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)