Embedding maximum entropy models in algebraic varieties by Gröbner bases methods

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.

Simulation and analysis of red-crowned crane habitat suitability using maximum entropy and information entropy models

Empirical evidence of scaling behaviors in neuronal avalanches suggests that neuronal populations in the brain operate near criticality. Departure from scaling in neuronal avalanches has been used as a measure of distance to criticality and linked to brain disorders. A distinct line of evidence for brain criticality has come from thermodynamic signatures in maximum entropy (ME) models. Both of these approaches have been widely applied to the analysis of neuronal data. However, the relationship between deviations from avalanche criticality and thermodynamics of ME models of neuronal populations remains poorly understood. To address this question, we study spontaneous activity of organotypic rat cortex slice cultures in physiological and drug-induced hypo- or hyper-excitable conditions, which are classified as critical, subcritical and supercritical based on avalanche dynamics. We find that ME models inferred from critical cultures show signatures of criticality in thermodynamic quantities, e.g. specific heat. However, such signatures are also present, and equally strong, in models inferred from supercritical cultures—despite their altered dynamics and poor functional performance. On the contrary, ME models inferred from subcritical cultures do not show thermodynamic hints of criticality. Importantly, we confirm these results using an interpretable neural network model that can be tuned to and away from avalanche criticality. Our findings indicate that maximum entropy models correctly distinguish subcritical from critical/supercritical systems. However, they may not be able to discriminate between avalanche criticality and supercriticality, although they may still capture a number of important features from neuronal data.

Hodge theory is pivotal in studying algebraic varieties' intricate geometry and topology: it provides essential insights into their structure. The Hodge decomposition theorem establishes a profound link between the geometry of varieties and their cohomology groups, helping to understand their underlying properties. Moreover, Hodge theory was crucial at the inception of the field of mirror symmetry, revealing deep connections among seemingly disparate algebraic varieties. It also sheds light on studying algebraic cycles and motives, crucial objects in algebraic geometry. This article explores Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate a diverse range of such manifolds, including (quasi-)Hopf, (quasi-)Calabi-Eckmann, and LVM manifolds, alongside a class of definable complex manifolds encompassing both algebraic varieties and the aforementioned special cases. Our research establishes the preservation of the motivic nature of Hodge polynomials inside this broader context. Through explicit calculations and thorough analyses, this work contributes to a deeper understanding of complex manifold geometry beyond the realm of algebraic varieties. The outcomes of this study have potential applications in various areas of mathematics and physics where complex manifolds play a significant role.

Macaulay is a system for computing in algebraic geometry and cummutative algebra; it is capable of a variety of computations which are tedious or impossible to perform by hand. The primitive types in the system are polynomials, matrices, ideals, polynomial rings, modules, maps between rings, and complexes of modules. The system performs algebraic manipulation on objects of these types. The possible manipulations include the computation of standard (Gröbner) bases, modules of syzygies, finite free resolutions, Hilbert polynomials and functions. Using these basic operations, a variety of derived operations are possible, such as projections, ideal intersections, and the computation of coherent sheaf cohomology groups. The algorithm used for constructing standard (Gröbner) bases and syzygies is described in ([Buc76], [Zac78], [Sch80], [Bay82], [BaSt86a]).

We present an extension to Jaynes’ maximum entropy principle that incorporates latent variables. The principle oflatent maximum entropywe propose is different from both Jaynes’ maximum entropy principle and maximum likelihood estimation, but can yield better estimates in the presence of hidden variables and limited training data. We first show that solving for a latent maximum entropy model poses a hard nonlinear constrained optimization problem in general. However, we then show that feasible solutions to this problem can be obtained efficiently for the special case of log-linear models---which forms the basis for an efficient approximation to the latent maximum entropy principle. We derive an algorithm that combines expectation-maximization with iterative scaling to produce feasible log-linear solutions. This algorithm can be interpreted as an alternating minimization algorithm in the information divergence, and reveals an intimate connection between the latent maximum entropy and maximum likelihood principles. To select a final model, we generate a series of feasible candidates, calculate the entropy of each, and choose the model that attains the highest entropy. Our experimental results show that estimation based on the latent maximum entropy principle generally gives better results than maximum likelihood when estimating latent variable models on small observed data samples.

We propose a probabilistic model for estimating word confidence by fusing predictor features. Starting from the maximum entropy (ME) method, we first prove that ME model is equivalent to the best model with certain form to the minimum expected cross entropy (MECE) criterion. Under the MECE criterion, We extend the form of ME model by introducing a hidden state. We call the new model hidden-state maximum entropy (HSME) model. In a keyword-spotting task, we combine predictor features from both phonetic and word-level systems. Compared to lattice posterior alone, recall at 80% precision is improved from 38.1% to 49.5% on voicemail and from 37.1% to 51.9% on Switchboard. Compared with other fusion methods, HSME consistently outperforms decision tree, and most cases SVM.

