Type:	Package
Title:	Functional Time Series Analysis
Version:	6.6
Date:	2025-02-22
Depends:	R (≥ 3.5.0), forecast, rainbow, sde
Suggests:	fds, R2jags, meboot
Imports:	colorspace, MASS, pcaPP, fda, pdfCluster, ecp, strucchange, e1071, psych, fGarch, KernSmooth, vars, boot, fdapace, LaplacesDemon, evgam, ROOPSD, glue, methods
LazyLoad:	yes
LazyData:	yes
LazyDataCompression:	xz
ByteCompile:	TRUE
Maintainer:	Han Lin Shang <hanlin.shang@mq.edu.au>
Description:	Functions for visualizing, modeling, forecasting and hypothesis testing of functional time series.
License:	GPL-3
NeedsCompilation:	no
Packaged:	2025-02-22 12:00:48 UTC; hanlinshang
Author:	Rob Hyndman [aut], Han Lin Shang [aut, cre, cph]
Repository:	CRAN
Date/Publication:	2025-02-22 13:00:02 UTC

Functional Time Series Analysis

Description

This package presents descriptive statistics of functional data; implements principal component regression and partial least squares regression to provide point and distributional forecasts for functional data; utilizes functional linear regression, ordinary least squares, penalized least squares, ridge regression, and moving block approaches to dynamically update point and distributional forecasts when partial data points in the most recent curve are observed; performs stationarity test for a functional time series; estimates a long-run covariance function by kernel sandwich estimator.

Author(s)

Rob J Hyndman and Han Lin Shang

Maintainer: Han Lin Shang <hanlin.shang@anu.edu.au>

References

########################### # References in Statistics ###########################

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series (with discussion)", Journal of the Korean Statistical Society, 38(3), 199-221.

R. J. Hyndman and H. L. Shang (2010) "Rainbow plots, bagplots, and boxplots for functional data", Journal of Computational and Graphical Statistics, 19(1), 29-45.

H. L. Shang and R. J. Hyndman (2011) "Nonparametric time series forecasting with dynamic updating", Mathematics and Computers in Simulation, 81(7), 1310-1324.

H. L. Shang (2011) "rainbow: an R package for visualizing functional time series, The R Journal, 3(2), 54-59.

H. L. Shang (2013) "Functional time series approach for forecasting very short-term electricity demand", Journal of Applied Statistics, 40(1), 152-168.

H. L. Shang (2013) "ftsa: An R package for analyzing functional time series", The R Journal, 5(1), 64-72.

H. L. Shang (2014) "A survey of functional principal component analysis", Advances in Statistical Analysis, 98(2), 121-142.

H. L. Shang (2014) "Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density", Journal of Nonparametric Statistics, 26(3), 599-615.

H. L. Shang (2014) "Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density", Computational Statistics, 29(3-4), 829-848.

H. L. Shang (2015) "Resampling techniques for estimating the distribution of descriptive statistics of functional data", Communications in Statistics - Simulation and Computation, 44(3), 614- 635.

H. L. Shang (2016) "Mortality and life expectancy forecasting for a group of populations in developed countries: A robust multilevel functional data method", in C. Agostinelli, A. Basu, P. Filzmoser, D. Mukherjee (ed.), Recent Advances in Robust Statistics: Theory and Applications, Springer, India, pp. 169-184.

H. L. Shang (2016) "Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method", Annals of Applied Statistics, 10(3), 1639-1672.

H. L. Shang (2016) "A Bayesian approach for determining the optimal semi-metric and bandwidth in scalar-on-function quantile regression with unknown error density and dependent functional data", Journal of Multivariate Analysis, 146, 95-104.

H. L. Shang (2017) "Functional time series forecasting with dynamic updating: An application to intraday particulate matter concentration", Econometrics and Statistics, 1, 184-200.

H. L. Shang (2017) "Forecasting Intraday S&P 500 Index Returns: A Functional Time Series Approach", Journal of Forecasting, 36(7), 741-755.

H. L. Shang and R. J. Hyndman (2017) "Grouped functional time series forecasting: An application to age-specific mortality rates", Journal of Computational and Graphical Statistics, 26(2), 330-343.

G. Rice and H. L. Shang (2017) "A plug-in bandwidth selection procedure for long-run covariance estimation with stationary functional time series", Journal of Time Series Analysis, 38(4), 591-609.

P. Reiss, J. Goldsmith, H. L. Shang and R. T. Ogden (2017) "Methods for scalar-on-function regression", International Statistical Review, 85(2), 228-249.

P. Kokoszka, G. Rice and H. L. Shang (2017) "Inference for the autocovariance of a functional time series under conditional heteroscedasticity", Journal of Multivariate Analysis, 162, 32-50.

Y. Gao, H. L. Shang and Y. Yang (2017) "High-dimensional functional time series forecasting", in G. Aneiros, E. Bongiorno, R. Cao and P. Vieu (ed.), Functional Statistics and Related Fields, Springer, Cham, pp. 131-136.

Y. Gao and H. L. Shang (2017) "Multivariate functional time series forecasting: An application to age-specific mortality rates", Risks, 5(2), Article 21.

H. L. Shang (2018) "Visualizing rate of change: An application to age-specific fertility rates", Journal of the Royal Statistical Society: Series A (Statistics in Society), 182(1), 249-262.

H. L. Shang (2018) "Bootstrap methods for stationary functional time series", Statistics and Computing, 28(1), 1-10.

Y. Gao, H. L. Shang and Y. Yang (2019) "High-dimensional functional time series forecasting: An application to age-specific mortality rates", Journal of Multivariate Analysis, 170, 232-243.

D. Li, P. M. Robinson and H. L. Shang (2020) "Long-range dependent curve time series", Journal of the American Statistical Association: Theory and Methods, 115(530), 957-971.

H. L. Shang (2020) "A comparison of Hurst exponent estimators in long-range dependent curve time series", Journal of Time Series Econometrics, 12(1).

D. Li, P. M. Robinson and H. L. Shang (2021) "Local Whittle estimation of long range dependence for functional time series", Journal of Time Series Analysis, 42(5-6), 685-695.

H. L. Shang and R. Xu (2021) "Functional time series forecasting of extreme values", Communications in Statistics: Case Studies, Data Analysis and Applications, 7(2), 182-199.

U. Beyaztas, H. L. Shang and Z. Yaseen (2021) "Development of functional autoregressive model based exogenous hydrometeorological variables for river flow prediction", Journal of Hydrology, 598, 126380.

U. Beyaztas and H. L. Shang (2022) "Machine learning-based functional time series forecasting: Application to age-specific mortality rates", Forecasting, 4(1), 394-408.

Y. Yang, Y. Yang and H. L. Shang (2022) "Feature extraction for functional time series: Theory and application to NIR spectroscopy data", Journal of Multivariate Analysis, 189, 104863.

A. E. Fernandez, R. Jimenez and H. L. Shang (2022) "On projection methods for functional time series forecasting", Journal of Multivariate Analysis, 189, 104890.

H. L. Shang (2022) "Not all long-memory estimators are born equal: A case of non-stationary curve time series", The Canadian Journal of Statistics, 50(1), 357-380.

X. Huang, H. L. Shang and D. Pitt (2022) "Permutation entropy and its variants for measuring temporal dependence", Australian and New Zealand Journal of Statistics, 64(4), 442-477.

H. L. Shang, J. Cao and P. Sang (2022) "Stopping time detection of wood panel compression: A functional time series approach", Journal of the Royal Statistical Society: Series C, 71(5), 1205-1224.

C. Tang, H. L. Shang and Y. Yang (2022) "Clustering and forecasting multiple functional time series", The Annals of Applied Statistics, 16(4), 2523-2553.

J. Trinka, H. Haghbin, M. Maadooliat and H. L. Shang (2023) "Functional time series forecasting: Functional singular spectrum analysis approaches", Stat, 12(1), e621.

D. Li, P. M. Robinson and H. L. Shang (2023) "Nonstationary fractionally integrated functional time series", Bernoulli, 29(2), 1505-1526.

X. Huang and H. L. Shang (2023) "Nonlinear autocorrelation function of functional time series", Nonlinear Dynamics: An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 111, 2537-2554.

H. L. Shang (2023) "Sieve bootstrapping memory parameter in long-range dependent stationary functional time series", AStA Advances in Statistical Analysis, 107, 421-441.

E. Paparoditis and H. L. Shang (2023) "Bootstrap prediction bands for functional time series", Journal of the American Statistical Association: Theory and Methods, 118(542), 972-986.

Y. Gao, H. L. Shang and Y. Yang (2024) "Factor-augmented smoothing model for functional data", Statistica Sinica, 34(1), 1-26.

############################# # References in Population Studies #############################

H. L. Shang, H. Booth and R. J. Hyndman (2011) "Point and interval forecasts of mortality rates and life expectancy: a comparison of ten principal component methods, Demographic Research, 25(5), 173-214.

H. L. Shang (2012) "Point and interval forecasts of age-specific fertility rates: a comparison of functional principal component methods", Journal of Population Research, 29(3), 249-267.

H. L. Shang (2012) "Point and interval forecasts of age-specific life expectancies: a model averaging", Demographic Research, 27, 593-644.

H. L. Shang, A. Wisniowski, J. Bijak, P. W. F. Smith and J. Raymer (2014) "Bayesian functional models for population forecasting", in M. Marsili and G. Capacci (eds), Proceedings of the Sixth Eurostat/UNECE Work Session on Demographic Projections, Istituto nazionale di statistica, Rome, pp. 313-325.

H. L. Shang (2015) "Selection of the optimal Box-Cox transformation parameter for modelling and forecasting age-specific fertility", Journal of Population Research, 32(1), 69-79.

H. L. Shang (2015) "Forecast accuracy comparison of age-specific mortality and life expectancy: Statistical tests of the results", Population Studies, 69(3), 317-335.

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.

H. L. Shang (2017) "Reconciling forecasts of infant mortality rates at national and sub-national levels: Grouped time-series method", Population Research and Policy Review, 36(1), 55-84.

R. J. Hyndman, Y. Zeng and H. L. Shang (2021) "Forecasting the old-age dependency ratio to determine the best pension age", Australian and New Zealand Journal of Statistics, 63(2), 241-256.

Y. Yang and H. L. Shang (2022) "Is the group structure important in grouped functional time series?", Journal of Data Science, 20(3), 303-324.

H. L. Shang and Y. Yang (2022) "Forecasting Australian subnational age-specific mortality rates", Journal of Population Research, 38, 1-24.

Y. Yang, H. L. Shang and J. Raymer (2024) "Forecasting Australian fertility by age, region, and birthplace", International Journal of Forecasting, in press.

########################### # References in Actuarial Studies ###########################

H. L. Shang and S. Haberman (2017) "Grouped multivariate and functional time series forecasting: An application to annuity pricing", Presented at the Living to 100 Symposium, Orlando Florida, January 4-6, 2017.

H. L. Shang and S. Haberman (2017) "Grouped multivariate and functional time series forecasting: An application to annuity pricing", Insurance: Mathematics and Economics, 75, 166-179.

H. L. Shang and S. Haberman (2018) "Model confidence sets and forecast combination: An application to age-specific mortality", Genus - Journal of Population Sciences, 74, Article number: 19.

H. L. Shang and S. Haberman (2020) "Forecasting multiple functional time series in a group structure: an application to mortality", ASTIN Bulletin, 50(2), 357-379.

H. L. Shang (2020) "Dynamic principal component regression for forecasting functional time series in a group structure", Scandinavian Actuarial Journal, 2020(4), 307-322.

