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Abstract— Stochastic computing (SC) uses streams of pseudo-random bits to perform low-cost and error-tolerant numerical processing for applications like neural networks and digital filtering. A key operation in these domains is the summation of many hundreds of bit-streams, but existing SC adders are inflexible and unpredictable. Basic mux adders have low area but poor accuracy while other adders like accumulative parallel counters (APCs) have good accuracy but high area. This work introduces parallel sampling adders (PSAs), a novel weighted adder family that offers a favorable area-accuracy trade-off and provides great flexibility to large-scale SC adder design. Our experiments show that PSAs can sometimes achieve the same high accuracy as APCs, but at half the area cost. We also examine the behavior of large-scale SC adders in depth and uncover some surprising results. First, APC accuracy is shown to be sensitive to input correlation despite the common belief that APCs are correlation insensitive. Then, we show that mux-based adders are sometimes more accurate than APCs, which contradicts most prior studies. Explanations for these anomalies are given and a decorrelation scheme is proposed to improve APC accuracy by 4x for a digital filtering application. 

ABSTRACT - High-cost stochastic number generators (SNGs) are the main source of stochastic numbers (SNs) in stochastic computing. Interacting SNs must usually be uncorrelated for satisfactory results, but deliberate correlation can sometimes dramatically reduce area and/or improve accuracy. However, very little is known about the correlation behavior of SNGs. In this work, a core SNG component, its probability conversion circuit (PCC), is analyzed to reveal important tradeoffs between area, correlation, and accuracy. We show that PCCs of the weighted binary generator (WBG) type cannot consistently generate correlated bitstreams, which leads to inaccurate outputs for some designs. In contrast, comparator-based PCCs (CMPs) can generate highly correlated bitstreams but are about twice as large as WBGs. To overcome these area-correlation limitations, a novel class of PCCs called multiplexer majority chains (MMCs) is introduced. Some MMCs are area efficient like WBGs but can generate highly correlated SNs like CMPs and can reduce the area of a filtering circuit by 30% while sacrificing only 7% accuracy. The large influence of PCC design on circuit area and accuracy is explored and suggestions are made for selecting the best PCC based on a target system’s correlation requirements. 

Stochastic computing (SC) is a low-cost computational paradigm that has promising applications in digital filter design, image processing, and neural networks. Fundamental to these applications is the weighted addition operation, which is most often implemented by a multiplexer (mux) tree. Mux-based adders have very low area but typically require long bitstreams to reach practical accuracy thresholds when the number of summands is large. In this work, we first identify the main contributors to mux adder error. We then demonstrate with analysis and experiment that two new techniques, precise sampling and full correlation, can target and mitigate these error sources. Implementing these techniques in hardware leads to the design of CeMux (Correlation-enhanced Multiplexer), a stochastic mux adder that is significantly more accurate and uses much less area than traditional weighted adders. We compare CeMux to other SC and hybrid designs for an electrocardiogram filtering case study that employs a large digital filter. One major result is that CeMux is shown to be accurate even for large input sizes. CeMux's higher accuracy leads to a latency reduction of 4× to 16× over other designs. Furthermore, CeMux uses about 35% less area than existing designs, and we demonstrate that a small amount of accuracy can be traded for a further 50% reduction in area. Finally, we compare CeMux to a conventional binary design and we show that CeMux can achieve a 50% to 73% area reduction for similar power and latency as the conventional design but at a slightly higher level of error. 

Designing low-cost filterbanks is important due to severe resource limitations imposed by hearing aid size. Here, we develop a novel FIR filterbank employing stochastic computing (SC). SC-based filters use (pseudo)-random bitstreams to efficiently perform the core filtering operation. We demonstrate that SC is well-suited to low-cost filterbank design and compare our SC filterbank to a conventional sequential binary (SB) design. We show that the SC design achieves the same accuracy and latency as the SB one, with an exceptionally large 70% reduction in chip area. The power consumption of our proposed SC filterbank is 38-96% that of the SB design. 

Understanding accuracy and the tradeoffs it entails is key to evaluating the growing list of stochastic computing (SC) circuit designs. Due to shortcomings of current SC error theory, simulation has become the standard way to estimate a circuit's accuracy. However, simulation can demand large computational resources and lead to uncertain, misleading, or unexplainable results. A soundly based analytic approach is therefore preferable to simulation. In this work, we first show the input value distribution's large influence on circuit accuracy. Then we develop a Bayesian error analysis methodology which uses the input value distribution as a prior to inform better accuracy estimates. This error formulation introduces concepts new to SC such as estimator dominance and points to ways of improving simulation-based accuracy estimates. Orthogonal to the Bayesian ideas, we also show how to use bias-variance decomposition to simplify and aggregate the effects of SC's many error sources. We present techniques that use the beta distribution to model the stochastic number value distribution. Finally, we demonstrate the use of these ideas to improve the accuracy and analysis of an SC-based neural network. 

