ORTHONORMAL REALIZATION OF FAST FIXED-ORDER RLS ADAPTIVE FILTERS
泊松融合原理和python代码
泊松融合原理和python代码【原创版】目录1.泊松融合原理概述2.Python 代码实现泊松融合原理3.泊松融合原理的应用正文1.泊松融合原理概述泊松融合原理是一种概率论中的经典理论，由法国数学家泊松提出。
泊松融合原理描述了一个事件在特定时间间隔内发生的概率与另一个事件在相同时间间隔内发生的次数之间的关系。
具体而言，泊松融合原理表明，一个事件在时间间隔Δt 内发生的次数服从泊松分布，即P(X=k)=e^(-λΔt) * (λΔt)^k / k!，其中λ是事件的平均发生率，X 是事件在时间间隔Δt 内发生的次数，k 是事件发生的次数。
2.Python 代码实现泊松融合原理为了验证泊松融合原理，我们可以使用 Python 编写代码模拟事件的发生过程。
以下是一个简单的 Python 代码示例：```pythonimport randomimport mathdef poisson_fusion(lambda_value, dt, num_trials):"""泊松融合原理模拟:param lambda_value: 事件的平均发生率:param dt: 时间间隔:param num_trials: 模拟次数:return: 事件在时间间隔内发生的次数"""count = 0for _ in range(num_trials):# 随机生成一个 0 到 dt 之间的时间间隔t = random.uniform(0, dt)# 计算在时间间隔内事件发生的次数k = int(lambda_value * t)# 计算泊松分布的概率poisson_prob = math.exp(-lambda_value * t) * (lambda_value * t) ** k / k# 根据泊松分布概率随机生成事件发生的次数count += random.choice([0, 1, 2, 3, 4],p=poisson_prob)return count# 示例lambda_value = 1 # 事件的平均发生率为 1dt = 1 # 时间间隔为 1um_trials = 1000 # 模拟次数为 1000# 模拟事件在时间间隔内发生的次数counts = [poisson_fusion(lambda_value, dt, num_trials) for _in range(num_trials)]# 计算平均值和方差mean = sum(counts) / num_trialsvariance = sum((count - mean) ** 2 for count in counts) / num_trialsprint("平均值:", mean)print("方差:", variance)```3.泊松融合原理的应用泊松融合原理在实际应用中有很多场景，例如统计学、保险、生物学等领域。
最近鲁棒优化进展Recent Advances in Robust Optimization and Robustness An Overview
Recent Advances in Robust Optimization and Robustness:An OverviewVirginie Gabrel∗and C´e cile Murat†and Aur´e lie Thiele‡July2012AbstractThis paper provides an overview of developments in robust optimization and robustness published in the aca-demic literature over the pastfive years.1IntroductionThis review focuses on papers identified by Web of Science as having been published since2007(included),be-longing to the area of Operations Research and Management Science,and having‘robust’and‘optimization’in their title.There were exactly100such papers as of June20,2012.We have completed this list by considering 726works indexed by Web of Science that had either robustness(for80of them)or robust(for646)in their title and belonged to the Operations Research and Management Science topic area.We also identified34PhD disserta-tions dated from the lastfive years with‘robust’in their title and belonging to the areas of operations research or management.Among those we have chosen to focus on the works with a primary focus on management science rather than system design or optimal control,which are broadfields that would deserve a review paper of their own, and papers that could be of interest to a large segment of the robust optimization research community.We feel it is important to include PhD dissertations to identify these recent graduates as the new generation trained in robust optimization and robustness analysis,whether they have remained in academia or joined industry.We have also added a few not-yet-published preprints to capture ongoing research efforts.While many additional works would have deserved inclusion,we feel that the works selected give an informative and comprehensive view of the state of robustness and robust optimization to date in the context of operations research and management science.∗Universit´e Paris-Dauphine,LAMSADE,Place du Mar´e chal de Lattre de Tassigny,F-75775Paris Cedex16,France gabrel@lamsade.dauphine.fr Corresponding author†Universit´e Paris-Dauphine,LAMSADE,Place du Mar´e chal de Lattre de Tassigny,F-75775Paris Cedex16,France mu-rat@lamsade.dauphine.fr‡Lehigh University,Industrial and Systems Engineering Department,200W Packer Ave Bethlehem PA18015,USA aure-lie.thiele@2Theory of Robust Optimization and Robustness2.1Definitions and BasicsThe term“robust optimization”has come to encompass several approaches to protecting the decision-maker against parameter ambiguity and stochastic uncertainty.At a high level,the manager must determine what it means for him to have a robust solution:is it a solution whose feasibility must be guaranteed for any realization of the uncertain parameters?or whose objective value must be guaranteed?or whose distance to optimality must be guaranteed? The main paradigm relies on worst-case analysis:a solution is evaluated using the realization of the uncertainty that is most unfavorable.The way to compute the worst case is also open to debate:should it use afinite number of scenarios,such as historical data,or continuous,convex uncertainty sets,such as polyhedra or ellipsoids?The answers to these questions will determine the formulation and the type of the robust counterpart.Issues of over-conservatism are paramount in robust optimization,where the uncertain parameter set over which the worst case is computed should be chosen to achieve a trade-off between system performance and protection against uncertainty,i.e.,neither too small nor too large.2.2Static Robust OptimizationIn this framework,the manager must take a decision in the presence of uncertainty and no recourse action will be possible once uncertainty has been realized.It is then necessary to distinguish between two types of uncertainty: uncertainty on the feasibility of the solution and uncertainty on its objective value.Indeed,the decision maker generally has different attitudes with respect to infeasibility and sub-optimality,which justifies analyzing these two settings separately.2.2.1Uncertainty on feasibilityWhen uncertainty affects the feasibility of a solution,robust optimization seeks to obtain a solution that will be feasible for any realization taken by the unknown coefficients;however,complete protection from adverse realiza-tions often comes at the expense of a severe deterioration in the objective.This extreme approach can be justified in some engineering applications of robustness,such as robust control theory,but is less advisable in operations research,where adverse events such as low customer demand do not produce the high-profile repercussions that engineering failures–such as a doomed satellite launch or a destroyed unmanned robot–can have.To make the robust methodology appealing to business practitioners,robust optimization thus focuses on obtaining a solution that will be feasible for any realization taken by the unknown coefficients within a smaller,“realistic”set,called the uncertainty set,which is centered around the nominal values of the uncertain parameters.The goal becomes to optimize the objective,over the set of solutions that are feasible for all coefficient values in the uncertainty set.The specific choice of the set plays an important role in ensuring computational tractability of the robust problem and limiting deterioration of the objective at optimality,and must be thought through carefully by the decision maker.A large branch of robust optimization focuses on worst-case optimization over a convex uncertainty set.The reader is referred to Bertsimas et al.(2011a)and Ben-Tal and Nemirovski(2008)for comprehensive surveys of robust optimization and to Ben-Tal et al.(2009)for a book treatment of the topic.2.2.2Uncertainty on objective valueWhen uncertainty affects the optimality of a solution,robust optimization seeks to obtain a solution that performs well for any realization taken by the unknown coefficients.While a common criterion is to optimize the worst-case objective,some studies have investigated other robustness measures.Roy(2010)proposes a new robustness criterion that holds great appeal for the manager due to its simplicity of use and practical relevance.This framework,called bw-robustness,allows the decision-maker to identify a solution which guarantees an objective value,in a maximization problem,of at least w in all scenarios,and maximizes the probability of reaching a target value of b(b>w).Gabrel et al.