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NASA NTRSDate not supplied

The Added Value of SMAP Soil Moisture in Crop Yield Forecasting Over Argentina

Argentina is one of the major producers and exporter of soybeans, corn, and wheat to the world market; therefore, the accurate and timely forecasting of those crops yield is crucial to national crop management and global food security. Previous studies have mainly focused on developing forecasting models for a specific crop type and location using a single source of data (e.g., vegetation indices), thus providing little insight into the forecasting models' performance on different crop types and regions. Besides, these models are based on traditional statistical regression algorithms, while more advanced machine learning approaches have not been explored. This study investigated the estimation of crop yields of three major crops (corn, soybean, and winter wheat) using Multiple Linear Regression (MLR) and Support Vector Machine (SVM), over major growing provinces in Argentina. Our models were trained and evaluated on data from 2015 to 2020, where three remote sensing products (Normalized difference vegetation index (NDVI), SMAP soil moisture, and MODIS evapotranspiration) were used as predictors. Our results indicated that accurate crop yield forecasts using the developed regression models could be made one to two months before harvest. The MLR and SVM model performance varied among different crop types, where soybean and corn exhibited better predictability compare to the wheat. In most cases, the SVM outperformed the multiple linear regression model due to its ability to capture the nonlinear and complex features of the crop-production process. The forecasted model that combines data from multiple sources outperformed single-source satellite data. The highest accuracy was obtained when the three data sources were all considered in the model development. Results also indicated that the inclusion of SMAP soil moisture improved crop yield forecasting in most provinces, and the most significant improvements occurred in the drier region.

Nazmus Shams Sazib↗

DOE OSTI2021

Predicting measures of soil health using the microbiome and supervised machine learning

Soil health encompasses a range of biological, chemical, and physical soil properties that sustain the commercial and ecological value of agroecosystems. Monitoring soil health requires a comprehensive set of diagnostics that can be cost-prohibitive for routine analyses. The soil microbiome provides a rich source of information about soil properties, which can be assayed in a high-throughput, cost-effective way. We evaluated the accuracy of random forest (RF) and support vector machine (SVM) regression and classification models in predicting 12 measures of soil health, tillage status, and soil texture from 16S rRNA gene amplicon data with an operationally relevant sample set. We validated the efficacy of the best performing models against independent datasets and also tested best practices for processing microbiome data for use in machine learning. Soil health metrics could be predicted from microbiome data with the best models achieving a Kappa value of ~0.65, for categorical assessments, and a R2 value of ~0.8, for numerical scores. Biological health ratings were better predicted than chemical or physical ratings. Validation with independent datasets revealed that models had general predictive value for soil properties, including yield. The ecological profiles of several taxa important for model accuracy matched the observed relationships with soil health, including Pyrinomonadaceae, Nitrososphaeraceae, and Candidatus Udeaobacter. Models trained at the highest taxonomic resolution proved most accurate, with losses in accuracy resulting from rarefying, sparsity filtering, and aggregating at higher taxonomic ranks. Furthermore, our study provides the groundwork for developing scalable technology to use microbiome-based diagnostics for the assessment of soil health.

16S rRNA gene↗

DOE OSTI2022

Estimating mass-absorption cross-section of ambient black carbon aerosols: theoretical, empirical, and machine learning models

The mass-absorption cross-section of black carbon (MAC BC ) is an essential parameter to link the atmospheric concentration of black carbon (BC) with its radiative forcing. When a direct calculation of MAC BC based on observations of aerosol light absorption and BC mass concentration is impossible, we rely on modeling and simulations to estimate MAC BC , but currently, there is no consensus model that can be relied on for accurate predictions across all atmospheric environments when BC particles have different coating thicknesses. Here, we applied five MAC BC prediction models (including three light scattering theories, an empirical model based on observations of particle mass concentrations, and a machine learning model developed in our previous work) to aerosols from three Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) field campaigns. While many studies have found that increasing the complexity of the models helps to constrain biases of the estimated MAC BC , our effort is to evaluate the models based on the criteria of simplicity and accuracy. We find that our machine learning model (support vector machine for regression, SVM) generally performs well across all DOE ARM field campaign data, while the accuracy of core-shell Mie theory depends on the bias correction algorithm applied to filter-based light absorption data. Generally, the empirical model for internally-mixed particles that we considered tends to over-predict MAC BC , while Mie theory for externally-mixed particles tends to under-predict MAC BC . An examination of the influence of coating material on BC cores suggests that the performance of our current SVM model is degraded when the BC is thickly-coated (e.g., it has undergone aging and mixing with other materials in the atmosphere).

