Forecasting time series through functional principal components analysis

Alessandra Carioli investigated whether heterogeneity in fertility trends across the different provinces of Spain would persist in the future.

A. Carioli

A. Carioli


Spain is a highly heterogeneous country with different historical, cultural, and linguistic identities throughout the country. Previous period and cohort analysis of the fertility trend highlighted the persistence of heterogeneity in fertility trends across the different provinces. This analysis prompted the idea of investigating whether such trends would persist in the future.

Data and Method

The data used for the analysis is a 1975-2011 time series of age-specific fertility rates (ASFR) for Spain, 17 Spanish regions - NUTS2 - and 52 Spanish provinces - NUTS3. The approach used is a Time Series Forecasting using Functional Principal Component Analysis [1] [2].  The method uses time series of period fertility schedules and consists of three stages. In the first stage we smooth the fertility schedules using [3]. We then apply the principal component analysis (PCA) to obtain PC, which is further used for the forecasting. The final results consist of forecasted ASFR and Total Fertility Rates 15 years into the future. We first forecast  Spain as a whole, then the various regions, and then the provinces. To check the reliability of the model we implemented three different methods. The first approach uses the forecast of Spanish fertility schedules using the 1975-2000 time series and forecasts  first for 5 years and then 11 years. We then measure the difference between the observed and the forecast schedules. The second approach, “top-down,”  explains the changes in sub-national TFR through the explanatory framework of the national trends. The third approach uses sub-national estimates of fertility and adds them up to measure the accuracy of the model. In this way we obtain an estimated ASFR that can be compared to the observed ASFR. The third approach is a “bottom-up” approach. We first used the truncated time series to forecast fertility up to 2011 and then used the estimated ASFR and compared them with the national ASFR.


Even though period and cohort fertility analysis suggests heterogeneity among different provinces and shows recuperation of fertility in some selected areas, the recent economic crisis tempo effects on TFR are quite severe. This is reflected in the forecasts of national, regional, and provincial fertility schedules, where fertility shows a stable and slightly declining pattern, with the exception of the northwestern area.


A further step in this project would be to investigate the effects of the economic crisis by employing cohort fertility schedules and parity ASFR. 


[1] Hyndman RJ, & Booth H (2008). Stochastic population forecasts using functional data models for mortality, fertility and migration. International Journal of Forecasting, 24(3), 323–342. doi:10.1016/j.ijforecast.2008.02.009.
[2] Hyndman RJ & Shahid Ullah M (2007). Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics & Data Analysis, 51(10), 4942–4956. doi:10.1016/j.csda. 2006.07.028.
[3] Ediev D (2013). Comparative importance of the fertility model, the total fertility, the mean age and the standard deviation of age at childbearing in population projections. In IUSSP (pp. 1–31).


Alessandra Carioli, of the University of Groningen and the Netherlands Interdisciplinary Demographic Institute, is a citizen of Italy and is resident in the Netherlands. She was funded by IIASA's Dutch NMO and during YSSP worked in the World Population (POP) Program.

Please note these Proceedings have received limited or no review from supervisors and IIASA program directors, and the views and results expressed therein do not necessarily represent IIASA, its National Member Organizations, or other organizations supporting the work.

Print this page

Last edited: 19 August 2015


Tanja Huber

YSSP Coordinator & Team Leader

Young Scientists Summer Program

T +43(0) 2236 807 344

International Institute for Applied Systems Analysis (IIASA)
Schlossplatz 1, A-2361 Laxenburg, Austria
Phone: (+43 2236) 807 0 Fax:(+43 2236) 71 313