Deriving CDF of Kolmogorov-Smirnov Test Statistic

In this review article, we revisit derivation of the cumulative density function (CDF) of the test statistic of the one-sample Kolmogorov-Smirnov test. Even though several such proofs already exist, they often leave out essential details necessary for proper understanding of the individual steps. Our goal is filling in these gaps, to make our presentation accessible to advanced undergraduates. We also propose a simple formula capable of approximating the exact distribution to a sufficient accuracy for any practical sample size.

Kolmogorov-Smirnov APF Test for Inhomogeneous Poisson Processes with Shift Parameter

In this article, we study the Kolmogorov-Smirnov type goodness-of-fit test for the inhomogeneous Poisson process with the unknown translation parameter as multidimensional parameter. The basic hypothesis and the alternative are composite and carry to the intensity measure of inhomogeneous Poisson process and the intensity function is regular. For this model of shift parameter, we propose test which is asymptotically partially distribution free and consistent. We show that under null hypothesis the limit distribution of this statistic does not depend on unknown parameter.

Goodness-of-Fit Test for Non-Stationary and Strongly Dependent Samples

In this article we improve a goodness-of-fit test, of the Kolmogorov-Smirnov type, for equally distributed- but not stationary-strongly dependent data. The test is based on the asymptotic behavior of the empirical process, which is much more complex than in the classical case. Applications to simulated data and discussion of the obtained results are provided. This is, to the best of our knowledge, the first result providing a general goodness of fit test for non-weakly dependent data.

异形大跨指廊屋盖脉动风压非高斯特性研究

基于刚性模型测压风洞试验,研究异形大跨屋盖的平均风压分布及脉动风压非高斯特征,分别采用高阶统计量法和柯尔莫哥洛夫-斯米尔诺夫(Kolmogorov-Smirnov,K-S)检验法对脉动风压的非高斯特性进行判别并划分其分布区域,最后通过5种概率分布模型对非高斯脉动风压的概率分布进行拟合。结果表明:屋盖表面平均风压以负压分布为主,且负压极值多出现于屋脊或屋面转角的气流分离区;异形大跨屋盖脉动风压的非高斯特性显著,非高斯脉动风压主要出现在屋面迎风前缘、指廊背风侧和屋面转角处,相较于高阶统计量法,K-S检验法的划分结果集中度更好,规律性更强;通过相关系数和均方根误差评价拟合效果,5种概率分布模型中的Weibull模型对非高斯脉动风压概率分布的拟合效果最佳。

A Harmful-Intrusion Detection Method Based on Background Reconstruction and Two-Dimensional K-S Test in an Optical Fiber Pre-Warning System

The key technology and main difficulty for optical fiber intrusion pre-warning systems （OFIPS） is the extraction of harmful-intrusion signals. After being processed by a phase-sensitive optical time-domain reflectometer （O-0TDR）, vibration signals can be preliminarily extracted. Generally, these include noises and intrusions. Here, intrusions can be divided into harmful and harmless intrusions. With respect to the close study of signal characteristics, an effective extraction method of harmful intrusion is proposed in the paper. Firstly, in the part of the background reconstruction, all intrusion signals are first detected by a constant false alarm rate （CFAR）. We then reconstruct the backgrounds by extracting two-part information of alarm points, time and amplitude. This ensures that the detection background consists of intrusion signals. Secondly, in the part of the two-dimensional Kolmogorov-Smirnov （K-S） test, in order to extract harmful ones from all extracted intrusions, we design a separation method. It is based on the signal characteristics of harmful intrusion, which are shorter time interval and higher amplitude. In the actual OFIPS, the detection method is used in some typical scenes, which includes a lot of harmless intrusions, for example construction sites and busy roads. Results show that we can effectively extract harmful intrusions.

