PACE package for Functional Data Analysis and Empirical Dynamics (written in Matlab)

Version 2.11 (released February 09, 2010) can be downloaded from PACE 2.11.

PACE is a versatile package that provides implementations of various methods of Functional Data Analysis (FDA) and Empirical Dynamics. The core program of this package is Functional Principal Component Analysis (FPCA), a key technique for functional data analysis, for sparsely or densely sampled random trajectories and time courses, via the Principal Analysis by Conditional Estimation (PACE) algorithm. PACE is useful for the analysis of data that have been generated by a sample of underlying (usually unobserved) random trajectories and does not use pre-smoothing of trajectories, which is problematic if functional data are sparsely sampled or measurements are corrupted with noise. PACE provides options for Longitudinal Data Analysis, the analysis of Stochastic Processes from samples of realized trajectories, and for the analysis of underlying Dynamics. The development of PACE has been supported by various NSF grants.

PACE 2.11 includes the following options for Functional Data Analysis. Numbers [1], [2] etc. refer to the references below.

(1) Fitting of both sparsely and densely sampled random functions by Functional Principal Component Analysis (FPCA), including Spaghetti plots to view the sample of functions (pace) [1] [2] [5] [18]

(2) Fitting of derivatives for Empirical Dynamics for both sparsely and densely sampled random functions (pace-der) [17]

(3) Functional linear regression, fitting functional linear regression models for both sparsely or densely sampled random trajectories, for cases where the predictor is a random function and the response is a scalar or a random function (pace-reg) [3] [13]

(4) Diagnostics and bootstrap inference for functional linear regression (pace-reg) [9]

(4) Assessing functional dependence through functional singular value decomposition (pace-svd)

(6) Generalized functional linear regression (GFLM), where the response is a scalar generalized variable such as binary or Poisson, can also be used for classification of functional data via binary regression (pace-glm) [4] [5] [7]

(7) Functional quadratic and polynomial regression (pace-quadreg) [20]

(8) Functional Additive Modeling (FAM), an additive generalization of functional linear regression, for more flexible functional regression, for the case of functional predictors and both functional and scalar responses (pace-fam) [12]

(9) Modeling longitudinal data with repeated generalized responses (binary, Poisson etc.), which are derived from a latent Gaussian process by a link function (pace-grm) [11]

(10) The functional variance process, a generalization of variance functions useful for functional volatility modeling (pace-fvp) [6]

(11) Time-synchronization based on pairwise warping (alignment, registration) for sparsely and densely sampled functions (pace-warp) [8] [15] [16]

(12) Generalized functional distance for sparse data (spadis), which can be used for functional clustering and other applications (pace) [14]

(13) Transfer functions for dynamic modeling (pace-dyn)

[19] A requirement for all methods is that the pooled measurement times are dense on the domain and their pooled pairs are dense on the domain squared (design plot can be used as a check).

If you use the program, please refer to the articles below where the core methodology is described.

[1] Yao, F., Müller, H.G., Clifford, A.J., Dueker, S.R., Follett, J., Lin, Y., Buchholz, B., Vogel, J.S. (2003). Shrinkage estimation for functional principal component scores, with application to the population kinetics of plasma folate. Biometrics 59, 676-685. (pdf)

[2] Yao, F., Müller, H.G., Wang, J.L. (2005). Functional data analysis for sparse longitudinal data. J. American Statistical Association 100, 577-590. (pdf)

[3] Yao, F., Müller, H.G., Wang, J.L. (2005). Functional Linear Regression Analysis for Longitudinal Data. The Annals of Statistics 33, 2873-2903. (pdf)

[4] Müller, H.G., Stadtmüller, U. (2005). Generalized functional linear models. Annals of Statistics 33, 774-805. (pdf)

[5] Müller, H.G. (2005). Functional modeling and classification of longitudinal data. Scandinavian J. Statistics 32, 223-240. (pdf)


[6] Müller, H.G., Stadtmüller, U., Yao, F. (2006). Functional variance processes. Journal of the American Statistical Association 101, 1007-1018. (pdf)

[7] Leng, X., Müller, H.G. (2006). Classification using functional data analysis for temporal gene expression data. Bioinformatics 22, 68-76. (pdf)

[8] Leng, X., Müller, H.G. (2006). Time ordering of gene co-expression. Biostatistics 7, 569-584. (pdf)

[9] Chiou, J., Müller, H.G. (2007). Diagnostics for functional regression via residual processes. Computational Statistics & Data Analysis 51, 4849-4863. (pdf)

[10] Müller, H.G. (2008). Functional modeling of longitudinal data. Ed. Fitzmaurice, G. et al., Wiley & Sons, Inc, 223-252.

[11] Hall, P., Müller, H.G., Yao, F. (2008). Modeling sparse generalized longitudinal observations via latent Gaussian processes. Journal of the Royal Statistical Society B 70, 703-723. (pdf)

[12] Müller, H.G., Yao, F. (2008). Functional additive models. Journal of the American Statistical Association 103, 426-437. (pdf)

[13] Müller, H.G., Chiou, J.M., Leng, X. (2008). Inferring gene expression dynamics via functional regression analysis. BMC Bioinformatics 9:60. (pdf)

[14] Peng, J., Müller, H.G. (2008). Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions. Annals of Applied Statistics 2, 1056-1077. (pdf)

[15] Tang, R., Müller, H.G. (2008). Pairwise curve synchronization for high-dimensional data. Biometrika 95, 875-889. (pdf)

[16] Tang, R., Müller, H.G. (2009). Time-synchronized clustering of gene expression trajectories. Biostatistics 10, 32-45. (pdf)

[17] Liu, B., Müller, H.G. (2009). Estimating derivatives for samples of sparsely observed functions, with application to on-line auction dynamics. J. American Statistical Association 104, 704-717. (pdf)

[18] Müller, H.G. (2009). Functional modeling of longitudinal data. In: Longitudinal Data Analysis (Handbooks of Modern Statistical Methods), Ed. Fitzmaurice, G., Davidian, M., Verbeke, G., Molenberghs, G., Wiley, New York, 223--252. (pdf)

[19] Müller, H.G., Yang, W. (2010). Dynamic relations for sparsely sampled Gaussian processes. Test (pdf)

[20] Müller, H.G., Yao, F. (2010). Functional quadratic regression. Biometrika (pdf)