H. L. Shang and S. Haberman (2020) "Forecasting age distribution of death counts: An application to annuity pricing", Annals of Actuarial Science, 14(1), 150-169.

H. L .Shang and S. Haberman and R. Xu (2022) "Multi-population modelling and forecasting age-specific life-table death counts", Insurance: Mathematics and Economics, 106, 239-253.

#################### # References in Finance ####################

F. Kearney and H. L. Shang (2020) "Uncovering predictability in the evolution of the WTI oil futures curve", European Financial Management, 26(1), 238-257.

H. L. Shang, K. Ji and U. Beyaztas (2021) "Granger causality of bivariate stationary curve time series", Journal of Forecasting, 40(4), 626-635.

S. Butler, P. Kokoszka, H. Miao and H. L. Shang (2021) "Neural network prediction of crude oil futures using B-splines", Energy Economics, 94, 105080.

H. L. Shang and F. Kearney (2022) "Dynamic functional time series forecasts of foreign exchange implied volatility surfaces", International Journal of Forecasting, 38(3), 1025-1049.

H. L. Shang and K. Ji (2023) "Forecasting intraday financial time series with sieve bootstrapping and dynamic updating", Journal of Forecasting, 42(8), 1973-1988.

The US female log-mortality rate from 1959-2020 and 3 states (New York, California, Illinois).

Description

We generate for the female population in the US. The functional time series corresponding to the log mortality data in each of the 3 states. Each functional time series comprises the ages from 0 to 100+.

Usage

data("all_hmd_male_data")

Format

A n x p matrix with n=186 observations on the following p=101 ages from 0 to 100+.

Details

The data generated corresponds to the FTS for the female US log-mortality. The matrix contains 186 FTS stacked by rows. They correspond to 62 (number of years) times 3 (states). Each FTS contains 101 functional values.

References

United States Mortality Database (2023). University of California, Berkeley (USA). Department of Demography at the University of California, Berkeley. Available at usa.mortality.org (data downloaded on March 15, 2023).

Examples

data(all_hmd_male_data)

The US male log-mortality rate from 1959-2020 and 3 states (New York, California, Illinois).

Description

We generate for the male population in the US. The functional time series corresponding to the log mortality data in each of the 3 states. Each functional time series comprises the ages from 0 to 100+.

Usage

data("all_hmd_male_data")

Format

A n x p matrix with n=186 observations on the following p=101 ages from 0 to 100+.

Details

The data generated corresponds to the FTS for the male US log-mortality. The matrix contains 186 FTS stacked by rows. They correspond to 62 (number of years) times 3 (states). Each FTS contains 101 functional values.

References

United States Mortality Database (2023). University of California, Berkeley (USA). Department of Demography at the University of California, Berkeley. Available at usa.mortality.org (data downloaded on March 15, 2023).

Examples

data(all_hmd_male_data)

Mean function, variance function, median function, trim mean function of functional data

Description

Mean function, variance function, median function, trim mean function of functional data

Usage

centre(x, type)

Arguments

Value

Return mean function, variance function, median function or trim mean function.

Author(s)

Han Lin Shang

See Also

pcscorebootstrapdata, mean.fts, median.fts, sd.fts, var.fts

Examples

# mean function is often removed in the functional principal component analysis.
# trimmed mean function is sometimes employed for robustness in the presence of outliers.
# In calculating trimmed mean function, several functional depth measures were employed.	
centre(x = ElNino_ERSST_region_1and2$y, type = "mean")
centre(x = ElNino_ERSST_region_1and2$y, type = "var")
centre(x = ElNino_ERSST_region_1and2$y, type = "median")
centre(x = ElNino_ERSST_region_1and2$y, type = "trimmed")

Compositional data analytic approach and nonparametric function-on-function regression for forecasting density

Description

Log-ratio transformation from constrained space to unconstrained space, where a standard nonparametric function-on-function regression can be applied.

Usage

CoDa_BayesNW(data, normalization, m = 5001, 
	band_choice = c("Silverman", "DPI"), 
	kernel = c("gaussian", "epanechnikov"))

Arguments

Details

1) Compute the geometric mean function 2) Apply the centered log-ratio transformation 3) Apply a nonparametric function-on-function regression to the transformed data 4) Transform forecasts back to the compositional data 5) Add back the geometric means, to obtain the forecasts of the density function

Value

Out-of-sample density forecasts

Author(s)

Han Lin Shang

References

Egozcue, J. J., Diaz-Barrero, J. L. and Pawlowsky-Glahn, V. (2006) ‘Hilbert space of probability density functions based on Aitchison geometry’, Acta Mathematica Sinica, 22, 1175-1182.

Ferraty, F. and Shang, H. L. (2021) ‘Nonparametric density-on-density regression’, working paper.

See Also

CoDa_FPCA

Examples

## Not run: 
CoDa_BayesNW(data = DJI_return, normalization = "TRUE", 
		band_choice = "DPI", kernel = "epanechnikov")

## End(Not run)	

Compositional data analytic approach and functional principal component analysis for forecasting density

Description

Log-ratio transformation from constrained space to unconstrained space, where a standard functional principal component analysis can be applied.

Usage

CoDa_FPCA(data, normalization, h_scale = 1, m = 5001, 
	band_choice = c("Silverman", "DPI"), 
	kernel = c("gaussian", "epanechnikov"), 
	varprop = 0.99, fmethod)

Arguments

Details

1) Compute the geometric mean function 2) Apply the centered log-ratio transformation 3) Apply FPCA to the transformed data 4) Forecast principal component scores 5) Transform forecasts back to the compositional data 6) Add back the geometric means, to obtain the forecasts of the density function

Value

Out-of-sample forecast densities

Author(s)

Han Lin Shang

References

Boucher, M.-P. B., Canudas-Romo, V., Oeppen, J. and Vaupel, J. W. (2017) ‘Coherent forecasts of mortality with compositional data analysis’, Demographic Research, 37, 527-566.

Egozcue, J. J., Diaz-Barrero, J. L. and Pawlowsky-Glahn, V. (2006) ‘Hilbert space of probability density functions based on Aitchison geometry’, Acta Mathematica Sinica, 22, 1175-1182.

See Also

Horta_Ziegelmann_FPCA, LQDT_FPCA, skew_t_fun

Examples

## Not run: 
CoDa_FPCA(data = DJI_return, normalization = "TRUE", band_choice = "DPI", 
	kernel = "epanechnikov", varprop = 0.9, fmethod = "ETS")

## End(Not run)

Differences of a functional time series

Description

Computes differences of a fts object at each variable.

Usage

## S3 method for class 'fts'
diff(x, lag = 1, differences = 1, ...)

Arguments

Value

An object of class fts.

Author(s)

Rob J Hyndman

Examples

# ElNino is an object of sliced functional time series.
# Differencing is sometimes used to achieve stationarity.	
diff(x = ElNino_ERSST_region_1and2)

Dow Jones Industrial Average (DJIA)

Description

Dow Jones Industrial Average (DJIA) is a stock market index that shows how 30 large publicly owned companies based in the United States have traded during a standard NYSE trading session. We consider monthly cross-sectional returns from April 2004 to December 2017. The data were obtained from the CRSP (Center for Research in Security Prices) database.

Usage

data("DJI_return")

Format

A data matrix

References

Kokoszka, P., Miao, H., Petersen, A. and Shang, H. L. (2019) ‘Forecasting of density functions with an application to cross-sectional and intraday returns’, International Journal of Forecasting, 35(4), 1304-1317.

Examples

data(DJI_return)

Dynamic multilevel functional principal component analysis

Description

Functional principal component analysis is used to decompose multiple functional time series. This function uses a functional panel data model to reduce dimensions for multiple functional time series.

Usage

dmfpca(y, M = NULL, J = NULL, N = NULL, tstart = 0, tlength = 1)

Arguments

Value

Author(s)

Chen Tang and Han Lin Shang

References

Rice, G. and Shang, H. L. (2017) "A plug-in bandwidth selection procedure for long-run covariance estimation with stationary functional time series", Journal of Time Series Analysis, 38, 591-609.

Shang, H. L. (2016) "Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method", The Annals of Applied Statistics, 10, 1639-1672.

Di, C.-Z., Crainiceanu, C. M., Caffo, B. S. and Punjabi, N. M. (2009) "Multilevel functional principal component analysis", The Annals of Applied Statistics, 3, 458-488.

See Also

mftsc

Examples

## The following takes about 10 seconds to run ##
## Not run: 
y <- do.call(rbind, sim_ex_cluster) 
MFPCA.sim <- dmfpca(y, M = length(sim_ex_cluster), J = nrow(sim_ex_cluster[[1]]), 
				    N = ncol(sim_ex_cluster[[1]]), tlength = 1)

## End(Not run)

Dynamic updates via functional linear regression

Description

A functional linear regression is used to address the problem of dynamic updating, when partial data in the most recent curve are observed.

Usage

dynamic_FLR(dat, newdata, holdoutdata, order_k_percent = 0.9, order_m_percent = 0.9, 
    pcd_method = c("classical", "M"), robust_lambda = 2.33, bootrep = 100, 
    	pointfore, level = 80)

Arguments

Details

This function is designed to dynamically update point and interval forecasts, when partial data in the most recent curve are observed.

Value

Author(s)

Han Lin Shang

References

H. Shen and J. Z. Huang (2008) "Interday forecasting and intraday updating of call center arrivals", Manufacturing and Service Operations Management, 10(3), 391-410.

H. Shen (2009) "On modeling and forecasting time series of curves", Technometrics, 51(3), 227-238.

H. L. Shang and R. J. Hyndman (2011) "Nonparametric time series forecasting with dynamic updating", Mathematics and Computers in Simulation, 81(7), 1310-1324.

J-M. Chiou (2012) "Dynamical functional prediction and classification with application to traffic flow prediction", Annals of Applied Statistics, 6(4), 1588-1614.

H. L. Shang (2013) "Functional time series approach for forecasting very short-term electricity demand", Journal of Applied Statistics, 40(1), 152-168.

H. L. Shang (2015) "Forecasting Intraday S&P 500 Index Returns: A Functional Time Series Approach", Journal of Forecasting, 36(7), 741-755.

H. L. Shang (2017) "Functional time series forecasting with dynamic updating: An application to intraday particulate matter concentration", Econometrics and Statistics, 1, 184-200.

See Also

dynupdate

Examples

dynamic_FLR_point = dynamic_FLR(dat = ElNino_ERSST_region_1and2$y[,1:68], 
	newdata = ElNino_ERSST_region_1and2$y[1:4,69], 
	holdoutdata = ElNino_ERSST_region_1and2$y[5:12,69], pointfore = TRUE)

dynamic_FLR_interval = dynamic_FLR(dat = ElNino_ERSST_region_1and2$y[,1:68], 
	newdata = ElNino_ERSST_region_1and2$y[1:4,69], 
	holdoutdata = ElNino_ERSST_region_1and2$y[5:12,69], pointfore = FALSE)

Dynamic updates via BM, OLS, RR and PLS methods

Description

Four methods, namely block moving (BM), ordinary least squares (OLS) regression, ridge regression (RR), penalized least squares (PLS) regression, were proposed to address the problem of dynamic updating, when partial data in the most recent curve are observed.

Usage

dynupdate(data, newdata = NULL, holdoutdata, method = c("ts", "block", 
 "ols", "pls", "ridge"), fmethod = c("arima", "ar", "ets", "ets.na", 
  "rwdrift", "rw"), pcdmethod = c("classical", "M", "rapca"), 
   ngrid = max(1000, ncol(data$y)), order = 6, 
    robust_lambda = 2.33, lambda = 0.01, value = FALSE, 
     interval = FALSE, level = 80, 
      pimethod = c("parametric", "nonparametric"), B = 1000)

Arguments

Details

This function is designed to dynamically update point and interval forecasts, when partial data in the most recent curve are observed.

