterns of variation, explore the structure of the data, and that of correlation between the two phenomena. The data, collected from the weather stations across the country, were discrete and associated with the monthly mean of temperature and precipitation recorded at each station. However, we have first fitted appropriate curves to them in which we have taken smoothing methods into account. Then, we have started analyzing the data using FPCA, and interpreting the results. When estimating the eigenvalues, we have found that the first estimated eigenvalue $hat {theta}$ shows a strong domination of its associated variation on all other kinds. Furthermore, the first two eigenvalues explain more than 98% of the total variation, inwhich their contributions individually were 93.7 and 4.3 percent, respectively. Contributions from others, however, were less than 2 percent. Thus, we have only considered the first two components.

The first estimated principal component (PC) shows that the majority of variability among the data can be attributed to differences between summer and winter temperatures. The second PC shows regularity of temperature when moving from winter to summer. In other words, it reflects the variation from the average of the difference between the winter and summer temperatures. Furthermore, bootstrap confidence bands for eigenvalues and eigenfunctions of the real data were obtained. They contain both individual and simultaneous confidence intervals for the eigenvalues. We have also obtained single and double bootstrap bands for the first two eigenfunctions, and seen that they are extremely close to each other, reflecting the high degree of accuracy of the bands that are obtained by the single bootstrap methods.