With a view toward a fracton theory in condensed matter, we introduce a higher-moment polynomial degree-p global symmetry, acting on complex scalar/vector/tensor fields (e.g., ordinary or vector global symmetry for p$=0$ and p$=1$ respectively). We relate this higher-moment global symmetry of $n$-dimensional space, to a lower degree (either ordinary or higher-moment, e.g., degree-(p-$\ell$)) subdimensional or subsystem global symmetry on layers of $(n-\ell)$-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions of polynomials). The structure of layers of submanifolds as subvarieties can be studied via mathematical tools of embedding, foliation, and algebraic geometry. We also generalize Noether's theorem for this higher-moment polynomial global symmetry. We can promote the higher-moment global symmetry to a local symmetry, and derive a new family of higher-rank-m symmetric tensor gauge theory by gauging, with m = p$+1$. By further gauging a discrete $\mathbb{Z}_2^C$ charge conjugation (particle-hole) symmetry, we derive a new general class of rank-m tensor non-abelian gauge field theory (the gauge structure is non-commutative thus non-abelian but not an ordinary group): a hybrid class of (symmetric or non-symmetric) higher-rank-m tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879], interplaying between gapless and gapped sectors.

Climate change impacts on tropical forests and tree species have been documented as changes in distribution, growing period, phenology, habitat, productivity, species composition, and migration. This study attempts to assess the current and future habitat suitability distribution of two dominant species of Central India, teak (Tectona grandis Linn. f.) and sal (Shorea robusta Gaertn. f.) using a maximum entropy (MaxEnt) model to predict species occurrences by finding the distribution that has the most spread. The future suitable habitat ranges of the species were modeled for two time periods (2050 and 2070) and two representative concentration pathways (RCP 2.6 and 8.5). Representative concentration pathways are scenarios that describe alternative trajectories for carbon dioxide emissions and the resulting atmospheric concentration from 2000 to 2100. We collected and modeled the spatially well-dispersed species occurrence points of teak and sal. The results suggested an increase in suitable habitat range for teak and a decrease for sal for both time periods and RCPs. Furthermore, the jackknife analysis identified temperature and precipitation seasonality as the major variables that influence the distribution of teak. In the case of sal, minimum temperature makes the maximum contribution to distribution changes. The suitable silvicultural strategies for forest management are proposed based on the future distribution of species in relation to the climate variables responsible for the change in their distribution range. These findings and strategies will help forest departments build future management plans for teak and sal forest with a focus on minimizing the impact of climate change. Study Implications: Our study used maximum entropy (MaxEnt) modeling to understand the impact of changing climate on the distribution of teak and sal forests of central India and to propose future silvicultural strategies. The study used MaxEnt model for two time periods and two climate change scenarios at highest resolution. An increase in future suitable habitat for teak and a decrease for sal is predicted by the model. Temperature for teak and precipitation for sal were identified as the major influencing climatic variables. We recommend forest and other related government departments to commission focused research to understand the changing patterns of species with climate change and design appropriate silviculture strategies for effective management.

Intrusion detection is a critical component of secure information systems. Current intrusion detection systems (IDS) especially NIDS (Network Intrusion Detection System) examine all data features to detect intrusions. However, some of the features may be redundant or contribute little to the detection process and therefore they have great impact on the system performance. This paper proposes a lightweight intrusion detection model that is computationally efficient and effective based on feature selection and Maximum Entropy (ME) model. Firstly, the issue of identifying important input features is addressed. Since elimination of the insignificant and/or useless inputs leads to a simplification of the problem, therefore results to faster and more accurate detection. Secondly, classic ME model is used to learn and detect intrusions using the selected important features. Experimental results on the well-known KDD 1999 dataset show the proposed model is effective and can be applied to real-time intrusion detection environments.