If method = "classical", then standard functional principal component decomposition is used, as described by Ramsay and Dalzell (1991).

If method = "rapca", then the robust principal component algorithm of Hubert, Rousseeuw and Verboven (2002) is used.

If method = "M", then the hybrid algorithm of Hyndman and Ullah (2005) is used.

Value

Author(s)

Han Lin Shang

References

J. O. Ramsay and C. J. Dalzell (1991) "Some tools for functional data analysis (with discussion)", Journal of the Royal Statistical Society: Series B, 53(3), 539-572.

M. Hubert and P. J. Rousseeuw and S. Verboven (2002) "A fast robust method for principal components with applications to chemometrics", Chemometrics and Intelligent Laboratory Systems, 60(1-2), 101-111.

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

H. Shen and J. Z. Huang (2008) "Interday forecasting and intraday updating of call center arrivals", Manufacturing and Service Operations Management, 10(3), 391-410.

H. Shen (2009) "On modeling and forecasting time series of curves", Technometrics, 51(3), 227-238.

H. L. Shang and R. J. Hyndman (2011) "Nonparametric time series forecasting with dynamic updating", Mathematics and Computers in Simulation, 81(7), 1310-1324.

H. L. Shang (2013) "Functional time series approach for forecasting very short-term electricity demand", Journal of Applied Statistics, 40(1), 152-168.

H. L. Shang (2017) "Forecasting Intraday S&P 500 Index Returns: A Functional Time Series Approach", Journal of Forecasting, 36(7), 741-755.

H. L. Shang (2017) "Functional time series forecasting with dynamic updating: An application to intraday particulate matter concentration", Econometrics and Statistics, 1, 184-200.

See Also

ftsm, forecast.ftsm, plot.fm, residuals.fm, summary.fm

Examples

  
# ElNino is an object of sliced functional time series, constructed from a univariate time series. 
# When we observe some newly arrived information in the most recent time period, this function  
# allows us to update the point and interval forecasts for the remaining time period. 
dynupdate(data = ElNino_ERSST_region_1and2, newdata = ElNino_ERSST_region_1and2$y[1:4,69], 
	holdoutdata = ElNino_ERSST_region_1and2$y[5:12,57], method = "block", interval = FALSE) 

Selection of the number of principal components

Description

Eigenvalue ratio and growth ratio

Usage

ER_GR(data)

Arguments

Value

Author(s)

Han Lin Shang

References

Lam, C. and Yao, Q. (2012). Factor modelling for high-dimensional time series: Inference for the number of factors. The Annals of Statistics, 40, 694-726.

Ahn, S. and Horenstein, A. (2013). Eigenvalue ratio test for the number of factors. Econometrica, 81, 1203-1227.

See Also

ftsm

Examples

ER_GR(pm_10_GR$y)

Forecast error measure

Description

Computes the forecast error measure.

Usage

error(forecast, forecastbench, true, insampletrue, method = c("me", "mpe", "mae", 
 "mse", "sse", "rmse", "mdae", "mdse", "mape", "mdape", "smape", 
  "smdape", "rmspe", "rmdspe", "mrae", "mdrae", "gmrae", 
   "relmae", "relmse", "mase", "mdase", "rmsse"), giveall = FALSE)

Arguments

Details

Bias measure:

If method = "me", the forecast error measure is mean error.

If method = "mpe", the forecast error measure is mean percentage error.

Forecast accuracy error measure:

If method = "mae", the forecast error measure is mean absolute error.

If method = "mse", the forecast error measure is mean square error.

If method = "sse", the forecast error measure is sum square error.

If method = "rmse", the forecast error measure is root mean square error.

If method = "mdae", the forecast error measure is median absolute error.

If method = "mape", the forecast error measure is mean absolute percentage error.

If method = "mdape", the forecast error measure is median absolute percentage error.

If method = "rmspe", the forecast error measure is root mean square percentage error.

If method = "rmdspe", the forecast error measure is root median square percentage error.

Forecast accuracy symmetric error measure:

If method = "smape", the forecast error measure is symmetric mean absolute percentage error.

If method = "smdape", the forecast error measure is symmetric median absolute percentage error.

Forecast accuracy relative error measure:

If method = "mrae", the forecast error measure is mean relative absolute error.

If method = "mdrae", the forecast error measure is median relative absolute error.

If method = "gmrae", the forecast error measure is geometric mean relative absolute error.

If method = "relmae", the forecast error measure is relative mean absolute error.

If method = "relmse", the forecast error measure is relative mean square error.

Forecast accuracy scaled error measure:

If method = "mase", the forecast error measure is mean absolute scaled error.

If method = "mdase", the forecast error measure is median absolute scaled error.

If method = "rmsse", the forecast error measure is root mean square scaled error.

Value

A numeric value.

Author(s)

Han Lin Shang

References

P. A. Thompson (1990) "An MSE statistic for comparing forecast accuracy across series", International Journal of Forecasting, 6(2), 219-227.

C. Chatfield (1992) "A commentary on error measures", International Journal of Forecasting, 8(1), 100-102.

S. Makridakis (1993) "Accuracy measures: theoretical and practical concerns", International Journal of Forecasting, 9(4), 527-529.

R. J. Hyndman and A. Koehler (2006) "Another look at measures of forecast accuracy", International Journal of Forecasting, 22(3), 443-473.

Examples

# Forecast error measures can be categorized into three groups: (1) scale-dependent, 
# (2) scale-independent but with possible zero denominator, 
# (3) scale-independent with non-zero denominator.
error(forecast = 1:2, true = 3:4, method = "mae")
error(forecast = 1:5, forecastbench = 6:10, true = 11:15, method = "mrae")
error(forecast = 1:5, forecastbench = 6:10, true = 11:15, insampletrue = 16:20, 
	giveall = TRUE)

Extract variables or observations

Description

Creates subsets of a fts object.

Usage

extract(data, direction = c("time", "x"), timeorder, xorder)

Arguments

Value

When xorder is specified, it returns a fts object with same argument as data but with a subset of x variables.

When timeorder is specified, it returns a fts object with same argument as data but with a subset of time variables.

Author(s)

Han Lin Shang

Examples

# ElNino is an object of class sliced functional time series.
# This function truncates the data series rowwise or columnwise.	
extract(data = ElNino_ERSST_region_1and2, direction = "time", 
	timeorder = 1980:2006) # Last 27 curves
extract(data = ElNino_ERSST_region_1and2, direction = "x", 
	xorder = 1:8) # First 8 x variables

Functional autocorrelation function

Description

Compute functional autocorrelation function at various lags

Usage

facf(fun_data, lag_value_range = seq(0, 20, by = 1))

Arguments

Details

The autocovariance at lag i is estimated by the function \widehat{\gamma}_i(t,s), a functional analog of the autocorrelation is defined as

\widehat{\rho}_i = \frac{\|\widehat{\gamma}_i\|}{\int \widehat{\gamma}_0(t,t)dt}.

Value

A vector of functional autocorrelation function at various lags

Author(s)

Han Lin Shang

References

L. Horv\'ath, G. Rice and S. Whipple (2016) Adaptive bandwidth selection in the long run covariance estimator of functional time series, Computational Statistics and Data Analysis, 100, 676-693.

Examples

facf_value = facf(fun_data = t(ElNino_ERSST_region_1and2$y))

Functional data forecasting through functional principal component autoregression

Description

The coefficients from the fitted object are forecasted using a multivariate time-series forecasting method. The forecast coefficients are then multiplied by the functional principal components to obtain a forecast curve.

Usage

farforecast(object, h = 10, var_type = "const", Dmax_value, Pmax_value,
	level = 80, PI = FALSE)

Arguments

Details

1. Decompose the smooth curves via a functional principal component analysis (FPCA).

2. Fit a multivariate time-series model to the principal component score matrix.

3. Forecast the principal component scores using the fitted multivariate time-series models. The order of VAR is selected optimally via an information criterion.

4. Multiply the forecast principal component scores by estimated principal components to obtain forecasts of f_{n+h}(x).

5. Prediction intervals are constructed by taking quantiles of the one-step-ahead forecast errors.

Value

Author(s)

Han Lin Shang

References

A. Aue, D. D. Norinho and S. Hormann (2015) "On the prediction of stationary functional time series", Journal of the American Statistical Association, 110(509), 378-392.

J. Klepsch, C. Kl\"uppelberg and T. Wei (2017) "Prediction of functional ARMA processes with an application to traffic data", Econometrics and Statistics, 1, 128-149.

See Also

forecast.ftsm, forecastfplsr

Examples

sqrt_pm10 = sqrt(pm_10_GR$y)
multi_forecast_sqrt_pm10 = farforecast(object = fts(seq(0, 23.5, by = 0.5), sqrt_pm10),
	h = 1, Dmax_value = 5, Pmax_value = 3)

Bootstrap independent and identically distributed functional data

Description

Computes bootstrap or smoothed bootstrap samples based on independent and identically distributed functional data.

Usage

fbootstrap(data, estad = func.mean, alpha = 0.05, nb = 200, suav = 0,
 media.dist = FALSE, graph = FALSE, ...)

Arguments

Value

A list containing the following components is returned.

Author(s)

Han Lin Shang

References

A. Cuevas and M. Febrero and R. Fraiman (2006), "On the use of the bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

A. Cuevas and M. Febrero and R. Fraiman (2007), "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2007) "A functional analysis of NOx levels: location and scale estimation and outlier detection", Computational Statistics, 22(3), 411-427.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2008) "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2010) "Measures of influence for the functional linear model with scalar response", Journal of Multivariate Analysis, 101(2), 327-339.

J. A. Cuesta-Albertos and A. Nieto-Reyes (2010) "Functional classification and the random Tukey depth. Practical issues", Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, 77, 123-130.

D. Gervini (2012) "Outlier detection and trimmed estimation in general functional spaces", Statistica Sinica, 22(4), 1639-1660.

H. L. Shang (2015) "Re-sampling techniques for estimating the distribution of descriptive statistics of functional data", Communication in Statistics–Simulation and Computation, 44(3), 614-635.

H. L. Shang (2018) Bootstrap methods for stationary functional time series, Statistics and Computing, 28(1), 1-10.

See Also

pcscorebootstrapdata

Examples

# Bootstrapping the distribution of a summary statistics of functional data.
fbootstrap(data = ElNino_ERSST_region_1and2)

Forecast functional time series

Description

The coefficients from the fitted object are forecasted using either an ARIMA model (method = "arima"), an AR model (method = "ar"), an exponential smoothing method (method = "ets"), a linear exponential smoothing method allowing missing values (method = "ets.na"), or a random walk with drift model (method = "rwdrift"). The forecast coefficients are then multiplied by the principal components to obtain a forecast curve.

Usage

## S3 method for class 'ftsm'
forecast(object, h = 10, method = c("ets", "arima", "ar", "ets.na", 
 "rwdrift", "rw", "struct", "arfima"), level = 80, jumpchoice = c("fit", 
  "actual"), pimethod = c("parametric", "nonparametric"), B = 100, 
   usedata = nrow(object$coeff), adjust = TRUE, model = NULL,
    damped = NULL, stationary = FALSE, ...)

Arguments

Details

1. Obtain a smooth curve f_t(x) for each t using a nonparametric smoothing technique.

2. Decompose the smooth curves via a functional principal component analysis.

3. Fit a univariate time series model to each of the principal component scores.

4. Forecast the principal component scores using the fitted time series models.

5. Multiply the forecast principal component scores by fixed principal components to obtain forecasts of f_{n+h}(x).

6. The estimated variances of the error terms (smoothing error and model residual error) are used to compute prediction intervals for the forecasts.

