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Functional classwise principal component analysis: a classification framework for functional data analysis

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Abstract

In recent times, functional data analysis has been successfully applied in the field of high dimensional data classification. In this paper, we present a classification framework using functional data and classwise Principal Component Analysis (PCA). Our proposed method can be used in high dimensional time series data which typically suffers from small sample size problem. Our method extracts a piecewise linear functional feature space and is particularly suitable for hard classification problems. The proposed framework converts time series data into functional data and uses classwise functional PCA for feature extraction followed by classification using a Bayesian linear classifier. We demonstrate the efficacy of our proposed method by applying it to both synthetic data sets and real time series data from diverse fields including but not limited to neuroscience, food science, medical sciences and chemometrics.

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Availability of data and materials

Available on request.

Code availability

https://github.com/ChatterjeeAvishek/FCPCA.

Notes

  1. \(A=\{f_1,f_2,\ldots , f_n\}\) is said to be linearly independent (LI) if \(c_1f_1+c_2f_2+\ldots +c_nf_n=\textbf{0}\), where \(c_i\)’s are scalars and \(\textbf{0}\) denotes zero function, has only one solution i.e., \(c_1=c_2=\ldots =c_n=0\) (Kreyszig 1991). A set that is not LI is called linearly dependent (LD). Hence a linearly dependent set of functions has at least one function \(f_j\) that can be written as a linear combination of other elements of that set. Usually, the Gram–Schmidt orthonormalization process is applied to LI set. If we try to apply the Gram–Schmidt orthonormalization process on the LD set, then some \(g_j\), for \(j\in \{1, \ldots ,n\}\) will become zero function, and hence we cannot obtain an orthonormal set.

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Acknowledgements

A. Chatterjee is supported by an INSPIRE fellowship from the Department of Science and Technology (DST), Government of India. We sincerely thank the anonymous reviewers for their insightful comments which have significantly improved the manuscript.

Funding

This work was supported by INSPIRE fellowship from the Department of Science and Technology (DST), Government of India (INSPIRE Code: IF170367)

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  1. Department of Mathematics and Statistics, IISER Kolkata, Mohanpur, Nadia, West Bengal, 741246, India

    Avishek Chatterjee, Satyaki Mazumder & Koel Das

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  1. Avishek Chatterjee
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  2. Satyaki Mazumder
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  3. Koel Das
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AC and KD designed the research. AC and SM performed the research. AC wrote the manuscript. KD edited the manuscript and supervised the entire work.

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Correspondence to Koel Das.

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Chatterjee, A., Mazumder, S. & Das, K. Functional classwise principal component analysis: a classification framework for functional data analysis. Data Min Knowl Disc 37, 552–594 (2023). https://doi.org/10.1007/s10618-022-00898-1

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