Research

Functional Data Classification via Deep Learning

A fundamental problem in functional data analysis is to classify a data function based on training samples.

A typical 1D example is the speech recognition data extracted from the TIMIT database, in which the training samples are digitized speech curves of American English speakers from different phoneme groups, and the task is to predict the phoneme of a new speech curve.

Typical 2D and 3D examples include the brain imaging data extracted from Early Mild Cognitive Impairment (EMCI) or Alzeheimer's Disease (AD), in which the training samples are digitized brain images, and the task is to predict the stage of a new patient.

We propose a unified deep neural networks (DNN) based classifiers for functional data (for example, 2D & 3D imaging data).

• Wang, S., Cao, G. (2022) Deep Neural Network Classifier for Multi-dimensional Functional Data (PDF) (R package)

Functional Data Regression via Deep Learning

Although considerable advances have been achieved in deep learning research, from the statistical perspective its application and theoretical research is still in its infancy stage.

We propose a unified deep neural networks (DNN) based estimator for multi-dimensional functional data (for example, 2D & 3D imaging data).

• Wang, S., Cao, G. and Shang, Z.(2021) Estimation of the mean function of functional data via deep neural networks. Stat, 10 (1), e393. (PDF) (R package)

(Python package)

• Wang, S., Cao, G. (2022) Robust Deep Neural Network Estimation for Multi-dimensional Functional Data. (PDF)(R package)

Simultaneous Inference for Functional Characteristics

Advancements in modern technology have enabled the collection of sophisticated, high-dimensional data sets, such as curves, 3D images, and other objects living in a functional space, thus boosting the investigation of functional data.

We established asymptotic correctness of the proposed simultaneous confidence band for mean functions using various properties of spline smoothing.

• Cao, G., Yang, L. and Todem, D. (2012) Simultaneous inference for the mean function based on dense functional data. Journal of Nonparametric Statistics, 24(2), 359-377.(PDF)

An immediate goal after focusing on mean curves is to extend theoretical findings for the derivative of mean curves.

• Cao, G., Wang, J., Wang, L. and Todem, D. (2012) Spline confidence bands for functional derivatives. Journal of Statistical Planning and Inference. 142(6), 1557-1570. (PDF)

We proposed a new simultaneous confidence envelope for the covariance function, which can be used to visualize the variability of the covariance estimator and to make global inferences on the shape and other properties of the true covariance.

• Cao, G., Wang, L., Li, Y. and Yang, L. (2016) Oracle-efficient confidence envelopes for covariance functions in dense functional data. Statistics Sinica, 26, 359-383. (PDF)

• Wang, J., Cao, G., Wang, L. and Yang, L. (2020) Simultaneous confidence band for stationary covariance function of dense functional data. Journal of Multivariate Analysis, 176, 104584. (PDF)

We developed novel procedures to construct simultaneous confidence bands for

mean and derivative functions of repeatedly observed functional data

• Cao, G., (2014) Simultaneous confidence bands for derivatives of dependent functional data. Electronic Journal of Statistics, 8(2), 2639-2663. (PDF)

• Cao, G. and Wang, L. (2018) Simultaneous inference for repeated functional data. Journal of Multivariate Analysis, 165, 279-295.(PDF)

In the presence of outliers, we aim at developing outlying-resistant methods for the mean function in the functional setting, that can provide valid statistical inference even in the presence of a significant proportion of outlier curves.

• Lima, R.I., Cao, G. and Billor, R. (2019) Robust simultaneous inference for the mean function of functional data Test. 28 (3), 785-803. (PDF)

• Lima, R.I., Cao, G. and Billor, R. (2018) M-Based simultaneous inference for the mean function of functional data. Annals of the Institute of Statistical Mathematics, 176, 104584. (PDF)

Statistical Algorithms for Modern Power Systems

A smart grid is an electrical power grid that is enhanced with communications and networking, computing, control, and signal processing technologies. The key to the success of the smart grid, we argue, lies in developing effective techniques to make it more secure, with respect to detecting anomalies and attacks, and more profitable, with respect to more efficient energy management. To fully harvest its potential, various theoretical tools have been developed and applied to smart grid problems. However, statistical techniques for smart grid research are rather sparse, and have begun to emerge as important and practical tools for smart grid modeling and analysis, but with many challenges.

• Wang, Y., Shen, Y., Mao, S., Cao, G. and Nelms, R. (2018) Adaptive learning hybrid model for solar intensity forecasting. IEEE Transactions on Industrial Informatics, 14, 1635-1645. (PDF)

• Wang, Y., Cao, G., Mao, S. and Nelms, R. (2015) Analysis of solar generation and weather data in smart grid with simultaneous inference of nonlinear time series. International Workshop on Smart Cities and Urban Informatics in conjunction with IEEE INFOCOM 2015. (PDF)