Value

List with the following components:

Author(s)

Rob J Hyndman

References

H. Booth and R. J. Hyndman and L. Tickle and P. D. Jong (2006) "Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions", Demographic Research, 15, 289-310.

B. Erbas and R. J. Hyndman and D. M. Gertig (2007) "Forecasting age-specific breast cancer mortality using functional data model", Statistics in Medicine, 26(2), 458-470.

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. Booth (2008) "Stochastic population forecasts using functional data models for mortality, fertility and migration", International Journal of Forecasting, 24(3), 323-342.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series" (with discussion), Journal of the Korean Statistical Society, 38(3), 199-221.

H. L. Shang (2012) "Functional time series approach for forecasting very short-term electricity demand", Journal of Applied Statistics, 40(1), 152-168.

H. L. Shang (2013) "ftsa: An R package for analyzing functional time series", The R Journal, 5(1), 64-72.

H. L. Shang, A. Wisniowski, J. Bijak, P. W. F. Smith and J. Raymer (2014) "Bayesian functional models for population forecasting", in M. Marsili and G. Capacci (eds), Proceedings of the Sixth Eurostat/UNECE Work Session on Demographic Projections, Istituto nazionale di statistica, Rome, pp. 313-325.

H. L. Shang (2015) "Selection of the optimal Box-Cox transformation parameter for modelling and forecasting age-specific fertility", Journal of Population Research, 32(1), 69-79.

H. L. Shang (2015) "Forecast accuracy comparison of age-specific mortality and life expectancy: Statistical tests of the results", Population Studies, 69(3), 317-335.

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.

See Also

ftsm, forecastfplsr, plot.ftsf, plot.fm, residuals.fm, summary.fm

Examples

# ElNino is an object of class sliced functional time series.
# Via functional principal component decomposition, the dynamic was captured 
# by a few principal components and principal component scores. 
# By using an exponential smoothing method, 
# the principal component scores are forecasted.
# The forecasted curves are constructed by forecasted principal components 
# times fixed principal components plus the mean function.	
forecast(object = ftsm(ElNino_ERSST_region_1and2), h = 10, method = "ets")              
forecast(object = ftsm(ElNino_ERSST_region_1and2, weight = TRUE))

Forecasting via a high-dimensional functional principal component regression

Description

Forecast high-dimensional functional principal component model.

Usage

## S3 method for class 'hdfpca'
forecast(object, h = 3, level = 80, B = 50, ...)

Arguments

Details

The low-dimensional factors are forecasted with autoregressive integrated moving average (ARIMA) models separately. The forecast functions are then calculated using the forecast factors. Bootstrap prediction intervals are constructed by resampling from the forecast residuals of the ARIMA models.

Value

Author(s)

Y. Gao and H. L. Shang

References

Y. Gao, H. L. Shang and Y. Yang (2018) High-dimensional functional time series forecasting: An application to age-specific mortality rates, Journal of Multivariate Analysis, forthcoming.

See Also

hdfpca, hd_data

Examples

## Not run: 
hd_model = hdfpca(hd_data, order = 2, r = 2)
hd_model_fore = forecast.hdfpca(object = hd_model, h = 1)

## End(Not run)

Forecast functional time series

Description

The decentralized response is forecasted by multiplying the estimated regression coefficient with the new decentralized predictor

Usage

forecastfplsr(object, components, h)

Arguments

Value

A fts class object, containing forecasts of responses.

Author(s)

Han Lin Shang

References

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series" (with discussion), Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

forecast.ftsm, ftsm, plot.fm, plot.ftsf, residuals.fm, summary.fm

Examples

# A set of functions are decomposed by functional partial least squares decomposition.	
# By forecasting univariate partial least squares scores, the forecasted curves are 
# obtained by multiplying the forecasted scores by fixed functional partial least 
# squares function plus fixed mean function.
forecastfplsr(object = ElNino_ERSST_region_1and2, components = 2, h = 5)

Functional partial least squares regression

Description

Fits a functional partial least squares (PLSR) model using nonlinear partial least squares (NIPALS) algorithm or simple partial least squares (SIMPLS) algorithm.

Usage

fplsr(data, order = 6, type = c("simpls", "nipals"), unit.weights =
 TRUE, weight = FALSE, beta = 0.1, interval = FALSE, method =
  c("delta", "boota"), alpha = 0.05, B = 100, adjust = FALSE,
   backh = 10)

Arguments

Details

Point forecasts:

The NIPALS function implements the orthogonal scores algorithm, as described in Martens and Naes (1989). This is one of the two classical PLSR algorthms, the other is the simple partial least squares regression in DeJong (1993). The difference between these two approaches is that the NIPALS deflates the original predictors and responses, while the SIMPLS deflates the covariance matrix of original predictors and responses. Thus, SIMPLS is more computationally efficient than NIPALS.

In a functional data set, the functional PLSR can be performed by setting the functional responses to be 1 lag ahead of the functional predictors. This idea has been adopted from the Autoregressive Hilbertian processes of order 1 (ARH(1)) of Bosq (2000).

Distributional forecasts:

Parametric method:

Influenced by the works of Denham (1997) and Phatak et al. (1993), one way of constructing prediction intervals in the PLSR is via a local linearization method (also known as the Delta method). It can be easily understood as the first two terms in a Taylor series expansion. The variance of coefficient estimators can be approximated, from which an analytic-formula based prediction intervals are constructed.

Nonparametric method:

After discretizing and decentralizing functional data f_t(x) and g_s(y), a PLSR model with K latent components is built. Then, the fit residuals o_s(y_i) between g_s(y_i) and \hat{g}_s(y_i) are calculated as

o_s(y_i)=g_s(y_i)-\hat{g}_s(y_i), i=1,...,p.

The next step is to generate B bootstrap samples o_s^b(y_i) by randomly sampling with replacement from [o_1(y_i),...,o_n(y_i)]. Adding bootstrapped residuals to the original response variables in order to generate new bootstrap responses,

g_s^b(y_i)=g_s(y_i)+o_s^b(y_i).

Then, the PLSR models are constructed using the centered and discretized predictors and bootstrapped responses to obtain the boostrapped regression coefficients and point forecasts, from which the empirical prediction intervals and kernel density plots are constructed.

Value

A list containing the following components is returned.

Author(s)

Han Lin Shang

References

S. Wold and A. Ruhe and H. Wold and W. J. Dunn (1984) "The collinearity problem in linear regression. The partial least squares (PLS) approach to generalized inverses", SIAM Journal of Scientific and Statistical Computing, 5(3), 735-743.

S. de Jong (1993) "SIMPLS: an alternative approach to partial least square regression", Chemometrics and Intelligent Laboratory Systems, 18(3), 251-263.

C J. F. Ter Braak and S. de Jong (1993) "The objective function of partial least squares regression", Journal of Chemometrics, 12(1), 41-54.

B. Dayal and J. MacGregor (1997) "Recursive exponentially weighted PLS and its applications to adaptive control and prediction", Journal of Process Control, 7(3), 169-179.

B. D. Marx (1996) "Iteratively reweighted partial least squares estimation for generalized linear regression", Technometrics, 38(4), 374-381.

L. Xu and J-H. Jiang and W-Q. Lin and Y-P. Zhou and H-L. Wu and G-L. Shen and R-Q. Yu (2007) "Optimized sample-weighted partial least squares", Talanta, 71(2), 561-566.

A. Phatak and P. Reilly and A. Penlidis (1993) "An approach to interval estimation in partial least squares regression", Analytica Chimica Acta, 277(2), 495-501.

M. Denham (1997) "Prediction intervals in partial least squares", Journal of Chemometrics, 11(1), 39-52.

D. Bosq (2000) Linear Processes in Function Spaces, New York: Springer.

N. Faber (2002) "Uncertainty estimation for multivariate regression coefficients", Chemometrics and Intelligent Laboratory Systems, 64(2), 169-179.

J. A. Fernandez Pierna and L. Jin and F. Wahl and N. M. Faber and D. L. Massart (2003) "Estimation of partial least squares regression prediction uncertainty when the reference values carry a sizeable measurement error", Chemometrics and Intelligent Laboratory Systems, 65(2), 281-291.

P. T. Reiss and R. T. Ogden (2007), "Functional principal component regression and functional partial least squares", Journal of the American Statistical Association, 102(479), 984-996.

C. Preda, G. Saporta (2005) "PLS regression on a stochastic process", Computational Statistics and Data Analysis, 48(1), 149-158.

C. Preda, G. Saporta, C. Leveder (2007) "PLS classification of functional data", Computational Statistics, 22, 223-235.

A. Delaigle and P. Hall (2012), "Methodology and theory for partial least squares applied to functional data", Annals of Statistics, 40(1), 322-352.

M. Febrero-Bande, P. Galeano, W. Gonz\'alez-Manteiga (2017), "Functional principal component regression and functional partial least-squares regression: An overview and a comparative study", International Statistical Review, 85(1), 61-83.

See Also

ftsm, forecast.ftsm, plot.fm, summary.fm, residuals.fm, plot.fmres

Examples

# When weight = FALSE, all observations are assigned equally.
# When weight = TRUE, all observations are assigned geometrically decaying weights.
fplsr(data = ElNino_ERSST_region_1and2, order = 6, type = "nipals")
fplsr(data = ElNino_ERSST_region_1and2, order = 6)
fplsr(data = ElNino_ERSST_region_1and2, weight = TRUE)
fplsr(data = ElNino_ERSST_region_1and2, unit.weights = FALSE)
fplsr(data = ElNino_ERSST_region_1and2, unit.weights = FALSE, weight = TRUE)

# The prediction intervals are calculated numerically.
fplsr(data = ElNino_ERSST_region_1and2, interval = TRUE, method = "delta")

# The prediction intervals are calculated by bootstrap method.
fplsr(data = ElNino_ERSST_region_1and2, interval = TRUE, method = "boota")

Fit functional time series model

Description

Fits a principal component model to a fts object. The function uses optimal orthonormal principal components obtained from a principal components decomposition.

Usage

ftsm(y, order = 6, ngrid = max(500, ncol(y$y)), method = c("classical", 
 "M", "rapca"), mean = TRUE, level = FALSE, lambda = 3, 
  weight = FALSE, beta = 0.1, ...)

Arguments

Details

If method = "classical", then standard functional principal component decomposition is used, as described by Ramsay and Dalzell (1991).

If method = "rapca", then the robust principal component algorithm of Hubert, Rousseeuw and Verboven (2002) is used.

If method = "M", then the hybrid algorithm of Hyndman and Ullah (2005) is used.

Value

Object of class “ftsm” with the following components:

Author(s)

Rob J Hyndman

References

J. O. Ramsay and C. J. Dalzell (1991) "Some tools for functional data analysis (with discussion)", Journal of the Royal Statistical Society: Series B, 53(3), 539-572.

M. Hubert and P. J. Rousseeuw and S. Verboven (2002) "A fast robust method for principal components with applications to chemometrics", Chemometrics and Intelligent Laboratory Systems, 60(1-2), 101-111.

B. Erbas and R. J. Hyndman and D. M. Gertig (2007) "Forecasting age-specific breast cancer mortality using functional data model", Statistics in Medicine, 26(2), 458-470.