(2011)extend this criterion from afinite set of scenarios to the case of an uncertainty set modeled using intervals.Kalai et al.(2012)suggest another criterion called lexicographicα-robustness,also defined over afinite set of scenarios for the uncertain parameters,which mitigates the primary role of the worst-case scenario in defining the solution.Thiele(2010)discusses over-conservatism in robust linear optimization with cost uncertainty.Gancarova and Todd(2012)studies the loss in objective value when an inaccurate objective is optimized instead of the true one, and shows that on average this loss is very small,for an arbitrary compact feasible region.In combinatorial optimization,Morrison(2010)develops a framework of robustness based on persistence(of decisions)using the Dempster-Shafer theory as an evidence of robustness and applies it to portfolio tracking and sensor placement.2.2.3DualitySince duality has been shown to play a key role in the tractability of robust optimization(see for instance Bertsimas et al.(2011a)),it is natural to ask how duality and robust optimization are connected.Beck and Ben-Tal(2009) shows that primal worst is equal to dual best.The relationship between robustness and duality is also explored in Gabrel and Murat(2010)when the right-hand sides of the constraints are uncertain and the uncertainty sets are represented using intervals,with a focus on establishing the relationships between linear programs with uncertain right hand sides and linear programs with uncertain objective coefficients using duality theory.This avenue of research is further explored in Gabrel et al.(2010)and Remli(2011).2.3Multi-Stage Decision-MakingMost early work on robust optimization focused on static decision-making:the manager decided at once of the values taken by all decision variables and,if the problem allowed for multiple decision stages as uncertainty was realized,the stages were incorporated by re-solving the multi-stage problem as time went by and implementing only the decisions related to the current stage.As thefield of static robust optimization matured,incorporating–ina tractable manner–the information revealed over time directly into the modeling framework became a major area of research.2.3.1Optimal and Approximate PoliciesA work going in that direction is Bertsimas et al.(2010a),which establishes the optimality of policies affine in the uncertainty for one-dimensional robust optimization problems with convex state costs and linear control costs.Chen et al.(2007)also suggests a tractable approximation for a class of multistage chance-constrained linear program-ming problems,which converts the original formulation into a second-order cone programming problem.Chen and Zhang(2009)propose an extension of the Affinely Adjustable Robust Counterpart framework described in Ben-Tal et al.(2009)and argue that its potential is well beyond what has been in the literature so far.2.3.2Two stagesBecause of the difficulty in incorporating multiple stages in robust optimization,many theoretical works have focused on two stages.Regarding two-stage problems,Thiele et al.(2009)presents a cutting-plane method based on Kelley’s algorithm for solving convex adjustable robust optimization problems,while Terry(2009)provides in addition preliminary results on the conditioning of a robust linear program and of an equivalent second-order cone program.Assavapokee et al.(2008a)and Assavapokee et al.(2008b)develop tractable algorithms in the case of robust two-stage problems where the worst-case regret is minimized,in the case of interval-based uncertainty and scenario-based uncertainty,respectively,while Minoux(2011)provides complexity results for the two-stage robust linear problem with right-hand-side uncertainty.2.4Connection with Stochastic OptimizationAn early stream in robust optimization modeled stochastic variables as uncertain parameters belonging to a known uncertainty set,to which robust optimization techniques were then applied.An advantage of this method was to yield approaches to decision-making under uncertainty that were of a level of complexity similar to that of their deterministic counterparts,and did not suffer from the curse of dimensionality that afflicts stochastic and dynamic programming.Researchers are now making renewed efforts to connect the robust optimization and stochastic opti-mization paradigms,for instance quantifying the performance of the robust optimization solution in the stochastic world.The topic of robust optimization in the context of uncertain probability distributions,i.e.,in the stochastic framework itself,is also being revisited.2.4.1Bridging the Robust and Stochastic WorldsBertsimas and Goyal(2010)investigates the performance of static robust solutions in two-stage stochastic and adaptive optimization problems.The authors show that static robust solutions are good-quality solutions to the adaptive problem under a broad set of assumptions.They provide bounds on the ratio of the cost of the optimal static robust solution to the optimal expected cost in the stochastic problem,called the stochasticity gap,and onthe ratio of the cost of the optimal static robust solution to the optimal cost in the two-stage adaptable problem, called the adaptability gap.Chen et al.(2007),mentioned earlier,also provides a robust optimization perspective to stochastic programming.Bertsimas et al.(2011a)investigates the role of geometric properties of uncertainty sets, such as symmetry,in the power offinite adaptability in multistage stochastic and adaptive optimization.Duzgun(2012)bridges descriptions of uncertainty based on stochastic and robust optimization by considering multiple ranges for each uncertain parameter and setting the maximum number of parameters that can fall within each range.The corresponding optimization problem can be reformulated in a tractable manner using the total unimodularity of the feasible set and allows for afiner description of uncertainty while preserving tractability.It also studies the formulations that arise in robust binary optimization with uncertain objective coefficients using the Bernstein approximation to chance constraints described in Ben-Tal et al.(2009),and shows that the robust optimization problems are deterministic problems for modified values of the coefficients.While many results bridging the robust and stochastic worlds focus on giving probabilistic guarantees for the solutions generated by the robust optimization models,Manuja(2008)proposes a formulation for robust linear programming problems that allows the decision-maker to control both the probability and the expected value of constraint violation.Bandi and Bertsimas(2012)propose a new approach to analyze stochastic systems based on robust optimiza-tion.The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory,with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem.The authors show that the performance analysis questions become highly structured optimization problems for which there exist efficient algorithms that are capable of solving problems in high dimensions.They also demonstrate that the proposed approach achieves computationally tractable methods for(a)analyzing queueing networks,(b)designing multi-item,multi-bidder auctions with budget constraints,and (c)pricing multi-dimensional options.2.4.2Distributionally Robust OptimizationBen-Tal et al.