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A zero-inflated non-negative matrix factorization for the deconvolution of mixed signals of biological data

  • Yixin Kong , Ariangela Kozik , Cindy H. Nakatsu , Yava L. Jones-Hall and Hyonho Chun EMAIL logo
Published/Copyright: March 30, 2021
The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 18 Issue 1

Abstract

A latent factor model for count data is popularly applied in deconvoluting mixed signals in biological data as exemplified by sequencing data for transcriptome or microbiome studies. Due to the availability of pure samples such as single-cell transcriptome data, the accuracy of the estimates could be much improved. However, the advantage quickly disappears in the presence of excessive zeros. To correctly account for this phenomenon in both mixed and pure samples, we propose a zero-inflated non-negative matrix factorization and derive an effective multiplicative parameter updating rule. In simulation studies, our method yielded the smallest bias. We applied our approach to brain gene expression as well as fecal microbiome datasets, illustrating the superior performance of the approach. Our method is implemented as a publicly available R-package, iNMF.

Keywords: deconvolution; latent factor model; non-negative matrix factorization; zero-inflation
Corresponding author: Hyonho Chun, Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, South Korea, Email:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A10073887) for H.C.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

A.1 Conditional posterior distribution

The conditional posterior distributions for the Gibbs sampling are given as follows:

  1. ω i j | ω − ( i j ) , S , X , κ , τ ∼  PG  ( 1 , ψ i j )

  2. ( κ j , τ j ) | ω , S , X ∼ N ( m j , V j − 1 ) , w h e r e   V j = ∑ i = 1 N ω i j + σ κ 2 ∑ i = 1 N ω i j ( A W ) i j ∑ i = 1 N ω i j ( A W ) i j ∑ i = 1 N ω i j ( A W ) i j 2 + σ τ 2

  3. m j = V j − 1 ∑ i = 1 N S i j − N / 2 + μ κ / σ κ 2 ∑ i = 1 N S i j ( A W ) i j − 1 / 2 + μ τ / σ τ 2

  4. S i j | S − ( i j ) , ψ i j ∼ Bernoulli ( b i j ) where

  5. b i j = 1  if  X i j > 0 expit ( ψ i j + R j log ∑ n ≠ i S n j ( A W ) n j ( A W ) i j + ∑ n ≠ i S n j ( A W ) n j )  if  X i j = 0 .

A.2 Simulation study for checking the robustness of the regularization method

In order for checking the robustness of the regularization, we further investigated the approach by generating the data from a Negative Binomial distribution with a mean μ ij = R j p ij and a variance σ i j 2 = μ i j + 0.5 μ i j 2 with zero-inflation. The results are in Figure A.2.1. The same conclusion to Subsection 3.3.3 is made from this study, where the regularization helps to estimate W more accurately.

Figure A.2.1: Four different values of α0 (10, 50, 100, and 500) were used for checking its effects, when the data is generated under the negative binomial model. As α0 gets larger, the W estimates become more accurate, whereas the A estimates are less accurate in both 30 and 50% missing rates.
Figure A.2.1:

Four different values of α0 (10, 50, 100, and 500) were used for checking its effects, when the data is generated under the negative binomial model. As α0 gets larger, the W estimates become more accurate, whereas the A estimates are less accurate in both 30 and 50% missing rates.

A.3 Simulation study for using the log likelihood function for choosing K

We perform a simulation study to show how the likelihood method works for choosing a proper K under the simulation scenario of 3.2 with a zero-rate of 30%. Here, the true K is 3. From the likelihood method (Figure A.3.1), we are able to find K = 3 as the increase is reduced from K = 4.

Figure A.3.1: As K increases, the log-likelihood function increases. But the rate of increase is dropped when K = 4, hence we can infer K = 3 from the likelihood function.
Figure A.3.1:

As K increases, the log-likelihood function increases. But the rate of increase is dropped when K = 4, hence we can infer K = 3 from the likelihood function.

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Received: 2020-03-30
Revised: 2021-02-07
Accepted: 2021-02-23
Published Online: 2021-03-30

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Integrating additional knowledge into the estimation of graphical models
  4. Asymptotic properties of the two one-sided t-tests – new insights and the Schuirmann-constant
  5. Bayesian optimization design for finding a maximum tolerated dose combination in phase I clinical trials
  6. A Bayesian mixture model for changepoint estimation using ordinal predictors
  7. Power prior for borrowing the real-world data in bioequivalence test with a parallel design
  8. Bayesian approaches to variable selection: a comparative study from practical perspectives
  9. Bayesian adaptive design of early-phase clinical trials for precision medicine based on cancer biomarkers
  10. More than one way: exploring the capabilities of different estimation approaches to joint models for longitudinal and time-to-event outcomes
  11. Designing efficient randomized trials: power and sample size calculation when using semiparametric efficient estimators
  12. Power formulas for mixed effects models with random slope and intercept comparing rate of change across groups
  13. The effect of data aggregation on dispersion estimates in count data models
  14. A zero-inflated non-negative matrix factorization for the deconvolution of mixed signals of biological data
  15. Multiple scaled symmetric distributions in allometric studies
  16. Estimation of semi-Markov multi-state models: a comparison of the sojourn times and transition intensities approaches
  17. Regularized bidimensional estimation of the hazard rate
  18. The effect of random-effects misspecification on classification accuracy
  19. The area under the generalized receiver-operating characteristic curve
https://doi.org/10.1515/ijb-2020-0039
Keywords for this article
deconvolution; latent factor model; non-negative matrix factorization; zero-inflation

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Integrating additional knowledge into the estimation of graphical models
  4. Asymptotic properties of the two one-sided t-tests – new insights and the Schuirmann-constant
  5. Bayesian optimization design for finding a maximum tolerated dose combination in phase I clinical trials
  6. A Bayesian mixture model for changepoint estimation using ordinal predictors
  7. Power prior for borrowing the real-world data in bioequivalence test with a parallel design
  8. Bayesian approaches to variable selection: a comparative study from practical perspectives
  9. Bayesian adaptive design of early-phase clinical trials for precision medicine based on cancer biomarkers
  10. More than one way: exploring the capabilities of different estimation approaches to joint models for longitudinal and time-to-event outcomes
  11. Designing efficient randomized trials: power and sample size calculation when using semiparametric efficient estimators
  12. Power formulas for mixed effects models with random slope and intercept comparing rate of change across groups
  13. The effect of data aggregation on dispersion estimates in count data models
  14. A zero-inflated non-negative matrix factorization for the deconvolution of mixed signals of biological data
  15. Multiple scaled symmetric distributions in allometric studies
  16. Estimation of semi-Markov multi-state models: a comparison of the sojourn times and transition intensities approaches
  17. Regularized bidimensional estimation of the hazard rate
  18. The effect of random-effects misspecification on classification accuracy
  19. The area under the generalized receiver-operating characteristic curve
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