54 ENVIRONMENTAL SCIENCES↗

NASA NTRSDate not supplied

Algorithmic Detection of Elemental Biosignatures

Machine learning models that classify a sample as indicative or non-indicative of life could play an important role in life-detection missions. Their predictions result from agnostic algorithms and thereby add redundancy to judgements resulting from human expertise. Additionally, their important features can reveal the most informative measurements within the operational constraints of a life-detection mission. The Ladder of Life Detection (Neveu 2018) identifies the need for an understanding of how combinations of multiple biosignatures affect overall confidence. The present work provides a starting point to answer this need, and future work will expand the data types to obtain even more predictive combinations of features. Elemental abundance was chosen as a starting set of features due to its availability in diverse sample types, which are needed to train a generalizable model. A standardized dataset was collected, including 35 non-indicative, e.g., lunar rock, basalt; 19 indicative mixed, e.g., seawater, agricultural soil; 46 indicative non-alive, e.g., coal, chalk; and 10 indicative alive, e.g., biofilm, bacteria. This dataset could be valuable for complementary biosignature research. The samples were standardized to the same limit of detection of a simulated mission scenario. Four classification models were used: k-nearest neighbors (KNN), logistic regression (LR), linear support vector machines (SVM), and Gaussian naïve Bayes (GNB). To obtain feature importances, KNN was run on three principal components of the training data and LR and SVM were run with L1 and L2 regularization. The performances and feature importances of the six model variants on 40:60 train to validation ratios were assessed with Monte Carlo simulations. ROC AUC and mean accuracy scores ranged between 82% - 94%, with sensitivity greater than specificity. For indicative of life predictors, all models had C and Ca as strong and Cl as medium; a majority of models had N, K, and P as medium. For non-indicative of life predictors, all models had Si as strong, and a majority of models had Mg, Al, and Ti as medium. Varied elements were Fe (slightly non-indicative), H (slightly indicative), O (widely varied), Na, Mn, and S. These results serve as a proof of concept and suggest important elemental signals beyond merely the CHNOPS of Earth-based life.

Algorithmic↗

NASA NTRSDate not supplied

Algorithmic Classification of Raman Spectra Biosignatures: Improving Life Detection Confidence

“Agnostic” biosignatures – indicators of life (or the absence of life), independent of a particular biochemistry – are increasingly considered a high standard for life detection. The Ladder of Life Detection (2018) called for investigating how combinations of independent and different potential biosignatures affect confidence. To address this gap, statistical classification of elemental abundances, isotopic fractionation, and reflectance spectroscopy (VNIR) has been implemented. Raman spectroscopy, highly desirable due to its wide availability, has the potential to improve this predictive power. This work implemented biosignature classification algorithms on Raman data alone, in preparation for combination with the other data types. Raman spectroscopy data was collected from published databases and papers as part of a manually curated dataset of “indicative” and “non-indicative of life” samples. These currently include 61 non-indicative samples (meteorites, magnetite); 3 indicative living samples (bacteria); 20 indicative non-living samples (chalk, bone); and 12 indicative mixed (with non-indicative material) samples (soil, microbial mats). Laboratory work is ongoing to characterize additional samples, particularly a greater breadth of mixed systems. Spectra were interpolated, filtered with the Savitzsky-Golay filter, and de-noised. For a preliminary examination, agnostic features were manually extracted including mean intensity, number of peaks, and mean peak width. Different peak prominences and filtering polynomials were used to refine features. Classification algorithms were implemented: k-nearest neighbors (KNN), logistic regression (LR), linear support vector machines (SVM), random forest (RF), Gaussian naïve bayes (GNB). Lastly, Monte Carlo simulations on 1,000 50%-train-test-splits were used to validate classification performance and feature significance. The preliminary feature set achieved its highest AUC of 0.52 with LR, with no strongly discriminatory features. Work to improve feature extraction, such as through deep learning with back propagation, is planned. In future work, the Raman data will be combined with the other data types, and potentially new data types such as enantiomeric excess. This project was partially supported through the NASA Ames Project EXcellence (APEX) incubator program.