Tree growth and longevity trade-offs fundamentally shape the terrestrial carbon balance. Yet, we lack a unified understanding of how such trade-offs vary across the world’s forests. By mapping life history traits for a wide range of species across the Americas, we reveal considerable variation in life expectancies from 10 centimeters in diameter (ranging from 1.3 to 3195 years) and show that the pace of life for trees can be accurately classified into four demographic functional types. We found emergent patterns in the strength of trade-offs between growth and longevity across a temperature gradient. Furthermore, we show that the diversity of life history traits varies predictably across forest biomes, giving rise to a positive relationship between trait diversity and productivity. Our pan-latitudinal assessment provides new insights into the demographic mechanisms that govern the carbon turnover rate across forest biomes. 

Abstract The emergence of alternative stable states in forest systems has significant implications for the functioning and structure of the terrestrial biosphere, yet empirical evidence remains scarce. Here, we combine global forest biodiversity observations and simulations to test for alternative stable states in the presence of evergreen and deciduous forest types. We reveal a bimodal distribution of forest leaf types across temperate regions of the Northern Hemisphere that cannot be explained by the environment alone, suggesting signatures of alternative forest states. Moreover, we empirically demonstrate the existence of positive feedbacks in tree growth, recruitment and mortality, with trees having 4–43% higher growth rates, 14–17% higher survival rates and 4–7 times higher recruitment rates when they are surrounded by trees of their own leaf type. Simulations show that the observed positive feedbacks are necessary and sufficient to generate alternative forest states, which also lead to dependency on history (hysteresis) during ecosystem transition from evergreen to deciduous forests and vice versa. We identify hotspots of bistable forest types in evergreen-deciduous ecotones, which are likely driven by soil-related positive feedbacks. These findings are integral to predicting the distribution of forest biomes, and aid to our understanding of biodiversity, carbon turnover, and terrestrial climate feedbacks. 

Abstract AimEcological and anthropogenic factors shift the abundances of dominant and rare tree species within local forest communities, thus affecting species composition and ecosystem functioning. To inform forest and conservation management it is important to understand the drivers of dominance and rarity in local tree communities. We answer the following research questions: (1) What are the patterns of dominance and rarity in tree communities? (2) Which ecological and anthropogenic factors predict these patterns? And (3) what is the extinction risk of locally dominant and rare tree species? LocationGlobal. Time period1990–2017. Major taxa studiedTrees. MethodsWe used 1.2 million forest plots and quantified local tree dominance as the relative plot basal area of the single most dominant species and local rarity as the percentage of species that contribute together to the least 10% of plot basal area. We mapped global community dominance and rarity using machine learning models and evaluated the ecological and anthropogenic predictors with linear models. Extinction risk, for example threatened status, of geographically widespread dominant and rare species was evaluated. ResultsCommunity dominance and rarity show contrasting latitudinal trends, with boreal forests having high levels of dominance and tropical forests having high levels of rarity. Increasing annual precipitation reduces community dominance, probably because precipitation is related to an increase in tree density and richness. Additionally, stand age is positively related to community dominance, due to stem diameter increase of the most dominant species. Surprisingly, we find that locally dominant and rare species, which are geographically widespread in our data, have an equally high rate of elevated extinction due to declining populations through large‐scale land degradation. Main conclusionsBy linking patterns and predictors of community dominance and rarity to extinction risk, our results suggest that also widespread species should be considered in large‐scale management and conservation practices. 

Abstract Determining the drivers of non-native plant invasions is critical for managing native ecosystems and limiting the spread of invasive species^1,2. Tree invasions in particular have been relatively overlooked, even though they have the potential to transform ecosystems and economies^3,4. Here, leveraging global tree databases^5–7, we explore how the phylogenetic and functional diversity of native tree communities, human pressure and the environment influence the establishment of non-native tree species and the subsequent invasion severity. We find that anthropogenic factors are key to predicting whether a location is invaded, but that invasion severity is underpinned by native diversity, with higher diversity predicting lower invasion severity. Temperature and precipitation emerge as strong predictors of invasion strategy, with non-native species invading successfully when they are similar to the native community in cold or dry extremes. Yet, despite the influence of these ecological forces in determining invasion strategy, we find evidence that these patterns can be obscured by human activity, with lower ecological signal in areas with higher proximity to shipping ports. Our global perspective of non-native tree invasion highlights that human drivers influence non-native tree presence, and that native phylogenetic and functional diversity have a critical role in the establishment and spread of subsequent invasions.