(2010)considers the optimization of a worst-case expected-value criterion,where the worst case is computed over all probability distributions within a set.The contribution of the work is to define a notion of robustness that allows for different guarantees for different subsets of probability measures.The concept of distributional robustness is also explored in Goh and Sim(2010),with an emphasis on linear and piecewise-linear decision rules to reformulate the original problem in aflexible manner using expected-value terms.Xu et al.(2012) also investigates probabilistic interpretations of robust optimization.A related area of study is worst-case optimization with partial information on the moments of distributions.In particular,Popescu(2007)analyzes robust solutions to a certain class of stochastic optimization problems,using mean-covariance information about the distributions underlying the uncertain parameters.The author connects the problem for a broad class of objective functions to a univariate mean-variance robust objective and,subsequently, to a(deterministic)parametric quadratic programming problem.The reader is referred to Doan(2010)for a moment-based uncertainty model for stochastic optimization prob-lems,which addresses the ambiguity of probability distributions of random parameters with a minimax decision rule,and a comparison with data-driven approaches.Distributionally robust optimization in the context of data-driven problems is the focus of Delage(2009),which uses observed data to define a”well structured”set of dis-tributions that is guaranteed with high probability to contain the distribution from which the samples were drawn. Zymler et al.(2012a)develop tractable semidefinite programming(SDP)based approximations for distributionally robust individual and joint chance constraints,assuming that only thefirst-and second-order moments as well as the support of the uncertain parameters are given.Becker(2011)studies the distributionally robust optimization problem with known mean,covariance and support and develops a decomposition method for this family of prob-lems which recursively derives sub-policies along projected dimensions of uncertainty while providing a sequence of bounds on the value of the derived policy.Robust linear optimization using distributional information is further studied in Kang(2008).Further,Delage and Ye(2010)investigates distributional robustness with moment uncertainty.Specifically,uncertainty affects the problem both in terms of the distribution and of its moments.The authors show that the resulting problems can be solved efficiently and prove that the solutions exhibit,with high probability,best worst-case performance over a set of distributions.Bertsimas et al.(2010)proposes a semidefinite optimization model to address minimax two-stage stochastic linear problems with risk aversion,when the distribution of the second-stage random variables belongs to a set of multivariate distributions with knownfirst and second moments.The minimax solutions provide a natural distribu-tion to stress-test stochastic optimization problems under distributional ambiguity.Cromvik and Patriksson(2010a) show that,under certain assumptions,global optima and stationary solutions of stochastic mathematical programs with equilibrium constraints are robust with respect to changes in the underlying probability distribution.Works such as Zhu and Fukushima(2009)and Zymler(2010)also study distributional robustness in the context of specific applications,such as portfolio management.2.5Connection with Risk TheoryBertsimas and Brown(2009)describe how to connect uncertainty sets in robust linear optimization to coherent risk measures,an example of which is Conditional Value-at-Risk.In particular,the authors show the link between polyhedral uncertainty sets of a special structure and a subclass of coherent risk measures called distortion risk measures.Independently,Chen et al.(2007)present an approach for constructing uncertainty sets for robust opti-mization using new deviation measures that capture the asymmetry of the distributions.These deviation measures lead to improved approximations of chance constraints.Dentcheva and Ruszczynski(2010)proposes the concept of robust stochastic dominance and shows its applica-tion to risk-averse optimization.They consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint and develop necessary and sufficient conditions of optimality for such optimization problems in the convex case.In the nonconvex case,they derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.2.6Nonlinear OptimizationRobust nonlinear optimization remains much less widely studied to date than its linear counterpart.Bertsimas et al.(2010c)presents a robust optimization approach for unconstrained non-convex problems and problems based on simulations.Such problems arise for instance in the partial differential equations literature and in engineering applications such as nanophotonic design.An appealing feature of the approach is that it does not assume any specific structure for the problem.The case of robust nonlinear optimization with constraints is investigated in Bertsimas et al.(2010b)with an application to radiation therapy for cancer treatment.Bertsimas and Nohadani (2010)further explore robust nonconvex optimization in contexts where solutions are not known explicitly,e.g., have to be found using simulation.They present a robust simulated annealing algorithm that improves performance and robustness of the solution.Further,Boni et al.(2008)analyzes problems with uncertain conic quadratic constraints,formulating an approx-imate robust counterpart,and Zhang(2007)provide formulations to nonlinear programming problems that are valid in the neighborhood of the nominal parameters and robust to thefirst order.Hsiung et al.(2008)present tractable approximations to robust geometric programming,by using piecewise-linear convex approximations of each non-linear constraint.Geometric programming is also investigated in Shen et al.(2008),where the robustness is injected at the level of the algorithm and seeks to avoid obtaining infeasible solutions because of the approximations used in the traditional approach.Interval uncertainty-based robust optimization for convex and non-convex quadratic programs are considered in Li et al.(2011).Takeda et al.(2010)studies robustness for uncertain convex quadratic programming problems with ellipsoidal uncertainties and proposes a relaxation technique based on random sampling for robust deviation optimization sserre(2011)considers minimax and robust models of polynomial optimization.A special case of nonlinear problems that are linear in the decision variables but convex in the uncertainty when the worst-case objective is to be maximized is investigated in Kawas and Thiele(2011a).In that setting,exact and tractable robust counterparts can be derived.A special class of nonconvex robust optimization is examined in Kawas and Thiele(2011b).Robust nonconvex optimization is examined in detail in Teo(2007),which presents a method that is applicable to arbitrary objective functions by iteratively moving along descent directions and terminates at a robust local minimum.3Applications of Robust OptimizationWe describe below examples to which robust optimization has been applied.While an appealing feature of robust optimization is that it leads to models that can be solved using off-the-shelf software,it is worth pointing the existence of algebraic modeling tools that facilitate the formulation and subsequent analysis of robust optimization problems on the computer(Goh and Sim,2011).3.1Production,Inventory and Logistics3.1.1Classical logistics problemsThe capacitated vehicle routing problem with demand uncertainty is studied in Sungur et al.