We address causal reasoning in multivariate time series data generated by stochastic processes. Existing approaches are largely restricted to static settings, ignoring the continuity and emission of variations across time. In contrast, we propose a learning paradigm that directly establishes causation between events in the course of time. We present two key lemmas to compute causal contributions and frame them as reinforcement learning problems. Our approach offers formal and computational tools for uncovering and quantifying causal relationships in diffusion processes, subsuming various important settings such as discrete-time Markov decision processes. Finally, in fairly intricate experiments and through sheer learning, our framework reveals and quantifies causal links, which otherwise seem inexplicable.

Machine learning models achieve state-of-the-art performance on many supervised learning tasks. However, prior evidence suggests that these models may learn to rely on "shortcut" biases or spurious correlations (intuitively, correlations that do not hold in the test as they hold in train) for good predictive performance. Such models cannot be trusted in deployment environments to provide accurate predictions. While viewing the problem from a causal lens is known to be useful, the seamless integration of causation techniques into machine learning pipelines remains cumbersome and expensive. In this work, we study and extend a causal pre-training debiasing technique called causal bootstrapping (CB) under five practical confounded-data generation-acquisition scenarios (with known and unknown confounding). Under these settings, we systematically investigate the effect of confounding bias on deep learning model performance, demonstrating their propensity to rely on shortcut biases when these biases are not properly accounted for. We demonstrate that such a causal pre-training technique can significantly outperform existing base practices to mitigate confounding bias on real-world domain generalization benchmarking tasks. This systematic investigation underlines the importance of accounting for the underlying data-generating mechanisms and fortifying data-preprocessing pipelines with a causal framework to develop methods robust to confounding biases.

In this paper, we study the effect of radio frequency (RF) impairments and channel estimation error (CEE) in multi-user two-way and one-way amplify-and-forward relay network. In particular, for both the networks, we derive the approximate signal-to-noise-plus-distortion-and-error ratio expressions and then use them to evaluate the overall outage probability and their asymptotic analysis. Based on the analytical results, it is shown that the RF impairments and CEE can severely degrade the outage performance, and even reduce the diversity order to zero. Unexpectedly, in two-way relay system, we notice that the system performance decreases with increasing number of users in a multi-user environment. Finally, numerical illustrations are done to validate our analytical expressions with the help of Monte-Carlo simulations.

In this paper, the impact of HWI on channel estimation error (CEE) and their combined impact on the performance of amplify-and-forward variable-gain (AF-VG) two-way relay network (TWRN) has been investigated. The expression for equivalent signal-to-noise plus distortion- anderror ratio (SNDER) at user terminals is derived. This is used to deduce closed-form expressions for the exact analysis of outage probabilities (OP) and tractable formulations for average symbol error rate (SER) taking a Rayleigh block fading channel. For further insights, we perform asymptotic analysis of OP and observe irreducible outage floors which act as fundamental performance limits. This SNDER ceiling acts as an upper limit for practical transmission rates. These derived results are validated using Monte Carlo simulations. Finally, we provide insights into the trade-off between the training length and pilot transmission power to obtain better performance by analyzing the symbol error rate (SER) curves under different training conditions.

In this paper, the performance of variable-gain (VG) amplify-and-forward (AF) relaying is studied considering both the practical problems of hardware impairments (HWI) and imperfect channel estimation (ICE). End-to-end signal-to-noise plus distortion-and-error ratio (SNDER) and outage probability (OP) expressions for this model are found over Nakagami-m fading channels. Upper bound of the system performance with respect to the SNDER is also found out in the presence of HWI and ICE. High SNR analysis of the derived expressions is performed to gain deeper insights into the performance. The results of fixed-gain (FG) AF and VG-AF are compared with the lower bound of the OP. To verify the theoretical results, Monte Carlo simulation is used. Further, there are plots which show the system behavior with improving channel conditions.

Hardware impairments (HWI) in transceivers have been found to cause severe performance degradation in wireless relay networks. However, literature investigating their impact on networks with imperfect channel estimation (ICE) is limited. In this paper, the impact of HWI on channel estimation error (CEE) and their combined impact on the performance of amplify-and-forward variable-gain (AF-VG) two-way relay network (TWRN) and one-way relay network (OWRN) have been investigated. The expressions for equivalent signal-to-noise plus distortion-and-error ratio (SNDER) and a closed-form expression for the exact analysis of outage probabilities (OP) are derived by taking a Rayleigh block fading channel. In addition, the combined impacts of HWI and CEE on outage performance of TWRN and OWRN systems for low and high transmission rates are evaluated. For further insights, we perform asymptotic analysis of OP and observe irreducible outage floors which act as fundamental performance limits. This SNDER ceiling acts as an upper limit for practical transmission rates. From this we infer that at high transmit powers, HWI and CEE are largely responsible in elevating the outage floors. Finally, the derived results are validated using Monte Carlo simulations.