Group lasso with overlap and graph lasso
基于3D增强CT影像组学的肾癌亚型三分类预测模型
肾细胞癌（RCC ）是人类十大最常见的恶性肿瘤之一，也是尿路最常见的恶性肿瘤，约占肾恶性肿瘤的85%和整体恶性肿瘤的3%［1］。
根据2016年世界卫生组织分类标准,肾透明细胞癌（ccRCC ）是最常见的肾癌亚型，约占全部RCC 的75%，也是最具侵入性和预后最差的一种亚型［2］。
第二、三位最常见的RCC 亚型是乳头状癌（pRCC ）和嫌色细胞癌（cRCC ），分别占10%~15%和5%。
RCC 的其他亚型包括集合管癌、多房囊性肾癌、髓样癌和未分类癌等［3-4］。
不同亚型肾癌的的生物学行为和侵袭性不同，其治疗方法和预后也不同，所以在治疗前进行RCC 亚型的鉴别很重要［5］。
此外，晚期肿瘤的靶向药物治疗和免疫治疗的选择也基于RCC 亚型［6-7］。
A three categories prediction model for renal cell carcinoma subtype based on 3D enhanced CT radiomicsZHANG Haijie 1,3,YIN Fu 2,CHEN Menglin 3,QI Anqi 3,YANG Liyang 3,CUI Weiwei 3,YANG Shanshan 3,WEN Ge 31PET/CT Center,First Affiliated Hospital of Shenzhen University,Shenzhen 518052,China;2School of Information Engineering,Shenzhen University,Shenzhen 518052,China;3Department of Imaging,Nanfang Hospital,Southern Medical University,Guangzhou 510515,China摘要：目的探讨可靠的基于3D 多期增强CT 影像组学特征的肾癌亚型三分类预测模型。
稀疏GroupLasso高维统计分析
^ = arg min｛ 1 ‖y － Xθ‖2 θ 2 + λ ‖ θ‖ G， 2｝ 。 θ∈Ｒ p 2 n 其中 λ n ＞ 0 是正则化参数。 由于 ‖θ G i ‖2 的奇点 在 θ G i = 0 （ 适当的正则参数可以使整个参数向量 θGi = 0 ） ， 从而会从拟合模型中将第 i 组剔除。 若
1021 收稿日期： 2013基金项目： 国家自然科学基金资助项目 （ 11171272 ） 作者简介： 丁毅涛， 男， 陕西礼泉人， 从事机器学习研究。
关注处理高维数据变量选择 （ 特征提取 ） 的有效 方法。而正则化方法为上述问题的求解提供了一 条途径。一般地， 正则化方法可以写成如下形式：
n
min｛ ∑ l（ f（ y i ， x i ） ） + λ ‖θ‖ q ｝ 。
n
有力工具， 并利用二次规划问题进行求解， 但该方 ［9 ］ 法比较复杂。 后来 Fu 提出了“shooting ” 算法， 等发现 Lasso 回归的解的路径是 分片线性的并提出相应的同伦算法。 但是， 目前 比较流行的方法是 Efron 等提出的最小角回归 （ Lars） 。有效算法的提出使得 L1 正则化成为机 器学习和稀疏信号处理领域的热点。 但是 Lasso Lasso 方 方法在理论上也存在一些不足， 一般地， 法的解是不相合的， 变量之间相关性不能太强。
［8 ］ 时， 其本身为 NP 组合优化问题。 Tibshirani 提 出的 L1 正则化（ Lasso） 是变量选择和特征提取的
更一般的稀疏 Group Lasso 罚函数， 该罚函数能使 Group 组和每个特征水平都能得到稀疏性 。 然而， Lasso 的提出为高维、 海量数据的分析， 并从中选 择重要因子， 得到准确估计提供了一种新的方法 。 对于高维、 海量数据而言， 当参数个数 p 大于样本 大小 n 时， 解是不相合的。 此时需要求解一个正 Negahban 等［20］就给出了在 则凸规划问题。近期， 高维情形下， 用于确立正则估计的相合性和收敛 速率的统一框架， 为稀疏 Group Lasso 的理论分析 提供可能。 本文开展稀疏 Group Lasso 的理论分析， 重点 稀疏 Group Lasso 的 研究在组内无重叠的情况下， 误差界估计。确定损失函数和正则子的两个重要 性质， 并选择适当的正则参数 λ n ， 从而得到稀疏 ^ λ 和未知参数 θ * 误差的概率 Group Lasso 的估计 θ
graph lasso的用法
graph lasso的用法Graph Lasso（Graphical Lasso）是一种用于估计具有稀疏精度矩阵（逆协方差矩阵）的统计方法。
这个方法在图论和统计学中都有应用，特别是在处理高维数据时，比如通过网络或传感器收集到的数据。
Graph Lasso 主要用于以下两个方面：1. 精度矩阵估计：给定一个数据集，Graph Lasso 估计数据的精度矩阵，它是协方差矩阵的逆。
精度矩阵描述了变量之间的关系，而且Graph Lasso 的优势在于它能够推断这种关系的稀疏性。
2. 图的估计：通过精度矩阵，可以构建一个图，其中节点表示变量，边表示变量之间的关系。
Graph Lasso 通过稀疏性，使得图中的边数目较少，这有助于理解和解释数据中的关系。
以下是使用Graph Lasso 的一般步骤：1. 数据准备：收集和准备数据集，确保数据是高维的，例如，包含多个变量。
2. 正则化参数选择：Graph Lasso 中有一个正则化参数，通常表示为alpha。
选择适当的alpha 对于获得良好的估计是重要的。
你可以使用交叉验证或其他模型选择方法来确定最佳的alpha 值。
3. 应用Graph Lasso：使用选择的alpha 值应用Graph Lasso 算法，估计数据的精度矩阵。
4. 图构建：基于估计的精度矩阵，构建表示变量关系的图。
在Python 中，你可以使用`sklearn.covariance.GraphicalLasso` 类来实现Graph Lasso。
以下是一个简单的示例：```pythonfrom sklearn.covariance import GraphicalLassoimport numpy as np# 准备数据，假设X 是你的数据矩阵X = np.random.rand(100, 5)# 选择正则化参数alphaalpha = 0.01# 应用Graph Lassomodel = GraphicalLasso(alpha=alpha)model.fit(X)# 获取估计的精度矩阵precision_matrix = model.precision_```请注意，这只是一个简单的示例，你可能需要根据你的数据和具体问题进行调整。
CorelDRAW(2024)
• Grouping and Ungrouping: Combine multiple objects into a single group for individual manipulation or ungrouping them to edit individual components
2024/1/29
Suitable for creating illustrations, cartons, comics, and other visual works
Can be used to design web graphics, icons, and other web elements
Fill and Outline
Apply colors, gradients, patterns, or textures to the fill or outline of selected objects using the Fill tool and Outline color picker
application skills
2
目录
2024/1/29
• Layer management and special effects production
• Symbol library and automation functions
3
Overview of CorelDRAW 01 software
Art+实验室共识刘平
共识刘平在学术界获得了广泛的认可 和赞誉，其研究成果多次获得国内外 学术奖项的肯定。
THANK YOU
强调技术创新的适度与审慎
共识刘平同时强调在技术创新的运用中要保持适度与审慎。技术只是手段，不应过分追求技术的炫酷而牺牲艺术 的本质。
对未来发展的展望
倡导跨学科合作与交流
共识刘平认为，未来的艺术发展需要打破学科界限，加强与其他领域的合作与交流。通过跨学科的碰 撞与融合，可以产生更多创新的艺术观念和形式。
计算机视觉
实验室在计算机视觉领域开展深 入研究，涉及目标检测、图像识 别、图像生成等方面的研究。
数据挖掘与机器学
习
实验室关注数据挖掘和机器学习 算法的研究，探索如何从大量数 据中提取有价值的信息和知识。
实验室研究成果
发表高水平论文
01
实验室成员在人工智能领域的国际顶级会议和期刊上发表多篇
高水平的学术论文。
丰富艺术表现形式
共识刘平在艺术领域的研究与实践，为艺术表现形式的探索提供了 新的可能性，为观众带来了更加丰富的艺术体验。
提高艺术地位
共识刘平在学术界的贡献和影响力，提高了艺术领域在社会中的地 位和认知度。
对学术界的贡献与评价
学术贡献
共识刘平在学术界的研究成果丰硕， 为相关领域的发展做出了重要贡献。
去中心化金融领域的发展具有重要影响。
02
区块链技术创新
共识刘平在区块链技术领域做出了多项创新，包括共识机制、智能合约
、去中心化应用等方面。他的研究成果推动了区块链技术的实际应用和
发展。
03
金融科技研究
共识刘平在金融科技领域也有深入研究，致力于将区块链技术和金融业
务相结合，为金融行业带来更多创新和价值。
An Overview of Recent Progress in the Study of Distributed Multi-agent Coordination
An Overview of Recent Progress in the Study of Distributed Multi-agent CoordinationYongcan Cao,Member,IEEE,Wenwu Yu,Member,IEEE,Wei Ren,Member,IEEE,and Guanrong Chen,Fellow,IEEEAbstract—This article reviews some main results and progress in distributed multi-agent coordination,focusing on papers pub-lished in major control systems and robotics journals since 2006.Distributed coordination of multiple vehicles,including unmanned aerial vehicles,unmanned ground vehicles and un-manned underwater vehicles,has been a very active research subject studied extensively by the systems and control community. The recent results in this area are categorized into several directions,such as consensus,formation control,optimization, and estimation.After the review,a short discussion section is included to summarize the existing research and to propose several promising research directions along with some open problems that are deemed important for further investigations.Index Terms—Distributed coordination,formation control,sen-sor networks,multi-agent systemI.I NTRODUCTIONC ONTROL theory and practice may date back to thebeginning of the last century when Wright Brothers attempted theirfirst testflight in1903.Since then,control theory has gradually gained popularity,receiving more and wider attention especially during the World War II when it was developed and applied tofire-control systems,missile nav-igation and guidance,as well as various electronic automation devices.In the past several decades,modern control theory was further advanced due to the booming of aerospace technology based on large-scale engineering systems.During the rapid and sustained development of the modern control theory,technology for controlling a single vehicle, albeit higher-dimensional and complex,has become relatively mature and has produced many effective tools such as PID control,adaptive control,nonlinear control,intelligent control, This work was supported by the National Science Foundation under CAREER Award ECCS-1213291,the National Natural Science Foundation of China under Grant No.61104145and61120106010,the Natural Science Foundation of Jiangsu Province of China under Grant No.BK2011581,the Research Fund for the Doctoral Program of Higher Education of China under Grant No.20110092120024,the Fundamental Research Funds for the Central Universities of China,and the Hong Kong RGC under GRF Grant CityU1114/11E.The work of Yongcan Cao was supported by a National Research Council Research Associateship Award at AFRL.Y.Cao is with the Control Science Center of Excellence,Air Force Research Laboratory,Wright-Patterson AFB,OH45433,USA.W.Yu is with the Department of Mathematics,Southeast University,Nanjing210096,China and also with the School of Electrical and Computer Engineering,RMIT University,Melbourne VIC3001,Australia.W.Ren is with the Department of Electrical Engineering,University of California,Riverside,CA92521,USA.G.Chen is with the Department of Electronic Engineering,City University of Hong Kong,Hong Kong SAR,China.Copyright(c)2009IEEE.Personal use of this material is permitted. However,permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@.and robust control methodologies.In the past two decades in particular,control of multiple vehicles has received increas-ing demands spurred by the fact that many benefits can be obtained when a single complicated vehicle is equivalently replaced by multiple yet simpler vehicles.In this endeavor, two approaches are commonly adopted for controlling multiple vehicles:a centralized approach and a distributed approach. The centralized approach is based on the assumption that a central station is available and powerful enough to control a whole group of vehicles.Essentially,the centralized ap-proach is a direct extension of the traditional single-vehicle-based control philosophy and strategy.On the contrary,the distributed approach does not require a central station for control,at the cost of becoming far more complex in structure and organization.Although both approaches are considered practical depending on the situations and conditions of the real applications,the distributed approach is believed more promising due to many inevitable physical constraints such as limited resources and energy,short wireless communication ranges,narrow bandwidths,and large sizes of vehicles to manage and control.Therefore,the focus of this overview is placed on the distributed approach.In distributed control of a group of autonomous vehicles,the main objective typically is to have the whole group of vehicles working in a cooperative fashion throughout a distributed pro-tocol.Here,cooperative refers to a close relationship among all vehicles in the group where information sharing plays a central role.The distributed approach has many advantages in achieving cooperative group performances,especially with low operational costs,less system requirements,high robustness, strong adaptivity,andflexible scalability,therefore has been widely recognized and appreciated.The study of distributed control of multiple vehicles was perhapsfirst motivated by the work in distributed comput-ing[1],management science[2],and statistical physics[3]. In the control systems society,some pioneering works are generally referred to[4],[5],where an asynchronous agree-ment problem was studied for distributed decision-making problems.Thereafter,some consensus algorithms were studied under various information-flow constraints[6]–[10].There are several journal special issues on the related topics published af-ter2006,including the IEEE Transactions on Control Systems Technology(vol.15,no.4,2007),Proceedings of the IEEE (vol.94,no.4,2007),ASME Journal of Dynamic Systems, Measurement,and Control(vol.129,no.5,2007),SIAM Journal of Control and Optimization(vol.48,no.1,2009),and International Journal of Robust and Nonlinear Control(vol.21,no.12,2011).In addition,there are some recent reviewsand progress reports given in the surveys[11]–[15]and thebooks[16]–[23],among others.This article reviews some main results and recent progressin distributed multi-agent coordination,published in majorcontrol systems and robotics journals since2006.Due to space limitations,we refer the readers to[24]for a more completeversion of the same overview.For results before2006,thereaders are referred to[11]–[14].Specifically,this article reviews the recent research resultsin the following directions,which are not independent but actually may have overlapping to some