PDF HTML阅读 XML下载 导出引用 引用提醒 气候变化情景下基于最大熵模型的青海云杉潜在分布格局模拟 DOI: 10.5846/stxb201809151999 作者: 作者单位: 甘肃农业大学林学院,兰州大学生命科学学院,甘肃农业大学林学院,甘肃农业大学林学院 作者简介: 通讯作者: 中图分类号: 基金项目: 甘肃农业大学科技创新基金-公招博士科研启动基金项目（GSAU-RCZX01707）；学科建设专项基金项目（GAU-XKJS-2018-099）；国家自然基金项目（31860197） Simulation of the potential distribution patterns of Picea crassifolia in climate change scenarios based on the maximum entropy (Maxent) model Author: Affiliation: Gansu Agricultural University, Lanzhou 730070 Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:青海云杉（Picea crassifolia）是我国青藏高原东北缘特有树种，在维系我国西北地区生态平衡、水土保持、水源涵养和生物多样性等方面发挥着重要作用。基于其分布范围内的69个地理分布样点，利用最大熵（Maxent）模型对现实气候条件下青海云杉的潜在分布及其分布的主导气候因子进行分析，同时结合3种大气环流模型模拟青海云杉在3种气候变化情景（温室气候排放量不同）下未来2050s和2080s潜在分布区的变化。结果表明：Maxent模型对青海云杉潜在分布区的预测具有极高的准确度，所有模型的平均受试者工作特征曲线下面积（AUC测试值）均高于0.99；Jackknife检验和气候因子响应曲线表明年最低降雨量是限制青海云杉分布的主导因子；当前青海云杉的潜在分布区主要集中于青海东部、甘肃东南部、宁夏大部分地区、西藏东部、四川西部山区以及陕西、新疆和内蒙古部分地区。在未来3种增温情景下，青海云杉在2050s和2080s的潜在分布总面积与当前相比变化不明显，但不同适生等级的潜在分布面积变化较大，其中，中度适生区和低度适生区受气候增温影响显著，中度增温下这些区域在2080s的面积明显增大，而高度适生区（核心分布）则在所有增温情景下均呈缩小趋势。同时，在未来3种增温情景下，青海云杉在2050s和2080s的潜在分布区有向北移动趋势，但其心分布区域（高度适生区）仍然以青海东部、甘肃北部为主，无明显变迁趋势。从气候因素角度考虑，本研究表明未来气候变化情景下，青海云杉依然在西部高山地区，特别是作为我国重要生态屏障的祁连山、贺兰山等山区具有重要的经济价值并将持续其生态服务功能。 Abstract:Qinghai spruce (Picea crassifolia) is the endemic tree species in the northeastern margin of the Qinghai-Tibet Plateau (QTP) in China. It plays an important role in maintaining ecological balance, biodiversity, and soil and water conservation in northwestern China. In this study, the current potential distribution of Qinghai spruce and its dominant climatic factors were analyzed firstly based on 69 distributional records via the maximum entropy model (Maxent). Then the distribution patterns of Qinghai spruce under three climate change scenarios (i.e., the lowest, a moderate, and the highest greenhouse gas emission scenarios; RCP2.6, RCP4.5 and RCP8.5) with three general atmospheric circulation models were predicted by Maxent for two future decades:the 2050s and 2080s. The results showed that the accuracy of the Maxent model was pretty high for modeling potential distribution regions of Qinghai spruce, and the average values of the area under the receiver operator characteristic curve (AUC) were higher than 0.99. The results from the Jackknife test and climate factor response curves revealed that the annual lowest precipitation is the key factor that limited the distribution of Qinghai spruce; the current potential distribution regions of Qinghai spruce are mainly concentrated in eastern Qinghai, southeastern Gansu, most of Ningxia, eastern Tibet, mountainous areas of western Sichuan, and parts of Shaanxi, Xinjiang, and Inner Mongolia. The total potential distribution area of Qinghai spruce in the 2050s and 2080s was changed little compared to the current distribution in all three scenarios, but its potential distribution area at different levels of fitness was changed a lot. Among them, the regions with fitness of less than 0.1 and in the range of 0.1-0.3 were significantly affected by the climate changes, and their area was largely increased in the medium emission scenario. In contrast, the regions with fitness in the range of 0.5-0.8 were shrunk in all scenarios. At the same time, the distribution patterns of Qinghai spruce in the 2050s and 2080s had a tendency to move northward in all scenarios, whereas its central distributional regions (more suitable regions) were still kept in eastern Qinghai and northern Gansu, with no obvious trend to change. From the perspective of climatic factors, this study showed that Qinghai spruce would continue to have important economic values and ecological service functions in the western mountains, especially in the Qilian Mountains and Helan Mountains, which are the important ecological barriers in China. 参考文献 相似文献 引证文献

While radiologists regularly issue follow-up recommendations, our preliminary research has shown that anywhere from 35 to 50% of patients who receive follow-up recommendations for findings of possible cancer on abdominopelvic imaging do not return for follow-up. As such, they remain at risk for adverse outcomes related to missed or delayed cancer diagnosis. In this study, we develop an algorithm to automatically detect free text radiology reports that have a follow-up recommendation using natural language processing (NLP) techniques and machine learning models. The data set used in this study consists of 6000 free text reports from the author's institution. NLP techniques are used to engineer 1500 features, which include the most informative unigrams, bigrams, and trigrams in the training corpus after performing tokenization and Porter stemming. On this data set, we train naive Bayes, decision tree, and maximum entropy models. The decision tree model, with an F1 score of 0.458 and accuracy of 0.862, outperforms both the naive Bayes (F1 score of 0.381) and maximum entropy (F1 score of 0.387) models. The models were analyzed to determine predictive features, with term frequency of n-grams such as "renal neoplasm" and "evalu with enhanc" being most predictive of a follow-up recommendation. Key to maximizing performance was feature engineering that extracts predictive information and appropriate selection of machine learning algorithms based on the feature set.