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. Booth (2008) "Stochastic population forecasts using functional data models for mortality, fertility and migration", International Journal of Forecasting, 24(3), 323-342.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series (with discussion)", Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

ftsmweightselect, forecast.ftsm, plot.fm, plot.ftsf, residuals.fm, summary.fm

Examples

# ElNino is an object of class sliced functional time series, constructed 
# from a univariate time series. 
# By default, all observations are assigned with equal weighting. 	
ftsm(y = ElNino_ERSST_region_1and2, order = 6, method = "classical", weight = FALSE)
# When weight = TRUE, geometrically decaying weights are used.
ftsm(y = ElNino_ERSST_region_1and2, order = 6, method = "classical", weight = TRUE)

Forecast functional time series

Description

The coefficients from the fitted object are forecasted using either an ARIMA model (method = "arima"), an AR model (method = "ar"), an exponential smoothing method (method = "ets"), a linear exponential smoothing method allowing missing values (method = "ets.na"), or a random walk with drift model (method = "rwdrift"). The forecast coefficients are then multiplied by the principal components to obtain a forecast curve.

Usage

ftsmiterativeforecasts(object, components, iteration = 20)

Arguments

Details

1. Obtain a smooth curve f_t(x) for each t using a nonparametric smoothing technique.

2. Decompose the smooth curves via a functional principal component analysis.

3. Fit a univariate time series model to each of the principal component scores.

4. Forecast the principal component scores using the fitted time series models.

5. Multiply the forecast principal component scores by fixed principal components to obtain forecasts of f_{n+h}(x).

6. The estimated variances of the error terms (smoothing error and model residual error) are used to compute prediction intervals for the forecasts.

Value

List with the following components:

Author(s)

Han Lin Shang

References

H. Booth and R. J. Hyndman and L. Tickle and P. D. Jong (2006) "Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions", Demographic Research, 15, 289-310.

B. Erbas and R. J. Hyndman and D. M. Gertig (2007) "Forecasting age-specific breast cancer mortality using functional data model", Statistics in Medicine, 26(2), 458-470.

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. Booth (2008) "Stochastic population forecasts using functional data models for mortality, fertility and migration", International Journal of Forecasting, 24(3), 323-342.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series" (with discussion), Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

ftsm, plot.ftsf, plot.fm, residuals.fm, summary.fm

Examples

# Iterative one-step-ahead forecasts via functional principal component analysis.	
ftsmiterativeforecasts(object = Australiasmoothfertility, components = 2, iteration = 5)

Selection of the weight parameter used in the weighted functional time series model.

Description

The geometrically decaying weights are used to estimate the mean curve and functional principal components, where more weights are assigned to the more recent data than the data from the distant past.

Usage

ftsmweightselect(data, ncomp = 6, ntestyear, errorcriterion = c("mae", "mse", "mape"))

Arguments

Details

The data set is split into a fitting period and forecasting period. Using the data in the fitting period, we compute the one-step-ahead forecasts and calculate the forecast error. Then, we increase the fitting period by one, and carry out the same forecasting procedure until the fitting period covers entire data set. The forecast accuracy is determined by the averaged forecast error across the years in the forecasting period. By using an optimization algorithm, we select the optimal weight parameter that would result in the minimum forecast error.

Value

Optimal weight parameter.

Note

Can be computational intensive, as it takes about half-minute to compute. For example, ftsmweightselect(ElNinosmooth, ntestyear = 1).

Author(s)

Han Lin Shang

References

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series (with discussion)", Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

ftsm, forecast.ftsm

Fit a generalized additive extreme value model to the functional data with given basis numbers

Description

One-step-ahead forecast for any given quantile(s) of functional time sereies of extreme values using a generalized additive extreme value (GAEV) model.

Usage

GAEVforecast(data, q, d.loc.max = 10, d.logscale.max = 10)

Arguments

Details

For the functional time seres \{X_t(u),t=1,...,T,u\in \mathcal{I}\}, the GAEV model is given as

X_{t}(u) ~ GEV[\mu_{t}(u),\sigma_t(u),\xi_t],

where

\mu_t(u) = \beta^{(\mu)}_{t,0} + \sum_{i=1}^{d_1}\beta^{(\mu)}_{t,i}b^{(\mu)}_{i}(u),

\ln(\sigma_t(u)) = \beta^{(\sigma)}_{t,0} + \sum_{i=1}^{d_2}\beta^{(\sigma)}_{t,i}b^{(\sigma)}_{i}(u), \xi_t \in [0,\infty),

where d_{j},j=1,2 are positive integers of basis numbers, \{b^{(\mu)}_{i}(u),i=1,\dots,d_{1}\} and \{b^{(\sigma)}_{i}(u),i=1,\dots,d_{2}\} are the cubic regression spline basis functions.

The optimal number of basis functions (d_1,d_2) are chosen by minimizing the Kullback-Leibler divergence on the test set using a leave-one-out cross-validation technique.

The one-step-ahead forecast of the joint coefficients (\widehat{\beta^{(\mu)}}_{T+1,i},\widehat{\beta^{(\sigma)}}_{T+1,j},\widehat{\xi}_{T+1},i=0,...,d_1,j=0,...,d_2) are produced using a vector autoregressive model, whose order is selected via the corrected Akaike information criterion. Then the one-step-ahead forecast of the GEV parameter (\widehat{\mu}_{T+1}(u),\widehat{\sigma}_{T+1}(u),\widehat{\xi}_{T+1}) can be computed accordingly.

The one-step-ahead forecast for the \tau-th quantile of the extreme values \widehat{X}_{T+1}(u) is computed by

Q_{\tau}(u|\widehat{\mu}_{T+1},\widehat{\sigma}_{T+1},\widehat{\xi}_{T+1})

=

\widehat{\mu}_{T+1}(u) + \frac{\widehat{\sigma}_{T+1}(u) \big[(-\ln(\tau))^{-\widehat{\xi}_{T+1}}-1\big]}{\widehat{\xi}_{T+1}}, \xi > 0, \tau\in [0,1);\ \xi < 0, \tau\in (0,1], \\ \widehat{\mu}_{T+1}(u) - \widehat{\sigma}_{T+1}(u) \cdot \ln[-\ln\big(\tau)], \xi=0, \tau \in (0,1).

Value

Author(s)

Ruofan Xu and Han Lin Shang

References

Shang, H. L. and Xu, R. (2021) ‘Functional time series forecasting of extreme values’, Communications in Statistics Case Studies Data Analysis and Applications, in press.

Examples

## Not run: 
library(evd)
data = matrix(rgev(1000),ncol=50) 
GAEVforecast(data = data, q = c(0.02,0.7), d.loc.max = 5, d.logscale.max = 5)

## End(Not run)

Simulated high-dimensional functional time series

Description

We generate N populations of functional time series. For each i\in \{1,\dots, N\}, the ith function at time t\in \{1,\dots, T\} is given by

X_t^{(i)}(u) = \sum^2_{p=1}\beta_{p,t}^{(i)}\gamma_p^{(i)}(u) + \theta_t^{(i)}(u),

where \theta_t^{(i)}(u) = \sum^{\infty}_{p=3}\beta_{p,t}^{(i)}\gamma_p^{(i)}(u).

Usage

data("hd_data")

Details

The coefficients \beta_{p,t}^{(i)} for all N populations are combined and generated, for all p\in N, by

\bm{\beta}_{p,t} = \bm{A}_p\bm{f}_{p,t},

where \bm{\beta}_{p,t}=\{\beta_{p,t}^{1},\dots,\beta_{p,t}^N\}. Here, \bm{A}_p is an N\times N matrix, and \bm{f}_{p,t} is an N\times 1 vector. Furthermore, we assume that the \beta_{p,t}^{(i)}s have mean 0 and variance 0 when p>3, so we only construct the coefficients \bm{\beta}_{p,t} for p\in\{1, 2, 3\}.

The first set of coefficients \bm{\beta}_{1,t} for N populations are generated with \bm{\beta}_{1,t}=\bm{A}_1\bm{f}_{1,t}. Each element in the matrix \bm{A}_1 is generated by a_{ij}=N^{-1/4}\times b_{ij}, where b_{ij}\sim N(2,4).

The factors \bm{f}_{1,t} are generated using an autoregressive model of order 1, i.e., AR(1). Define the ith element in vector \bm{f}_{1,t} as f_{1,t}^{(i)}. Then, f_{1,t}^{1} is generated by f_{1,t}^{1}=0.5\times f_{1,t-1}^{1}+\omega_t, where \omega_t are independent N(0,1) random variables. We generate f_{1,t}^{(i)} for all i\in \{2,\dots, N\} by f_{1,t}^{(i)}=(1/N) \times g_t^{(i)}, where g_t^{(2)},\dots,g_t^{(N)} are also AR(1) and follow g_t^{(i)} = 0.2\times g_{t-1}^{(i)}+\omega_t. It is then ensured that most of the variance of \bm{\beta}_{1,t} can be explained by one factor. The second coefficient \bm{\beta}_{2,t} are constructed the same way as \bm{\beta}_{1,t}.

We also generate the third functional principal component scores \bm{\beta}_{3,t} but with small values. Moreover, \bm{A}_3 is generated by a_{ij}=N^{-1/4}\times b_{ij}, where b_{ij}\sim N(0, 0.04). The factors bm{f}_{3,t} are generated as \bm{f}_{1,t}.

The three basis functions are constructed by \gamma_1^{(i)}(u) = \sin(2\pi u + \pi i/2), \gamma_2^{(i)}(u) = \cos(2\pi u + \pi i/2) and \gamma_3^{(i)}(u) = \sin(4\pi u + \pi i/2), where u\in [0,1]. Finally, the functional time series for the ith population is constructed by

\bm{X}_t^{(i)}(u) = \bm{\beta}_{1,t}\gamma_1^{(i)}(u) + \bm{\beta}_{2,t}\gamma_2^{(i)}(u) + \bm{\beta}_{3,t}\gamma_3^{(i)}(u),

where (\cdot)_i denotes the ith element of the vector.

References

Y. Gao, H. L. Shang and Y. Yang (2018) High-dimensional functional time series forecasting: An application to age-specific mortality rates, Journal of Multivariate Analysis, forthcoming.

See Also

hdfpca, forecast.hdfpca

Examples

data(hd_data)

High-dimensional functional principal component analysis

Description

Fit a high dimensional functional principal component analysis model to a multiple-population of functional time series data.

Usage

hdfpca(y, order, q = sqrt(dim(y[[1]])[2]), r)

Arguments

Details

In the first step, dynamic functional principal component analysis is performed on each population and then in the second step, factor models are fitted to the resulting principal component scores. The high-dimensional functional time series are thus reduced to low-dimensional factors.

Value

Author(s)

Y. Gao and H. L. Shang

References

Y. Gao, H. L. Shang and Y. Yang (2018) High-dimensional functional time series forecasting: An application to age-specific mortality rates, Journal of Multivariate Analysis, forthcoming.

See Also

forecast.hdfpca, hd_data

Examples

hd_model = hdfpca(hd_data, order = 2, r = 2)

Dynamic functional principal component analysis for density forecasting

Description

Implementation of a dynamic functional principal component analysis to forecast densities.

Usage

Horta_Ziegelmann_FPCA(data, gridpoints, h_scale = 1, p = 5, m = 5001, 
	kernel = c("gaussian", "epanechnikov"), band_choice = c("Silverman", "DPI"), 
	VAR_type = "both", lag_maximum = 6, no_boot = 1000, alpha_val = 0.1, 
	ncomp_select = "TRUE", D_val = 10)

Arguments

Details

1) Compute a kernel covariance function 2) Via eigen-decomposition, a density can be decomposed into a set of functional principal components and their associated scores 3) Fit a vector autoregressive model to the scores with the order selected by Akaike information criterion 4) By multiplying the estimated functional principal components with the forecast scores, obtain forecast densities 5) Since forecast densities may neither be non-negative nor sum to one, normalize the forecast densities accordingly

Value

Author(s)

Han Lin Shang

References

Horta, E. and Ziegelmann, F. (2018) ‘Dynamics of financial returns densities: A functional approach applied to the Bovespa intraday index’, International Journal of Forecasting, 34, 75-88.

