密度估计函数的收敛速度及重对数率

设ｆＦ为（－∞，∞）上的一族概率密度，ｘ１，ｘ２，…，ｘｎ为取自ｆ的样本。记Ｊｎｉ＝（（ｉ－１）ｈｎ，ｉｈｎ），ｈｎ∞（ｎ→∞），又记Ｒｉ＝＃｛ｔ：ｔ＝１，２，…，ｎ｝，当ｘＪｎｉ时，讨论了ｆ（ｘ）的密度估计函数。并且在Ｌｉｐｓｈｉｔｚ条件下研究了密度估计函数ｆｎ（ｘ）的渐近正态性，最佳可能收敛速度和一致收敛的重对数率。当０＜α＜１，β＜１－α２时，ｆｎ（ｘ）－ｆ（ｘ）＝Ｏ（ｌｎｎｎ－β）ａ．ｓ．；当－１４＜α＜１２时，ｓｕｐｘ｜ｆｎ（ｘ）－ｆ（ｘ）｜＝Ｏ（ｎα－１２ｌｎｌｎｎ）ａ．ｓ等．

高斯核密度估计背景建模及噪声与阴影抑制

提出了一种多模态非参数背景模型,用于背景减方法检测运动目标。针对户外监控系统存在背景局部运动以及摄像机抖动、活动阴影等问题,利用像素邻域相关性信息进行多模态高斯核密度估计,并采用HMMD色彩值抑制阴影。通过抖动噪声去除以及阴影抑制处理,降低了目标检测的虚警率。实验结果表明该算法在运动目标检测中具有对噪声和阴影的鲁棒性,可用于户外复杂场景监控系统。

湖北省财政性科研投入效率测度及时空演化

文章利用SBM模型测度了2011—2020年湖北省12个地级市财政性科研投入产出综合效率、纯技术效率和规模效率的变化趋势,借用莫兰指数与Kernel核密度估计函数刻画了湖北省财政性科研投入产出效率的时空动态演化特征。研究发现:从整体看,湖北省地级市财政性科研产出效率整体较高,且呈增长趋势,纯技术效率波动与综合效率升降趋势较为同步,区域效率差异主要源于规模效率;在时序上,地级市相继由低值区向高值区跃迁,处于低值区的地级市数量在减少,处于高值区的地级市数量在增加,技术进步与技术扩散在加快;在空间上,高效率区域和低效率区域间存在趋同效应,具有明显的区块分布特征,高值区域可以拉动低值区域的绩效水平,区域间的差距呈现缩小的趋势。

中国式现代化的统计测度及推进路径

本文构建由社会服务现代化、经济发展现代化、技术创新现代化、文化建设现代化、绿色发展现代化、政治治理现代化6个维度组成的指标体系,采用熵值法、Dagum基尼系数、Kernel密度估计函数和空间Markov链方法,对中国式现代化发展水平进行测度。研究结果表明:目前中国式现代化水平不高,但整体呈现良好上升趋势,其中绿色发展现代化水平相对较低;东部地区中国式现代化综合水平明显高于中部和西部地区;中国式现代化水平区域差异主要来源于区域间差异,且这一差异呈现扩大趋势;中国式现代化水平具有俱乐部收敛特点,发生状态转移概率较小。鉴于此,提出构建绿色现代化发展体系、建立区域差异化发展机制、释放中国式现代化高水平地区辐射带动效应的推进路径,希冀为中国式现代化发展提供有益借鉴。

长江经济带第二产业绿色技术创新效率测度及时空演变

采用SBM的Max-min-DEA模型以及Kernel密度估计函数,对2001—2018年长江经济带沿线11省市第二产业绿色技术创新效率进行测度,并分析其时空差异规律。结果表明,长江经济带沿线11省市第二产业绿色技术创新效率较高,但仍有上升空间,需做好节能减排工作,以提升效率。在观察期内,效率整体呈上升趋势,但长江经济带子地区间和各省市间第二产业绿色技术创新效率的演变规律和趋势存在明显的时空差异。

Calculation of One-Valued Control Limits by Control Chart of Angles

The data we use to express angle or direction are entitled directional data. In a plan right angled coordinate system the traditional control chart can’t solve the quality control problem which the characteristic value is angle. This paper analyses and calculates the one valued control limits by control chart of angles.