extent:1.Consensus and the like(synchronization,rendezvous).Consensus refers to the group behavior that all theagents asymptotically reach a certain common agreementthrough a local distributed protocol,with or without predefined common speed and orientation.2.Distributed formation and the like(flocking).Distributedformation refers to the group behavior that all the agents form a pre-designed geometrical configuration throughlocal interactions with or without a common reference.3.Distributed optimization.This refers to algorithmic devel-opments for the analysis and optimization of large-scaledistributed systems.4.Distributed estimation and control.This refers to dis-tributed control design based on local estimation aboutthe needed global information.The rest of this article is organized as follows.In Section II,basic notations of graph theory and stochastic matrices are introduced.Sections III,IV,V,and VI describe the recentresearch results and progress in consensus,formation control, optimization,and estimation.Finally,the article is concludedby a short section of discussions with future perspectives.II.P RELIMINARIESA.Graph TheoryFor a system of n connected agents,its network topology can be modeled as a directed graph denoted by G=(V,W),where V={v1,v2,···,v n}and W⊆V×V are,respectively, the set of agents and the set of edges which directionallyconnect the agents together.Specifically,the directed edgedenoted by an ordered pair(v i,v j)means that agent j can access the state information of agent i.Accordingly,agent i is a neighbor of agent j.A directed path is a sequence of directed edges in the form of(v1,v2),(v2,v3),···,with all v i∈V.A directed graph has a directed spanning tree if there exists at least one agent that has a directed path to every other agent.The union of a set of directed graphs with the same setof agents,{G i1,···,G im},is a directed graph with the sameset of agents and its set of edges is given by the union of the edge sets of all the directed graphs G ij,j=1,···,m.A complete directed graph is a directed graph in which each pair of distinct agents is bidirectionally connected by an edge,thus there is a directed path from any agent to any other agent in the network.Two matrices are used to represent the network topology: the adjacency matrix A=[a ij]∈R n×n with a ij>0if (v j,v i)∈W and a ij=0otherwise,and the Laplacian matrix L=[ℓij]∈R n×n withℓii= n j=1a ij andℓij=−a ij,i=j, which is generally asymmetric for directed graphs.B.Stochastic MatricesA nonnegative square matrix is called(row)stochastic matrix if its every row is summed up to one.The product of two stochastic matrices is still a stochastic matrix.A row stochastic matrix P∈R n×n is called indecomposable and aperiodic if lim k→∞P k=1y T for some y∈R n[25],where 1is a vector with all elements being1.III.C ONSENSUSConsider a group of n agents,each with single-integrator kinematics described by˙x i(t)=u i(t),i=1,···,n,(1) where x i(t)and u i(t)are,respectively,the state and the control input of the i th agent.A typical consensus control algorithm is designed asu i(t)=nj=1a ij(t)[x j(t)−x i(t)],(2)where a ij(t)is the(i,j)th entry of the corresponding ad-jacency matrix at time t.The main idea behind(2)is that each agent moves towards the weighted average of the states of its neighbors.Given the switching network pattern due to the continuous motions of the dynamic agents,coupling coefficients a ij(t)in(2),hence the graph topologies,are generally time-varying.It is shown in[9],[10]that consensus is achieved if the underlying directed graph has a directed spanning tree in some jointly fashion in terms of a union of its time-varying graph topologies.The idea behind consensus serves as a fundamental principle for the design of distributed multi-agent coordination algo-rithms.Therefore,investigating consensus has been a main research direction in the study of distributed multi-agent co-ordination.To bridge the gap between the study of consensus algorithms and many physical properties inherited in practical systems,it is necessary and meaningful to study consensus by considering many practical factors,such as actuation,control, communication,computation,and vehicle dynamics,which characterize some important features of practical systems.This is the main motivation to study consensus.In the following part of the section,an overview of the research progress in the study of consensus is given,regarding stochastic network topologies and dynamics,complex dynamical systems,delay effects,and quantization,mainly after2006.Several milestone results prior to2006can be found in[2],[4]–[6],[8]–[10], [26].A.Stochastic Network Topologies and DynamicsIn multi-agent systems,the network topology among all vehicles plays a crucial role in determining consensus.The objective here is to explicitly identify necessary and/or suffi-cient conditions on the network topology such that consensus can be achieved under properly designed algorithms.It is often reasonable to consider the case when the network topology is deterministic under ideal communication chan-nels.Accordingly,main research on the consensus problem was conducted under a deterministicfixed/switching network topology.That is,the adjacency matrix A(t)is deterministic. Some other times,when considering random communication failures,random packet drops,and communication channel instabilities inherited in physical communication channels,it is necessary and important to study consensus problem in the stochastic setting where a network topology evolves according to some random distributions.That is,the adjacency matrix A(t)is stochastically evolving.In the deterministic setting,consensus is said to be achieved if all agents eventually reach agreement on a common state. In the stochastic setting,consensus is said to be achieved almost surely(respectively,in mean-square or in probability)if all agents reach agreement on a common state almost surely (respectively,in mean-square or with probability one).Note that the problem studied in the stochastic setting is slightly different from that studied in the deterministic setting due to the different assumptions in terms of the network topology. Consensus over a stochastic network topology was perhaps first studied in[27],where some sufficient conditions on the network topology were given to guarantee consensus with probability one for systems with single-integrator kinemat-ics(1),where the rate of convergence was also studied.Further results for consensus under a stochastic network topology were reported in[28]–[30],where research effort was conducted for systems with single-integrator kinematics[28],[29]or double-integrator dynamics[30].Consensus for single-integrator kine-matics under stochastic network topology has been exten-sively studied in particular,where some general conditions for almost-surely consensus was derived[29].Loosely speaking, almost-surely consensus for single-integrator kinematics can be achieved,i.e.,x i(t)−x j(t)→0almost surely,if and only if the expectation of the network topology,namely,the network topology associated with expectation E[A(t)],has a directed spanning tree.It is worth noting that the conditions are analogous to that in[9],[10],but in the stochastic setting. In view of the special structure of the closed-loop systems concerning consensus for single-integrator kinematics,basic properties of the stochastic matrices play a crucial role in the convergence analysis of the associated control algorithms. Consensus for double-integrator dynamics was studied in[30], where the switching network topology is assumed to be driven by a Bernoulli process,and it was shown that consensus can be achieved if the union of all the graphs has a directed spanning tree.Apparently,the requirement on the network topology for double-integrator dynamics is a special case of that for single-integrator kinematics due to the difference nature of thefinal states(constantfinal states for single-integrator kinematics and possible dynamicfinal states for double-integrator dynamics) caused by the substantial dynamical difference.It is still an open question as if some general conditions(corresponding to some specific algorithms)can be found for consensus with double-integrator dynamics.In addition to analyzing the conditions on the network topology such that consensus can be achieved,a special type of consensus algorithm,the so-called gossip algorithm[31],[32], has been used to achieve consensus in the stochastic setting. The gossip algorithm can always guarantee consensus almost surely if the available pairwise communication channels satisfy certain conditions(such as a connected graph).The way of network topology switching does not play any role in the consideration of consensus.The current study on consensus over stochastic network topologies has shown some interesting results regarding:(1) consensus algorithm design for various multi-agent systems,(2)conditions of the network topologies on consensus,and(3)effects of the stochastic network topologies on the con-vergence rate.Future research on this topic includes,but not limited to,the following two directions:(1)when the network topology itself is stochastic,how to determine the probability of reaching consensus almost surely?