Astrobiology↗

NASA NTRS2021

Estimation of Snow Mass Information via Assimilation of C-Band Synthetic Aperture Radar Backscatter Observations Into an Advanced and Surface Model

This study assimilated Sentinel-1 C-band backscatter observations over snow-covered terrain into the Noah-Multiparameterization land surface model using support vector machine (SVM) regression and an ensemble Kalman filter to improve the modeled terrestrial snow mass estimates. The data assimilation (DA) experiment was conducted across Western Colorado from September 2016 to August 2017. As part of the DA experiments, the impact of a rule-based update was evaluated by comparing snow water equivalent (SWE) estimates via DA (with [ DAv1 ] and without [ DAv2 ] the rule-based update) against SNOTEL SWE measurements. Results confirmed that rule-based update helped minimize SVM controllability issues, and in turn, improved the accuracy of SWE estimates relative to both open loop (OL) and DAv2 . Comparison of SWE estimates from Sentinel-1 DAv1 against SNOTEL SWE revealed that 75% of stations showed improvements in bias and correlation coefficient relative to the OL. Assimilated SWE estimates also showed statistical improvements during both the snow accumulation and snow ablation periods. However, unbiased root mean square error showed a slight increase during the snow ablation period due to the large variability in the electromagnetic response of C-band backscatter over deep and/or wet snow. Improvement of the SWE estimates also resulted in improving river discharge estimates compared to in situ measurements. River discharge using Sentinel-1 DAv1 improved the Nash–Sutcliffe efficiency at all available stations. These results suggest that physically constrained SVM can serve as an efficient observation operator for snow mass DA through explicit consideration of the first-order C-band scattering mechanisms over different terrestrial snow conditions.

Jongmin Park↗

NASA NTRSDate not supplied

Statistical Classification of Biosignature Information using Multiple Instrument Observations

The accurate identification of biosignatures (indications of life) from data taken from remote or in situ planetary exploration is one of the most important challenges in astrobiology, the interdisciplinary field examining habitability and the potential for extraterrestrial life. This study employs machine learning algorithms to optimize the identification of biosignatures, with an emphasis on those which are agnostic to a specific biochemical basis. We exploit the wealth of terrestrial data available from biogenic and abiogenic systems to enhance efficient feature prioritization. Our dataset, pulled from public databases and laboratory recorded measurements, includes elemental abundance, isotopic fractionation, and VNIR/Raman spectra The data curation process included standardization for detection limits and ranges. Subsequent feature extraction yielded detailed inputs for machine learning, including combinations of elemental content, isotopic ratios, and parameters of spectral peaks and troughs. Feature significance was evaluated across diverse machine learning methodologies, such as k-nearest neighbors, logistic regression, Random Forest, support vector machines, and Gaussian Naïve Bayes, along with a combined voting classifier. We utilized Receiver Operating Characteristic Area Under the Curve (ROC AUC) across 2,000 50% test-train splits as a robust metric of model performance. Results revealed a promising ROC AUC of 0.853 for the combined voting classifier. Removing elemental abundance data notably reduced model accuracy (13% decrease in AUC), highlighting its critical role in biosignature detection. Several other individual data features exhibited significance within their respective data types, offering additional granularity. This research fortifies the relevance of machine learning to astrobiology, potentially enhancing life detection missions by allowing algorithmic prioritization of high-interest samples for further investigation. Future work will refine data standardization, expand the dataset to include more terrestrial systems, and incorporate convolutional neural networks for spectral feature extraction. The potential for public data sharing is also under exploration, reinforcing our commitment to collective scientific advancement.