(2008),with a more extensive treatment in Sungur(2007),and the robust traveling salesman problem with interval data in Montemanni et al.(2007).Remli and Rekik(2012)considers the problem of combinatorial auctions in transportation services when shipment volumes are uncertain and proposes a two-stage robust formulation solved using a constraint gener-ation algorithm.Zhang(2011)investigates two-stage minimax regret robust uncapacitated lot-sizing problems with demand uncertainty,in particular showing that it is polynomially solvable under the interval uncertain demand set.3.1.2SchedulingGoren and Sabuncuoglu(2008)analyzes robustness and stability measures for scheduling in a single-machine environment subject to machine breakdowns and embeds them in a tabu-search-based scheduling algorithm.Mittal (2011)investigates efficient algorithms that give optimal or near-optimal solutions for problems with non-linear objective functions,with a focus on robust scheduling and service operations.Examples considered include parallel machine scheduling problems with the makespan objective,appointment scheduling and assortment optimization problems with logit choice models.Hazir et al.(2010)considers robust scheduling and robustness measures for the discrete time/cost trade-off problem.3.1.3Facility locationAn important question in logistics is not only how to operate a system most efficiently but also how to design it. Baron et al.(2011)applies robust optimization to the problem of locating facilities in a network facing uncertain demand over multiple periods.They consider a multi-periodfixed-charge network location problem for which they find the number of facilities,their location and capacities,the production in each period,and allocation of demand to facilities.The authors show that different models of uncertainty lead to very different solution network topologies, with the model with box uncertainty set opening fewer,larger facilities.?investigate a robust version of the location transportation problem with an uncertain demand using a2-stage formulation.The resulting robust formulation is a convex(nonlinear)program,and the authors apply a cutting plane algorithm to solve the problem exactly.Atamt¨u rk and Zhang(2007)study the networkflow and design problem under uncertainty from a complexity standpoint,with applications to lot-sizing and location-transportation problems,while Bardossy(2011)presents a dual-based local search approach for deterministic,stochastic,and robust variants of the connected facility location problem.The robust capacity expansion problem of networkflows is investigated in Ordonez and Zhao(2007),which provides tractable reformulations under a broad set of assumptions.Mudchanatongsuk et al.(2008)analyze the network design problem under transportation cost and demand uncertainty.They present a tractable approximation when each commodity only has a single origin and destination,and an efficient column generation for networks with path constraints.Atamt¨u rk and Zhang(2007)provides complexity results for the two-stage networkflow anddesign plexity results for the robust networkflow and network design problem are also provided in Minoux(2009)and Minoux(2010).The problem of designing an uncapacitated network in the presence of link failures and a competing mode is investigated in Laporte et al.(2010)in a railway application using a game theoretic perspective.Torres Soto(2009)also takes a comprehensive view of the facility location problem by determining not only the optimal location but also the optimal time for establishing capacitated facilities when demand and cost parameters are time varying.The models are solved using Benders’decomposition or heuristics such as local search and simulated annealing.In addition,the robust networkflow problem is also analyzed in Boyko(2010),which proposes a stochastic formulation of minimum costflow problem aimed atfinding network design andflow assignments subject to uncertain factors,such as network component disruptions/failures when the risk measure is Conditional Value at Risk.Nagurney and Qiang(2009)suggests a relative total cost index for the evaluation of transportation network robustness in the presence of degradable links and alternative travel behavior.Further,the problem of locating a competitive facility in the plane is studied in Blanquero et al.(2011)with a robustness criterion.Supply chain design problems are also studied in Pan and Nagi(2010)and Poojari et al.(2008).3.1.4Inventory managementThe topic of robust multi-stage inventory management has been investigated in detail in Bienstock and Ozbay (2008)through the computation of robust basestock levels and Ben-Tal et al.(2009)through an extension of the Affinely Adjustable Robust Counterpart framework to control inventories under demand uncertainty.See and Sim (2010)studies a multi-period inventory control problem under ambiguous demand for which only mean,support and some measures of deviations are known,using a factor-based model.The parameters of the replenishment policies are obtained using a second-order conic programming problem.Song(2010)considers stochastic inventory control in robust supply chain systems.The work proposes an inte-grated approach that combines in a single step datafitting and inventory optimization–using histograms directly as the inputs for the optimization model–for the single-item multi-period periodic-review stochastic lot-sizing problem.Operation and planning issues for dynamic supply chain and transportation networks in uncertain envi-ronments are considered in Chung(2010),with examples drawn from emergency logistics planning,network design and congestion pricing problems.3.1.5Industry-specific applicationsAng et al.(2012)proposes a robust storage assignment approach in unit-load warehouses facing variable supply and uncertain demand in a multi-period setting.The authors assume a factor-based demand model and minimize the worst-case expected total travel in the warehouse with distributional ambiguity of demand.A related problem is considered in Werners and Wuelfing(2010),which optimizes internal transports at a parcel sorting center.Galli(2011)describes the models and algorithms that arise from implementing recoverable robust optimization to train platforming and rolling stock planning,where the concept of recoverable robustness has been defined in。
强化学习算法中的最优化方法详解(五)
强化学习算法中的最优化方法详解强化学习是一种让智能体在与环境互动中学习行为策略的机器学习方法。
在强化学习中，最优化方法是非常重要的，因为它能够帮助智能体在复杂、不确定的环境中学习到最优的策略。
本文将详细介绍强化学习算法中的最优化方法，包括值函数、策略函数以及基于模型和无模型的最优化方法。
值函数值函数是强化学习中最常用的一种最优化方法。
它用来评估某个状态或行为的价值，帮助智能体做出最优的决策。
值函数可以分为状态值函数和动作值函数。
状态值函数V(s)表示在状态s下智能体能够获得的长期奖励的期望值，而动作值函数Q(s, a)表示在状态s下选择动作a后能够获得的长期奖励的期望值。
值函数的更新通常通过贝尔曼方程来进行，贝尔曼方程是强化学习中最重要的方程之一。
它描述了值函数之间的递归关系，帮助智能体在不断与环境互动中更新值函数，从而得到最优的策略。
策略函数除了值函数，策略函数也是强化学习中常用的最优化方法。
策略函数π(a|s)表示在状态s下选择动作a的概率。
在强化学习中，智能体的目标是找到一个最优的策略函数，使得在与环境互动中能够获得最大的长期奖励。
在确定性策略中，策略函数直接映射状态到动作。
而在随机性策略中，策略函数会输出一个动作的概率分布。
确定性策略通常更容易优化，但是随机性策略在某些情况下能够带来更好的探索能力。
基于模型和无模型的最优化方法在强化学习中，最优化方法可以分为基于模型和无模型的方法。
基于模型的方法通过对环境建模，预测状态转移和奖励函数，从而寻找最优的策略。
这种方法需要对环境有一定的先验知识，并且在环境模型不准确或无法建模时会出现问题。
无模型的方法则直接在与环境互动中学习策略，不需要对环境进行建模。
这种方法可以更好地适应不确定的环境，并且在一些复杂的情况下能够得到更好的效果。
无模型的方法包括值迭代、策略迭代、蒙特卡洛方法、时序差分学习等。
结语强化学习中的最优化方法是智能体学习最优策略的关键。
泊松融合原理和python代码
泊松融合原理与Python代码实现1. 前言泊松融合是一种图像处理技术，用于将源图像的内容无缝地融合到目标图像中。
它的基本原理是通过解决一个泊松方程，将源图像的梯度与目标图像的梯度相匹配，从而实现融合效果。
在本文中，我们将详细介绍泊松融合的基本原理，并使用Python编写代码来实现泊松融合。
2. 泊松方程首先，让我们来了解一下泊松方程。
泊松方程是一个偏微分方程，通常用于描述物理系统中的平衡状态。
在图像处理中，我们可以使用泊松方程来实现无缝融合。
泊松方程可以表示为：∇²f = g其中，∇²表示拉普拉斯算子（二阶导数），f表示未知函数，g表示已知函数。
在泊松融合中，我们希望通过求解这个方程来得到未知函数f。
3. 泊松融合原理泊松融合的基本原理是将源图像的梯度与目标图像的梯度相匹配。
通过匹配这些梯度，我们可以确保融合后的图像在边界处具有连续性。