Gumbel分布的基本风压计算与分析

统计了中国159个代表性城市在1951—2008年的历年最大风速值,采用Gumbel分布进行统计分析并对其参数进行估算。用矩法和耿贝尔法分别计算出重现期为10、50、100a的基本风压值,并采用柯尔莫哥洛夫检验法进行检验,最后与规范取值进行对比分析。结果表明,当采用Gumbel分布对中国各地区的最大风速年极值进行统计时,耿贝尔法比矩法拟合效果好;重现期为10、50、100a的基本风压值对比分析表明,《建筑结构荷载规范》(GB 50009)的基本风压值与采用近58a的历年最大风速值计算的基本分压值相比存在较大差异,表明规范计算的采样数据少,取值已经不能真实反映中国的基本风压情况,建议进行相应修订。

SVD滤波的串联故障电弧特征分析

为进一步研究故障电弧特征,针对三相电动机及变频器负载开展了串联故障电弧实验。首先将电流信号经过一阶差分预处理,再通过奇异值分解SVD(singular value decomposition)对信号进行两级滤波,剔除信号中的工频和噪声成分。采用柯尔莫可洛夫-斯米洛夫K-S(Kolmogorov-Smirnov)检验法分析SVD滤波信号的正态分布情况。采用线性调频Z变换CZT(chirp-Z transform)对SVD滤波信号0~500 Hz频段进行频谱细化分析。提取时域峭度和特征频段幅值平均值组成特征向量,并构建故障电弧区矩形。通过大量数据测试表明:该方法可有效识别三相电动机及变频器负载回路中发生的故障电弧。

基于站点降雨量最优拟合函数的SPI指数计算

标准化降水指数（SPI）能较好地反映不同时间尺度和区域的旱涝状况,时空适应性较强,成为科学研究和实际应用最广泛的描述多时间尺度的气象干旱指标,采用不同的概率分布函数拟合降雨量系列,将会得到不同的SPI值。利用淮河流域30个气象站点1951-2013年间约60 a的逐月降雨资料,分别采用7种不同分布函数（Gamma分布、皮尔逊III型分布、正态分布、对数正态分布、广义Pareto分布、广义Logistic分布和Weibull分布）对各站点不同时间尺度（1、3、6、12和24 mon）各月份降雨系列进行拟合,以K-S方法检验了各分布函数的拟合效果,并以均方根误差（RMSE）最小为准则选取的站点各月份最优分布函数OPT作为参考,分析比较上述7种分布函数模拟的不同时间尺度下的SPI时间序列。结果表明：在各不同时间尺度上,采用皮尔逊III型分布作为计算SPI指数的分布函数误差最小,其模拟的SPI值各站点总偏差平均值均在5%以下,Weibull分布和对数正态分布计算得到的SPI时间系列结果次之,它们模拟的SPI值各站点总偏差平均值介于5%-10%之间;当前我国气象部门统一采用的Gamma分布函数,并非计算SPI指数的最佳选择,各不同时间尺度下,采用Gamma分布计算的SPI值各站点总偏差平均值均在10%左右。其他函数,如正态分布、广义Pareto分布和广义Logistic分布,其误差介于10%-30%之间,均超过采用Gamma分布计算的结果。

PERCENTAGE POINTS AND POWER OF AK-S TYPE TEST FOR LINEARITY IN AUTOREGRESSIVE TIME SERIES

The empirical upper percentage points of the null distribution of a Kolmogorov-Smirnov type test for checking linearity in autoregressive models are tabulated in this paper, and the good power property of the test is demonstrated.

频率选择信道下的空时分组码盲识别

在单接收天线下,针对频率选择性衰落信道下空时分组码(STBC)的盲识别问题,提出了一种基于Kolmogorov-Smirnov(K-S)检测的有效算法。该算法以经验累积分布函数作为特征函数,通过K-S检测经验累积分布函数之间的距离,达到识别空时分组码的目的。在不同调制方式、采样因子和置信区间的条件下分别对算法进行仿真并讨论其性能,结果表明,该算法性能较好,在信噪比大于6 d B时可达到90%以上的正确识别概率,在非合作通信方面具有一定的实用价值。

基于BB算法的左截断威布尔分布可靠性分析

针对寿命分布类型和参数估计方法问题,对国内某航空公司现役机型B737-800飞机主起落架减振支柱漏油故障数据进行可靠性分析。结合数据类型特点,初步预判其寿命分布类型可能为左截断两参数威布尔分布、对数正态分布、指数分布。基于极大似然估计法分别估计三种分布的未知参数,并利用根据BB算法改进的柯尔莫哥洛夫-斯米尔诺夫假设检验(K-S检验)法判断其寿命分布类型。实证分析结果表明,采用左截断两参数威布尔分布来拟合是合适的,并提出了具体维修建议。