(2)compared with the deterministic network topology,what are the advantages and disadvantages of the stochastic network topology,regarding such as robustness and convergence rate?As is well known,disturbances and uncertainties often exist in networked systems,for example,channel noise,commu-nication noise,uncertainties in network parameters,etc.In addition to the stochastic network topologies discussed above, the effect of stochastic disturbances[33],[34]and uncertain-ties[35]on the consensus problem also needs investigation. Study has been mainly devoted to analyzing the performance of consensus algorithms subject to disturbances and to present-ing conditions on the uncertainties such that consensus can be achieved.In addition,another interesting direction in dealing with disturbances and uncertainties is to design distributed localfiltering algorithms so as to save energy and improve computational efficiency.Distributed localfiltering algorithms play an important role and are more effective than traditional centralizedfiltering algorithms for multi-agent systems.For example,in[36]–[38]some distributed Kalmanfilters are designed to implement data fusion.In[39],by analyzing consensus and pinning control in synchronization of complex networks,distributed consensusfiltering in sensor networks is addressed.Recently,Kalmanfiltering over a packet-dropping network is designed through a probabilistic approach[40]. Today,it remains a challenging problem to incorporate both dynamics of consensus and probabilistic(Kalman)filtering into a unified framework.plex Dynamical SystemsSince consensus is concerned with the behavior of a group of vehicles,it is natural to consider the system dynamics for practical vehicles in the study of the consensus problem. Although the study of consensus under various system dynam-ics is due to the existence of complex dynamics in practical systems,it is also interesting to observe that system dynamics play an important role in determining thefinal consensus state.For instance,the well-studied consensus of multi-agent systems with single-integrator kinematics often converges to a constantfinal value instead.However,consensus for double-integrator dynamics might admit a dynamicfinal value(i.e.,a time function).These important issues motivate the study of consensus under various system dynamics.As a direct extension of the study of the consensus prob-lem for systems with simple dynamics,for example,with single-integrator kinematics or double-integrator dynamics, consensus with general linear dynamics was also studied recently[41]–[43],where research is mainly devoted tofinding feedback control laws such that consensus(in terms of the output states)can be achieved for general linear systems˙x i=Ax i+Bu i,y i=Cx i,(3) where A,B,and C are constant matrices with compatible sizes.Apparently,the well-studied single-integrator kinematics and double-integrator dynamics are special cases of(3)for properly choosing A,B,and C.As a further extension,consensus for complex systems has also been extensively studied.Here,the term consensus for complex systems is used for the study of consensus problem when the system dynamics are nonlinear[44]–[48]or with nonlinear consensus algorithms[49],[50].Examples of the nonlinear system dynamics include:•Nonlinear oscillators[45].The dynamics are often as-sumed to be governed by the Kuramoto equation˙θi=ωi+Kstability.A well-studied consensus algorithm for(1)is given in(2),where it is now assumed that time delay exists.Two types of time delays,communication delay and input delay, have been considered in the munication delay accounts for the time for transmitting information from origin to destination.More precisely,if it takes time T ij for agent i to receive information from agent j,the closed-loop system of(1)using(2)under afixed network topology becomes˙x i(t)=nj=1a ij(t)[x j(t−T ij)−x i(t)].(7)An interpretation of(7)is that at time t,agent i receives information from agent j and uses data x j(t−T ij)instead of x j(t)due to the time delay.Note that agent i can get its own information instantly,therefore,input delay can be considered as the summation of computation time and execution time. More precisely,if the input delay for agent i is given by T p i, then the closed-loop system of(1)using(2)becomes˙x i(t)=nj=1a ij(t)[x j(t−T p i)−x i(t−T p i)].(8)Clearly,(7)refers to the case when only communication delay is considered while(8)refers to the case when only input delay is considered.It should be emphasized that both communication delay and input delay might be time-varying and they might co-exist at the same time.In addition to time delay,it is also important to consider packet drops in exchanging state information.Fortunately, consensus with packet drops can be considered as a special case of consensus with time delay,because re-sending packets after they were dropped can be easily done but just having time delay in the data transmission channels.Thus,the main problem involved in consensus with time delay is to study the effects of time delay on the convergence and performance of consensus,referred to as consensusabil-ity[52].Because time delay might affect the system stability,it is important to study under what conditions consensus can still be guaranteed even if time delay exists.In other words,can onefind conditions on the time delay such that consensus can be achieved?For this purpose,the effect of time delay on the consensusability of(1)using(2)was investigated.When there exists only(constant)input delay,a sufficient condition on the time delay to guarantee consensus under afixed undirected interaction graph is presented in[8].Specifically,an upper bound for the time delay is derived under which consensus can be achieved.This is a well-expected result because time delay normally degrades the system performance gradually but will not destroy the system stability unless the time delay is above a certain threshold.Further studies can be found in, e.g.,[53],[54],which demonstrate that for(1)using(2),the communication delay does not affect the consensusability but the input delay does.In a similar manner,consensus with time delay was studied for systems with different dynamics, where the dynamics(1)are replaced by other more complex ones,such as double-integrator dynamics[55],[56],complex networks[57],[58],rigid bodies[59],[60],and general nonlinear dynamics[61].In summary,the existing study of consensus with time delay mainly focuses on analyzing the stability of consensus algo-rithms with time delay for various types of system dynamics, including linear and nonlinear dynamics.Generally speaking, consensus with time delay for systems with nonlinear dynam-ics is more challenging.For most consensus algorithms with time delays,the main research question is to determine an upper bound of the time delay under which time delay does not affect the consensusability.For communication delay,it is possible to achieve consensus under a relatively large time delay threshold.A notable phenomenon in this case is that thefinal consensus state is constant.Considering both linear and nonlinear system dynamics in consensus,the main tools for stability analysis of the closed-loop systems include matrix theory[53],Lyapunov functions[57],frequency-domain ap-proach[54],passivity[58],and the contraction principle[62]. Although consensus with time delay has been studied extensively,it is often assumed that time delay is either constant or random.However,time delay itself might obey its own dynamics,which possibly depend on the communication distance,total computation load and computation capability, etc.Therefore,it is more suitable to represent the time delay as another system variable to be considered in the study of the consensus problem.In addition,it is also important to consider time delay and other physical constraints simultaneously in the study of the consensus problem.D.QuantizationQuantized consensus has been studied recently with motiva-tion from digital signal processing.Here,quantized consensus refers to consensus when the measurements are digital rather than analog therefore the information received by each agent is not continuous and might have been truncated due to digital finite precision constraints.Roughly speaking,for an analog signal s,a typical quantizer with an accuracy parameterδ, also referred to as quantization step size,is described by Q(s)=q(s,δ),where Q(s)is the quantized signal and q(·,·) is the associated quantization function.For instance[63],a quantizer rounding a signal s to its nearest integer can be expressed as Q(s)=n,if s∈[(n−1/2)δ,(n+1/2)δ],n∈Z, where Z denotes the integer set.Note that the types of quantizers might be different for different systems,hence Q(s) may differ for different systems.Due to the truncation of the signals received,consensus is now considered achieved if the maximal state difference is not larger than the accuracy level associated with the whole system.A notable feature for consensus with quantization is that the time to reach consensus is usuallyfinite.That is,it often takes afinite period of time for all agents’states to converge to an accuracy interval.Accordingly,the main research is to investigate the convergence time associated with the proposed consensus algorithm.Quantized consensus was probablyfirst studied in[63], where a quantized gossip algorithm was proposed and its convergence was analyzed.In particular,the bound of theconvergence time for a complete graph was shown to be poly-nomial in the network size.In[64],coding/decoding strate-gies were introduced to the quantized consensus algorithms, where it was shown that the convergence rate depends on the accuracy of the quantization but not the coding/decoding schemes.In[65],quantized consensus was studied via the gossip algorithm,with both lower and upper bounds of the expected convergence time in the worst case derived in terms of the principle submatrices of the Laplacian matrix.Further results regarding quantized consensus were reported in[66]–[68],where the main research was also on the convergence time for various proposed quantized consensus algorithms as well as the quantization effects on the convergence time.It is intuitively reasonable that the convergence time depends on both the quantization level and the network topology.It is then natural to ask if and how the quantization methods affect the convergence time.This is an important measure of the robustness of a quantized consensus algorithm(with respect to the quantization method).Note that it is interesting but also more challenging to study consensus for general linear/nonlinear systems with quantiza-tion.Because the difference between the truncated signal and the original signal is bounded,consensus with quantization can be considered as a special case of one without quantization when there exist bounded disturbances.Therefore,if consensus can be achieved for a group of vehicles in the absence of quantization,it might be intuitively correct to say that the differences among the states of all vehicles will be bounded if the quantization precision is small enough.However,it is still an open question to rigorously describe the quantization effects on consensus with general linear/nonlinear systems.E.RemarksIn summary,the existing research on the consensus problem has covered a number of physical properties for practical systems and control performance analysis.However,the study of the consensus problem covering multiple physical properties and/or control performance analysis has been largely ignored. In other words,two or more problems discussed in the above subsections might need to be taken into consideration simul-taneously when studying the consensus problem.In addition, consensus algorithms normally guarantee the agreement of a team of agents on some common states without taking group formation into consideration.To reflect many practical applications where a group of agents are normally required to form some preferred geometric structure,it is desirable to consider a task-oriented formation control problem for a group of mobile agents,which motivates the study of formation control presented in the next section.IV.F ORMATION C ONTROLCompared with the consensus problem where thefinal states of all agents typically reach a singleton,thefinal states of all agents can be more diversified under the formation