具体来说，泊松融合的步骤如下：1.将源图像和目标图像进行对齐。
对齐可以使用特征点匹配或其他图像对齐算法来实现。
2.在目标图像上选择一个感兴趣区域（ROI），并在该区域内将源图像的内容融合进去。
3.计算源图像和目标图像的梯度。
我们可以使用Sobel算子等滤波器来计算梯度。
4.在ROI中，将泊松方程应用于目标图像的梯度和源图像的梯度之间的差异。
这相当于求解泊松方程中的未知函数f。
5.将求解得到的未知函数f与目标图像中ROI外的部分进行融合，得到最终的融合结果。
4. Python代码实现接下来，让我们使用Python编写代码来实现泊松融合。
首先，我们需要导入所需的库：import numpy as npimport cv2from scipy import sparsefrom scipy.sparse.linalg import spsolve然后，我们定义一个名为poisson_blend的函数来执行泊松融合：def poisson_blend(source, target, mask):# 获取源图像和目标图像的尺寸height, width, _ = source.shape# 将源图像和目标图像转换为灰度图像source_gray = cv2.cvtColor(source, cv2.COLOR_BGR2GRAY)target_gray = cv2.cvtColor(target, cv2.COLOR_BGR2GRAY)# 计算源图像和目标图像的梯度source_grad_x = cv2.Sobel(source_gray, cv2.CV_64F, 1, 0, ksize=3)source_grad_y = cv2.Sobel(source_gray, cv2.CV_64F, 0, 1, ksize=3)target_grad_x = cv2.Sobel(target_gray, cv2.CV_64F, 1, 0, ksize=3)target_grad_y = cv2.Sobel(target_gray, cv2.CV_64F, 0, 1, ksize=3)# 创建稀疏矩阵A和向量bA = sparse.lil_matrix((height * width, height * width))b = np.zeros((height * width,))# 填充矩阵A和向量bfor y in range(1,height-1):for x in range(1,width-1):if mask[y,x] == 255:index = y * width + xA[index,index] = -4A[index,index-1] = 1A[index,index+1] = 1A[index,index-width] = 1A[index,index+width] = 1b[index] = target_grad_x[y,x] - target_grad_x[y,x-1] + target_ grad_y[y,x] - target_grad_y[y-1,x]# 解泊松方程f = spsolve(A.tocsr(), b)# 将求解得到的未知函数f转换为图像f_image = np.uint8(np.clip(f.reshape(height, width), 0, 255))# 在目标图像中将ROI外的部分与源图像进行融合blended = target.copy()for y in range(height):for x in range(width):if mask[y,x] == 255:blended[y,x] = source[y,x]return blended最后，我们加载源图像、目标图像和蒙版，并调用poisson_blend函数来执行泊松融合：source = cv2.imread('source.jpg')target = cv2.imread('target.jpg')mask = cv2.imread('mask.jpg', cv2.IMREAD_GRAYSCALE)result = poisson_blend(source, target, mask)cv2.imwrite('result.jpg', result)5. 总结在本文中，我们详细介绍了泊松融合的基本原理，并使用Python编写了相应的代码实现。
线性系统理论-中英文对照
[Linear system theory and design] Absolutely integrable 绝对可积Adder 加法器Additivity 可加性Adjoint 伴随Aeronautical航空的Arbitrary 任意的Asymptotic stability渐近稳定Asymptotic tracking 渐近跟踪Balanced realization 平衡实现Basis 基BIBO stability 有界输入有界输出稳定Black box 黑箱Blocking zero 阻塞零点Canonical decomposition 规分解Canonical规Capacitor 电容Causality 因果性Cayley-Hamilton theorem 凯莱-哈密顿定理Characteristic polynominal 特征多项式Circumflex 卷积Coefficient 系数Cofactor 余因子Column degree 列次数Column-degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Common Divisor公共因式Companion-form matrix 规型矩阵Compensator 调节器，补偿器Compensator equation补偿器方程Control configuration 控制构型Controllability 能控性Convolution 卷积Conventional常规的Coprimeness互质Corollary推论Cyclic matrix 循环矩阵Dead beat design 有限拍设计Decoupling 解耦Degree of rational function有理矩阵的次数Description of system系统描述Derivative 导数Determinant 行列式Diagonal对角型Discretization 离散化Disturbance rejectionDivisor 因式Diverge分叉Duality 对偶Eigenvalue特征值Eigenvextor 特征向量Empirical 经验Equivalence 等价Equivalence transformation等价变换Exhaustive 详细的Exponential function 指数函数Extensively广泛地Filter 滤波Finite time有限时间Finite 有限的Fraction 分式Polynomial fraction 多项式分式Fundamental cutest 基本割集Fundamental loop 基本回路Fundamental matrix 基本解阵Gramian 格拉姆Geometric几何的Hankel matrix Hankel 矩阵Hankel singular valueHomogeneity奇次性Hurwitz polynomial Hurwitz 多项式Implementable transfer function 可实现的传递函数Impulse response 脉冲响应Impulse response matrix 脉冲响应矩阵Impulse response sequence 脉冲响应序列Index指数Inductance 自感系数Inductor 电感Integrator 积分器Internal model principle 模原理Intuitively 直观的Inverse 逆Jacobian 雅可比Jordan block约当块Jordan form matrix 约当型矩阵Lapalace expansion 拉普拉斯展开Lapalace transform 拉普拉斯变换Left multiple 左倍式Linear algebraic equation线性代数方程Linear space 线性空间Linearity 线性性Linearization 线性化Linearly dependent 线性相关Linearly independent 线性无关Lumpedness 集中（参数）性Lyapunov equation Lyapunov方程Lyapunov theorem Lyapunov定理Lyapunov transform Lyapunov变换Markov parameter 马尔科夫参数Magnitude 模Manuscript 手稿，原稿Marginal stability 稳定Matrix 矩阵Minimal polynomial 最小多项式Minimal realization 最小实现Minimum energy control 最小能量控制Minor 子行列式Model matching 模型匹配Model reduction 模型降维Monic 首1的Monic polynomial首1的多项式Multiple 倍式Multiplier 乘法器Nilpotent matrix 幂为零的矩阵Nominal equation 标称（名义）方程Norm 数Nonsingular非奇异Null space 零空间Nullity 零度（零空间的维数）Observability 能观性Observer 观测器Op-amp circuit 运算发大器Orthogonal 正交的Orthogonal matrix正交矩阵Orthogonality of vectors 向量的正交性Orthonormal set 规正交集Orthonormalization 规正交化Oscillator振荡器Parenthesis圆括号Parameterization 参数化Pendulum钟摆Periodic 周期的Periodic state system 周期状态系统Pertinent 相关的Plant被控对象Pole 极点Pole placement 极点配置Pole-zero cancellation 零极点对消Pole-zero excess inequalityPolynomial多项式Polynomial matrix 多项式矩阵Column degree 列次数Column degree 列次数Column-degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Echelon formLeft divisor 左因式Left multiple 左倍式Right divisor 右因式Right multiple 右倍式Row degree 行次数Row-degree-coefficient matrix 行次数系数矩阵Row reduced行既约Unimodular 幺模Positive正的Positive definite matrix正定矩阵Positive semidefinite matrix正定矩阵Power series 幂级数Primary dependent column 主相关列Primary dependent column 主相关行Prime素数的Principal minor 主子行列式Pseudoinverse 伪逆Pseudostate 伪状态QR decomposition QR 分解Quadratic二次的Quadratic form二次型Range space 值域空间Rank秩Rational function 有理函数Biproper双真Improper 不真Proper 真Strictly proper 严格真Rational matrix 有理矩阵Biproper双真Improper 不真Proper 真Strictly proper 严格真Reachability 可达性Realization 状态空间可实现性Balanced 平衡的（实现）Companion form 规型（实现）Controllable form能控型（实现）Input-normal 输入规（实现）Minimal 最小（实现）Observable-form能控型（实现）Output-normal 输出规（实现）Time-varying 时变（实现）Reduced 既约的Regulator problem 调节器问题Relaxedness 松弛Response 响应Impulse 脉冲Zero-input 零输入Zero-state 零状态Remainder 余数Resistor 电阻Resultant 结式Generalized Resultant 广义结式Sylvester Resultant Sylvester（西尔维斯特）结式Right divisor右因子Robust design 鲁棒设计Robust tracking 鲁棒跟踪Row indices 行指数集Saturate 使饱和Scalar 标量Schmidt orthonormalizationSemidefinite 半正定的Similar matrices 相似矩阵Separation property 分离原理Servomechanism 司服机制Similar transformation 相似变换Singular value 奇异值Singular value decomposition奇异值分解Spectrum 谱Stability 稳定Asymptotic stability 渐近稳定BIBO stability 有界输入有界输出稳定Stability in the sense of Lyapunov 雅普诺夫意义下的稳定Marginal stability 临界稳定Total stability 整体稳定Stabilization镇定（稳定化）State 状态State variable 状态变量State equation 状态方程Discrete-time State equation 离散时间状态方程Discretization State equation 离散化状态方程Equivalent State equation 等价状态方程Periodic State equation 周期状态方程Reduction State equation 状态方程Solution of State equation 状态方程的解Time-varying State equation 时变状态方程State estimator 状态估计器Asymptotic State estimator 渐近状态估计器Closed-loop State estimator 闭环状态估计器Full-dimentional State estimator 全维状态估计器Open-loop State estimator 开环状态估计器Reduced-dimentional State estimator 降维状态估计器State feedback 状态反馈State-space equation 状态空间模型State transition matrix 状态转移矩阵Superposition property 叠加性Sylvester resultant （西尔维斯特）结式Sylvester’s inequality （西尔维斯特）不等式System 系统Superposition叠加Terminology 术语Total stability 整体稳定Trace of matrix 矩阵的迹Tracking 跟踪Transfer function 传递函数Discrete Transfer function离散传递函数Pole of Transfer function传递函数的极点Zero of Transfer function传递函数的零点Transfer matrix 传递函数矩阵Blocking zero of matrix 传递函数矩阵的阻塞零点Pole of Transfer function传递函数矩阵的极点Transmission Zero of Transfer function传递函数矩阵的传输零点Transistor 晶体管Transmission zero 传输零点Tree 树Truncation operator 截断算子Trajectory 轨迹Transpose 转置Triangular 三角形的Unimodular matrix 么模阵Unit-time-delay element 单位时间延迟单元Unity-feedback 单位反馈Vandermonde matrix Vandermonde矩阵Vector 向量well posed 适定的well-posedness 适定性Yield 等于，得出Zero 零点Blocking zero 阻塞零点Minimum-phase zero 最小相位零点nonminimum-phase zero 非最小相位零点transmission zero 传输零点zero-input response 零输入响应zero-pole-gain form 零极点增益形式zero-state equivalence零状态等价zero-state response零状态响应z-transform z-变换。
随机有限集扩展目标跟踪算法研究
随机有限集扩展目标跟踪算法研究随机有限集扩展目标跟踪算法研究摘要：目标跟踪是计算机视觉和机器人领域的研究热点之一。
随机有限集（Random Finite Set，RFS）扩展目标跟踪算法是一类基于概率和统计的目标跟踪方法。
本文通过研究RFS扩展目标跟踪算法的基本理论和关键技术，总结了该算法的发展历程和应用情况，并对其未来发展进行了展望。
