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In recent years, using orthogonal matrices has been shown to be a promising approach to improving recurrent neural networks (RNNs) with training, stability, and convergence, particularly to control gradients. While gated recurrent unit (GRU) and long short-term memory (LSTM) architectures address the vanishing gradient problem by using a variety of gates and memory cells, they are still prone to the exploding gradient problem. In this work, we analyze the gradients in GRU and propose the use of orthogonal matrices to prevent exploding gradient problems and enhance long-term memory. We study where to use orthogonal matrices and propose a Neumann series–based scaled Cayley transformation for training orthogonal matrices in GRU, which we call Neumann-Cayley orthogonal GRU (NC-GRU). We present detailed experiments of our model on several synthetic and real-world tasks, which show that NC-GRU significantly outperforms GRU and several other RNNs.
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This content will become publicly available on September 8, 2026
Predicting Software Defect Discovery Incorporating Covariates with Recurrent Neural Networks
ABSTRACT Traditional software reliability growth models (SRGM) characterize defect discovery with the Non‐Homogeneous Poisson Process (NHPP) as a function of testing time or effort. More recently, covariate NHPP SRGM models have substantially improved tracking and prediction of the defect discovery process by explicitly incorporating discrete multivariate time series on the amount of each underlying testing activity performed in successive intervals. Both classes of NHPP models with and without covariates are parametric in nature, imposing assumptions on the defect discovery process, and, while neural networks have been applied to SRGM models without covariates, no such studies have been applied in the context of covariate SRGM models. Therefore, this paper assesses the effectiveness of neural networks in predicting the software defect discovery process, incorporating covariates. Three types of neural networks are considered, including (i) recurrent neural networks (RNNs), (ii) long short‐term memory (LSTM), and (iii) gated recurrent unit (GRU), which are then compared with covariate models to validate tracking and predictive accuracy. Our results suggest that GRU achieved better overall goodness‐of‐fit, such as approximately 3.22 and 1.10 times smaller predictive mean square error, and 5.33 and 1.22 times smaller predictive ratio risk in DS1G and DS2G data sets, respectively, compared to covariate models when of the data is used for training. Moreover, to provide an objective comparison, three different proportions for training data splits were employed to illustrate the advancements between the top‐performing covariate NHPP model and the neural network, in which GRU illustrated a better performance over most of the scenarios. Thus, the neural network model with gated recurrent units may be a suitable alternative to track and predict the number of defects based on covariates associated with the software testing process.
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- Award ID(s):
- 1749635
- PAR ID:
- 10638610
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Quality and Reliability Engineering International
- ISSN:
- 0748-8017
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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