The convergence rates of generalized iterative scaling (GIS) and improved iterative scaling (IIS) algorithms for fitting maximum entropy (ME) models are investigated and also a particular linear dynamic system monitoring with partial active measurements is studied. An information-theoretic based measurement scheme is derived to select informative hidden states, which is validated on a problem of origin-destination matrix estimation for Internet traffic.

To predict the potential geographic distribution of Oncomelania hupensis in Yunnan Province using random forest (RF) and maximum entropy (MaxEnt) models, so as to provide insights into O. hupensis surveillance and control in Yunnan Province. The O. hupensis snail survey data in Yunnan Province from 2015 to 2016 were collected and converted into O. hupensis snail distribution site data. Data of 22 environmental variables in Yunnan Province were collected, including twelve climate variables (annual potential evapotranspiration, annual mean ground surface temperature, annual precipitation, annual mean air pressure, annual mean relative humidity, annual sunshine duration, annual mean air temperature, annual mean wind speed, ≥ 0 ℃ annual accumulated temperature, ≥ 10 ℃ annual accumulated temperature, aridity and index of moisture), eight geographical variables (normalized difference vegetation index, landform type, land use type, altitude, soil type, soil textureclay content, soil texture-sand content and soil texture-silt content) and two population and economic variables (gross domestic product and population). Variables were screened with Pearson correlation test and variance inflation factor (VIF) test. The RF and MaxEnt models and the ensemble model were created using the biomod2 package of the software R 4.2.1, and the potential distribution of O. hupensis snails after 2016 was predicted in Yunnan Province. The predictive effects of models were evaluated through cross-validation and independent tests, and the area under the receiver operating characteristic curve (AUC), true skill statistics (TSS) and Kappa statistics were used for model evaluation. In addition, the importance of environmental variables was analyzed, the contribution of environmental variables output by the models with AUC values of > 0.950 and TSS values of > 0.850 were selected for normalization processing, and the importance percentage of environmental variables was obtained to analyze the importance of environmental variables. Data of 148 O. hupensis snail distribution sites and 15 environmental variables were included in training sets of RF and MaxEnt models, and both RF and MaxEnt models had high predictive performance, with both mean AUC values of > 0.900 and all mean TSS values and Kappa values of > 0.800, and significant differences in the AUC (t = 19.862, P < 0.05), TSS (t = 10.140, P < 0.05) and Kappa values (t = 10.237, P < 0.05) between two models. The AUC, TSS and Kappa values of the ensemble model were 0.996, 0.954 and 0.920, respectively. Independent data verification showed that the AUC, TSS and Kappa values of the RF model and the ensemble model were all 1, which still showed high performance in unknown data modeling, and the MaxEnt model showed poor performance, with TSS and Kappa values of 0 for 24%(24/100) of the modeling results. The modeling results of 79 RF models, 38 MaxEnt models and their ensemble models with AUC values of > 0.950 and TSS values of > 0.850 were included in the evaluation of importance of environmental variables. The importance of annual sunshine duration (SSD) was 32.989%, 37.847% and 46.315% in the RF model, the MaxEnt model and their ensemble model, while the importance of annual mean relative humidity (RHU) was 30.947%, 15.921% and 28.121%, respectively. Important environment variables were concentrated in modeling results of the RF model, dispersed in modeling results of the MaxEnt model, and most concentrated in modeling results of the ensemble model. The potential distribution of O. hupensis snails after 2016 was predicted to be relatively concentrated in Yunnan Province by the RF model and relatively large by the MaxEnt model, and the distribution of O. hupensis snails predicted by the ensemble model was mostly the joint distribution of O. hupensis snails predicted by RF and MaxEnt models. Both RF and MaxEnt models are effective to predict the potential distribution of O. hupensis snails in Yunnan Province, which facilitates targeted O. hupensis snail control.

Computation in commutative algebra and algebraic geometry has taken a major step forward in the last decade. This is due to the development of Grobner bases, computer algebra systems which are readily available to researchers, and the great increase in power of computers in this last decade.