1. 引言目标跟踪是计算机视觉和机器人领域的重要课题，广泛应用于目标检测、自动驾驶、智能监控等领域。
传统的目标跟踪算法主要基于滤波或粒子滤波方法，其在处理高维状态空间和多目标的情况下存在一定的局限性。
为了克服这些问题，研究人员提出了随机有限集扩展目标跟踪算法。
2. 随机有限集扩展目标跟踪算法概述RFS扩展目标跟踪算法是一类基于RFS理论的目标跟踪方法，该方法通过描述目标状态的概率密度函数来进行目标跟踪。
RFS概率密度函数描述了目标的存在概率以及目标的状态分布，能够更准确地表示目标的不确定性。
3. RFS扩展目标跟踪算法的基本理论RFS扩展目标跟踪算法的核心理论包括卡尔曼滤波器、粒子滤波器、卡尔曼-粒子滤波器等。
卡尔曼滤波器是一种基于线性状态空间模型的目标跟踪方法，其通过线性近似的方式估计目标状态。
粒子滤波器是一种基于重采样的蒙特卡洛方法，能够处理非线性和非高斯的目标模型。
卡尔曼-粒子滤波器是卡尔曼滤波器和粒子滤波器的结合，用于解决高维状态空间和多目标的跟踪问题。
4. RFS扩展目标跟踪算法的关键技术RFS扩展目标跟踪算法的关键技术包括目标测量、目标建模、数据关联和滤波更新等。
目标测量用于从视频序列中提取目标特征，如位置、速度、外观等。
目标建模用于描述目标的运动模型和外观模型，常见的建模方法包括卡尔曼滤波器、高斯混合模型等。
数据关联用于将每个测量与目标进行关联，常见的关联方法有最小二乘关联、相似度度量等。
滤波更新用于根据最新的目标测量信息更新目标的状态估计值和不确定度。
《2024年度二维弹性快速多极边界元算法及截断误差分析》范文
《二维弹性快速多极边界元算法及截断误差分析》篇一一、引言在计算力学领域，边界元法（Boundary Element Method, BEM）是一种重要的数值技术，尤其适用于解决弹性力学问题。
随着计算机技术的发展，二维弹性快速多极边界元算法成为了研究的热点。
该算法结合了多极展开与边界元法，可大幅提高计算效率并降低存储需求。
本文将详细介绍二维弹性快速多极边界元算法的原理，并对其截断误差进行分析。
二、二维弹性快速多极边界元算法原理二维弹性快速多极边界元算法基于边界元法的基本原理，通过多极展开技术来提高计算效率。
该算法的主要步骤包括：问题定义、离散化处理、多极展开、快速求解和后处理。
1. 问题定义在二维弹性问题中，我们需定义问题的几何形状、材料属性及边界条件。
这些信息将作为后续计算的基础。
2. 离散化处理将问题域离散化为一系列的子域，每个子域的边界上设置节点。
这些节点将用于后续的边界元法计算。
3. 多极展开多极展开是该算法的核心部分。
通过将场源项的多极展开，可以将计算复杂度从O(N^2)降低到O(NlogN)，其中N为节点的数量。
这一步显著提高了计算效率。
4. 快速求解利用快速求解技术，如迭代法或直接法，对离散化后的系统进行求解。
通过多极展开和快速求解的结合，可大幅提高计算速度。
5. 后处理对求解结果进行后处理，如提取应力、位移等数据，并进行可视化展示。
三、截断误差分析在二维弹性快速多极边界元算法中，截断误差主要来源于多极展开的截断和数值求解的误差。
本部分将对这两部分误差进行分析。
1. 多极展开的截断误差多极展开的截断会导致一定的计算误差。
这种误差主要与展开的阶数、场源项的复杂性及问题的规模有关。
一般来说，阶数越高，截断误差越小，但计算复杂度也越高。
因此，在实际应用中，需要根据问题的规模和精度要求来选择合适的阶数。
2. 数值求解的误差数值求解的误差主要来源于迭代法或直接法的求解过程。
这种误差可以通过选择合适的求解器、优化算法参数等方式来减小。
潜类别增长 组基轨迹模型-概述说明以及解释
潜类别增长组基轨迹模型-概述说明以及解释1.引言1.1 概述概述部分的内容可以是对整篇文章的简要介绍和背景提要。
下面是一个示例：概述潜类别增长和组基轨迹模型是机器学习领域中一种重要的研究方向。
随着数据规模的快速增长和应用需求的提升，潜类别增长的概念引起了广泛的关注。
在传统的机器学习任务中，数据被划分为已知的类别，预定义的分类模型可以对其进行准确的分类。
然而，实际应用中常常遇到新的类别出现的情况，传统的分类模型无法处理这些未知的类别。
潜类别增长方法的提出就是为了解决这一问题。
该方法通过动态地扩展已有的类别集合，允许系统在学习过程中接收新的未知类别，并不断更新分类模型。
这种方法的核心是利用数据中的潜在信息来推断新类别的存在。
潜类别增长方法的应用范围非常广泛，例如在物体识别、图像分类和语音识别等领域都有重要的应用。
与此同时，组基轨迹模型也是一类重要的机器学习模型。
在许多实际问题中，数据往往具有一定的时序性质，例如时间序列数据、视频数据等。
传统的机器学习模型往往无法充分利用数据中的时序信息。
组基轨迹模型通过将数据表示为一组子序列（组基）的线性组合，更好地描述了数据的时序特征。
这种方法不仅可以提高数据的表示能力，还可以减少特征的维度，从而提高模型的泛化能力。
本文将详细介绍潜类别增长和组基轨迹模型的原理和方法，并通过实验验证其性能。
接下来，我们将首先介绍潜类别增长方法的基本概念和原理，然后重点阐述组基轨迹模型的建模过程。
最后，我们将对实验结果进行分析和总结，并展望这两个方法在未来的发展方向。
1.2文章结构文章结构部分的内容可以描述本文的组织结构和各个章节的内容安排。
可以使用以下内容作为参考：本文主要围绕着“潜类别增长”和“组基轨迹模型”展开讨论。
文章分为引言、正文和结论三个部分。
引言部分将首先给出对整篇文章的概述，介绍潜类别增长和组基轨迹模型的背景和重要性。
接着，给出文章的整体结构以及各个章节的简要内容介绍。
导数法求解最优化模型代码
导数法求解最优化模型代码最优化模型是一种数学模型，用于在满足一定约束条件的情况下，寻找一组变量的值，使目标函数达到最优值。
使用导数法求解最优化模型需要对目标函数进行求导，并通过导数的正负性来确定函数的单调性，从而找到最优解。
以下是一个简单的 Python 代码示例，演示了如何使用导数法求解一个一元函数的最优化问题：pythondef find_maxima(f, df, x0, tol=1e-6):"""使用导数法求解函数的最大值。
参数：f (callable)：目标函数。
df (callable)：目标函数的导数。
x0 (float)：初始点。
tol (float)：迭代停止的容差范围，默认为 1e-6。
返回：float：函数的最大值。
"""x = x0fx = f(x)dfx = df(x)while abs(dfx) > tol:if dfx > 0:x = x - tol * dfxelse:x = x + tol * dfxfx = f(x)dfx = df(x)return fx# 定义目标函数和导数def f(x):return x ** 2 - 2 * xdef df(x):return 2 * x - 2# 调用 find_maxima 函数求解最大值x0 = 0max_value = find_maxima(f, df, x0)print("最大值为：", max_value)```在上述代码中，`find_maxima` 函数接受目标函数`f`、目标函数的导数 `df`、初始点 `x0` 和容差范围 `tol` 作为参数。
在函数内部，通过不断迭代更新`x` 的值，使目标函数的导数趋近于 0，从而找到函数的最大值。
请注意，上述代码仅演示了求解一元函数最大值的情况。
如果是多元函数的最优化问题，需要使用更复杂的方法，如牛顿法、梯度下降法等。
openpcdet nuscenes 预测格式
一、背景介绍openpcdet是一个开源的目标检测与跟踪工具，是基于点云数据进行3D目标检测的工具包。
而nuscenes是一个用于自动驾驶的开放数据库，包含了丰富的标记数据和传感器采集的场景信息。
在自动驾驶领域，准确地预测周围环境中的障碍物是至关重要的。
针对nuscenes 数据库设计的openpcdet nuscenes预测格式具有重要的意义。
二、openpcdet nuscenes预测格式介绍1. openpcdet nuscenes预测格式是指在nuscenes数据库中使用openpcdet工具进行目标检测的数据输出格式。
该格式通常包含以下信息：2. 目标类别：通常包括行人、车辆、自行车等不同的目标类别。
3. 目标位置：目标在3D空间中的位置坐标，通常以(x, y, z)的形式表示。
4. 目标尺寸：目标在3D空间中的长宽高尺寸，通常以(length, width, height)的形式表示。
5. 目标朝向：目标的朝向角度，通常以yaw角表示。
6. 目标置信度：系统对目标预测的置信度，通常为一个0到1之间的值。
三、在openpcdet中使用nuscenes数据库进行目标检测的步骤1. 数据准备：从nuscenes数据库中获取需要的场景数据，包括点云数据、相机图像、激光雷达数据等。
2. 数据预处理：对获取的数据进行预处理，包括点云数据的滤波、坐标变换、数据配准等操作。
3. 模型加载：在openpcdet中加载预训练好的目标检测模型，通常是基于深度学习的3D目标检测模型。
4. 目标检测：对预处理后的数据进行目标检测，获取目标的位置、尺寸、朝向等信息。
5. 结果输出：将目标检测的结果按照openpcdet nuscenes预测格式进行输出，以便后续的自动驾驶系统可以使用。
四、openpcdet nuscenes预测格式的优势1. 适用性广：openpcdet nuscenes预测格式适用于nuscenes数据库中的各种场景数据，包括城市道路、高速公路、乡村道路等不同环境下的目标检测。
线性系统理论-中英文对照解析
[Linear system theory and design] Absolutely integrable 绝对可积Adder 加法器Additivity 可加性Adjoint 伴随Aeronautical航空的Arbitrary 任意的Asymptotic stability渐近稳定Asymptotic tracking 渐近跟踪Balanced realization 平衡实现Basis 基BIBO stability 有界输入有界输出稳定Black box 黑箱Blocking zero 阻塞零点Canonical decomposition 规范分解Canonical规范Capacitor 电容Causality 因果性Cayley-Hamilton theorem 凯莱-哈密顿定理Characteristic polynominal 特征多项式Circumflex 卷积Coefficient 系数Cofactor 余因子Column degree 列次数Column-degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Common Divisor公共因式Companion-form matrix 规范型矩阵Compensator 调节器，补偿器Compensator equation补偿器方程Control configuration 控制构型Controllability 能控性Convolution 卷积Conventional常规的Coprimeness互质Corollary推论Cyclic matrix 循环矩阵Dead beat design 有限拍设计Decoupling 解耦Degree of rational function有理矩阵的次数Description of system系统描述Derivative 导数Determinant 行列式Diagonal对角型Discretization 离散化Disturbance rejectionDivisor 因式Diverge分叉Duality 对偶Eigenvalue特征值Eigenvextor 特征向量Empirical 经验Equivalence 等价Equivalence transformation等价变换Exhaustive 详细的Exponential function 指数函数Extensively广泛地Filter 滤波Finite time有限时间Finite 有限的Fraction 分式Polynomial fraction 多项式分式Fundamental cutest 基本割集Fundamental loop 基本回路Fundamental matrix 基本解阵Gramian 格拉姆Geometric几何的Hankel matrix Hankel 矩阵Hankel singular valueHomogeneity奇次性Hurwitz polynomial Hurwitz 多项式Implementable transfer function 可实现的传递函数Impulse response 脉冲响应Impulse response matrix 脉冲响应矩阵Impulse response sequence 脉冲响应序列Index指数Inductance 自感系数Inductor 电感Integrator 积分器Internal model principle 内模原理Intuitively 直观的Inverse 逆Jacobian 雅可比Jordan block约当块Jordan form matrix 约当型矩阵Lapalace expansion 拉普拉斯展开Lapalace transform 拉普拉斯变换Left multiple 左倍式Linear algebraic equation线性代数方程Linear space 线性空间Linearity 线性性Linearization 线性化Linearly dependent 线性相关Linearly independent 线性无关Lumpedness 集中（参数）性Lyapunov equation Lyapunov方程Lyapunov theorem Lyapunov定理Lyapunov transform Lyapunov变换Markov parameter 马尔科夫参数Magnitude 模Manuscript 手稿，原稿Marginal stability 稳定Matrix 矩阵Minimal polynomial 最小多项式Minimal realization 最小实现Minimum energy control 最小能量控制Minor 子行列式Model matching 模型匹配Model reduction 模型降维Monic 首1的Monic polynomial首1的多项式Multiple 倍式Multiplier 乘法器Nilpotent matrix 幂为零的矩阵Nominal equation 标称（名义）方程Norm 范数Nonsingular非奇异Null space 零空间Nullity 零度（零空间的维数）Observability 能观性Observer 观测器Op-amp circuit 运算发大器Orthogonal 正交的Orthogonal matrix正交矩阵Orthogonality of vectors 向量的正交性Orthonormal set 规范正交集Orthonormalization 规范正交化Oscillator振荡器Parenthesis圆括号Parameterization 参数化Pendulum钟摆Periodic 周期的Periodic state system 周期状态系统Pertinent 相关的Plant被控对象Pole 极点Pole placement 极点配置Pole-zero cancellation 零极点对消Pole-zero excess inequalityPolynomial多项式Polynomial matrix 多项式矩阵Column degree 列次数Column degree 列次数Column-degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Echelon formLeft divisor 左因式Left multiple 左倍式Right divisor 右因式Right multiple 右倍式Row degree 行次数Row-degree-coefficient matrix 行次数系数矩阵Row reduced行既约Unimodular 幺模Positive正的Positive definite matrix正定矩阵Positive semidefinite matrix正定矩阵Power series 幂级数Primary dependent column 主相关列Primary dependent column 主相关行Prime素数的Principal minor 主子行列式Pseudoinverse 伪逆Pseudostate 伪状态QR decomposition QR 分解Quadratic二次的Quadratic form二次型Range space 值域空间Rank秩Rational function 有理函数Biproper双真Improper 不真Proper 真Strictly proper 严格真Rational matrix 有理矩阵Biproper双真Improper 不真Proper 真Strictly proper 严格真Reachability 可达性Realization 状态空间可实现性Balanced 平衡的（实现）Companion form 规范型（实现）Controllable form能控型（实现）Input-normal 输入规范（实现）Minimal 最小（实现）Observable-form能控型（实现）Output-normal 输出规范（实现）Time-varying 时变（实现）Reduced 既约的Regulator problem 调节器问题Relaxedness 