Statistical↗

DOE OSTI2021

Predicting nepheline precipitation in waste glasses using ternary submixture model and machine learning

Nepheline precipitation in nuclear waste glasses during vitrification can be detrimental due to its negative effect on chemical durability. Developing models to accurately predict nepheline precipitation from compositions is important to increase waste loading since existing models can be overly conservative. In this study, an expanded dataset containing 955 glasses was compiled from literature data, where 355 glasses are for high-level waste (HLW). Previously developed submixture models were refitted using the new dataset, where a misclassification rate of 7.8% was achieved. Nine machine learning (ML) algorithms (e.g., k-nearest neighbor, Gaussian process regression, artificial neural network, support vector machine, decision tree, etc.) were applied to evaluate their ability of predicting nepheline precipitation from compositions. Model accuracy, precision, recall/sensitivity, and F1 score were systemically compared between different ML algorithms and modeling protocols. Good model prediction with an accuracy ~0.9 (misclassification rate of ~10%) was observed with different algorithms under certain protocol. This study evaluated various ML models to predict nepheline precipitations in waste glasses, highlighting the importance of data preparation, modeling protocol, and their effect on model stability and reproducibility. The results provide insights into applying ML to predict glass properties and suggest areas for future research on modeling nepheline precipitations.

Lu, Xiaonan↗

DOE OSTI2021

Deep Learning Estimation of Daily Ground–Level NO 2 Concentrations from Remote Sensing Data

The limited number of nitrogen dioxide (NO 2 ) surface measurements calls for the development of highly accurate approaches to estimating surface NO 2 concentrations. In this study, we leverage a new satellite instrument, the TROPOspheric Monitoring Instrument (TROPOMI), along with other predictor variables, to estimate daily surface NO 2 concentrations over Texas in 2019. We use the deep convolutional neural network (Deep-CNN), an advanced deep learning algorithm, to obtain estimates and achieve a correlation coefficient (R) of 0.91, an index of agreement (IOA) of 0.95, and a mean absolute bias (MAB) of 1.75 ppb in surface NO 2 estimation. Additionally, we leverage a novel approach, SHapley Additive exPlanations (SHAP), to describe how Deep-CNN understands each predictor variable. The SHAP results show that the Deep-CNN model has an advanced understanding of the dataset, revealing that TROPOMI closely captures levels of NO 2 . In addition, we show the superiority of our Deep-CNN model at estimating surface NO 2 over other well-known machine learning and regression models in the field, including the support vector machines (SVM), random forest (RF), and multiple linear regression (MLR). Although SVM and RF show strong capabilities at estimating surface NO 2 concentrations, their accuracy is inferior to that of the Deep-CNN model, ranking second and third in model accuracy in this study. The MLR, however, shows a poor ability at NO 2 estimation and ranks last among all models. Furthermore, testing the impact of sample size on model performance, we also show that, compared to other models, Deep-CNN needs more samples to trigger its strength at surface NO 2 estimation.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Predicting melt pool depth and grain length using multiple signatures from in-situ single camera two-wavelength imaging pyrometry for laser powder bed fusion

In laser powder bed fusion (LPBF), the in-situ process signatures are known to have a direct correlation with the microstructural properties of the solidified melt pool (MP). It is known that the MP cooling and heating rates, and laser processing parameters can critically determine the grain structure and thereby affect the part properties. The objective of this work is to study the feasibility of using in-process, high-speed imaging pyrometry for evaluating the solidified MP properties “below” the surface, such as depth and microstructural properties. To accomplish this, we employ an in-house single camera-based two-wavelength imaging pyrometry (STWIP) system for monitoring the printing of single-scan tracks with Inconel 718 on a commercial LPBF printer (EOS M290). Further, the lab designed STWIP system is a coaxial high-speed (>10,000 fps) imaging system capable of monitoring MP temperature, morphology, and intensity profiles. The temperature measurements from STWIP are emissivity independent. The STWIP measured MP signatures of the printed tracks are correlated with the ex-situ microscopy characterized MP depth and the average grain lengths. From the data analysis, using support vector machine (SVM)-based regression models, we found that the MP temperature signatures are crucial for an accurate prediction of MP depth and the grain length, thus validating the novelty and necessity of the developed in-situ monitoring methods and analysis.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Machine learning prediction of incidence of Alzheimer’s disease using large-scale administrative health data