The Catch Distribution of Ommastrephes batramii in Squid Jigging Fishery and the Relationship between Fishing Ground and SST in the North Pacific Ocean in 2004

Neon flying squid Ommastrephes batramii is widely distributed in the North Pacific Ocean, which has become the main fishing species for Chinese squid jigging fleets since 1993. Many authors have made the studies on the fields of fishing ground and its environment conditions. However, the squid catch per fishing vessel attained the highest level of about 550 t in 2004. In this paper, the catch and its distribution in 2004 would be compared with the previous year. Based on the catch data from Chinese squid jigging vessels and sea surface temperature with the format of 1 °latitude by 1 °longitude from May to November in 2004, the distribution maps were drawn by Marine explorer 4.0. The results show that the production in the east waters to 160°E was low during May and July. During October and November, the production in the waters from 150°E to 160°E was relatively higher, which occupied 62.5 percent of the total catch. During November, the production in the west waters to 150°E was also low. The highest CPUE area located in the west waters to 150°E, the next was the area from 150°E to 160°E and the lowest CPUE area located in the east waters to 160°E. The SST in the fishing ground seems to change seasonally. The suitable SST for each month is as follows： 12-14 ℃ in May, 15 ℃ - 16 ℃ in June, 14 ℃ - 16 ℃ in July, 18 ℃ - 19 ℃ in August, 16 ℃ -17 ℃ in September, 15 ℃- 16 ℃ in October and 12 ℃ - 13 ℃ in November. The result of K-S test shows that the above monthly suitable SST is considered as the indicator of looking for the main fishing ground.

基于威布尔分布的谷物干燥机可靠性分析

为了提高谷物干燥机的使用寿命,减少其故障的维修次数,运用可靠性分析理论对其使用寿命进行了分析研究。首先,对某型号谷物干燥机的维修数据进行了分析整理,根据维修记录得到的故障间隔时间拟合出了威布尔概率(WPP)图,基于图形法初步确定了其服从二参数威布尔分布;然后,利用点估计的优良性判别准则,采用粒子群算法求解似然函数得出了估计值,将其作为威布尔分布模型的参数;最后,采用K-S检验法对威布尔分布模型进行了检验,并在此基础上求得了谷物干燥机故障间隔时间的概率密度函数、概率分布函数和可靠度函数。研究结果表明:谷物干燥机的故障间隔时间服从二参数威布尔分布,并计算得到干燥机的平均故障间隔时间为249.8 h,且在可靠度为0.9时,预防性维修周期为137.6 h;该方法具有较高的精确度和易用性,对企业制定合理的维修决策具有一定的参考价值。

一种裂隙岩体力学参数的评价方法及应用

针对裂隙岩体的力学参数选取问题,提出了一种两阶段评价方法.通过裂隙岩体力学参数的三维空间分布情况,评价其各向同性与各向异性状态,对两类情况采用不同的计算模型处理;采用Kolmogorov-Smirnov(K-S)检验法对力学参数的概率分布模型进行假设检验,对检验合格的试验数据分布进行置信区间选取,在给定的置信区间内获取计算参数.在某地下硐室工程中运用该方法进行了开挖模拟计算,计算位移与实测位移吻合较好.

我国上市公司资本结构与行业关系的实证分析

本文利用One-SampleKolmogorov-SmirnovTest对我国上市公司资本结构与行业关系进行了实证分析,结果发现:我国上市公司资产负债率近似服从正态分布,不同行业各总体的方差无显著性差异,不同行业上市公司资本结构存在显著性差异且在1999-2002年中都保持了相当的稳定性,波动较小。

Spatial and temporal variability of daily precipitation in Haihe River basin, 1958-2007

