The optimal number of people in each group of nucleic acid detection is mathematically determined in this paper. The number is related to the proportion of healthy people, whose nucleic acid test results are negative. When the proportion is not larger than 0.6922, person-by-person detection is the best solution. Otherwise, it is more appropriate to adopt a hybrid detection scheme. The optimal number of people in each group and the proportion of healthy people can be expressed by a parameter t perfectly. Some useful and interesting results are presented by the numerical analysis results. The determination of the optimal number of people in the mixed detection has certain practical significance for improving the detection efficiency, saving cost and controlling the spread of the epidemic as soon as possible. The optimal grouping number method given in this paper can be applied to mixed detection of other viruses and is thus of certain universality.

It has a good outlier detection effect and satellite clock error prediction accuracy based on the time series model to eliminate the influence of AO (Additive Outlier) and carry out the satellite clock error prediction. However, there are difference, inverse difference and model order determination operations when a time series model is used. In this paper, the AR model with trend item is combined with the EM algorithm, and an algorithm for AO detection and satellite clock error prediction are proposed to avoid difference, inverse difference and model order determination operations. When the AO in the clock error is successfully detected, the algorithm can obtain an accurate AR model with trend item, and then fit the growth or decline trend of the satellite clock error accurately. The algorithm also has good satellite clock error prediction accuracy. Finally, using the measured data of the BDS satellite clock error to calculate and analyze, the results verify the correctness and effectiveness of the algorithm.

The precise satellite clock bias prediction is critical in improving the positioning, navigation and timing (PNT) service capabilities of the global navigation satellite system (GNSS). Due to the influence of satellite signal path and the observation environment, the satellite clock bias data usually contain outliers that heavily affect the accuracy of satellite clock bias prediction. Based on the time series ARMA model and Bayes statistical theory, we propose a method to precisely predict satellite clock bias and detect outliers in the historical sequence of satellite clock bias. At first, considering the effects of an additive outlier (AO) and innovative outlier (IO), a labeling model for robustly fitting the time series ARMA model and detecting AOs and IOs simultaneously is constructed based on the labeling method of classification variables. Second, the Bayes method for robustly fitting time series ARMA model is proposed based on the Bayes statistical theory. Furthermore, it develops an algorithm to precisely predict satellite clock bias using the Bayes method for robustly fitting the time series ARMA model mentioned above. Finally, in order to illustrate the performance of the method for precisely predicting satellite clock bias that we presented, three examples are designed based on the real GPS data come from the IGS official website, and the prediction results of the method are compared with that of original ARMA model (oARMA), quadratic polynomial model (QP) and gray model (GM). It is found that the method can precisely predict the satellite clock bias as well as accurately detect the outliers in the historical sequence.

由于各种不确定因素的干扰,人们获取的卫星钟差数据中经常会出现异常扰动,降低了卫星钟性能分析的可靠性,破坏了钟差建模和预报的有效性,影响了导航定位结果的精准度.对此,以求和自回归移动平均模型为基础,建立了钟差时间序列异常值探测模型;基于Bayes统计原理,将异常值的定位和定值问题转化为模型选择问题;通过模型后验概率的近似计算,构建了模型选择的度量标准,避免了复杂的迭代计算问题.通过全球定位系统和北斗导航卫星系统不同卫星钟差数据的仿真试验,验证了所提出的方法对于卫星钟差序列中异常影响的定位和定值的正确性和有效性.

将复共线性对参数估计危害的度量结果与截断奇异值估计相结合,提出了基于信噪比检验的双截断奇异值估计.利用信噪比检验,根据每个参数最小二乘估计信噪比估值的大小将待估参数分为受复共线性危害较大和较小的两部分,并对这两部分参数的截断奇异值估计进行不同强度的截断.对受复共线性危害较大的部分参数,使其截断参数相对较小,对受复共线性危害较小的部分参数,使其截断参数较大.这种精细化的处理在有效降低参数估计方差的同时减少了偏差的引入.将基于信噪比检验的双截断奇异值估计应用于GEO卫星定轨仿真算例中,实验结果表明,新方法的解算精度较高.