Nationwide population-based cohort provides a new opportunity to build an automated risk prediction model based on individuals’ history of health and healthcare beyond existing risk prediction models. We tested the possibility of machine learning models to predict future incidence of Alzheimer’s disease (AD) using large-scale administrative health data. From the Korean National Health Insurance Service database between 2002 and 2010, we obtained de-identified health data in elders above 65 years (N = 40,736) containing 4,894 unique clinical features including ICD-10 codes, medication codes, laboratory values, history of personal and family illness and socio-demographics. To define incident AD we considered two operational definitions: “definite AD” with diagnostic codes and dementia medication (n = 614) and “probable AD” with only diagnosis (n = 2026). We trained and validated random forest, support vector machine and logistic regression to predict incident AD in 1, 2, 3, and 4 subsequent years. For predicting future incidence of AD in balanced samples (bootstrapping), the machine learning models showed reasonable performance in 1-year prediction with AUC of 0.775 and 0.759, based on “definite AD” and “probable AD” outcomes, respectively; in 2-year, 0.730 and 0.693; in 3-year, 0.677 and 0.644; in 4-year, 0.725 and 0.683. The results were similar when the entire (unbalanced) samples were used. Important clinical features selected in logistic regression included hemoglobin level, age and urine protein level. This study may shed a light on the utility of the data-driven machine learning model based on large-scale administrative health data in AD risk prediction, which may enable better selection of individuals at risk for AD in clinical trials or early detection in clinical settings.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Application of machine learning approaches in the analysis of mass absorption cross-section of black carbon aerosols: Aerosol composition dependencies and sensitivity analyses

Physics-based models typically require an in-depth understanding of a phenomenon and assumptions of the underlying process(es), which are often hard to obtain in practice, whereas data-driven machine learning models learn the structure and patterns in the training data without any prior theoretical assumptions and then use inference to develop useful predictions. A novel machine learning-based algorithm has been previously developed for the prediction of black carbon mass absorption cross-section (MAC BC ) and applied to a variety of different atmospheric environments. In contrast to light scattering theories which require assumptions about the underlying physics, this algorithm uses time-series data of aerosol properties to estimate the temporally-varying MAC BC at 870 nm. Here, we analyze our algorithm and discuss the influence of aerosol optical properties (such as Ångström exponents and single scattering albedo) and chemical composition on the model outputs and the associated accuracy. Additionally, we conduct sensitivity analyses on our models to understand how the predictions change in response to different sets of input variables. Our support vector machine (SVM) for regression model is the least sensitive to variations in the input variables, although all models tend to exhibit a degradation to their accuracy when scattering Ångström exponents are less than one.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRSDate not supplied

Document Classification Techniques for Aviation Letters of Agreement

Often when working with technical documents, it is helpful to classify them into specific categories. In this paper, we conduct a thorough review of natural language processing techniques to perform this classification task on Letters of Agreement (LOAs), technical aviation documents outlining rules for utilizing US airspace. We evaluate multiple techniques, including Transfer Learning, for representing the text in the documents as embeddings: unigram and bigram Term Frequency Inverse Document Frequency (TFIDF), Word2Vec, Doc2Vec, GloVe and RoBERTa. We investigate a wide range of classification models: K-Nearest Neighbors, Random Forest, Support Vector Machines (SVM), Logistic Regression, Naive Bayes, Feed-Forward Neural Network, Convolutional Neural Networks (CNNs) and Long-Short Term Memory (LSTM). By comparing the different methods, we found the best overall approach for our task was to use unigram TFIDF representations with SVM while also gaining insight into how the other methodologies performed on a small technical datasets.