control scenario.Indeed,formation control is more desirable in many practical applications such as formationflying,co-operative transportation,sensor networks,as well as combat intelligence,surveillance,and reconnaissance.In addition,theperformance of a team of agents working cooperatively oftenexceeds the simple integration of the performances of all individual agents.For its broad applications and advantages,formation control has been a very active research subject inthe control systems community,where a certain geometric pattern is aimed to form with or without a group reference.More precisely,the main objective of formation control is to coordinate a group of agents such that they can achievesome desired formation so that some tasks can befinished bythe collaboration of the agents.Generally speaking,formation control can be categorized according to the group reference.Formation control without a group reference,called formationproducing,refers to the algorithm design for a group of agents to reach some pre-desired geometric pattern in the absenceof a group reference,which can also be considered as the control objective.Formation control with a group reference,called formation tracking,refers to the same task but followingthe predesignated group reference.Due to the existence of the group reference,formation tracking is usually much morechallenging than formation producing and control algorithmsfor the latter might not be useful for the former.As of today, there are still many open questions in solving the formationtracking problem.The following part of the section reviews and discussesrecent research results and progress in formation control, including formation producing and formation tracking,mainlyaccomplished after2006.Several milestone results prior to 2006can be found in[69]–[71].A.Formation ProducingThe existing work in formation control aims at analyzingthe formation behavior under certain control laws,along with stability analysis.1)Matrix Theory Approach:Due to the nature of multi-agent systems,matrix theory has been frequently used in thestability analysis of their distributed coordination.Note that consensus input to each agent(see e.g.,(2))isessentially a weighted average of the differences between the states of the agent’s neighbors and its own.As an extensionof the consensus algorithms,some coupling matrices wereintroduced here to offset the corresponding control inputs by some angles[72],[73].For example,given(1),the controlinput(2)is revised as u i(t)= n j=1a ij(t)C[x j(t)−x i(t)], where C is a coupling matrix with compatible size.If x i∈R3, then C can be viewed as the3-D rotational matrix.The mainidea behind the revised algorithm is that the original controlinput for reaching consensus is now rotated by some angles. The closed-loop system can be expressed in a vector form, whose stability can be determined by studying the distribution of the eigenvalues of a certain transfer matrix.Main research work was conducted in[72],[73]to analyze the collective motions for systems with single-integrator kinematics and double-integrator dynamics,where the network topology,the damping gain,and C were shown to affect the collective motions.Analogously,the collective motions for a team of nonlinear self-propelling agents were shown to be affected by。
Analysisofrepresentationsfordomainadaptation
AnalysisofrepresentationsfordomainadaptationShai Ben-David School of Computer Science University of Waterlooshai@cs.uwaterloo.ca John Blitzer,Koby Crammer,and Fernando Pereira Department of Computer and Information Science University of Pennsylvania{blitzer,crammer,pereira}@/doc/d3bd993987c24028915fc339.htmlAbstractDiscriminative learning methods for classi?cation perform well when training andtest data are drawn from the same distribution.In many situations,though,wehave labeled training data for a source domain,and we wish to learn a classi?erwhich performs well on a target domain with a different distribution.Under whatconditions can we adapt a classi?er trained on the source domain for use in thetarget domain?Intuitively,a good feature representation is a crucial factor in thesuccess of domain adaptation.We formalize this intuition theoretically with ageneralization bound for domain adaption.Our theory illustrates the tradeoffs in-herent in designing a representation for domain adaptation and gives a new justi?-cation for a recently proposed model.It also points toward a promising new modelfor domain adaptation:one which explicitly minimizes the difference between thesource and target domains,while at the same time maximizing the margin of thetraining set.1IntroductionWe are all familiar with the situation in which someone learns to perform a task on training examples drawn from some domain(the source domain),but then needs to perform the same task on a related domain(the target domain).In this situation,we expect the task performance in the target domain to depend on both the performance in the source domain and the similarity between the two domains. This situation arises often in machine learning.For example,we might want to adapt for a new user(the target domain)a spam?lter trained on the email of a group of previous users(the source domain),under the assumption that users generally agree on what is spam and what is not.Then,the challenge is that the distributions of emails for the?rst set of users and for the new user are different. Intuitively,one might expect that the closer the two distributions are,the better the?lter trained on the source domain will do on the target domain.Many other instances of this situation arise in natural language processing.In general,labeled data for tasks like part-of-speech tagging,parsing,or information extraction are drawn from a limited set of document types and genres in a given language because of availability,cost,and project goals. However,applications for the trained systems often involve somewhat different document types and genres.Nevertheless,part-of-speech,syntactic structure,or entity mention decisions are to a large extent stable across different types and genres since they depend on general properties of the language under consideration.Discriminative learning methods for classi?cation are based on the assumption that training and test data are drawn from the same distribution.This assumption underlies both theoretical estimates of generalization error and the many experimental evaluations of learning methods.However,the as-sumption does not hold for domain adaptation[5,7,13,6].For the situations we outlined above,the challenge is the difference in instance distribution between the source and target domains.We will approach this challenge by investigating how a common representation between the two domainscan make the two domains appear to have similar distributions,enabling effective domain adapta-tion.We formalize this intuition with a bound on the target generalization error of a classi?er trained from labeled data in the source domain.The bound is stated in terms of a representation function,and it shows that a representation function should be designed to minimize domain divergence,as well as classi?er error. While many authors have analyzed adaptation from multiple sets of labeled training data[3,5,7, 13],our theory applies to the setting in which the target domain has no labeled training data,butplentiful unlabeled data exists for both target and source domains.As we suggested above,this setting realistically captures the problems widely encountered in real-world applications of machine learning.Indeed recent empirical work in natural language processing[11,6]has been targeted atexactly this setting.We show experimentally that the heuristic choices made by the recently proposed structural corre-spondence learning algorithm[6]do lead to lower values of the relevant quantities in our theoreticalanalysis,providing insight as to why this algorithm achieves its empirical success.Our theory also points to an interesting new algorithm for domain adaptation:one which directly minimizes a trade-off between source-target similarity and source training error.The remainder of this paper is structured as follows:In the next section we formally de?ne domainadaptation.Section3gives our main theoretical results.We discuss how to compute the bound in section4.Section5shows how the bound behaves for the structural correspondence learning representation[6]on natural language data.We discuss our? ndings,including a new algorithm fordomain adaptation based on our theory,in section6and conclude in section7.2Background and Problem SetupLet X be an instance set.In the case of[6],this could be all English words,together with the possible contexts in which they occur.Let Z be a feature space(R d is a typical choice)and{0,1} be the label set for binary classi?cation1.A learning problem is speci?ed by two parameters:a distribution D over X and a(stochastic)targetfunction f:X→[0,1].The value of f(x)corresponds to the probability that the label of x is 1.A representation function R is a function which maps instances to features R:X→Z.A representation R induces a distribution over Z and a(stochastic)target function from Z to[0,1]asfollows:[B]def=Pr D R?1(B)Pr?Df(z)def=ED[f(x)|R(x)=z]for any A?Z such that R?1(B)is D-measurable.In words,the probability of an event B under ?D is the probability of the inverse image of B under R according to D,and the probability that the label of z is1according to?f is the mean of probabilities of instances x that z represents.Note that?f(z)may be a stochastic function even if f(x)is not.This is because the function R can map two instances with different f-labels to the same feature representation.In summary,our learning setting is de?ned by? xed but unknown D and f,and our choice of representation function R and hypothesis class H?