Bathia, N., Yao, Q. and Ziegelmann, F. (2010) ‘Identifying the finite dimensionality of curve time series’, The Annals of Statistics, 38, 3353-3386.

See Also

CoDa_FPCA, LQDT_FPCA, skew_t_fun

Examples

## Not run: 
Horta_Ziegelmann_FPCA(data = DJI_return, kernel = "epanechnikov", 
				band_choice = "DPI", ncomp_select = "FALSE")

## End(Not run)

Test for functional time series

Description

Tests whether an object is of class fts.

Usage

is.fts(x)

Arguments

Author(s)

Rob J Hyndman

Examples

# check if ElNino is the class of the functional time series.	
is.fts(x = ElNino_ERSST_region_1and2)

Integrated Squared Forecast Error for models of various orders

Description

Computes integrated squared forecast error (ISFE) values for functional time series models of various orders.

Usage

isfe.fts(data, max.order = N - 3, N = 10, h = 5:10, method = 
 c("classical", "M", "rapca"), mean = TRUE, level = FALSE, 
  fmethod = c("arima", "ar", "ets", "ets.na", "struct", "rwdrift", 
   "rw", "arfima"), lambda = 3, ...)

Arguments

Value

Numeric matrix with (max.order+1) rows and length(h) columns containing ISFE values for models of orders 0:(max.order).

Note

This function can be very time consuming for data with large dimensionality or large sample size. By setting max.order small, computational speed can be dramatically increased.

Author(s)

Rob J Hyndman

References

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

See Also

ftsm, forecast.ftsm, plot.fm, plot.fmres, summary.fm, residuals.fm

Estimating long-run covariance function for a functional time series

Description

Bandwidth estimation in the long-run covariance function for a functional time series, using different types of kernel function

Usage

long_run_covariance_estimation(dat, C0 = 3, H = 3)

Arguments

Value

An estimated covariance function of size (p by p)

Author(s)

Han Lin Shang

References

L. Horvath, G. Rice and S. Whipple (2016) Adaptive bandwidth selection in the long run covariance estimation of functional time series, Computational Statistics and Data Analysis, 100, 676-693.

G. Rice and H. L. Shang (2017) A plug-in bandwidth selection procedure for long run covariance estimation with stationary functional time series, Journal of Time Series Analysis, 38(4), 591-609.

D. Li, P. M. Robinson and H. L. Shang (2018) Long-range dependent curve time series, Journal of the American Statistical Association: Theory and Methods, under revision.

See Also

fts

Examples

dum = long_run_covariance_estimation(dat = ElNino_OISST_region_1and2$y[,1:5])

Log quantile density transform

Description

Probability density function, cumulative distribution function and quantile density function are three characterizations of a distribution. Of these three, quantile density function is the least constrained. The only constrain is nonnegative. By taking a log transformation, there is no constrain.

Usage

LQDT_FPCA(data, gridpoints, h_scale = 1, M = 3001, m = 5001, lag_maximum = 4, 
		no_boot = 1000, alpha_val = 0.1, p = 5, 
		band_choice = c("Silverman", "DPI"), 
		kernel = c("gaussian", "epanechnikov"), 
		forecasting_method = c("uni", "multi"), 
		varprop = 0.85, fmethod, VAR_type)

Arguments

Details

1) Transform the densities f into log quantile densities Y and c specifying the value of the cdf at 0 for the target density f. 2) Compute the predictions for future log quantile density and c value. 3) Transform the forecasts in Step 2) into the predicted density f.

Value

Author(s)

Han Lin Shang

References

Petersen, A. and Muller, H.-G. (2016) ‘Functional data analysis for density functions by transformation to a Hilbert space’, The Annals of Statistics, 44, 183-218.

Jones, M. C. (1992) ‘Estimating densities, quantiles, quantile densities and density quantiles’, Annals of the Institute of Statistical Mathematics, 44, 721-727.

See Also

CoDa_FPCA, Horta_Ziegelmann_FPCA, skew_t_fun

Examples

## Not run: 
LQDT_FPCA(data = DJI_return, band_choice = "DPI", kernel = "epanechnikov", 
			forecasting_method = "uni", fmethod = "ets")

## End(Not run)		

Maximum autocorrelation factors

Description

Dimension reduction via maximum autocorrelation factors

Usage

MAF_multivariate(data, threshold)

Arguments

Value

Author(s)

Han Lin Shang

References

M. A. Haugen, B. Rajaratnam and P. Switzer (2015). Extracting common time trends from concurrent time series: Maximum autocorrelation factors with applications, arXiv paper https://arxiv.org/abs/1502.01073.

See Also

ftsm

Examples

MAF_multivariate(data = pm_10_GR_sqrt$y, threshold = 0.85)

Mean functions for functional time series

Description

Computes mean of functional time series at each variable.

Usage

## S3 method for class 'fts'
mean(x, method = c("coordinate", "FM", "mode", "RP", "RPD", "radius"), 
 na.rm = TRUE, alpha, beta, weight, ...)

Arguments

Details

If method = "coordinate", it computes the coordinate-wise functional mean.

If method = "FM", it computes the mean of trimmed functional data ordered by the functional depth of Fraiman and Muniz (2001).

If method = "mode", it computes the mean of trimmed functional data ordered by h-modal functional depth.

If method = "RP", it computes the mean of trimmed functional data ordered by random projection depth.

If method = "RPD", it computes the mean of trimmed functional data ordered by random projection derivative depth.

If method = "radius", it computes the mean of trimmed functional data ordered by the notion of alpha-radius.

Value

A list containing x = variables and y = mean rates.

Author(s)

Rob J Hyndman, Han Lin Shang

References

O. Hossjer and C. Croux (1995) "Generalized univariate signed rank statistics for testing and estimating a multivariate location parameter", Journal of Nonparametric Statistics, 4(3), 293-308.

A. Cuevas and M. Febrero and R. Fraiman (2006) "On the use of bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

A. Cuevas and M. Febrero and R. Fraiman (2007), "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2007) "A functional analysis of NOx levels: location and scale estimation and outlier detection", Computational Statistics, 22(3), 411-427.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2008) "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2010) "Measures of influence for the functional linear model with scalar response", Journal of Multivariate Analysis, 101(2), 327-339.

J. A. Cuesta-Albertos and A. Nieto-Reyes (2010) "Functional classification and the random Tukey depth. Practical issues", Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, 77, 123-130.

D. Gervini (2012) "Outlier detection and trimmed estimation in general functional spaces", Statistica Sinica, 22(4), 1639-1660.

See Also

median.fts, var.fts, sd.fts, quantile.fts

Examples

# Calculate the mean function by the different depth measures.	
mean(x = ElNino_ERSST_region_1and2, method = "coordinate")
mean(x = ElNino_ERSST_region_1and2, method = "FM")
mean(x = ElNino_ERSST_region_1and2, method = "mode")
mean(x = ElNino_ERSST_region_1and2, method = "RP")
mean(x = ElNino_ERSST_region_1and2, method = "RPD")
mean(x = ElNino_ERSST_region_1and2, method = "radius", 
	alpha = 0.5, beta = 0.25, weight = "hard")
mean(x = ElNino_ERSST_region_1and2, method = "radius", 
	alpha = 0.5, beta = 0.25, weight = "soft")

Median functions for functional time series

Description

Computes median of functional time series at each variable.

Usage

## S3 method for class 'fts'
median(x, na.rm, method = c("hossjercroux", "coordinate", "FM", "mode", 
 "RP", "RPD", "radius"), alpha, beta, weight, ...)

Arguments

Details

If method = "coordinate", it computes a coordinate-wise median.

If method = "hossjercroux", it computes the L1-median using the Hossjer-Croux algorithm.

If method = "FM", it computes the median of trimmed functional data ordered by the functional depth of Fraiman and Muniz (2001).

If method = "mode", it computes the median of trimmed functional data ordered by h-modal functional depth.

If method = "RP", it computes the median of trimmed functional data ordered by random projection depth.

If method = "RPD", it computes the median of trimmed functional data ordered by random projection derivative depth.

If method = "radius", it computes the mean of trimmed functional data ordered by the notion of alpha-radius.

Value

A list containing x = variables and y = median rates.

Author(s)

Rob J Hyndman, Han Lin Shang

References

O. Hossjer and C. Croux (1995) "Generalized univariate signed rank statistics for testing and estimating a multivariate location parameter", Journal of Nonparametric Statistics, 4(3), 293-308.

A. Cuevas and M. Febrero and R. Fraiman (2006) "On the use of bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

A. Cuevas and M. Febrero and R. Fraiman (2007), "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2007) "A functional analysis of NOx levels: location and scale estimation and outlier detection", Computational Statistics, 22(3), 411-427.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2008) "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2010) "Measures of influence for the functional linear model with scalar response", Journal of Multivariate Analysis, 101(2), 327-339.

J. A. Cuesta-Albertos and A. Nieto-Reyes (2010) "Functional classification and the random Tukey depth. Practical issues", Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, 77, 123-130.

D. Gervini (2012) "Outlier detection and trimmed estimation in general functional spaces", Statistica Sinica, 22(4), 1639-1660.

See Also

mean.fts, var.fts, sd.fts, quantile.fts

Examples

# Calculate the median function by the different depth measures.	
median(x = ElNino_ERSST_region_1and2, method = "hossjercroux")
median(x = ElNino_ERSST_region_1and2, method = "coordinate")
median(x = ElNino_ERSST_region_1and2, method = "FM")
median(x = ElNino_ERSST_region_1and2, method = "mode")
median(x = ElNino_ERSST_region_1and2, method = "RP")
median(x = ElNino_ERSST_region_1and2, method = "RPD")
median(x = ElNino_ERSST_region_1and2, method = "radius", 
	alpha = 0.5, beta = 0.25, weight = "hard")
median(x = ElNino_ERSST_region_1and2, method = "radius", 
	alpha = 0.5, beta = 0.25, weight = "soft")

Multilevel functional data method

Description

Fit a multilevel functional principal component model. The function uses two-step functional principal component decompositions.

Usage

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)

Arguments

Details

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.

Value

Note

It can be quite time consuming, especially when MCMCiter is large.

Author(s)

Han Lin Shang

References

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.

See Also

ftsm, forecast.ftsm, fplsr, forecastfplsr

Multilevel functional principal component analysis for clustering

Description

A multilevel functional principal component analysis for performing clustering analysis

Usage

MFPCA(y, M = NULL, J = NULL, N = NULL)

Arguments

Value

Author(s)

Chen Tang, Yanrong Yang and Han Lin Shang

See Also

mftsc

Multiple funtional time series clustering

Description

Clustering the multiple functional time series. The function uses the functional panel data model to cluster different time series into subgroups

Usage

mftsc(X, alpha)

Arguments

Details

As an initial step, conventional k-means clustering is performed on the dynamic FPC scores, then an iterative membership updating process is applied by fitting the MFPCA model.

Value

Author(s)

Chen Tang, Yanrong Yang and Han Lin Shang

See Also

MFPCA

Examples

## Not run: 
data(sim_ex_cluster)
cluster_result<-mftsc(X=sim_ex_cluster, alpha=0.99)
cluster_result$member.final

## End(Not run)

Bootstrap independent and identically distributed functional data or functional time series

Description

Computes bootstrap or smoothed bootstrap samples based on either independent and identically distributed functional data or functional time series.

