ABSTRACT

As the field of study expands, or as the number of observations in a sample increases, data observed at discrete points is generally assumed to be sampled from an underlying real function. As the number of observation points increases, those observations are likely to be sampled from a real-valued function. In this case, the derived data will be examples of a functional structure. We analyzed the daily average temperature data at 65 discrete points in 18 cities in Turkey's Black Sea Region. Fourier basis functions were used as a basis function approach because the temperature data had a periodic structure. The data were then transformed into a continuous function using the basis function and roughness penalty approach. Functional data were obtained using the roughness penalty approach. The generalized cross-validation method was used to determine the smoothing parameter of the variable (temperature variable). Finally, smoothed functional principal components analysis was applied to the functional data. In this way, changes in temperature functions, which seem hard to tackle, were evaluated on the same graph using the mean function generated for the principal component function and using the functions and the mean function obtained using the principal component function.

摘要

随着研究领域的扩大，或者随着样本中观测值数量的增加，在离散点观察到的数据通常被假定为从底层实函数中采样。随着观察点数量的增加，这些观察结果很可能是从实值函数中采样的。在这种情况下，派生数据将是功能结构的示例。我们分析了土耳其黑海地区 18 个城市 65 个离散点的日平均气温数据。傅立叶基函数被用作基函数方法，因为温度数据具有周期性结构。然后使用基函数和粗糙度惩罚方法将数据转换为连续函数。使用粗糙度惩罚方法获得功能数据。广义交叉验证方法用于确定变量（温度变量）的平滑参数。最后，对函数数据应用平滑函数主成分分析。以这种方式，使用为主成分函数生成的均值函数和使用主成分函数获得的函数和均值函数在同一张图上评估似乎难以解决的温度函数的变化。