{g:Z→{0,1}}of deterministic hypotheses to be used to approximate the function f.2.1Domain AdaptationWe now formalize the problem of domain adaptation.A domain is a distribution D on the instance set X.Note that this is not the domain of a function.To avoid confusion,we will always mean a speci?c distribution over the instance set when we say domain.Unlike in inductive transfer,where the tasks we wish to perform may be related but different,in domain adaptation we perform the same task in multiple domains.This is quite common in natural language processing,where we might be performing the same syntactic analysis task,such as tagging or parsing,but on domains with very different vocabularies[6,11]. We assume two domains,a source domain and a target domain.We denote by D S the source distribution of instances and?D S the induced distribution over the feature space Z.We use parallel notation,D T,?D T,for the target domain.f:X→[0,1]is the labeling rule,common to both domains,and?f is the induced image of f under R.A predictor is a function,h,from the feature space,Z to[0,1].We denote the probability,according the distribution D S,that a predictor h disagrees with f byS(h)=Ez～?D S E y～?f(z)[y=h(z)]=Ez～?D S ?f(z)?h(z) .Similarly,?T(h)denotes the expected error of h with respect to D T.3Generalization Bounds for Domain AdaptationWe now proceed to develop a bound on the target domain generalization performance of a classi?er trained in the source domain.As we alluded to in section1,the bound consists of two terms.The?rst term bounds the performance of the classi?er on the source domain.The second term is a measure of the divergence between the induced source marginal?D S and the induced target marginal?D T.A natural measure of divergence for distributions is the L1or variational distance.This is de?ned asd L1(D,D′)=2supB∈B|Pr D[B]?Pr D′[B]|where B is the set of measureable subsets under D and D′.Unfortunately the variational distance between real-valued distributions cannot be computed from?nite samples[2,9]and therefore is not useful to us when investigating representations for domain adaptation on real-world data.A key part of our theory is the observation that in many realistic domain adaptation scenarios,we do not need such a powerful measure as variational distance.Instead we can restrict our notion of domain distance to be measured only with respect to function in our hypothesis class.3.1The A-distance and labeling function complexityWe make use of a special measure of distance between probability distributions,the A-distance,as introduced in[9].Given a domain X and a collection A of subsets of X,let D,D′be probability distributions over X,such that every set in A is measurable with respect to both distributions.the A-distance between such distributions is de?ned asd A(D,D′)=2supA∈A|Pr D[A]?Pr D′[A]|In order to use the A-distance,we need to limit the complexity of the true function f in terms of our hypothesis class H.We say that a function?f:Z→[0,1]isλ-close to a function class H with respect to distributions?D S and?D T ifinfh∈H[?S(h)+?T(h)]≤λ.A function?f isλ-close to H when there is a single hypothesis h∈H which performs well on both domains.This embodies our domain adaptation assumption,and we will assume will assume that our induced labeling function?f isλ-close to our hypothesis class H for a smallλ.We brie?y note that in standard learning theory,it is possible to achieve bounds with no explicit as-sumption on labeling function complexity.If H has bounded capacity(e.g.,a?nite VC-dimension), then uniform convergence theory tells us that whenever?f is notλ-close to H,large training samples have poor empirical error for every h∈H.This is not the case for domain adaptation.If the training data is generated by some D S and we wish to use some H as a family of predictors for labels in the target domain,T,then one can construct a function which agrees with some h∈H with respect to?D S and yet is far from H with respect to?D T.Nonetheless we believe that such examples do not occur for realistic domain adaptation problems whenthe hypothesis class H is suf?ciently rich, since for most domain adaptation problems of interest the labeling functionis’similarly simple’for both the source and target domains.3.2Bound on the target domain errorWe require one last piece of notation before we state and prove the main theorems of this work:the correspondence between functions and characteristic subsets.For a binary-valued function g(z),we let Z g?Z be the subset whose characteristic function is gZ g={z∈Z:g(z)=1}.In a slight abuse of notation,for a binary function class H we will write d H(·,·)to indicate the A-distance on the class of subsets whose characteristic functions are functions in H.Now we can state our main theoretical result.Theorem1Let R be a?xed representation function from X to Z and H be a hypothesis space of VC-dimension d.If a random labeled sample of size m is generated by applying R to a D S-i.i.d. sample labeled according to f,then with probability at least1?δ,for every h∈H:T(h)≤S(h)+ m d log2emδ +d H(?D S,?D T)+λwhere e is the base of the natural logarithm.Proof:Let h?=argmin h∈H(?T(h)+?S(h)),and letλT andλS be the errors of h?with respect to D T and D S respectively.Notice thatλ=λT+λS.T(h)≤λT+Pr D[Z h?Z h?]T[Z h?Z h?]+|Pr D S[Z h?Z h?]?Pr D T[Z h?Z h?]|≤λT+Pr DS[Z h?Z h?]+d H(?D S,?D T)≤λT+Pr DS≤λT+λS+?S(h)+d H(?D S,?D T)≤λ+?S(h)+d H(?D S,?D T)The theorem now follows by a standard application Vapnik-Chervonenkis theory[14]to bound the true?S(h)by its empirical estimate??S(h).Namely,if S is an m-size.i.i.d.sample,then with probability exceeding1?δ,S(h)≤S(h)+ m d log2emδm′m d+log4d log(2m′)+log(4Let us brie?y examine the bound from theorem2,with an eye toward feature representations,R. Under the assumption of subsection3.1,we assume thatλis small for reasonable R.Thus the two main terms of interest are the?rst and fourthterms,since the representation R directly affects them. The?rst term is the empirical training error.The fourth term is the sample A-distance between domains for hypothesis class H.Looking at the two terms,we see that a good representation R is one which achieves low values for both training error and domain A-distance simultaneously.4Computing the A-distance for Signed Linear Classi?ersIn this section we discuss practical considerations in computing the A-distance on real data.Ben-David et al.[9]show that the A-distance can be approximated arbitrarily well with increasing sample size.Recalling the relationship between sets and their characteristic functions,it should be clear thatcomputing the A-distance is closely related to learning a classi?er.In fact they are identical.The set A h∈H which maximizes the H-distance between?D S and?D T has a characteristic function h.Then h is the classi?er which achieves minimum error on the binary classi?cation problem ofdiscriminating between points generated by the two distributions.To see this,suppose we have two samples?U S and?U T,each of size m′from?D S and?D T respectively. De?ne the error of a classi?er h on the task of discriminating between points sampled from different distributions as1err(h)=theorem2that random projections approximate well distances in the original high dimensional space,as long as d is suf?ciently large.Arriaga and Vempala[1]show that one can achieve good prediction with random projections as long as the margin is suf?ciently large.5.2Structural Correspondence LearningBlitzer et al.[6]describe a heuristic method for domain adaptation that they call structural corre-spondencelearning(henceforth also SCL).SCL uses unlabeled data from both domains to induce correspondences among features in the two domains.Its?rst step is to identify a small set of domain-independent“pivot”features which occur frequently in the unlabeled data of both domains.Other features are then represented using their relative co-occurrence counts with these pivot features.Fi-nally they use a low-rank approximation to the co-occurence count matrix as a projection matrix P. The intuition is that by capturing these important correlations,features from the source and target domains which behave similarly for PoS tagging will be represented similarly in the projected space.5.3ResultsWe use as our source data set100sentences(about2500words)of PoS-tagged Wall Street Journal text.The target domain test set is the same set as in[6].We use one million words(500thousand from each domain)of unlabeled data to estimate the A-distance between the?nancial and biomedi-cal domains.The results in this section are intended to illustrate the different parts of theorem2and how they can affect the target domain generalization error.We give two types of results.The?rst are pictorial and appear in?gures1(a),1(b)and2(a).These are intended to illustrate either the A-distance(?gures 1(a)and2(a))or the empirical error(?gure1(b))for different representations.The second type are empirical and appear in2(b).In this case we use the Huber loss as a proxy from the empirical training error.Figure1(a)shows one hundred random instances projected onto the space spanned by the best two discriminating projections from the SCL projection matrix for part of the?nancial and biomedical dataset.Instances from the WSJ are depicted as?lled red squares,whereas those from MEDLINE are depicted as empty blue circles.An approximating linear discrimnator is also shown.Note, however,that the discriminator performs poorly,and recall that if the best discriminator performs poorly the A-distance is low.On the other hand,?gure1(b)shows the best two discriminating components for the task of discriminating between nouns and verbs.Note that in this case,a good discriminating divider is easy to?nd,even in such a low-dimensional space.Thus these pictures lead us to believe that SCL?nds a representation which results both in small empirical classi?cation error and small A-distance.In this case theorem2predicts good performance.(a)Plot of random projections repre-sentation for ?nancial (squares)vs.(b)Comparison of bound terms vs.target domain error for different choices of representation.Reprentations linear projections of the original feature space.Hu-loss is the labeled training loss after training,andA -distance is approximated as described in thesubsection.Error refers to tagging error forfull tagset on the target domain.Representation A -distance Error0.003Random Proj 0.2230.5610.077ConclusionsWe presented an analysis of representations for domain adaptation.It is reasonable to think that agood representation is the key to effective domain adaptation,and our theory backs up that intuition.Theorem2gives an upper bound on the generalization of a classi?er trained on a source domain and applied in a target domain.The bound depends on the representation and explicitly demonstrates thetradeoff between low empirical source domain error and a small difference between distributions. Under the assumption that the labeling function?f is close to our hypothesis class H,we can compute the bound from?nite samples.The relevant distributional divergence term can be written as the A-distance of Kifer et al[9].Computing the A-distance is equivalent to?nding the minimum-errorclassi?er.For hyperplane classi?ers in R d,this is an NP-hard problem,but we give experimental evidence that minimizing a convex upper bound on the error,as in normal classi?cation,can give a reasonable approximation to the A-distance.Our experiments indicate that the heuristic structural correspondence learning method[6]does infact simultaneously achieve low A-distance as well as a low margin-based loss.This provides a justi?cation for the heuristic choices of SCL“pivots”.Finally we note that our theory points to an interesting new algorithm for domain adaptation.Instead of making heuristic choices,we are investigating algorithms which directly minimize a combination of the A-distance and the empirical training margin.References[1]R.Arriaga and S.Vempala.An algorithmic theory of learning robust concepts and randomprojection.In FOCS,volume40,1999.