辛普森悖论是大数据分析中的"陷阱".为研究这一现象,首先提炼出辛普森悖论的数学模型;其次对建立的模型进行较为全面的分析,从理论上揭示辛普森悖论产生的原因,推导辛普森悖论发生的概率;再次通过数据分析,验证辛普森悖论出现的合理性;最后说明数据粗糙和精细程度对所得结论的重要性.

In geodesy and geophysics,many large-scale over-determined linear equations need to be solved which are often illconditioned. When the conjugate gradient method is used,their ill-conditioning effects to the solutions must be overcome,which is studied in this paper. Through the regularization ideas,the conjugate gradient method is improved,and the regularization iterative solution based on controlling condition number is put forward. Firstly by constructing the interference source vector,a new equation is derived with ill-condition diminished greatly,which has the same solution to the original normal equation. Then the new equation is solved by conjugate gradient method. Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface. In the numerical experiments the new method is compared with LS,CG and Tikhonov methods,and its accuracy is the highest.

北斗卫星导航系统(BDS)卫星钟差异常值处理过程中,由于成片异常值的存在,往往会产生掩盖与淹没现象,致使异常值探测效率不高甚至失败.基于求和自回归移动平均模型,分析了时间序列中成片加性AO(Additive Outlier)异常值探测时易产生掩盖与淹没现象的原因;考虑了差分及逆差分对异常值探测的影响,提出了成片AO类异常值探测的抗掩盖与淹没新算法.通过仿真算例,验证了新算法对于序列中成片AO类异常值探测的准确性和有效性.将算法应用于BDS卫星钟差异常值探测和钟差预报中,较好地克服了数据中掩盖和淹没现象产生的影响,对于进一步提高卫星钟差的预报精度具有重要作用.

Based on the EM algorithm, an algorithm for detecting additive outlier in an autoregressive (AR) time series is proposed. The algorithm can fit the AR model and detect the additive outlier at the same time, and it can efficiently prevent the occurrence of masking and swamping.At last, the proposed algorithm is applied to process the data of GPS satellite clock error prediction. The examples verify the effectiveness of the algorithm in detecting the additive outlier and predicting the satellite clock error.

现有卫星导航系统的PNT服务无法提供全空域服务,LEO卫星星座、星间链路是未来多源导航信息源的重要组成部分.本文根据极轨道星座设计理论,在LEO卫星个数较少的条件下,设计了一个LEO导航增强星座,并分析了该星座的全球覆盖性.然后,根据卫星间的几何特性,在星座中建立了星间链路拓扑结构,并采用图论知识讨论了该星间链路拓扑结构的连通性和稳健性.最后,分析增加5颗GEO卫星后的星座星间链路性能.

Based on the time series analysis method, this article develops a Bayesian method of detecting and repairing the cycle slips in the GNSS carrier-phase data. Firstly, this article analyses the characteristics of the cycle slips in the GNSS carrier-phase observations and establishes the relationships between the cycle slips and the additive outliers (AOs) in the stationary time series. When the ARMA (autoregressive moving-average) model is used to fit the stationary time series obtained by differencing the GNSS carrier-phase observations, the detection of cycle slips in the GNSS carrier-phase observations can be transformed to the detection of AOs in the ARMA model. Then, this article proposes a Bayesian method of detecting the AOs in the ARMA model, and the implementation of detecting the cycle slips in the GNSS carrier-phase observations is also developed. Finally, the new Bayesian method of detecting the cycle slips is used to the real GNSS carrier-phase data. From the comparison among the Bayesian method, the high-order differences method and ionospheric residual method, we can find that the Bayesian method has a better detection efficiency for several kinds of cycle slips in the GNSS carrier-phase observations than other methods.