Aayushi Batra↗

NASA NTRSDate not supplied

Document Classification Techniques for Aviation Letters of Agreement

Often when working with historic air traffic management (ATM) documents, it is helpful to classify them into specific categories. In this paper, we conduct a thorough review of natural language processing techniques to perform this classification task on Letters of Agreement (LOAs), technical aviation documents outlining rules for utilizing US airspace. We evaluate multiple techniques for representing the text in the documents as embeddings: unigram and bigram Term Frequency Inverse Document Frequency (TFIDF), Word2Vec, Doc2Vec, GloVe and RoBERTa. We investigate a wide range of classification models: K-Nearest Neighbors, Random Forest, Support Vector Machines (SVM), Logistic Regression, Naive Bayes, Feed-Forward Neural Network, Convolutional Neural Networks (CNNs) and Long-Short Term Memory (LSTM). By comparing the different methods, we found the best overall approach for our task was to use unigram TFIDF representations with SVM while also gaining insight into how the other methodologies performed on a small technical datasets.

ATM↗

NASA NTRSDate not supplied

Document Classification Techniques for Aviation Letters of Agreement

ATM↗

DOE OSTI2021

Analysis and prediction of intersection traffic violations using automated enforcement system data

We report that the automated enforcement system (AES) is an effective way of supplementing traditional traffic enforcement, and the traffic violation data from AES can also be effectively used for safety research. In this study, traffic violation data were used to analyze the influencing factors associated with traffic violations and to predict the probability of violations at intersections. The potential factors influencing violations include 24 independent factors related to time, space, traffic and weather. Results from a logistic model showed that the midday period, weekends, residential districts, collector roads, congested traffic conditions, high traffic flow, lower wind speed and low temperature would increase the probability of traffic violations. The probability of violations was predicted by the random forest algorithm, which was proven to be the best traffic violation prediction model among logistic regression, Gaussian naive Bayes, and support vector machine. Moreover, the proximity weighted synthetic oversampling technique (ProWSyn) method was applied to reduce the impact of the imbalance ratio (IR) and improve the model’s prediction performance. The receiver operating characteristics (ROC) curves and Precision-Recall (PR) curves illustrated that the random forest algorithm using oversampling data had the best classifier prediction performance than undersampling data. The area under curve (AUC) and out-of-bag (OOB) error with IR = 1 reached 0.914 and 0.0787, which showed the better performance of the random forest algorithm using ProWSyn in dealing with imbalanced traffic violation data.

42 ENGINEERING↗

DOE OSTI2022

Machine learning for design principles for single atom catalysts towards electrochemical reactions

Machine learning (ML) integrated density functional theory (DFT) calculations have recently been used to accelerate the design and discovery of heterogeneous catalysts such as single atom catalysts (SACs) through the establishment of deep structure–activity relationships. Here, this review provides recent progress in the ML-aided rational design of heterogeneous catalysts with the focus on SACs in terms of structure–activity relationships, feature importance analysis, high-throughput screening, stability, and metal–support interactions for electrochemistry. Support vector machine (SVM), random forest regression (RFR), and deep neural networks (DNN) along with atomic properties are mainly used for the design of SACs. The ML results have shown that the number of electrons in the d orbital, oxide formation enthalpy, ionization energy, Bader charge, d-band center, and enthalpy of vaporization are mainly the most important parameters for the defining of the structure–activity relationships for electrochemistry. However, the black-box nature of ML techniques occasionally makes a physical interpretation of descriptors, such as the Bader charge, d-band center, and enthalpy of vaporization, non-trivial. At the current stage, ML application is limited by the lack of a large and high-quality database. Future prospects for the development of a large database and a generalized ML algorithm for SAC design are discussed to give insights for further studies in this field.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ 2 -norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Computer Science↗