[2]T.Batu,L.Fortnow,R.Rubinfeld,W.Smith,and P.White.Testing that distributions are close.In FOCS,volume41,pages259–269,2000.[3]J.Baxter.Learning internal representations.In COLT’95:Proceedings of the eighth annualconference on Computational learning theory,pages311–320,New York,NY,USA,1995. [4]S.Ben-David,N.Eiron,and P.Long.On the dif?culty of approximately maximizing agree-ments.Journal of Computer and System Sciences,66:496–514,2003.[5]S.Ben-David and R.Schuller.Exploiting task relatedness for multiple task learning.InCOLT2003:Proceedings of the sixteenth annual conference on Computational learning the-ory,2003.[6]J.Blitzer,R.McDonald,and F.Pereira.Domain adaption with structural correspondencelearning.In EMNLP,2006.[7]K.Crammer,M.Kearns,and J.Wortman.Learning from data of variable quality.In NeuralInformation Processing Systems(NIPS),Vancouver,Canada,2005.[8]W.Johnson and J.Lindenstrauss.Extension of lipschitz mappings to hilbert space.Contem-porary Mathematics,26:189–206,1984.[9]D.Kifer,S.Ben-David,and J.Gehrke.Detecting change in data streams.In Very LargeDatabases(VLDB),2004.[10]C.Manning.Foundations of Statistical Natural Language Processing.MIT Press,Boston,1999.[11]D.McClosky,E.Charniak,and M.Johnson.Reranking and self-training for parser adaptation.In ACL,2006.[12]M.Sugiyama and K.Mueller.Generalization error estimation under covariate shift.In Work-shop on Information-Based Induction Sciences,2005.[13]Y.W.Teh,M.I.Jordan,M.J.Beal,and D.M.Blei.Sharing clusters among related groups:Hierarchical Dirichlet processes.In Advances in Neural Information Processing Systems,vol-ume17,2005.[14]V.Vapnik.Statistical Learning Theory.John Wiley,New York,1998.[15]T.Zhang.Solving large-scale linear prediction problems with stochastic gradient descent.InICML,2004.。
稀疏总结
稀疏表示在目标检测方面的学习总结1，稀疏表示的兴起大量研究表明视觉皮层复杂刺激的表达采用的是稀疏编码原则，以稀疏编码为基础的稀疏表示方法能较好刻画人类视觉系统对图像的认知特性，已引起人们极大的兴趣和关注，在机器学习和图像处理领域得到了广泛应用，是当前国内外的研究热点之一．［１］Vinje W E ，Gallant J L ．Sparse coding and decorrelation in pri- mary visual cortex during natural vision ［J］．Science ，２０００，２８７（５４５６）：１２７３－１２７６．［２］Nirenberg S ，Carcieri S ，Jacobs A ，et al ．Retinal ganglion cells act largely as independent encoders ［J ］．Nature ，２００１，４１１（６８３８）：６９８－７０１．［３］Serre T ，Wolf L ，Bileschi S ，et al ．Robust object recognition with cortex-like mechanisms［J］．IEEE Transactions on PatternAnalysis and Machine Intelligence ，２００７，２９（３）：４１１－４２６．［４］赵松年，姚力，金真，等．视像整体特征在人类初级视皮层上的稀疏表象：脑功能成像的证据［J］．科学通报，２００８，５３（１１）：１２９６－１３０４．图像稀疏表示研究主要沿着两条线展开：单一基方法和多基方法．前者主要是多尺度几何分析理论，认为图像具有非平稳性和非高斯性，用线性算法很难处理，应建立适合处理边缘及纹理各层面几何结构的图像模型，以脊波（Ridgelet）、曲波（Curvelet）等变换为代表的多尺度几何分析方法成为图像稀疏表示的有效途径；后者以Mallat 和Zhang 提出的过完备字典分解理论为基础，根据信号本身的特点自适应选取能够稀疏表示信号的冗余基。
group lasso的定义公式
group lasso的定义公式Group Lasso 是一种用于特征选择和稀疏建模的正则化技术，通常用于线性回归和相关的机器学习任务。
它通过对特征进行分组，以鼓励模型在每个特征组内选择一组相关的特征，并对不同的特征组应用不同的L1正则化，以实现特征选择和稀疏性。
Group Lasso 的数学定义如下：假设有m 个训练样本，n 个特征，以及k 个特征组（也称为分组）。
我们用X 表示一个m×n 的特征矩阵，其中每一行代表一个训练样本，每一列代表一个特征。
另外，我们用β 表示一个n 维的系数向量，表示特征的权重。
Group Lasso 的目标函数通常由两部分组成：数据拟合项（Least Squares Term）：这是最小化拟合数据误差的部分，通常用平方误差（Least Squares）或其他回归损失函数来表示。
它的目标是使模型能够拟合训练数据。
Group L1 正则化项：这是对系数向量β进行正则化的部分，它鼓励特征在特定的分组内共享权重，从而实现特征选择和稀疏性。
Group L1 正则化通常表示为各个特征组的L1 范数之和。
Group Lasso 的数学目标函数可以表示为：minimize: 1/2m * ||Y - Xβ||² + λ * Σᵢ||βᵢ||₂其中：Y 是目标变量的向量（m 维度）。
X 是特征矩阵（m×n 维度）。
β是待学习的系数向量（n 维度）。
λ是控制正则化强度的超参数。
βᵢ表示特征组i 内的系数向量。
Σᵢ表示对所有特征组i 进行求和。
||.||₂表示L2 范数（Euclidean 范数）。
1/2m 是归一化因子，用于确保数据拟合项的尺度与正则化项相匹配。
通过调整λ 的值，可以控制正则化的强度，从而影响模型选择哪些特征组以及在每个特征组内选择哪些特征。
这使得Group Lasso 成为一种强大的特征选择技术，尤其适用于具有分组特征的问题，如图像处理、生物信息学和自然语言处理。
group lasso示例
group lasso示例Group Lasso是一种线性模型的正则化方法，可以用于特征选择和变量分组。
下面是一个简单的Group Lasso示例，假设我们有一个数据集，其中包含10个特征，分为两个组，每个组有5个特征。
首先，我们需要生成一些模拟数据。
假设每个特征都服从标准正态分布，并且组内特征的相关系数为，组间特征的相关系数为0。
```pythonimport numpy as npfrom _model import LinearRegressionfrom import StandardScaler生成模拟数据(0)n_samples = 200X = (n_samples, 10)X_group1 = X[:, :5]X_group2 = X[:, 5:]添加组内和组间相关性X_group1 = (X_group1, rowvar=False)[:5, 5:] + X_group1X_group2 = (X_group2, rowvar=False)[:5, 5:] + X_group2添加噪声y = (X, (10)) + (n_samples)```接下来，我们可以使用Group Lasso进行模型拟合。
在这里，我们使用LinearRegression作为基础模型，并设置`penalty='l2'`和`solver='sag'`来启用Group Lasso。
我们还可以通过`group`参数来指定特征的分组信息。
```python标准化数据scaler = StandardScaler()X_scaled = _transform(X)定义Group Lasso模型model = LinearRegression(penalty='l2', solver='sag', max_iter=1000) (X_scaled, y)最后，我们可以查看模型中每个特征的系数估计值。
asgl的group lasso的方法
ASGL的Group Lasso的方法一、引言1.1 研究背景在机器学习和统计学中，特征选择是一个重要的问题。
在高维数据集中，选择最相关的特征能够提高模型的准确性、降低计算成本并增加可解释性。
Group Lasso是一种常用的特征选择方法，它能够结合特征间的相关性进行特征选择。
ASGL是Group Lasso的扩展方法，能够更好地处理高维数据集中的特征选择问题。
1.2 研究目的本文旨在介绍ASGL的Group Lasso方法的原理和应用。
通过深入探讨ASGL的Group Lasso方法，我们可以了解其优势和适用范围，为实际问题的特征选择提供参考。
二、Group Lasso方法简介2.1 Group Lasso的基本思想Group Lasso是一种结合特征间相关性的特征选择方法。
它将特征划分为若干组，每一组特征被视为一个整体。
Group Lasso通过对每个组的特征进行稀疏化，将一部分组的特征系数收缩为零，从而实现特征选择的目的。
2.2 Group Lasso的数学模型Group Lasso的数学模型可以表示为：min β12n∥y−Xβ∥22+λ∑√|βj|pj=1其中，y是观测变量，X是特征矩阵，β是特征系数，p是特征数，λ是正则化参数。
三、ASGL的Group Lasso方法原理3.1 ASGL的改进ASGL是对Group Lasso方法的改进，它引入了自适应稀疏度参数。
传统的Group Lasso方法中，每个组的稀疏度参数是固定的，而ASGL通过学习每个组的稀疏度参数，能够更好地适应不同组的特征。
3.2 ASGL的数学模型ASGL的数学模型可以表示为：min β12n∥y−Xβ∥22+λ∑√|βj|pj=1+γ∑ρgGg=1∑√|βj|j∈g其中，ρg是每个组的自适应稀疏度参数，G是组的个数，j∈g表示第g组中的特征。
四、ASGL的Group Lasso方法应用4.1 特征选择ASGL的Group Lasso方法可以应用于特征选择问题。
基于生物信息学筛选宫颈癌血管生成基因及构建预后模型
DOI:10.16605/ki.1007-7847.2022.10.0221基于生物信息学筛选宫颈癌血管生成基因及构建预后模型夏娜娜1,2,杨京蕊1,康敏1,余敏敏1*(1.南京市第二医院妇科,中国江苏南京210003;2.中国人民解放军总医院第七医学中心药剂科,中国北京100700)摘要:利用生物信息学方法筛选与宫颈癌发生、发展和预后相关的血管生成相关基因(angiogenesis related gene,ARG),并进行相关预后风险模型的构建与验证。
首先,从TCGA 数据库中检索宫颈癌患者的表达谱和临床特征,并提取差异表达的ARG;其次,采用Lasso Cox 回归筛选预后ARG,构建相关预后模型;再次,使用GSE52903和GSE44001数据集进行外部验证;最后,利用基因集富集分析(gene set enrichment analysis,GSEA)探讨宫颈癌预后机制。
筛选结果显示,共获得15个预后ARG,分别为EFNA 1、ITGA 5、EPHB 4、NRP 1、CDH 5、PLAU 、BMP 6、DLL 4、JUN 、CA 9、MMP 1、BAIAP 2L 1、SERPINF 1、F 2RL 1和FGFR 2。
GSE52903和GSE44001数据集的Ka-plan-Meier 生存曲线显示,高风险组的总生存期(overall survival,OS)(P =0.005)和无病生存期(disease-free sur-vival,DFS)(P <0.001)显著低于低风险组。
受试者操作特征(receiver operating characteristic,ROC)曲线分析结果显示,GSE52903验证集在1年、3年和5年的曲线下面积(area under the curve,AUC)值分别为0.84、0.77和0.73,C-指数为0.72;GSE44001验证集在1年、3年和5年的AUC 值分别为0.71、0.72和0.70,C-指数为0.70,说明该模型对患者预后具有很强的预测效能。
Revo Studio 用户手册说明书
REVO 3D SCANNER Revo Studio - User Manual2022.4 V.2.0.0ContentIntroduction (3)System Requirements (3)Main Menu Panel (3)Process Model (4)1. Edit (4)2. Point (4)3. Mesh (5)4. Alignment (6)5. View (6)6. Help (7)Warning (8)Support & Help (9)Contact Us (9)IntroductionRevo Studio is a post-processing application released by the Revopoint team to edit individual 3D models created by Revo Scan and to align multiple 3D models so that they can be merged into a new 3D model. In this instruction manual, we will introduce the details about the application and how to adjust parameters in it.System RequirementsMain Menu Panel① Studio Icon:Includes Import , Export , Recent Files and Exit . ② Undo & Redo: Click the Undo icon to reverse the action of the previous step(s) as many times as necessary. Asneeded, click Redo to move forward to the desired step.③ Function Buttons: Includes Edit , Point , Mesh , Alignment , View and Help .④ Toolbar: Includes shortcuts to Set Center , Orthogonal , Show/Hide Bounding Box , Show/Hide Track Ball ,Box Selection and Lasso Selection . These functions are described in their sections later.⑤ Center Window: Displays the name of the model being processed.⑥ Information of the 3D model: Provides details of the model (such as name & number of points) to verify identity. ⑦ Mouse Command Prompt: Displays the mouse’s function commands.⑧ Parameter Settings: Adjust the parameters for any function allowing custom configurations. Mac with Intel × 86 chip: Mac OS 10.15 and models after ； Mac with Apple M1 chip: Mac OS 11.0 and models after ； Memory: ≥ 8GWindows: Win 8/Win 10 (64 bit ） Memory ：≥ 8G *The Windows 7 is not supported.Process Model1. EditIn the Edit module, the available commands are: Box Selection, Lasso Selection and Set Center.(Note: The words or phrases on the following UI will be improved.)Box Selection: Select features of a model by drawing a rectangle around them.Lasso Selection: Create a completely free-form selection around the desired features by drawing a lasso.Set Center:Change the center coordinate of the 3D model. The object is re-positioned so that the selected center is in the middle of the work screen. Any rotations will be around this center point.2. PointIn the Point module, the available commands are: Clip, Smooth, Isolation, Simplify, Overlap Detection and Meshing to edit the point cloud of a 3D model.Clip: Clip the model plane. Use this function to remove unnecessary parts of the scan. Click the right mouse button and drag; the clipping plane will appear as a line (the plane is seen on-edge). The portion of the model in thedirection of the arrow will be saved. Click Apply to finish. If needed, click the right mouse button again to cancel the operation.Smooth: Smooth flat or curved surfaces; this function can be applied repeatedly as needed. Adjust the Parameters：There are two smoothing methods: Geometry and Normal.Geometry refers to the features in the scanned object.Normal refers to the calculated directions of each facet.In the BASIC and ADV ANCED parameters, set the amount of smoothing as needed. In ADV ANCED, highervalues for Radius and Iteration result in smoother surfaces.Isolation: Automatically remove point cloud data that is isolated from the main body of the object (for example, a portion of the arm when scanning a person’s head). For parameters in ADV ANCED, higher values for Radius, Angleand Isolation Rate remove more of the point cloud data.Simplify: Simplify the point cloud (remove data points that do not improve the accuracy of the scan).Overlap Detection: Detect and delete overlapping, and unnecessary, data points in the point cloud.Meshing: Convert the point cloud into a meshed 3D model.3. MeshIn the Mesh module, the available commands are Clip, Smooth, Isolation, Simplify, Enhance, Fill Hole and Convert to Point cloud.Clip: Clip the model plane. If there are unnecessary surfaces or meshed data that would not be excised by the Isolation command, use this function to manually remove it.Smooth: Smooth flat or curved surfaces; this function can be applied repeatedly as needed.In the parameters for BASIC and ADV ANCED, set the amount of smoothing as needed.There are two smoothing options for ADV ANCED: Geometry and Denoise.Isolation: Remove extraneous meshed data that is separated or only loosely connected to the main body.Simplify: Simplify the mesh file (reduce the number of polygons in the model while retaining as much detail as possible). Enhance: Sharpen