Leak Detection in Water Distribution Systems Using Bayesian Theory and Fisher’s Law

A leak detection method based on Bayesian theory and Fisher’s law was developed for water distribution systems. A hydraulic model was associated with the parameters of leaks (location, extent). The randomness of parameter values was quantified by probability density function and updated by Bayesian theory. Values of the parameters were estimated based on Fisher’s law. The amount of leaks was estimated by back propagation neural network. Based on flow characteristics in water distribution systems, the location of leaks can be estimated. The effectiveness of the proposed method was illustrated by simulated leak data of node pressure head and flow rate of pipelines in a test pipe network, and the leaks were spotted accurately and renovated on time.

An efficient approach for shadow detection based on Gaussian mixture model

An efficient approach was proposed for discriminating shadows from moving objects. In the background subtraction stage, moving objects were extracted. Then, the initial classification for moving shadow pixels and foreground object pixels was performed by using color invariant features. In the shadow model learning stage, instead of a single Gaussian distribution, it was assumed that the density function computed on the values of chromaticity difference or bright difference, can be modeled as a mixture of Gaussian consisting of two density functions. Meanwhile, the Gaussian parameter estimation was performed by using EM algorithm. The estimates were used to obtain shadow mask according to two constraints. Finally, experiments were carried out. The visual experiment results confirm the effectiveness of proposed method. Quantitative results in terms of the shadow detection rate and the shadow discrimination rate(the maximum values are 85.79% and 97.56%, respectively) show that the proposed approach achieves a satisfying result with post-processing step.

Probability density function and estimation for error of digitized map coordinates in GIS

Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in GIS does not obey the normal distribution but the p-norm distribution with a determinate parameter. Assuming that the error is random and has the same statistical properties, the probability density function of the normal distribution, Laplace distribution and p-norm distribution are derived based on the arithmetic mean axiom, median axiom and p-median axiom, which means that the normal distribution is only one of these distributions but not the least one. Based on this ideal distribution fitness tests such as Skewness and Kurtosis coefficient test, Pearson chi-square chi(2) test and Kolmogorov test for digitized data are conducted. The results show that the error in map digitization obeys the p-norm distribution whose parameter is close to 1.60. A least p-norm estimation and the least square estimation of digitized data are further analyzed, showing that the least p-norm adjustment is better than the least square adjustment for digitized data processing in GIS.

Fatigue Crack Initiation Sizing Using Acoustic Emission

One of the major concerns in structures is early detection of a growing crack to prevent fracture, predict remaining useful life, schedule maintenance and reduce costly downtimes. AE （acoustic emission） is a non-destructive testing method with potential applications for locating and monitoring fatigue cracks. This paper focuses on in-situ monitoring of structural health, specifically detection of small crack growth and crack initiation in structures using AE technology. A probabilistic AE-based model for small fatigue cracks was developed and the uncertainties of the model were estimated. The paper discusses the methodology used, experimental approach, results obtained and predictive models developed.The developed model can be used to evaluate the integrity of structures and assess structural health by estimating the probability density function of the length of detected cracks. The outcome of this research has significant potential to be used for in-situ monitoring and evaluation of structural integrity.

A New Approach to Estimating the T-Year Return-Period Wave Height

The paper introduces a new approach to estimating the T-year return-period wave height (TRPW), i.e. the wave height expected to occur in T-year, from two sets of observed extreme data and on the basis of the maximum entropy principle. The main points of the approach are as follows. 1) A maximum entropy probability density function (PDF) for the extreme wave height H is derived from a Euler equation subject to some necessary and rational constraints. 2) The parameters in the function are expressed in terms of the mth moment of H. 3) This PDF is convenient to theoretical and practical applications as it is simple and its four parameters are easy to be determined from observed extreme data. An example is given for estimating the TRPW in 50 and 100 years by the present approach and by some currently used methods using observed data at two hydrographic stations.The comparison of the estimated results shows that the present approach is quite similar to the Pearson-Ⅲ and Gumbel methods.