Kalman filter is one of the most common ways to deal with dynamic data and has been widely used in project fields. However, the accuracy of Kalman filter for discrete dynamic system is poor when the observation matrix is ill-conditioned. Therefore, the method for overcoming the harmful effect caused by ill-conditioned observation matrix in discrete dynamic system is studied in this paper. Firstly, Tikhonov regularized Kalman filter (TRKF) and its algorithm are proposed by combining Tikhonov regularization method and Kalman filter. Meanwhile, some excellent properties of TRKF are proved. Secondly, the methods of choosing regularization parameter and regularization matrix in TRKF are given. Thirdly, simulated examples are designed to evaluate the performance of TRKF and comparisons between TRKF and Ordinary Ridge-type Kalman Filter (ORKF) are given. Finally, TRKF is applied in autonomous orbit determination of BeiDou Navigation Satellite System (BDS) with cross-link ranging observations and ground tracking observations so as to prevent filter divergent which is caused by ill-conditioned observation matrix. Simulations and applications illustrate that TRKF can overcome the harmful effect caused by ill-conditioned observation matrix in discrete dynamic system and the accuracy is improved effectively

Clock offset measurements of satellite-ground time transfer are usually affected by outliers due to the impact of ionosphere errors,tropospheric errors,and muhipath effects.Therefore,in this paper,we propose an autoregressive model based on Bayesian methods for detecting outliers in the clock offset measurements with the classification variables.Furthermore,the model for estimating the magnitude of outliers is given to correct the clock offset measurements,and solves the problem of outlier estimation by transforming it into a simple least square problem.Different schemes based on the real BDS data were designed to evaluate the performance of the new Bayesian method.We applied the new method ito the fast recovery of the clock offset prediction.Test examples illustrate that the Bayesian methods can detect the outliers effectively and estimate the magnitudes of outliers accurately.

According to a puzzling probability theory exercise for student, the distribution law of random variable is given by multiplication formula and full probability formula respectively. By analysis, the reason that the probability theory exercises make student confused is that the exercise does not conform to real life.

从理论角度分析了观测矩阵的复共线性对卡尔曼滤波的影响,并在均方误差最小意义下,给出了一种有偏卡尔曼滤波算法.分别对观测矩阵和观测量施加扰动进行了试验和分析,证明观测矩阵的病态性会对卡尔曼滤波估计造成严重危害.数值模拟结果表明,本文算法能够有效改善观测矩阵病态性对卡尔曼滤波估计的影响,提高解算质量.

文章给出了线性矩阵方程中的一种新的解法.此方法利用初等行变换法化简常见的线性矩阵方程对应的特定矩阵,依据化简结果可同时得到非齐次线性矩阵方程的一个特解和对应的基础解系,从而可直接写出通解,并通过算例检验了初等行变换法的可行性、简便性和有效性.

从空间几何的角度分析了Kalman滤波中的病态性问题,提出了离散线性系统的岭型Kalman滤波及其算法,讨论了岭型Kalman滤波的性质,给出了选取岭参数的两种方法;数值模拟证明所给出的新算法能够有效改善观测矩阵的病态性对Kalman滤波的不良影响,提高了状态估计的精度.

Implementation of effective teaching is the responsibility of teachers. This paper describes the drawbacks of spoon-feeding education, and then discuss how to avoid cramming method of teaching through teacher-student interaction, followed by discussion of the relationship between"teaching"and"learning"in order to achieve long-term development of teachers in his career.

Within the next decade, there will be a number of GNSS (Global Navigation Satellite System) available, i.e. modernized GPS, Galileo, restored GLONASS, BeiDou and many other regional GNSS augmentation systems. Thus, measurement redundancies and geometry of the satellites can be improved. GDOP (Geometric Dilution of Precision) and PDOP (Position Dilution of Precision) are associated with the constellation geometry of satellites, and they are the geometrically determined factors that describe the effect of geometry on the relationship between measurement error and position error. GDOP and PDOP are often used as standards for selecting good satellites to meet the desired positioning precision. In this paper, the related conclusions of minimum of GDOP which was discussed are given, and it is used to study the minimum of PDOP for two cases that the receiver is on the earth’s surface and the receiver is on satellite. The corresponding theorem and constructive solutions of minimum of PDOP are given. Then, the rationality of the ISL (inter-satellite link) establishment criteria in Walker-δ constellation is discussed by using the theory of minimum of PDOP. Finally, the minimum of PDOP is calculated when the number of satellites is 4–10, and these results are verified by using Monte Carlo method.