the details of the 3D model.Fill Holes: Fill the missing part(s) on the model’s surface. Currently, this function only works when filling small holes because gross distortions typically result when filling holes that are a significant portion of the object’s volume. Convert to Point Cloud: Convert the meshed 3D model into a point cloud.4. AlignmentIn the Alignment module, multiple scans can be combined and merged into one object, two at a time.Select Add Float File to import every new scan that is to be merged with the current model. Merge Type allows for automatic (By Feature) and manual (By Mark Points) alignment. Automated alignment needs a minimum amount of overlapping areas with the same, unique features; 20% in common is a good starting point for most projects. Both point clouds and mesh objects can be aligned. Manual alignment requires a minimum of 3 pairs of corresponding points on the two scans.5. ViewIn the View module, the available commands support visual aids for reviewing the model.Icon: Switch between Perspective and Orthogonal.Icon: Display or hide the bounding box.Icon: Display or hide the track ball.6. HelpThe Help section contains the About information and Feedback popup.The About popup includes the Revo Studio version and links to the official website & user forum.The Feedback popup supports bug reporting and general feedback to our developers.WarningThe product cannot be returned if the "Warranty Void If Seal Is Broken" label is damagedor removed.Follow Revopoint 3D TechnologiesThis content is subject to change.Download the latest version from https:///downloadIf you have any questions about this document, please contact ***********************Support & HelpIf you need any help, please visit our official website or official Forum:Contact UsTel (US): Toll-free +1 (888) 807-3339Tel(China):+86181****6779Email:************************ Skype: +1 323 892 0859Our customer service team provides 24-hour o n li n e services support . If you have any questions and feedback, please don’t hesitate to contact us!/support/https:///。
lasso几何解释
lasso几何解释
Lasso回归是一种线性模型，它通过引入一个正则化项来对模型的复杂度进行惩罚。
这个正则化项通常是一个绝对值和系数的乘积，其中系数是回归模型中的特征的系数。
Lasso回归的目标是找到一个能够最小化预测误差和正则化项的系数。
在几何上，Lasso回归可以被解释为在特征空间中找到一个超平面，这个超平面能够最小化预测误差，并且将特征的系数压缩到0或某个固定值。
这个超平面可以被看作是一个带有一个“弹簧”的直线，这个“弹簧”会压缩系数较大的特征，使其接近于0。
Lasso回归的几何解释可以帮助我们更好地理解它的工作原理和特点。
例如，当我们在二维空间中绘制Lasso回归的解时，我们可以看到系数较大的特征被压缩到原点附近，而系数较小的特征则被保留在较远的位置。
这种压缩效应使得Lasso回归在处理高维数据时具有稀疏性，即它能够自动地选择出对预测结果最重要的特征。
此外，Lasso回归的几何解释还可以帮助我们更好地理解正则化的作用。
正则化是一种对模型复杂度的惩罚，它有助于防止过拟合和欠拟合。
在Lasso回归中，正则化项会压缩系数较大的特征，从而使得模型更加简单和可解释性更强。
这种简化模型的过程可以帮助我们更好地理解数据的本质和规律。
总之，Lasso回归的几何解释是一种直观、形象的方式，它有助于我们更好地理解线性模型的工作原理和特点。
同时，这种解释也有助于我们在实际应用中更好地选择和使用线性模型。
1。
多任务Sparse Group Lasso特征提取与支持向量机回归在恒星大气物理参量估计中的应用
多任务Sparse Group Lasso特征提取与支持向量机回归在恒星大气物理参量估计中的应用∗高伟;李乡儒【摘要】The multi-task learning puts the multiple tasks together to analyse and calculate for discovering the correlation between them, which can improve the accu-racy of analysis results. This kind of methods have been widely studied in machine learning, pattern recognition, computer vision, and other related fields. This paper investigates the application of multi-task learning in estimating the effective tempera-ture (Teff ), surface gravity (lg g), and chemical abundance ([Fe/H]). Firstly, the spectral characteristics of the three atmospheric physical parameters are extracted by using the multi-task Sparse Group Lasso algorithm, and then the support vector machine is used to estimate the atmospheric physical parameters. The proposed scheme is evaluated on both Sloan stellar spectra and theoretical spectra computed from Kurucz’s New Opacity Distribution Function (NEWODF) model. The mean absolute errors (MAEs) on the Sloan spectra are: 0.0064 for lg (Te ff/K), 0.1622 for lg (g/(cm · s−2)), and 0.1221 dex for [Fe/H];The MAEs on synthetic spectra are 0.0006 for lg (Teff/K), 0.0098 for lg (g/(cm · s−2)), and 0.0082 dex for [Fe/H]. Experimental results show that the proposed scheme is excellent for atmospheric parameter estimation.%多任务学习(Multi-task Learning, MTL)就是把多个问题一起进行分析、计算,以发掘不同问题之间的相关性,提高分析结果的精度,该类方法已被广泛地应用于机器学习、模式识别、计算机视觉等领域.使用多任务学习方案研究了恒星大气物理参数中表面温度(Teff )、表面重力加速度(lg g)、化学丰度([Fe/H])的估计问题.首先使用多任务Sparse Group Lasso算法提取对3个大气物理参数均有预测能力的光谱特征；然后使用支持向量机估计恒星大气物理参数.该方案在Sloan实测恒星光谱和理论光谱上均做了测试.在实测光谱上的平均绝对误差分别为：0.0064(lg (Teff/K)),0.1622(lg (g/(cm·s−2))),0.1221 dex ([Fe/H]).在由Kurucz的New Opacity Distribution Function(NEWODF)模型得到的理论光谱上也做了同样的特征提取和恒星大气物理参数估计测试,相应的平均绝对误差分别为：0.0006(lg (Teff/K))),0.0098(lg (g/(cm · s−2))),0.0082 dex ([Fe/H]).通过与文献中的同类研究比较表明,多任务Sparse Group Lasso特征提取与支持向量机回归(support vector machine regression, SVR)两者结合的方案有较高的恒星大气物理参量估计精度.【期刊名称】《天文学报》【年(卷),期】2016(057)004【总页数】13页(P389-401)【关键词】恒星:基本参数;方法:数据分析;方法:统计;方法:其他诸多方面【作者】高伟;李乡儒【作者单位】华南师范大学数学科学学院广州510631;华南师范大学数学科学学院广州510631【正文语种】中文【中图分类】P144随着现代科技的飞速发展,美国的Sloan数字巡天望远镜[1]获得了大量的光谱数据,而我国的郭守敬望远镜——大天区面积多目标光纤光谱天文望远镜(LAMOST)[2]更是目前世界上光谱获取率最高的望远镜,一次观测可同时获得多达4000个天体的光谱,使人类观测天体光谱的数目提高到千万数量级.海量恒星光谱数据的获得,使精确、快速地从中估计恒星的3个大气物理参数——表面温度(Teff)、表面重力加速度(lg g)与化学丰度([Fe/H])成为一个很值得探讨的研究课题.我们知道,来自遥远太空的天体光谱的数据量巨大,在传输的过程中会受到大量的噪声干扰,比如大气环境、杂散光、宇宙射线等,在接收时还会受到仪器的不稳定、系统误差等噪声干扰.这些因素会严重影响我们对恒星大气物理参数估计的精度和速度,所以应先对光谱数据进行预处理,降低数据量,提高速度,并减少噪声干扰,然后用提取出的光谱特征估计天体的大气物理参数.前述数据预处理在模式识别、数据挖掘等领域称之为特征提取.典型的特征提取方法有基于神经网络的多层自编码、主成分分析(Principle Com ponent Analysis, PCA)[3−4]、Lasso[5−6](Least Absolute Shrinkage Selection and Operator)等.特别是, Tibshirani在1996年提出的Lasso算法,该方法是通过对未知的系数向量施加一个l1范数约束,使得绝对值较小的系数自动缩小到0,达到变量选择和特征提取目的.不过, Lasso方法有两个局限性:第一,同一个光谱数据中包含着所有的光谱物理参量的信息量,而不同的光谱物理参量之间又有潜在的关系,所以上述方法把3个恒星大气物理参量分开考虑,对光谱数据进行降维提取光谱特征,会造成各个不同物理参量之间的信息量的损失,进而导致预测精度降低;第二,分开处理3个物理参量的方式繁琐耗时、效率低.实际上要把3个恒星大气物理参量一起考虑对光谱数据提取特征就是一个多任务学习的问题.多任务学习(M u lti-task learning,MTL)就是在多个任务一起学习中挖掘不同任务之间的关系信息量,同时又能区分不同任务之间的差别,进而能够提高预测模型的预测精度及泛化性能的一种方法.本文使用的多任务Sparse G roup Lasso[7–11],实际上也是改进于上述的Lasso方法及研究组变量选择的Group Lasso[12].它不仅继承了Lasso的优势,能够有效地剔除不重要的组,还克服了Group Lasso不具有组内稀疏性的弊端,可以灵活地选择组内变量,更重要的是还具备做多任务的特征提取的能力,也弥补了上述方法的两点不足,所以可以提高恒星大气物理参数的估计精度.在光谱数据的多任务学习中,假设有N条恒星光谱,每条光谱由P个流量描述,且有M个恒星大气物理参数需要估计(本文中M=3).记X为一个N×P维输入的光谱数据变量,xj=(,···,)T是X的第j列的流量变量.记Y为响应的N×M维的大气物理参量,ym=(,···,)T是Y的第m列的大气物理参量.对于每一列响应的大气物理参量,假设一个线性模型为其中cm=(,···,)T是P维的回归系数,εm=(,···,)T是N维的对应误差.为了同时计算M个任务的回归系数向量,即C=(c1,···,cM),需要优化的多任务Sparse Group Lasso模型为其中‖C‖l1/l2=∑‖(,···,)‖2,‖·‖1表示向量的1范数:求向量所有元素的绝对值和,‖·‖2表示向量的2范数:求向量所有元素的平方和,再开方.在这里C的每一行形成一个组.当M=1时,λ2=0,上式就是Lasso;λ1=0,上式就是Group Lasso.该方法的正则化参数λ1不仅控制整个模型的稀疏性,且控制着任务内的稀疏性,正则化参数λ2不仅控制着任务间的稀疏性,且控制着不同任务的信息保留程度.支持向量机是一种典型的统计学习算法,广泛地应用在文本识别、人脸识别、语音识别、时间序列预测等领域.它是建立在Vapnik等人提出的统计学习理论、结构风险最小准则之上,最初是作为一个分类机器提出来的学习方法.支持向量机回归支持向量用于回归问题中的情况,其核心思想是通过核函数间接进行非线性变换来实现非线性的支持向量函数拟合.由于高维度光谱数据的结构复杂性、非线性,本文中采用的是非线性支持向量机回归.假设待估计的恒星大气物理参数有效温度、表面重力或化学丰度用y表示,相应的多任务Sparse Group Lasso特征用x表示,则支持向量机回归模型为:其中K(∗,∗)是非线性的高斯核函数,即系数βi=−αi,i=1,···,l是以下优化问题的解:其中,ε是控制拟合误差的一个精度,即误差限,常数C控制着对超出误差限样本的惩罚与函数的平坦性之间的折中.本文在Sloan的实测数据和理论恒星光谱数据上均做了实验,验证了方案的可行性,共两个实验.实验1:数据是美国大型巡天项目Sloan发布的SDSS实测光谱数据中的50000条光谱及每条光谱对应的3个光谱物理参量,分别是表面温度(Teff)、表面重力加速度(lg g)和化学丰度([Fe/H]).每条光谱具有3821个流量特征.3个物理参量的范围分别为:Teff:[4088,9740]K,lg(g/(cm·s−2)):[1.015,4.998],[Fe/H]:[−3.497,0.268]dex.其中20000条光谱数据作为训练数据,剩下30000条光谱数据作为测试数据.设训练集为其中xi=(,···,)T∈Rp×1代表第i条光谱数据,yi=(,···,)T∈Rm×1代表第i条光谱数据所对应的m个光谱物理参量.令(Xtr,Ytr)代表训练光谱数据及每条光谱所对应的物理参量,其中这里n=20000,p=3821,m=3.m=1时,即(i=1,2,···,n)代表表面温度(Teff);m=2时,即(i=1,2,···,n)代表表面重力加速度(lg g);m=3时,即(i= 1,2,···,n)代表化学丰度([Fe/H]).同上设测试集为Ste.实验2:数据是由Kurucz的NEWODF模型得到的理论光谱中的18969条光谱数据.每条光谱具有3821个流量特征.3个物理参量的范围分别为:Teff:[4000,9750]K,lg (g/(cm·s−2)):[1,5],[Fe/H]:[−3.6,0.3]dex.其中8000条作训练数据,另外10969条作测试数据,对其也做同实验1数据的对应记法及下面的数据预处理.4.1 数据预处理(1)为了减小波动范围,精确地描述表面温度(Teff),实验中用温度参量的以10为底的对数lg Teff代替温度参量(Teff).记其中,(i=1,2,···,n)表示温度参量(Teff).于是(8)式转变为(2)对光谱数据Xtr的每列求均值和标准差,然后中心化,最后标准化.记其中i=1,···,n,j=1,···,p.则(7)式转变为训练集(6)式转变为测试集Ste也做同上处理为te={(i,i),i=1,2,···,n},同时得到te,te.4.2 估计准则为了更好地评价恒星大气物理参量估计的效果,把估计值与观测值作平均绝对误差(Mean Absolute Error,MAE)、平均误差(Mean Error,ME)和标准偏差(Standard Deviation,SD):其中en是第n条光谱的物理参量的估计值与观测值之差.5.1 实测光谱的结果与分析在实验1中,主要有3个操作步骤:第1步对光谱数据进行预处理,第2步用多任务Sparse Group Lasso提取光谱流量特征,第3步用支持向量机回归(SVR)对表面温度(Teff)、表面重力加速度(lg g)与化学丰度([Fe/H])进行估计.本实验不仅估计出3个主要物理参量的平均绝对误差(MAE)、平均误差(ME)与标准偏差(SD),而且列出一些相关文献方法的结果作比较,具体实验与相关文献的结果见表1.更重要的是分别检测出3个物理参量的特征:表面温度(Teff)36个、表面重力加速度(lg g)109个、化学丰度([Fe/H])136个,具体见图1及其具体位置见表2.另外,画出参量误差随Teff、lg g、[Fe/H]的变化情况,具体见图2.在表1中,相关文献中的研究方法SVRG、ANN与MAχ都是非线性拟合方法, SVRl与OLS是线性拟合方法,明显非线性拟合的结果要比线性拟合的结果好,这也说明恒星光谱数据与3个物理参量之间的函数关系更可能是非线性关系,尤其是与表面重力加速度(lg g)、化学丰度([Fe/H])的函数关系.这也是本文采用非线性的支持向量机回归(SVR),核函数为高斯函数作估计的原因之一.在实验过程中,发现对表面温度(Teff)的预测是最容易的,化学丰度([Fe/H])次之,表面重力加速度(lg g)是最难预测的,表1中3个物理参量相应的预测结果刚好也印证了这一点.从平均绝对误差(MAE)相较很小可以看出,本文采用的多任务Sparse Group Lasso 特征提取与支持向量机回归(SVR)相结合的方案的预测结果要优于相关文献中的线性与非线性方法的预测结果,特别是对表面重力加速度(lg g)与化学丰度([Fe/H])的预测;从平均误差(ME)几乎趋于0能够看出本文方法的系统误差要比相关文献方法的小;实测光谱本身含有相当多的各种噪声,而本文方法预测结果的标准偏差(SD)却相对很小,说明预测结果波动很小,此方法抗噪能力不错,鲁棒性好.在文献[6]中SVRG方法与本文的拟合方法是一样的,而特征提取的方法不同.在文献[6]中采用系数压缩法Lasso,是把3个物理参量分开单独进行特征提取,而未对3个物理参量同时提取特征,这样导致不同的光谱物理参量之间潜在的关系信息量的损失,进而影响预测结果.本文使用的多任务Sparse G roup Lasso是系数压缩法Lasso的改进方法,克服了上述弊端,另外多任务方法同时作特征提取也节省了科研时间,提高了效率.从表1中的预测结果也可以看到,此法确实比系数压缩法Lasso适合光谱特征的提取.特别在文献[15-16]中多任务Lasso回归法与本文的多任务Sparse Group Lasso 算法和支持向量机回归结合的方案有很大区别.在光谱特征提取方面,文献[15-16]的多任务Lasso算法与本文的多任务Sparse Group Lasso算法不同的是:第一,多任务Lasso算法中第2个惩罚项其中m是任务数,n是样本数,很显然惩罚项D严重受到样本数的限制,处理上万级的大样本,D的计算量随之增大.原本模型中的第1项计算量就与样本数相关,这样一来处理大数据的速度会很慢,效率降低,而本文算法中的第2个惩罚项的l1/l2范数的计算量只是与特征数有关,对于上万级的大数据来说,特征数要远远小于样本数;第二,D用的Frobenius范数没有本文中l1/l2范数压缩系数的效率高,导致提取出特征数目会很多.在估计物理参量方面,文献[15-16]用的是最简单的线性回归,不能够很好地拟合物理参量与光谱数据之间的非线性关系,而本文使用的高斯核函数的支持向量机回归可以弥补这一点.从表1中的多任务Lasso回归法的结果看,只是化学丰度([Fe/H])的结果相对好点,这可能是提取的特征数目多的缘故,而且其总共使用了4000条SDSS数据,其中75%的数据作为训练数据,而测试数据只用剩下的25%数据,有这样的结果也很正常,更不能表现出其方法的泛化能力强.而本实验使用了50000条SDSS数据,40%的数据作为训练集,60%的作为测试集,预测结果也很不错,足以说明本方案的泛化能力强,要比文献[15-16]的方法优越.观察图1及表2,可以明显看出3个物理参量的被检测到的特征不仅数目不相同,而且波长位置不都一样,这充分体现出了多任务Sparse Group Lasso方法不仅可以提取到单个物理参量的光谱信息,还能够挖掘到不同物理参量之间的潜在关系信息. 观察图2,可以看出3个参量误差的变化情况各不相同,但整体上误差都在0的附近变化,其中Teff的误差偏离最小,lg(g/(cm·s−2))的[1,2.8)、[Fe/H]的(−4,−2.2)dex 区间误差偏离非常大,主要因为这些区间的光谱分布比较稀疏分散,其误差偏离度会随着训练数据的增多而改善[17].其次,Teff的整个区间误差精度都很高,而lg(g/(cm·s−2))的(3, 5)、[Fe/H]的(−2,1)dex区间误差精度稍高,其他区间的误差精度很低.另外,高估了巨星(lg(g/(cm·s−2))～2–3)的表面重力加速度,而低估了矮星(lg(g/(cm·s−2))～4)的表面重力加速度;高估了贫金属恒星([Fe/H]～−3–−2 dex)的化学丰度,而低估了太阳丰度恒星([Fe/H]～0 dex)的化学丰度.5.2 理论光谱的结果与分析在实验2中,同样有3个主要的操作步骤:首先对光谱数据进行预处理,然后用多任务Sparse Group Lasso提取光谱流量特征,最后用支持向量机回归(SVR)对光谱的3个物理参量进行估计.3个主要物理参量的平均绝对误差(MAE)、平均误差(ME)、标准偏差(SD),与一些相关文献方法的结果具体见表3.更重要的是分别检测出3个物理参量的特征:表面温度(Teff)21个、表面重力加速度(lg g)24个、化学丰度([Fe/H])24个,具体见图3及其具体位置见表4.另外,画出参量误差随Teff、lg g、[Fe/H]的变化情况,具体见图4.观察表3,结合表1,可以看出本文采用的多任务Sparse Group Lasso特征提取与支持向量机回归(SVR)相结合的方案在理论光谱实验中预测效果比在实测光谱实验中预测效果更好.在表3中,不仅可以发现对表面温度(Teff)的预测最容易,化学丰度([Fe/H])次之,表面重力加速度(lg g)预测最难这个规律,还可以从平均绝对误差(MAE)看出本文方法预测结果的精度要比相关文献中非线性方法ANN及线性方法OLS高得多;平均误差(ME)都几乎为0也说明系统误差非常小;标准偏差(SD)很小也说明此方法对理论光谱物理参量的预测结果波动非常小.查看图3及表4,可以看到在理论光谱数据上检测到3个物理参量的特征数目不仅很接近而且明显要比实测光谱的少;表面温度(Teff)的特征与另外2个物理参量的特征位置很相近,甚至表面重力加速度(lg g)与化学丰度([Fe/H])的特征位置完全一样,这很可能是由Kurucz的NEWODF模型得到的理论光谱数据,没有各种噪声干扰的缘故.