The seasonal variability and spatial distribution of precipitation are the main cause of flood and drought events. The study of spatial distribution and temporal trend of precipitation in river basins has been paid more and more attention. However, in China, the precipitation data are measured by weather stations （WS） of China Meteorological Administration and hydrological rain gauges （RG） of national and local hydrology bureau. The WS data usually have long record with fewer stations, while the RG data usually have short record with more stations. The consistency and correlation of these two data sets have not been well understood. In this paper, the precipitation data from 30 weather stations for 1958-2007 and 248 rain gauges for 1995-2004 in the Haihe River basin are examined and compared using linear regression, 5-year moving average, Mann-Kendall trend analysis, Kolmogorov-Smirnov test, Z test and F test methods. The results show that the annual precipitation from both WS and RG records are normally distributed with minor difference in the mean value and variance. It is statistically feasible to extend the precipitation of RG by WS data sets. Using the extended precipitation data, the detailed spatial distribution of the annual and seasonal precipitation amounts as well as their temporal trends are calculated and mapped. The various distribution maps produced in the study show that for the whole basin the precipitation of 1958-2007 has been decreasing except for spring season. The decline trend is significant in summer, and this trend is stronger after the 1980s. The annual and seasonal precipitation amounts and changing trends are different in different regions and seasons. The precipitation is decreasing from south to north, from coastal zone to inland area.

Parameter Estimations for Some Modifications of the Weibull Distribution

Proposed by the Swedish engineer and mathematician Ernst Hjalmar Waloddi Weibull (1887-1979), the Weibull distribution is a probability distribution that is widely used to model lifetime data. Because of its flexibility, some modifications of the Weibull distribution have been made from several researches in order to best adjust the non-monotonic shapes. This paper gives a study on the performance of two specific modifications of the Weibull distribution which are the exponentiated Weibull distribution and the additive Weibull distribution.

Analysis of inhibition of concrete steel-rebar corrosion by Na_2Cr_2O_7 concentrations:Implications for conflicting reports on inhibitor effectiveness

Corrosion test data were measured using non-destructive electrochemical techniques and analysed for studying inhibition effectiveness by different concentrations of NazCr207 on the corrosion of concrete steel-rehar in NaC1 and in H2SO4 media. For these, specifications of ASTM G16-95 R04 were combined with the normal and the Gumbel probability density functions as model analytical methods for addressing issues of conflicting reports of inhibitor effectiveness that had generated concerns. Results show that reinforced concrete samples admixed with concentrations having 4 g （0.012 7 tool）, 8 g （0.025 4 mol） and 6 g （0.019 l tool） NaaCr207 exhibited, in that order, high inhibition effectiveness, with respective efficiency, r/, of （90.46±1.30）%, （88.41＋2.24）% and （84.87±4.74）%, in the NaC1 medium. These exhibit good agreements within replicates and statistical methods for the samples. Also, optimal inhibition effectiveness model in the H2SO4 medium was exhibited by 8 g （0.025 4 mol） Na2Cr207 concentration having r/=（78.44±1.10）%. These bear implications for addressing conflicting test data in the study of effective inhibitors for mitigating steel-rebar corrosion in aggressive environments.

Power Law for the Rates of Different Numbers of Chronic Diseases among Elderly Chinese People

Chronic disease is an important factor that affect the health of elderly people. We analyzed the 2006 and 2010 data from the Chinese Urban and Rural Elderly Population Surveys, which are nationally representative surveys of elderly people aged 60 years and above. We found that there existed a typical power-law distribution for the rates of different numbers of chronic diseases among elderly Chinese people. A Kolmogorov-Smirnov test indicated that the result was robust, and the power exponents were approximately ?2.5. In addition, a paired t-test was conducted, which demonstrated that the rates of different numbers of chronic diseases did not have significant urban-rural differences, time differences or gender differences.

RAYLEIGH-DISTRIBUTION BASED MINIMUM ERROR THRESHOLDING FOR SAR IMAGES

This paper presents a minimum error thresholding (MET) algorithm under the hypothesis that the gray level histogram of SAR image fits to a mixture model of shifted Rayleigh distribution. This algorithm is applied to real SAR images and compared with traditional Otsu algorithm and other MET algorithms based on various models of histogram. The hypothesis of using Rayleigh distribution model is confirmed by Kolmogorov-Smirnov testing and the comparison results obtained show that the proposed new algorithm has good performance in thresholding SAR images.