Usage

pcscorebootstrapdata(dat, bootrep, statistic, bootmethod = c("st", "sm", 
	"mvn", "stiefel", "meboot"), smo)

Arguments

Details

We will presume that each curve is observed on a grid of T points with 0\leq t_1<t_2\dots<t_T\leq \tau. Thus, the raw data set (X_1,X_2,\dots,X_n) of n observations will consist of an n by T data matrix. By applying the singular value decomposition, X_1,X_2,\dots,X_n can be decomposed into X = ULR^{\top}, where the crossproduct of U and R is identity matrix.

Holding the mean and L and R fixed at their realized values, there are four re-sampling methods that differ mainly by the ways of re-sampling U.

(a) Obtain the re-sampled singular column matrix by randomly sampling with replacement from the original principal component scores.

(b) To avoid the appearance of repeated values in bootstrapped principal component scores, we adapt a smooth bootstrap procedure by adding a white noise component to the bootstrap.

(c) Because principal component scores follow a standard multivariate normal distribution asymptotically, we can randomly draw principal component scores from a multivariate normal distribution with mean vector and covariance matrix of original principal component scores.

(d) Because the crossproduct of U is identitiy matrix, U is considered as a point on the Stiefel manifold, that is the space of n orthogonal vectors, thus we can randomly draw principal component scores from the Stiefel manifold.

Value

Author(s)

Han Lin Shang

References

H. D. Vinod (2004), "Ranking mutual funds using unconventional utility theory and stochastic dominance", Journal of Empirical Finance, 11(3), 353-377.

A. Cuevas, M. Febrero, R. Fraiman (2006), "On the use of the bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

D. S. Poskitt and A. Sengarapillai (2013), "Description length and dimensionality reduction in functional data analysis", Computational Statistics and Data Analysis, 58, 98-113.

H. L. Shang (2015), "Re-sampling techniques for estimating the distribution of descriptive statistics of functional data", Communications in Statistics–Simulation and Computation, 44(3), 614-635.

H. L. Shang (2018), "Bootstrap methods for stationary functional time series", Statistics and Computing, 28(1), 1-10.

See Also

fbootstrap

Examples

# Bootstrapping the distribution of a summary statistics of functional data.	
boot1 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, 
	statistic = "mean", bootmethod = "st")
boot2 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, 
	statistic = "mean", bootmethod = "sm", smo = 0.05)
boot3 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, 
	statistic = "mean", bootmethod = "mvn")
boot4 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, 
	statistic = "mean", bootmethod = "stiefel")
boot5 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, 
	statistic = "mean", bootmethod = "meboot")

Plot fitted model components for a functional model

Description

When class(x)[1] = ftsm, plot showing the principal components in the top row of plots and the coefficients in the bottom row of plots.

When class(x)[1] = fm, plot showing the predictor scores in the top row of plots and the response loadings in the bottom row of plots.

Usage

## S3 method for class 'fm'
plot(x, order, xlab1 = x$y$xname, ylab1 = "Principal component", 
 xlab2 = "Time", ylab2 = "Coefficient", mean.lab = "Mean", 
  level.lab = "Level", main.title = "Main effects", interaction.title 
   = "Interaction", basiscol = 1, coeffcol = 1, outlier.col = 2, 
    outlier.pch = 19, outlier.cex = 0.5, ...)

Arguments

Value

Function produces a plot.

Author(s)

Rob J Hyndman

References

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. Booth (2008) "Stochastic population forecasts using functional data models for mortality, fertility and migration", International Journal of Forecasting, 24(3), 323-342.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series (with discussion)", Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

ftsm, forecast.ftsm, residuals.fm, summary.fm, plot.fmres, plot.ftsf

Examples

plot(x = ftsm(y = ElNino_ERSST_region_1and2))

Plot residuals from a fitted functional model.

Description

Functions to produce a plot of residuals from a fitted functional model.

Usage

## S3 method for class 'fmres'
plot(x, type = c("image", "fts", "contour", "filled.contour", 
 "persp"), xlab = "Year", ylab = "Age", zlab = "Residual", ...)

Arguments

Value

Produces a plot.

Author(s)

Rob J Hyndman

See Also

ftsm, forecast.ftsm, plot.fm, plot.fmres, residuals.fm, summary.fm

Examples

# colorspace package was used to provide a more coherent color option.	
plot(residuals(ftsm(y = ElNino_ERSST_region_1and2)), type = "filled.contour", xlab = "Month", 
     ylab = "Residual sea surface temperature")

Plot fitted model components for a functional time series model

Description

Plot fitted model components for a fts object.

Usage

## S3 method for class 'ftsf'
plot(x, plot.type = c("function", "components", "variance"), 
 components, xlab1 = fit$y$xname, ylab1 = "Basis function", 
  xlab2 = "Time", ylab2 = "Coefficient", mean.lab = "Mean", 
   level.lab = "Level", main.title = "Main effects", 
    interaction.title = "Interaction", vcol = 1:3, shadecols = 7, 
     fcol = 4, basiscol = 1, coeffcol = 1, outlier.col = 2,
      outlier.pch = 19, outlier.cex = 0.5,...)

Arguments

Details

When plot.type = "function", it produces a plot of the forecast functions;

When plot.type = "components", it produces a plot of the principla components and coefficients with forecasts and prediction intervals for each coefficient;

When plot.type = "variance", it produces a plot of the variance components.

Value

Function produces a plot.

Author(s)

Rob J Hyndman

References

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. Booth (2008) "Stochastic population forecasts using functional data models for mortality, fertility and migration", International Journal of Forecasting, 24(3), 323-342.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series (with discussion)", Journal of the Korean Statistical Society, 38(3), 199-221.

H. L. Shang, H. Booth and R. J. Hyndman (2011) "Point and interval forecasts of mortality rates and life expectancy: A comparison of ten principal component methods", Demographic Research, 25(5), 173-214.

See Also

ftsm, plot.fm, plot.fmres, residuals.fm, summary.fm

Examples

plot(x = forecast(object = ftsm(y = ElNino_ERSST_region_1and2)))

Plot fitted model components for a functional time series model

Description

Plot showing the basis functions in the top row of plots and the coefficients in the bottom row of plots.

Usage

## S3 method for class 'ftsm'
plot(x, components, components.start = 0, xlab1 = x$y$xname, ylab1 = "Basis function", 
 xlab2 = "Time", ylab2 = "Coefficient", mean.lab = "Mean", 
  level.lab = "Level", main.title = "Main effects", 
   interaction.title = "Interaction", basiscol = 1, coeffcol = 1, 
    outlier.col = 2, outlier.pch = 19, outlier.cex = 0.5, ...)   

Arguments

Value

None. Function produces a plot.

Author(s)

Rob J Hyndman

References

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series" (with discussion), Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

forecast.ftsm, ftsm, plot.fm, plot.ftsf, residuals.fm, summary.fm

Examples

# plot different principal components.	
plot.ftsm(ftsm(y = ElNino_ERSST_region_1and2, order = 2), components = 2)

Plot fitted model components for a functional time series model

Description

Plot showing the basis functions of the predictors in the top row, followed by the basis functions of the responses in the second row, then the coefficients in the bottom row of plots.

Usage

plotfplsr(x, xlab1 = x$ypred$xname, ylab1 = "Basis function", xlab2 = "Time", 
 ylab2 = "Coefficient", mean.lab = "Mean", interaction.title = "Interaction")

Arguments

Value

None. Function produces a plot.

Author(s)

Han Lin Shang

References

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. L. Shang (2009) "Forecasting functional time series" (with discussion), Journal of the Korean Statistical Society, 38(3), 199-221.

See Also

forecast.ftsm, ftsm, plot.fm, plot.ftsf, residuals.fm, summary.fm

Examples

# Fit the data by the functional partial least squares.	
ausfplsr = fplsr(data = ElNino_ERSST_region_1and2, order = 2)
plotfplsr(x = ausfplsr)

Particulate Matter Concentrations (pm10)

Description

This data set consists of half-hourly measurement of the concentrations (measured in ug/m3) of particular matter with an aerodynamic diameter of less than 10um, abbreviated PM10, in ambient air taken in Graz-Mitte, Austria from October 1, 2010 until March 31, 2011. To stabilise the variance, a square-root transformation can be applied to the data.

Usage

data(pm_10_GR)

Details

As epidemiological and toxicological studies have pointed to negative health effects, European Union (EU) regulation sets pollution standards for the level of the concentration. Policy makers have to ensure compliance with these EU rules and need reliable statistical tools to determine, and justify the public, appropriate measures such as partial traffic regulation (see Stadlober, Hormann and Pfeiler, 2008).

Source

Thanks Professor Siegfried. Hormann for providing this data set. The original data source is https://zenodo.org/records/7959116

References

A. Aue, D. D. Norinho, S. Hormann (2015) "On the prediction of stationary functional time series", Journal of the American Statistical Association, 110(509), 378-392.

E. Stadlober, S. Hormann, B. Pfeiler (2008) "Quality and performance of a PM10 daily forecasting model", Atmospheric Environment, 42, 1098-1109.

S. Hormann, B. Pfeiler, E. Stadlober (2005) "Analysis and prediction of particulate matter PM10 for the winter season in Graz", Austrian Journal of Statistics, 34(4), 307-326.

H. L. Shang (2017) "Functional time series forecasting with dynamic updating: An application to intraday particulate matter concentration", Econometrics and Statistics, 1, 184-200.

Examples

plot(pm_10_GR)

Quantile

Description

Generic functions for quantile.

Usage

quantile(x, ...)

Arguments

Value

Refer to specific methods. For numeric vectors, see the quantile functions in the stats package.

Author(s)

Han Lin Shang

See Also

quantile.fts

Quantile functions for functional time series

Description

Computes quantiles of functional time series at each variable.

Usage

## S3 method for class 'fts'
quantile(x, probs, ...)

Arguments

Value

Return quantiles for each variable.

Author(s)

Han Lin Shang

See Also

mean.fts, median.fts, var.fts, sd.fts

Examples

quantile(x = ElNino_ERSST_region_1and2)

Compute residuals from a functional model

Description

After fitting a functional model, it is useful to inspect the residuals. This function extracts the relevant information from the fit object and puts it in a form suitable for plotting with image, persp, contour, filled.contour, etc.

Usage

## S3 method for class 'fm'
residuals(object, ...)

Arguments

Value

Produces an object of class “fmres” containing the residuals from the model.

Author(s)

Rob J Hyndman

References

B. Erbas and R. J. Hyndman and D. M. Gertig (2007) "Forecasting age-specific breast cancer mortality using functional data model", Statistics in Medicine, 26(2), 458-470.

R. J. Hyndman and M. S. Ullah (2007) "Robust forecasting of mortality and fertility rates: A functional data approach", Computational Statistics and Data Analysis, 51(10), 4942-4956.

R. J. Hyndman and H. Booth (2008) "Stochastic population forecasts using functional data models for mortality, fertility and migration", International Journal of Forecasting, 24(3), 323-342.

H. L. Shang (2012) "Point and interval forecasts of age-specific fertility rates: a comparison of functional principal component methods", Journal of Population Research, 29(3), 249-267.

H. L. Shang (2012) "Point and interval forecasts of age-specific life expectancies: a model averaging", Demographic Research, 27, 593-644.

See Also

ftsm, forecast.ftsm, summary.fm, plot.fm, plot.fmres

Examples

plot(residuals(object = ftsm(y = ElNino_ERSST_region_1and2)), 
	xlab = "Year", ylab = "Month")

Standard deviation

Description

Generic functions for standard deviation.

Usage

sd(...)

Arguments

Details

The sd functions in the stats package are replaced by sd.default.