观察图4,可以看出3个参量误差的变化情况大致相同,呈沿着纵轴0值的一条直线, Teff、lg g、[Fe/H]的区间误差偏离都非常小且误差精度都很高,这是由于理论光谱训练数据在整个参数范围内比较集中且分布均匀.本文把对3个重要的光谱物理参量表面温度(Teff)、表面重力加速度(lg g)、化学丰度[Fe/H]的估计,作为3个任务,用多任务Sparse Group Lasso提取特征,然后再用支持向量机回归(SVR),不仅估计的结果精度高、鲁棒性好、泛化性能高,而且操作简单、计算速度快.同时,这种对大数据多任务学习的方式,不仅学习到了单个任务的信息,而且兼顾学习到了多个任务之间的关联信息.总的来说,本文采用的多任务Sparse Group Lasso特征提取与支持向量机回归(SVR)相结合的方案对恒星大气物理参量的估计结果要优于相关文献中方法的结果.另外此方案不仅适用于多任务光谱数据的处理,还适用于其他类型大数据的多任务学习,比如银行金融大数据,期货股票大数据,淘宝交易大数据等等.当然,在SDSS实测光谱数据上,对于本文多任务Sparse Group Lasso所提取光谱特征的数目可以探讨进一步减少,同时保证估计的准确度.比如说可以做探索性实验,使用多任务Sparse Group Lasso提取特征之后,然后使用主成分分析(PCA)再次剔除冗余噪声或者采用对提取的每个光谱特征的邻近区域的一些特征求均值,以达到特征数目的减少,可以深入做实验以检验经过这些方式处理后的估计值精度是否有所提高等等.致谢衷心感谢潘儒扬在论文修订、校对中提供的帮助.【相关文献】[1]Ahn C P,A lexand roff R,A llende P rieto C A,et a l.A p JS,2012,203:21[2]Cu i X Q,Zhao Y H,Chu Y Q,et a l.RAA,2012,12:1197[3]Fioren tin P R,Bailer-Jones C A L,Lee Y S,et al.A&A,2007,467:1373[4]李乡儒.天文学进展,2012,30:94[5]T ibsh irani R.Jou rna l of the Roya l Statistica l Society Series B,1996,58:267[6]Li X R,W u Q M J,Luo A L,et a l.A p J,2014,790:105[7]Sim on N,Friedm an J,Hastie T,et a l.Jou rnal of Com pu tational&G raph ica l Statistics,2013,22:231[8]Liu J,Ji S,Ye J.P roceed ings of the Tw enty-fifth Con ference on Uncertain ty in A rtificia l In telligence, 2009:339[9]V incen t M,Hansen N pu tationa l Statistics&Data A nalysis,2014,71:771[10]Liu J,Ye J.A dvances in Neu ral In form ation P rocessing System s,2010,23:1459[11]张吐辉,张海.纯粹数学与应用数学,2014,30:178[12]Yuan M,L in Y.Jou rna l of the Roya l Statistical Society Series B,2006,68:49[13]Jofr´e P,Panter B,Hansen C J,et al.A&A,2010,517:57[14]谭鑫,潘景昌,王杰,等.光谱学与光谱分析,2013,33:1397[15]常丽娜,张培爱.天文学报,2015,56:26[16]Chang L N,Zhang P A.ChA&A,2015,39:319[17]Lu Y,Li X R.M NRAS,2015,452:1394。
Econometric Analysis软件用户手册说明书
Contents Intro...........................................Introduction to treatment-effects manual Treatment effects..............................Introduction to treatment-effects commandsDID intro..............................Introduction to difference-in-differences estimation didregress..........................................Difference-in-differences estimation didregress postestimation................Postestimation tools for didregress and xtdidregresseteffects.......................................Endogenous treatment-effects estimation eteffects postestimation..................................Postestimation tools for eteffects etpoisson............................Poisson regression with endogenous treatment effects etpoisson postestimation................................Postestimation tools for etpoisson etregress..............................Linear regression with endogenous treatment effects etregress postestimation................................Postestimation tools for etregressstteffects....................Treatment-effects estimation for observational survival-time data stteffects intro...........Introduction to treatment effects for observational survival-time data stteffects ipw.................................Survival-time inverse-probability weighting stteffects ipwra.............Survival-time inverse-probability-weighted regression adjustment stteffects postestimation................................Postestimation tools for stteffects stteffects ra........................................Survival-time regression adjustment stteffects wra...............................Survival-time weighted regression adjustmenttebalance.............................Check balance after teffects or stteffects estimation tebalance box..................................................Covariate balance box tebalance density.............................................Covariate balance density tebalance overid.............................................Test for covariate balance tebalance summarize................................Covariate-balance summary statistics teffects................................Treatment-effects estimation for observational data teffects intro.........................Introduction to treatment effects for observational data teffects intro advanced.......Advanced introduction to treatment effects for observational data teffects aipw...................................Augmented inverse-probability weighting teffects ipw..............................................Inverse-probability weighting teffects ipwra...........................Inverse-probability-weighted regression adjustment teffects multivalued........................................Multivalued treatment effects teffects nnmatch............................................Nearest-neighbor matching teffects postestimation....................................Postestimation tools for teffects teffects psmatch.............................................Propensity-score matching teffects ra.....................................................Regression adjustment telasso.........................................Treatment-effects estimation using lasso telasso postestimation....................................Postestimation tools for telasso teoverlap.............................................................Overlap plots Glossary......................................................................... Subject and author index...........................................................i。
asgl的group lasso的方法
asgl的group lasso的方法
ASGL (Adaptive Sparse Group Lasso)是一种基于Group Lasso 的方法，用于选择相关性特征的预处理和特征选择方法。
在特征选择问题中，ASGL是一种对于高维数据的有效方法，尤其是在存在相关特征的情况下。
Group Lasso算法是一种在线性回归中进行特征选择的方法，它能够选取相关性特征。
该算法将特征分为若干组，其中每组的特征具有相同的特征权重。
这样，当某个组中的特征被选中时，该组中的所有特征都将被包含在模型中。
因此，Group Lasso算法可以选择所有组中的一些或全部特征。
ASGL算法是一种在Group Lasso的基础上进行改进的算法。
ASGL算法在Group Lasso的基础上增加了一个自适应惩罚系数，用于平衡不同组之间的权重。
该算法可以有效地选择相关性特征，同时减少非重要性特征的选择。
ASGL算法可以用于许多应用程序，例如图像处理、语音识别、数据挖掘等。
在这些应用程序中，ASGL算法可以选择关键性特征，识别潜在的模式和结构，并提高预测和分类的准确性。
总之，ASGL算法是一种有效的特征选择方法，它可以选择相关性特
征，减少不相关的特征，并在高维数据中提高模型的性能。
asgl的group lasso的方法
asgl的group lasso的方法
ASGL (Alternating Sparse Group Lasso) 是一种基于 Group Lasso 的稀疏化方法。
Group Lasso 的目的是找到一组相关变量，将它们一起收缩为零，从而使得模型更加简洁。
ASGL 方法则是使用交
替方向乘子法（Alternating Direction Method of Multipliers, ADMM）来求解 Group Lasso 问题。
ASGL 方法在许多实际应用中都得到了广泛应用，如信号处理、计算机视觉和生物信息学等领域。
ASGL 方法的核心思想是在 Group Lasso 正则化项中加入惩罚系数，并使用 ADMM 来优化。
具体而言，ASGL 方法采用了一种交替更新的
策略，在每一步中分别更新模型参数和 Lagrange 乘子。
该方法的优点在于能够有效地处理高维数据集，并且具有良好的收敛性和稳定性。
ASGL 方法的实现可以使用现有的优化库，如CVX、SPAMS或LIBLINEAR 等，从而简化了算法的实现和使用。
此外，该方法还可以与其他稀疏化算法相结合，以提高模型的性能和效果。
总之，ASGL 方法是一种有效的 Group Lasso 算法，具有广泛的应用前景。
它可以用于各种领域的数据分析和建模任务，为这些任务提供了更加简洁和易于理解的模型。
未来，ASGL 方法将继续被广泛地研
究和应用，以满足不断增长的数据分析需求。
lasso系数曲线
lasso系数曲线
Lasso回归是一种常用的线性回归方法，它可以在一定程度上解决数据中存在过多特征问题。
在Lasso回归中，最小化损失函数的同时，加上L1范数正则化项来惩罚过多特征，通过调整正则化参数，可以得到一系列的Lasso系数曲线。
Lasso系数曲线是描述Lasso回归中系数与正则化参数关系的图形。
在Lasso回归中，正则化参数被认为是Lasso模型中特征的数量，随着正则化参数的增加，Lasso回归的稀疏度也会增加，相关的参数会被惩罚，从而有效地降低模型复杂度，避免过拟合。
Lasso系数曲线的横坐标是正则化参数，纵坐标是Lasso模型中各个特征的系数值，曲线上的点表示对应特征的系数值，在正则化参数为0时，对应于无约束的线性回归。
对于给定的数据集，我们可以通过交叉验证等方法来确定最优的正则化参数，以得到最优的Lasso系数曲线。
通过Lasso系数曲线，可以分析各个特征在不同的正则化参数下的重要性，进而筛选出对预测结果贡献较大的特征集合。
在实际应用中，Lasso系数曲线可以帮助数据分析人员进行特征选择和模型调优，提高模型预测精度。
总之，Lasso系数曲线是Lasso回归中的一种重要分析工具，它可以帮助我们了解Lasso模型中各个特征的重要性，辅助特征选择和模型调优。
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g ∈G vg
supp (vg ) ⊆ g .
Properties
Resulting support is a union of groups in G. Possible to select one variable without selecting all the groups containing it. Setting one vg to 0 doesn’t necessarily set to 0 all its variables in w .
Jacob, Obozinski, Vert (ParisTech, INRIA)
Overlapping group lasso
16 juin 2009 6 / 17
Biological markers for cancer
Issue of using the group-lasso
Ωgroup(w ) = g wg 2 sets groups to 0. One variable is selected ⇔ all the groups to which it belongs are selected.
Better interpretability. Correlated genes typically in the same group, hence selected together. Robustness to spurious gene selection. Group lasso originally proposed for disjoint groups. For overlapping groups, Ωgroup(w ) = g∈G wg 2 is still a norm and has been considered for : Hierarchical variable selection (Zhao et al. 2006, Bach 2008). Structured sparsity (Jenatton et al. 2009).
Jacob, Obozinski, Vert (ParisTech, INRIA)
Overlapping group lasso
16 juin 2009 6 / 17
Biological markers for cancer
Overlapping groups
We have prior information under the form of groups of genes with functional meaning (e.g. pathways). We would like to favor directly w involving few groups
Better interpretability. Correlated genes typically in the same group, hence selected together. Robustness to spurious gene selection. Group lasso originally proposed for disjoint groups. For overlapping groups, Ωgroup(w ) = g∈G wg 2 is still a norm and has been considered for : Hierarchical variable selection (Zhao et al. 2006, Bach 2008). Structured sparsity (Jenatton et al. 2009).
Metastasizing tumors ?
Gene expression in tumor
...
Metastasis ?
...
?
Predict metastasis, identify few predictive genes.
Jacob, Obozinski, Vert (ParisTech, INRIA)
Group lasso with Overlap and Graph Lasso
Laurent Jacob1,2 Guillaume Obozinski3,4 Jean-Philippe Vert1,2
1Mines ParisTech, Centre for Computational Biology 2Institut Curie, INSERM U900 3Ecole Normale Sup´erieure 4INRIA – Willow project
Jacob, Obozinski, Vert (ParisTech, INRIA)
Overlapping group lasso
16 juin 2009 8 / 17
Overlap norm
Better interpretability. Correlated genes typically in the same group, hence selected together. Robustness to spurious gene selection. Group lasso originally proposed for disjoint groups. For overlapping groups, Ωgroup(w ) = g∈G wg 2 is still a norm and has been considered for : Hierarchical variable selection (Zhao et al. 2006, Bach 2008). Structured sparsity (Jenatton et al. 2009).
Jacob, Obozinski, Vert (ParisTech, INRIA)
Overlapping group lasso
16 juin 2009 6 / 17
Biological markers for cancer
Overlapping groups
We have prior information under the form of groups of genes with functional meaning (e.g. pathways). We would like to favor directly w involving few groups
Jacob, Obozinski, Vert (ParisTech, INRIA)
Overlapping ቤተ መጻሕፍቲ ባይዱroup lasso
16 juin 2009 6 / 17
Biological markers for cancer
Overlapping groups
We have prior information under the form of groups of genes with functional meaning (e.g. pathways). We would like to favor directly w involving few groups
of the gene is 0.
Overlapping group lasso
16 juin 2009 7 / 17
Overlap norm
Overlap norm
Introduce latent variables vg :
min L(w ) + λ
w ,v
vg 2
g ∈G
w=
Learning with a 1-penalty favors a linear classifier w ∈ Rp involving few genes.
Remark : may only select one of several correlated genes.
After this selection, people often try to find enriched functional groups.
Overlapping group lasso
16 juin 2009 4 / 17
Biological markers for cancer
Gene selection
X is the expression matrix of p genes for n tumors.
0 wk w= 0 wl
0
⇒
wg1 2= wg3 2=0
IGF selection ⇒ selection of unwanted groups
Jacob, Obozinski, Vert (ParisTech, INRIA)
Removal of any group
containing a gene ⇒ the weight
Jacob, Obozinski, Vert (ParisTech, INRIA)