松弛Response 响应Impulse 脉冲Zero-input 零输入Zero-state 零状态Remainder 余数Resistor 电阻Resultant 结式Generalized Resultant 广义结式Sylvester Resultant Sylvester（西尔维斯特）结式Right divisor右因子Robust design 鲁棒设计Robust tracking 鲁棒跟踪Row indices 行指数集Saturate 使饱和Scalar 标量Schmidt orthonormalizationSemidefinite 半正定的Similar matrices 相似矩阵Separation property 分离原理Servomechanism 司服机制Similar transformation 相似变换Singular value 奇异值Singular value decomposition奇异值分解Spectrum 谱Stability 稳定Asymptotic stability 渐近稳定BIBO stability 有界输入有界输出稳定Stability in the sense of Lyapunov 李雅普诺夫意义下的稳定Marginal stability 临界稳定Total stability 整体稳定Stabilization镇定（稳定化）State 状态State variable 状态变量State equation 状态方程Discrete-time State equation 离散时间状态方程Discretization State equation 离散化状态方程Equivalent State equation 等价状态方程Periodic State equation 周期状态方程Reduction State equation 状态方程Solution of State equation 状态方程的解Time-varying State equation 时变状态方程State estimator 状态估计器Asymptotic State estimator 渐近状态估计器Closed-loop State estimator 闭环状态估计器Full-dimentional State estimator 全维状态估计器Open-loop State estimator 开环状态估计器Reduced-dimentional State estimator 降维状态估计器State feedback 状态反馈State-space equation 状态空间模型State transition matrix 状态转移矩阵Superposition property 叠加性Sylvester resultant （西尔维斯特）结式Sylvester’s inequality （西尔维斯特）不等式System 系统Superposition叠加Terminology 术语Total stability 整体稳定Trace of matrix 矩阵的迹Tracking 跟踪Transfer function 传递函数Discrete Transfer function离散传递函数Pole of Transfer function传递函数的极点Zero of Transfer function传递函数的零点Transfer matrix 传递函数矩阵Blocking zero of matrix 传递函数矩阵的阻塞零点Pole of Transfer function传递函数矩阵的极点Transmission Zero of Transfer function传递函数矩阵的传输零点Transistor 晶体管Transmission zero 传输零点Tree 树Truncation operator 截断算子Trajectory 轨迹Transpose 转置Triangular 三角形的Unimodular matrix 么模阵Unit-time-delay element 单位时间延迟单元Unity-feedback 单位反馈Vandermonde matrix Vandermonde矩阵Vector 向量well posed 适定的well-posedness 适定性Yield 等于，得出Zero 零点Blocking zero 阻塞零点Minimum-phase zero 最小相位零点nonminimum-phase zero 非最小相位零点transmission zero 传输零点zero-input response 零输入响应zero-pole-gain form 零极点增益形式zero-state equivalence零状态等价zero-state response零状态响应z-transform z-变换。
强化学习求解组合最优化问题的研究综述
强化学习求解组合最优化问题的研究综述强化学习求解组合最优化问题的研究综述引言组合最优化问题是数学与计算领域中一个重要的研究领域。
传统的解决方法往往依赖于启发式算法或者精确算法。
然而，随着人工智能领域的不断发展，强化学习作为一种新兴的算法模型，为求解组合最优化问题带来了新的可能性。
本文将对强化学习求解组合最优化问题的研究进行深入综述。
强化学习概述强化学习是一种机器学习的方法，通过试错的方式学习如何做出决策以最大化累积奖励。
在一个强化学习系统中，有一个智能体(agent)与环境(environment)进行交互，通过观察环境的状态和奖励信号来学习，目标是使智能体能够学会在给定环境下做出最佳决策。
组合最优化问题组合最优化问题是在一定的约束条件下，从一组可行解中选择出最好的解的问题。
其中，可行解是指满足所有约束条件的解，最好的解则依据特定的目标函数进行定义。
组合最优化问题在实际生活中有广泛的应用，如货物调度、路径规划、资源分配等等。
强化学习求解组合最优化问题的方法1. 基于价值函数的方法基于价值函数的方法是将组合最优化问题转化为强化学习问题，并通过价值函数来表示状态的价值。
常用的方法包括Q-learning算法、Deep Q Network(DQN)等。
例如，在路径规划问题中，智能体根据当前位置和可选动作来更新状态的价值函数，从而学习出最优路径。
2. 基于策略梯度的方法基于策略梯度的方法是直接学习策略函数，通过优化策略函数来达到最优解。
其中，策略函数是指智能体在给定状态下选择动作的概率分布。
常见的算法包括Actor-Critic算法、Proximal Policy Optimization(PPO)等。
这些算法在组合最优化问题的求解中展现了较好的效果。
3. 基于深度学习的方法近年来，深度学习技术的快速发展为求解组合优化问题提供了新的思路。
通过结合深度学习和强化学习，可以将大规模的组合优化问题转化为一个端到端的学习问题。
套索模型的基本原理
套索模型的基本原理套索模型（Lasso Model），也称为L1正则化线性回归模型，是一种用于特征选择和回归分析的统计模型。
套索模型通过在损失函数中引入L1范数的罚项，将模型的复杂度进行约束，有效地实现对具有稀疏性的特征的选择。
相较于传统的线性回归模型，套索模型能够自动将无关紧要的特征的权重置为零，从而达到特征选择和降维的目的。
套索模型的基本原理是在普通的线性回归模型的基础上，引入L1范数的正则化项，使得原损失函数尽可能小的同时，提供尽可能少的非零系数。
L1范数即为参数向量中各分量绝对值的和，表示模型复杂度的度量。
而通过控制正则化参数λ的值，可以调整正则化项对模型的影响程度，从而实现对特征的选择。
具体而言，套索模型的数学表达式如下：minimize(1/2n) y-Xβ^2 + λβ_1其中，y表示因变量向量；X表示自变量矩阵；β表示回归系数向量；λ表示正则化参数；n表示样本的个数。
在这个公式中，第一项表示普通的最小二乘损失函数，用于拟合观测值与模型预测值之间的差异；第二项表示L1范数的正则化项，用于控制模型复杂度。
套索模型的求解通常使用迭代算法，其中最常用的算法是坐标下降法（Coordinate Descent）。
其基本思想是，先固定除了某个回归系数之外的所有回归系数，然后对该回归系数使用最小二乘法求解。
然后，依次对每个回归系数进行迭代，直至收敛。
套索模型的重要特性之一是可以实现特征选择。
L1范数的正则化惩罚项具有尖峰效应，随着正则化参数的增加，不重要的特征很容易被惩罚而系数趋向于零，从而实现特征的选择。
这个特性使得套索模型在高维数据集中非常有用，可以自动选择出对目标变量有较大影响的特征。
此外，套索模型还具有稳定性和解释性。
相对于L2正则化（岭回归）模型而言，套索模型所得到的参数估计值更接近于真实值。
这是因为L1范数的正则化项更容易产生零系数，减少了过拟合的风险。
另外，套索模型所得到的非零系数通常非常少，因此更容易解释和解读模型。
open3d 点云配准 fitness公式
《Open3D 点云配准中的 Fitness 公式解析与应用》在点云数据处理与计算机视觉领域，点云配准一直是一个重要而又复杂的问题。
而在 Open3D 中，配准算法的核心之一就是 Fitness 公式。
在本文中，我将以深度和广度的方式，对 Open3D 中的 Fitness 公式进行全面评估、解析，并探讨其在点云配准中的应用及意义。
1. Fitness 公式在 Open3D 中的定义与原理Fitness 公式是用来衡量两个点云之间配准程度的数值指标。
在Open3D 中，Fitness 公式通常采用基于点云距离的衡量方法，用来评估变换矩阵的优劣。
其定义如下：Fitness = Σ exp(-d_i^2 / σ^2)其中，d_i 表示两个点云中对应点之间的距离，σ 是高斯核的带宽参数。
Fitness 越接近于 0，表示两个点云之间的配准越好。
2. Fitness 公式的深入解析及应用在点云配准中，Fitness 公式起着至关重要的作用。
通过对 Fitness 公式的深入解析，我们可以发现它的应用不仅局限于点云配准的评估，还可以用在点云匹配、目标识别等领域。
而 Fitness 公式中的高斯核带宽参数σ 的选择，也直接影响着配准结果的准确性和稳定性。
3. Open3D 中 Fitness 公式的实际案例分析为了更好地理解 Fitness 公式在 Open3D 中的应用，我们可以通过实际案例来进行分析。
在点云配准中，我们可以通过调整σ 的取值来观察配准结果的变化，进而选择最适合的σ 值来获得最优的配准效果。
这种基于 Fitness 公式的参数优化方法，能够帮助我们更准确地实现点云配准。
总结与回顾通过对 Open3D 中 Fitness 公式的全面评估与深度解析，我们可以更好地理解其在点云配准中的重要性及应用价值。
了解 Fitness 公式的原理和参数选择对于优化点云配准算法具有重要意义。
在实际应用中，我们也可以根据Fitness 公式对点云进行特征提取、配准评估等操作，从而实现更精确、稳定的点云处理效果。
强化学习中的模型基础与实现细节
强化学习中的模型基础和实现细节涉及到许多不同的技术和方法，以下是强化学习的一些基本模型以及它们的实现细节：1. 值函数模型：值函数是一种将策略函数作为输入，以获得一种近似状态价值的方式。
在实际实现中，可以通过构建一个神经网络模型来捕捉这种映射关系。
常用的神经网络模型包括多层感知器（MLP）和深度神经网络（DNN）。
2. 策略模型：策略模型用于捕捉策略的内在表示，即从给定状态中选择最优动作的概率分布。
常见的策略模型包括马尔可夫决策过程（MDP）和强化学习中的神经网络模型。
这些模型通常用于学习一个合适的动作选择概率分布，以便在给定状态下选择最优动作。
3. 深度Q学习（DQN）：深度Q学习是一种强化学习算法，通过使用深度神经网络来学习Q 函数，以优化目标函数。
该算法的核心思想是通过在经验回放过程中训练神经网络，使其能够预测未来的价值并优化选择最佳动作。
在实践中，深度Q学习算法通常采用Q-learning 和深度强化学习框架相结合的方式来实现。
4. 策略梯度方法：策略梯度方法是一种强化学习算法，通过利用策略梯度方法来更新策略模型，以提高算法的效率和鲁棒性。
策略梯度方法基于一种期望目标函数的假设，通过对当前策略下的Q函数进行微调，并通过最大化对数似然比率进行参数更新。
该方法可以在具有大规模数据的情况下快速收敛并得到较优的结果。
至于强化学习的实现细节，以下是几个需要考虑的因素：1. 目标函数的定义：强化学习中需要选择一个目标函数，以便根据观察到的环境状态和行为奖励进行学习。
常用的目标函数包括Bellman方程、价值函数和经验回放等。
选择一个合适的目标函数将直接影响算法的学习效果和性能。
2. 数据收集和处理：在强化学习中，需要通过经验回放或其他方法从环境中收集数据。
这些数据包括观察、状态、动作、奖励和时间步等。
为了获得高质量的数据，需要仔细考虑数据收集和处理的方法，以确保数据的质量和多样性。
3. 模型的训练和优化：在强化学习中，需要使用合适的优化算法来更新模型的参数。
凸优化问题的不定线性化Peaceman-Rachford分裂方法
凸优化问题的不定线性化Peaceman-Rachford分裂方法凸优化问题的不定线性化Peaceman-Rachford分裂方法随着科学技术的发展，凸优化在数学、计算机和工程领域中发挥着重要的作用。
凸优化问题是指在约束条件下寻找一组使目标函数最小化的变量。
为了解决这类问题，研究人员提出了各种各样的优化算法。
其中一种被广泛应用的算法是Peaceman-Rachford分裂方法。
Peaceman-Rachford分裂方法最初是为了解决线性偏微分方程的求解问题，但后来被扩展应用于各种凸优化问题。
该方法基于将复杂的凸优化问题分解为若干子问题，并通过迭代求解这些子问题来逐步靠近最优解。
其中的关键点是在每一步迭代中，使用线性化的方式来近似求解非线性问题。
不定线性化的思想是Peaceman-Rachford分裂方法的核心。
在传统的线性优化方法中，通常通过将非线性约束线性化来近似求解问题。
而不定线性化方法则通过迭代地将线性化的约束部分改变，来逼近非线性问题的最优解。
这种方法的优势在于可以更好地利用目标函数的凹凸性质，从而加速求解过程。
具体来说，不定线性化Peaceman-Rachford分裂方法的步骤如下：1. 将原始的凸优化问题转化为等价的子问题，这些子问题可以通过线性化的方式来求解。
2. 使用Peaceman-Rachford分裂方法对每个子问题进行迭代求解。
在每一步迭代中，将非线性部分线性化，并通过求解线性方程组的方式来更新变量。
3. 检查迭代过程中的收敛性，如果满足预设的收敛条件，则停止迭代；否则，返回第2步。
通过不定线性化Peaceman-Rachford分裂方法，可以更高效地求解复杂的凸优化问题。
该方法的优势在于能够同时考虑目标函数的凸性和凹性，从而避免了传统线性化方法可能带来的局部最优解问题。
此外，由于采用了分裂的思想，该方法能够充分利用问题的结构性质，进一步提高求解效率。
然而，不定线性化Peaceman-Rachford分裂方法也存在一些局限性。
张量融合 lmf python 代码-概述说明以及解释
张量融合lmf python 代码-概述说明以及解释1.引言1.1 概述概述部分的内容如下：在当前的数据分析和机器学习领域中，张量（tensor）是一种重要的数据结构，它是一个多维数组或矩阵的推广。
张量融合（tensor fusion）是指将多个张量结合在一起，以产生一个新的张量。
在本文中，我们将探讨张量融合的概念、方法以及如何使用Python代码来实现张量融合操作。
通过本文的学习，读者将能够深入了解张量融合的原理和应用，为今后的数据分析和机器学习工作提供更多的灵感和方法。
1.2 文章结构文章结构部分主要介绍了全文的章节安排和内容框架。
本文总共分为引言、正文和结论三个部分。
引言部分包括概述、文章结构和目的三个小节，主要介绍了本文的主题和写作背景。
正文部分分为张量概念、融合方法和Python代码实现三个小节，详细解释了张量的定义和应用、融合方法的原理和具体操作步骤，以及使用Python语言实现该功能的代码示例。
结论部分包括总结、应用前景和展望三个小节，对本文的研究成果进行了总结和展望，指出了该技术在未来的应用前景和发展方向。
通过这样的章节安排和内容框架，读者可以清晰地了解全文的主要内容和结构安排。
1.3 目的:本文的目的是介绍张量融合在机器学习和深度学习领域的应用，以及如何利用LMF（Low-rank Matrix Factorization）方法实现张量融合。
通过本文的讲解，读者可以了解张量的基本概念和融合方法，以及如何使用Python代码来实现张量融合。
同时，本文还将探讨张量融合的应用前景和未来发展趋势，希望能够为相关领域的研究者和开发者提供一些参考和启发。
2.正文2.1 张量概念在数学和物理学中，张量是一个具有多个分量的对象，它在不同坐标系下具有相同形式。
张量可以描述物理量的大小和方向，并在各种物理学和工程学领域中发挥重要作用。
张量可以分为多种类型，如标量、向量、矩阵和高阶张量等。
标量是零阶张量，表示一个单一数值，如温度或密度。
python 零膨胀泊松模型
python 零膨胀泊松模型Python（简称为Py）是一种高级编程语言，强调代码的可读性和简洁性。
它广泛应用于各个领域，包括科学计算、数据分析、人工智能等。
其中，零膨胀泊松模型（Zero Inflated Poisson Model）是一种常用的统计模型，用于描述零膨胀数据。
一、概述零膨胀泊松模型是一种混合模型，用于处理存在过度零计数问题的数据。
过度零计数是指在实际观测中，某些事件以较高的概率发生为零次。
传统的泊松模型无法处理这种情况，而零膨胀泊松模型则能对此进行建模。
二、模型原理零膨胀泊松模型由两部分组成：一个二项分布用于建模零的概率，一个泊松分布用于建模非零计数的概率。
