Fit a multilevel functional principal component model. The function uses two-step functional principal component decompositions.
MFDM(mort_female, mort_male, mort_ave, percent_1 = 0.95, percent_2 = 0.95, fh,
level = 80, alpha = 0.2, MCMCiter = 100, fmethod = c("auto_arima", "ets"),
BC = c(FALSE, TRUE), lambda)Female mortality (p by n matrix), where p denotes the dimension and n denotes the sample size.
Male mortality (p by n matrix).
Total mortality (p by n matrix).
Cumulative percentage used for determining the number of common functional principal components.
Cumulative percentage used for determining the number of sex-specific functional principal components.
Forecast horizon.
Nominal coverage probability of a prediction interval.
1 - Nominal coverage probability.
Number of MCMC iterations.
Univariate time-series forecasting method.
If Box-Cox transformation is performed.
If BC = TRUE, specify a Box-Cox transformation parameter.
Percentage of total variation explained by the first common functional principal component.
Percentage of total variation explained by the first female functional principal component in the residual.
Percentage of total variation explained by the first male functional principal component in the residual.
Forecast female mortality in the original scale.
Forecast male mortality in the original scale.
The basic idea of multilevel functional data method is to decompose functions from different sub-populations into an aggregated average, a sex-specific deviation from the aggregated average, a common trend, a sex-specific trend and measurement error. The common and sex-specific trends are modelled by projecting them onto the eigenvectors of covariance operators of the aggregated and sex-specific centred stochastic process, respectively.
C. M. Crainiceanu and J. Goldsmith (2010) "Bayesian functional data analysis using WinBUGS", Journal of Statistical Software, 32(11).
C-Z. Di and C. M. Crainiceanu and B. S. Caffo and N. M. Punjabi (2009) "Multilevel functional principal component analysis", The Annals of Applied Statistics, 3(1), 458-488.
V. Zipunnikov and B. Caffo and D. M. Yousem and C. Davatzikos and B. S. Schwartz and C. Crainiceanu (2015) "Multilevel functional principal component analysis for high-dimensional data", Journal of Computational and Graphical Statistics, 20, 852-873.
H. L. Shang, P. W. F. Smith, J. Bijak, A. Wisniowski (2016) "A multilevel functional data method for forecasting population, with an application to the United Kingdom", International Journal of Forecasting, 32(3), 629-649.