Parameters Estimation and Evaluation for the Probability Density Function of Structural Fatigue Stress of a Container Vessel

Fatigue stress measurement has been playing a significant role in the ship structural health monitoring and ship structural safety assessment. The PDF （probability density function） of the measured stress is one of the essentials for the further studyin this domain. This paper, based on the strain-stress data collected from a container ship, focuses on the spectrum feature of the ship structural fatigue stress. A general analysis procedure for ship hull health estimation was firstly demonstrated. With the guidance of this procedure, the estimation and test of the parameters for the PDF of the stress were conducted, which showed that the stress spectrums fit well with the Weibull distribution. To review the fatigue state, the PDF and distribution function of fatigue damage increment were further developed and examined. The structural healthy assessment of this vessel shown the daily relative fatigue damage increment obeys log-normal or Weibull distribution and the increment of the fatigue damage on steel box girders of the ship hull was very low. Finally, the analyzing results yielded that the girder structure of the ship hull had a very low failure probability, matching well with the actual relative low working load of the ship.

Strong Consistency for the Kernal Estimates of the Random Window Width of the Density Function and its Derivatives Under Φ-Mixing Samples

In the paper,we study the strong uniform consistency for the kernal estimates of random window w■th of density function and its derivatives under the condition that the sequence{X_n}of the ■ are the identically Φ-mixing random variabks.

Study on Probability Distributions of Multi-Timescale Aerosol Optical Depth Using AERONET Data

The probability distribution analysis is per-formed for multi-timescale aerosol optical depth (AOD) using AErosol RObotic NETwork (AERONET) level 2.0 data.The maximum likelihood estimation is employed to determine the best-fit probability density function (PDF),and the statement that the fitting Weibull distribution will be light-tailed is proved true for these AOD samples.The best-fit PDF results for multi-site data show that the PDF of AOD samples with longer timescale in most sites tends to be stably represented by lognormal distribution,while Weibull distribution is a better fit for AOD samples with short timescales.The reason for this difference is ana-lyzed through tail characteristics of the two distributions,and an indicator for the selection between Weibull and lognormal distributions is suggested and validated.The result of this research is helpful for determining the most accurate AOD statistics for a given site and a given time-scale and for validating the retrieved AOD through its PDF.

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.

Supersmooth density estimations over L^p risk by wavelets

This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of Lprisk(1 p < ∞) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over Lprisks for p 2, and a nearly-optimal result for 1 < p < 2 under both noises. In the nearly-optimal cases, the ratios of upper and lower bounds are determined. When p = 1, we give a nearly-optimal estimation with moderately ill-posed noise by using the Meyer wavelet. Finally, the practical estimators are considered. Our results are motivated by the work of Pensky and Vidakovic(1999), Butucea and Tsybakov(2008), Comte et al.(2006), Lacour(2006) and Lounici and Nickl(2011).

Asymptotic properties of plug-in level set estimators for right censored data

We assume T1,..., Tn are i.i.d. data sampled from distribution function F with density function f and C1,...,Cn are i.i.d. data sampled from distribution function G. Observed data consists of pairs （Xi, δi）, em= 1,..., n, where Xi = min{Ti,Ci}, δi = I（Ti 6 Ci）, I（A） denotes the indicator function of the set A. Based on the right censored data {Xi, δi}, em=1,..., n, we consider the problem of estimating the level set {f 〉 c} of an unknown one-dimensional density function f and study the asymptotic behavior of the plug-in level set estimators. Under some regularity conditions, we establish the asymptotic normality and the exact convergence rate of the λg-measure of the symmetric difference between the level set {f ≥ c} and its plug-in estimator {fn ≥ c}, where f is the density function of F, and fn is a kernel-type density estimator of f. Simulation studies demonstrate that the proposed method is feasible. Illustration with a real data example is also provided.