具体而言，模型可以表示为：P(X=0) = θ + (1-θ)e^(-λ)P(X=k) = (1-θ) * (e^(-λ) * λ^k) / k!其中，X表示计数变量的取值，θ代表零计数的概率，λ代表非零计数的平均值。
三、Python实现在Python中，可以使用statsmodels和pymc3等库来实现零膨胀泊松模型。
以statsmodels为例，首先需要导入相关的库：import statsmodels.api as sm然后，可以使用ZeroInflatedPoisson函数创建零膨胀泊松模型的对象：model = sm.ZeroInflatedPoisson(y, X)其中，y表示观测到的计数数据，X表示自变量。
接下来，可以使用fit函数拟合模型：result = model.fit()最后，可以使用summary函数查看模型的拟合结果：print(result.summary())四、应用示例以下是一个简单的零膨胀泊松模型的应用示例：import pandas as pdimport statsmodels.api as sm# 创建示例数据data = {'count': [0, 1, 2, 0, 3, 0, 0, 4, 0, 0, 1, 2],'income': [1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 11000, 12000]}df = pd.DataFrame(data)# 划分自变量和因变量y = df['count']X = df['income']# 创建模型对象model = sm.ZeroInflatedPoisson(y, X)# 拟合模型result = model.fit()# 查看拟合结果print(result.summary())通过以上代码，我们可以得到零膨胀泊松模型的拟合结果。
非光滑非凸向量极值问题的真有效解
非光滑非凸向量极值问题的真有效解
刘三阳;游兆永
【期刊名称】《应用数学》
【年(卷),期】1990(3)4
【摘要】本文考虑非光滑非凸向量极值问题的真有效解,其主要结果如
下:(1)Borwein真有效解与Benson真有效解的等价性;(2)向量极值问题的真有效解与标量极值问题的最优解的等价性;(3)广义鞍点定理;(4)真有效解的必要和充分条件。
【总页数】6页(P34-39)
【关键词】向量极值问题;真有效解;最优解
【作者】刘三阳;游兆永
【作者单位】西安电子科技大学;西安交通大学
【正文语种】中文
【中图分类】O221
【相关文献】
1.非光滑向量极值问题的真有效解与最优性条件 [J], 陈修素
2.非凸非光滑向量优化弱有效解的充分条件 [J], 臧睿
3.非光滑向量极值问题的广义真有效解 [J], 肖成河;陈修素;隋如滨;李允
4.半序线性空间中的非凸非光滑向量极值问题 [J], 刘三阳
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ORTHONORMAL REALIZATION OF FAST FIXED-ORDER RLS ADAPTIVE FILTERSRicardo Merched and Ali H.SayedAdaptive Systems LaboratoryDepartment of Electrical EngineeringUniversity of CaliforniaLos Angeles,CA90095ABSTRACTThe existing derivations of fast RLS adaptivefilters are dependent on the shift structure in the input regression vectors.This structure arises when a tapped-delay line(FIR)filter is used as a model-ingfilter.In this paper,we show,unlike what original derivations may suggest,that fastfixed-order RLS adaptive algorithms are not limited to FIRfilter structures.We show that fast recursions in both explicit and array forms exist for more general data struc-tures,such as orthonormally-based models.One of the benefits of working with an orthonormal basis is that fewer parameters can be used to model long impulse responses.1.INTRODUCTIONFast RLS adaptivefiltering algorithms represent an attrac-tive way to compute the least squares solution of grow-ing length data efficiently,in computations per sam-ple,where is thefilter order.The low complexity that is achieved by these algorithms is a direct consequence of the shift structure that is characteristic of regression vec-tors in FIR adaptive implementations.Recently,the authors showed that the input data structure that arises from more general networks,such as Laguerrefilters,can be exploited to derive fast order-recursive[1]andfixed-orderfilters[2,3] as well.In this paper,we show that fastfixed-order RLS adap-tive algorithms can also be derived for general orthonormal bases(see e.g.,[4,5])in both explicit and array forms.2.THE EXTENDED FAST TRANSVERSAL FILTER Given a column vector C,the exponentially-weighted least squares prob-lem seeks the column vector CThis material was based on work supported in part by the National Science Foundation under awards CCR-9732376and ECS-9820765,and UC Core project CR98-19.The work of R.Merched was also supported by a fellowship from CAPES,Brazil.The matrix is a positive-definite regularization matrix, and.The symbol denotes complex conjugate transposition.The individual entries of will be denoted by,and the individual rows of the matrix will be denoted by.The RLS algorithm computes the optimal solution of problem(1)recursively as follows:(2)(3)(4)(5)with and.When the regression vec-tors possess shift structure,it is well known that these recur-sions can be replaced by more efficient ones.Now,consider the generalized orthonormal network of Fig.1with transfer function(the case of equal poles,which corresponds to a Laguerre network,is treated in[2]):where is the()()matrix(e.g.,for)(8) where we defined(10) where is the solution to the least-squares problemwhose minimum cost we denote by.It holds that.Now,the following equations constitute the update of the quantities of this problem:(11)(14) where is the normalized gain vector, and are the a priori and a posteriori for-ward prediction errors,related via. Note that no information on the data structure is needed in order to derive these equations(see[1]).2.2.Backward Estimation ProblemSimilarly to the forward estimation problem,assume that one more column is appended to from the right,i.e.,(15) and define the corresponding coefficient matrix aswhere(16)for some constant vector and scalar to be specified.In-verting both sides,we obtain:where.The quantityis referred to as the rescue variable and can be directly ob-tained as the last entry of(to be computed further ahead).Proceeding similarly to the derivation of(14),we obtainNote that the variables and play roles similar to the a priori and a posteriori backward prediction problems.However,although all the quantities related to the backward prediction problems satisfy identical recursive equations,here they have different interpretations.2.3.Exploiting Data StructureWe still need to evaluate.For this purpose,we need to identify the variable that is affected by the input data struc-ture.Thus,consider any invertible matrix such as in(8). From Eq.(7),it follows thatwhere and are the corresponding aug-mented input data matrices.We then getNote that if we could choose(22) we obtain a simpler relation between:(23)In order for this relation to hold,we need to show how to choose,,and in order to satisfy(22).Substituting (16)into(22),we get(24) Now,the matrix can be defined block-wise aswhereandis a small positive number;is the solution to(25);c is given by(26).:Table1:The extended fast transversalfilter for orthonor-mal bases.Expanding(24),wefind that(25) Hence,ifThis relation shows that the time update of the gain vec-tor,which is necessary to update the optimal solution can be efficiently performed in three main steps:(1)Or-der update;(2)Time-update;(3)Order downdate[i.e, Eq.(12),(27)and(21)].Table1shows the resulting gener-alized FTF algorithm.Note that when,we have and therefore,in which case the recursions col-lapse to the FTF algorithm[6].Equation(27)is the only recursion that uses the fact that the input data has structure. For the orthonormal basis considered here,this multiplica-tion is essentially a convolution,and can be performed with operations.The cost of the usual FIR FTF algorithm is known to be operations[6].The overall cost for our extendedfilter is operations.3.THE EXTENDED FAST ARRAY ALGORITHM Using the expressions for)in(10) and(17)(and ignoring the order index),the FTF al-gorithm can be further motivated in a different manner by noticing that its recursions perform at each iteration the low rank factorization(28) where is an signature matrix andMore generally,it can be shown(see,e.g.[7,3])that by forcing the initial difference to have low rank,and a certain inertia,we end up forcing all the successive differences to have a similar property.This fact is the basis for the exis-tence of a fast recursion that does not necessarily propagate the difference explicitly.That is,we need to find a matrix such that the difference has low rank.Expanding this difference,we obtain(29)where and.The matrix that results from solving(25),under the condition.Thus consider the matrix difference in Eq.(29).We proceed tofind a positive definite matrix such that this difference has a rank1matrix factorization of the form(30) Again,from the properties of Lyapunov equations,we know that this equation admits a unique Hermitian solution,since all the eigenvalues of are equal to or.Moreover, since。