Value

Refer to specific methods. For numeric vectors, see the sd functions in the stats package.

Author(s)

Han Lin Shang

See Also

sd.fts

Standard deviation functions for functional time series

Description

Computes standard deviation of functional time series at each variable.

Usage

## S3 method for class 'fts'
sd(x, method = c("coordinate", "FM", "mode", "RP", "RPD", "radius"), 
 trim = 0.25, alpha, weight,...)

Arguments

Details

If method = "coordinate", it computes coordinate-wise standard deviation functions.

If method = "FM", it computes the standard deviation functions of trimmed functional data ordered by the functional depth of Fraiman and Muniz (2001).

If method = "mode", it computes the standard deviation functions of trimmed functional data ordered by h-modal functional depth.

If method = "RP", it computes the standard deviation functions of trimmed functional data ordered by random projection depth.

If method = "RPD", it computes the standard deviation functions of trimmed functional data ordered by random projection with derivative depth.

If method = "radius", it computes the standard deviation function of trimmed functional data ordered by the notion of alpha-radius.

Value

A list containing x = variables and y = standard deviation rates.

Author(s)

Han Lin Shang

References

O. Hossjer and C. Croux (1995) "Generalized univariate signed rank statistics for testing and estimating a multivariate location parameter", Nonparametric Statistics, 4(3), 293-308.

A. Cuevas and M. Febrero and R. Fraiman (2006) "On the use of bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

A. Cuevas and M. Febrero and R. Fraiman (2007), "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2007) "A functional analysis of NOx levels: location and scale estimation and outlier detection", Computational Statistics, 22(3), 411-427.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2008) "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2010) "Measures of influence for the functional linear model with scalar response", Journal of Multivariate Analysis, 101(2), 327-339.

J. A. Cuesta-Albertos and A. Nieto-Reyes (2010) "Functional classification and the random Tukey depth. Practical issues", Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, 77, 123-130.

D. Gervini (2012) "Outlier detection and trimmed estimation in general functional spaces", Statistica Sinica, 22(4), 1639-1660.

See Also

mean.fts, median.fts, var.fts, quantile.fts

Examples

# Fraiman-Muniz depth was arguably the oldest functional depth.	
sd(x = ElNino_ERSST_region_1and2, method = "FM")
sd(x = ElNino_ERSST_region_1and2, method = "coordinate")
sd(x = ElNino_ERSST_region_1and2, method = "mode")
sd(x = ElNino_ERSST_region_1and2, method = "RP")
sd(x = ElNino_ERSST_region_1and2, method = "RPD")
sd(x = ElNino_ERSST_region_1and2, method = "radius", 
	alpha = 0.5, weight = "hard")
sd(x = ElNino_ERSST_region_1and2, method = "radius", 
	alpha = 0.5, weight = "soft")

Simulated multiple sets of functional time series

Description

We generate 2 groups of m functional time series. For each i in {1, ..., m} in a given cluster c, c in {1,2}, the t th function, t in {1,..., T}, is given by

Y_{it}^{(c)} (x)= \mu^{(c)}(x) + \sum_{k=1}^{2}\xi_{tk}^{(c)} \rho_k^{(c)} (x) + \sum_{l=1}^{2}\zeta_{itl}^{(c)} \psi_l^{(c)} (x) + \upsilon_{it}^{(c)} (x)

Usage

data("sim_ex_cluster")

Details

The mean functions for each of these two clusters are set to be \mu^{(1)}(x) = 2(x-0.25)^{2} and \mu^{(2)}(x) = 2(x-0.4)^{2}+0.1.

While the variates \mathbf{\xi_{tk}^{(c)}}=(\xi_{1k}^{(c)}, \xi_{2k}^{(c)}, \ldots, \xi_{Tk}^{(c)})^{\top} for both clusters, are generated from autoregressive of order 1 with parameter 0.7, while the variates \zeta_{it1}^{(c)} and \zeta_{it2}^{(c)} for both clusters, are generated from independent and identically distributed N(0,0.5) and N(0,0.25), respectively.

The basis functions for the common-time trend for the first cluster, \rho_k^{(1)} (x), for k in {1,2} are sqrt(2)*sin(\pi*(0:200/200)) and sqrt(2)*cos(\pi*(0:200/200)) respectively; and the basis functions for the common-time trend for the second cluster, \rho_k^{(2)} (x), for k in {1,2} are sqrt(2)*sin(2\pi*(0:200/200)) and sqrt(2)*cos(2\pi*(0:200/200)) respectively.

The basis functions for the residual for the first cluster, \psi_l^{(1)} (x), for l in {1,2} are sqrt(2)*sin(3\pi*(0:200/200)) and sqrt(2)*cos(3\pi*(0:200/200)) respectively; and the basis functions for the residual for the second cluster, \psi_l^{(2)} (x), for l in {1,2} are sqrt(2)*sin(4\pi*(0:200/200)) and sqrt(2)*cos(4\pi*(0:200/200)) respectively.

The measurement error \upsilon_{it} for each continuum x is generated from independent and identically distributed N(0, 0.2^2)

Examples

data(sim_ex_cluster)

Skewed t distribution

Description

Fitting a parametric skewed t distribution of Fernandez and Steel's (1998) method

Usage

skew_t_fun(data, gridpoints, M = 5001)

Arguments

Details

1) Fit a skewed t distribution to data, and obtain four latent parameters; 2) Transform the four latent parameters so that they are un-constrained; 3) Fit a vector autoregressive model to these transformed latent parameters; 4) Obtain their forecasts, and then back-transform them to the original scales; 5) Via the skewed t distribution in Step 1), we obtain forecast density using the forecast latent parameters.

Value

Note

This is a parametric approach for fitting and forecasting density.

Author(s)

Han Lin Shang

References

Fernandez, C. and Steel, M. F. J. (1998), ‘On Bayesian modeling of fat tails and skewness’, Journal of the American Statistical Association: Theory and Methods, 93(441), 359-371.

See Also

CoDa_FPCA, Horta_Ziegelmann_FPCA, LQDT_FPCA

Examples

skew_t_fun(DJI_return)

Detection of the optimal stopping time in a curve time series

Description

Detecting the optimal stopping time for the glue curing of wood panels in an automatic process environment.

Usage

stop_time_detect(data, forecasting_method = c("ets", "arima", "rw"))

Arguments

Value

Author(s)

Han Lin Shang

References

Bekhta, P., Ortynska, G. and Sedliacik, J. (2014). Properties of modified phenol-formaldehyde adhesive for plywood panels manufactured from high moisture content veneer. Drvna Industrija 65(4), 293-301.

Simulated functional time series from a functional autoregression of order one

Description

For detecting the optimal stopping time, we simulate a curve time series that follows a functional autoregression of order 1, with a breakpoint in the middle point of the entire sample.

Usage

stop_time_sim_data(sample_size, omega, seed_number)

Arguments

Value

An object of class fts

Author(s)

Han Lin Shang

See Also

stop_time_detect

Examples

stop_time_sim_data(sample_size = 401, omega = 0.1, seed_number = 123)

Summary for functional time series model

Description

Summarizes a basis function model fitted to a functional time series. It returns various measures of goodness-of-fit.

Usage

## S3 method for class 'fm'
summary(object, ...)

Arguments

Value

None.

Author(s)

Rob J Hyndman

See Also

ftsm, forecast.ftsm, residuals.fm, plot.fm, plot.fmres

Examples

summary(object = ftsm(y = ElNino_ERSST_region_1and2))

Testing stationarity of functional time series

Description

A hypothesis test for stationarity of functional time series.

Usage

T_stationary(sample, L = 49, J = 500, MC_rep = 1000, cumulative_var = .90,
		    Ker1 = FALSE, Ker2 = TRUE, h = ncol(sample)^.5, pivotal = FALSE,
		    use_table = FALSE, significance)

Arguments

Details

As in traditional (scalar and vector) time series analysis, many inferential procedures for functional time series assume stationarity. Stationarity is required for functional dynamic regression models, for bootstrap and resampling methods for functional time series and for the functional analysis of volatility.

Value

Author(s)

Greg. Rice and Han Lin Shang

References

L. Horvath and Kokoszka, P. (2012) Inference for Functional Data with Applications, Springer, New York.

L. Horvath, P. Kokoszka, G. Rice (2014) "Testing stationarity of functional time series", Journal of Econometrics, 179(1), 66-82.

D. N. Politis (2003) "Adaptive bandwidth choice", Journal of Nonparametric Statistics, 15(4-5), 517-533.

A. Aue, G. Rice, O. S\"onmez (2018) "Detecting and dating structural breaks in functional data without dimension reduction", Journal of the Royal Statistical Society: Series B, 80(3), 509-529.

See Also

farforecast

Examples

result = T_stationary(sample = pm_10_GR_sqrt$y)
result_pivotal = T_stationary(sample = pm_10_GR_sqrt$y, J = 100, MC_rep = 5000,
				h = 20, pivotal = TRUE)

Variance

Description

Generic functions for variance.

Usage

var(...)

Arguments

Details

The cor functions in the stats package are replaced by var.default.

Value

Refer to specific methods. For numeric vectors, see the cor functions in the stats package.

Author(s)

Rob J Hyndman and Han Lin Shang

See Also

var.fts

Variance functions for functional time series

Description

Computes variance functions of functional time series at each variable.

Usage

## S3 method for class 'fts'
var(x, method = c("coordinate", "FM", "mode", "RP", "RPD", "radius"), 
 trim = 0.25, alpha, weight, ...)

Arguments

Details

If method = "coordinate", it computes coordinate-wise variance.

If method = "FM", it computes the variance of trimmed functional data ordered by the functional depth of Fraiman and Muniz (2001).

If method = "mode", it computes the variance of trimmed functional data ordered by h-modal functional depth.

If method = "RP", it computes the variance of trimmed functional data ordered by random projection depth.

If method = "RPD", it computes the variance of trimmed functional data ordered by random projection derivative depth.

If method = "radius", it computes the standard deviation function of trimmed functional data ordered by the notion of alpha-radius.

Value

A list containing x = variables and y = variance rates.

Author(s)

Han Lin Shang

References

O. Hossjer and C. Croux (1995) "Generalized univariate signed rank statistics for testing and estimating a multivariate location parameter", Nonparametric Statistics, 4(3), 293-308.

A. Cuevas and M. Febrero and R. Fraiman (2006) "On the use of bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

A. Cuevas and M. Febrero and R. Fraiman (2007), "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2007) "A functional analysis of NOx levels: location and scale estimation and outlier detection", Computational Statistics, 22(3), 411-427.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2008) "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2010) "Measures of influence for the functional linear model with scalar response", Journal of Multivariate Analysis, 101(2), 327-339.

J. A. Cuesta-Albertos and A. Nieto-Reyes (2010) "Functional classification and the random Tukey depth. Practical issues", Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, 77, 123-130.

D. Gervini (2012) "Outlier detection and trimmed estimation in general functional spaces", Statistica Sinica, 22(4), 1639-1660.

See Also

mean.fts, median.fts, sd.fts, quantile.fts

Examples

# Fraiman-Muniz depth was arguably the oldest functional depth.	
var(x = ElNino_ERSST_region_1and2, method = "FM")
var(x = ElNino_ERSST_region_1and2, method = "coordinate")
var(x = ElNino_ERSST_region_1and2, method = "mode")
var(x = ElNino_ERSST_region_1and2, method = "RP")
var(x = ElNino_ERSST_region_1and2, method = "RPD")
var(x = ElNino_ERSST_region_1and2, method = "radius", 
alpha = 0.5, weight = "hard")
var(x = ElNino_ERSST_region_1and2, method = "radius", 
alpha = 0.5, weight = "soft")