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Using complex variables to estimate derivatives of real functions

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USING COMPLEX V ARIABLES TO ESTIMATE DERIV ATIVESOF REAL FUNCTIONS ∗WILLIAM SQUIRE †AND GEORGE TRAPP †SIAM R EV .c 1998Society for Industrial and Applied Mathematics Vol.40,No.1,pp.110–112,March 1998008Abstract.A method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described.Numerical examples illustrating the power of the approximation are presented.Key words.divided difference,subtractive cancellationAMS subject classifications.65D25,30E10,65-04PII.S003614459631241X1.Overview.A standard method to approximate the derivative of a real valued function F (x )at the point x 0is to use the central difference formulaF (x 0)∼(F (x 0+h )−F (x 0−h ))/2h.(1)The truncation error is O (h 2).With most derivative approximations one is faced with the dilemma of using a small h to minimize the truncation error vs.avoiding a small h because of the subtractive cancellation error.Work has been done to improve the answer obtained using equation 1by analyzing its monotonic behavior as h goes to zero.It turns out that when the roundofferror becomes significant,the estimate from equation (1)oscillates around the correct answer,and this behavior can be used to select an optimal h ;see [1,5].This approach provides only marginal improvement,whereas allowing h to take on a complex value results in an entirely different approximation with unexpected properties.Using complex variables to develop differentiation formulas originated with Ly-ness and Moler [2]and Lyness [3].Since most current numerical analysis textbooks do not normally cover complex variables,we thought a sample of this work with representative computations would be of interest.In equation (1)we replace h with ih (i =sqrt(−1)).If F is analytic then,letting Im (F )represent the imaginary part of the function F ,the approximation in (1)can be rewrittenF (x 0)∼Im (F (x 0+ih )/h.(2)Formula (2)involves the evaluation of the function at a complex argument,but it eliminates the subtractive cancellation error.One way to understand the approx-imation in (2)is to recast the problem in terms of functions of a complex variable.Let F (z )be an analytic function of the complex variable z ;also assume that F is real on the real axis.F may be expanded in a Taylor series about the real point x 0as follows:F (x 0+ih )=F (x 0)+ihF (x 0)−h 2F (x 0)/2!−ih 3F (3)(x 0)/3!+···.(3)∗Received by the editors July 4,1996;accepted for publication November 1,1996./journals/sirev/40-1/31241.html †West Virginia University,Morgantown,WV 26506-6330(trapp@).110CLASSROOM NOTES111T ABLE1F(x)=x9/2.h Equation1Equation20.1D-010.18602018344501897D+020.18599607128036329D+020.1D-020.18600824790342307D+020.18600800678177631D+020.1D-030.18600812854818738D+020.18600812613698936D+020.1D-040.18600812735480865D+020.18600812733054151D+020.1D-050.18600812734248517D+020.18600812734247702D+020.1D-060.18600812735081185D+020.18600812734259637D+020.1D-070.18600812723423843D+020.18600812734259757D+020.1D-080.18600812723423843D+020.18600812734259759D+020.1D-090.18600814222224926D+020.18600812734259759D+020.1D-100.18600815332447951D+020.18600812734259759D+020.1D-110.18600898599174798D+020.18600812734259759D+020.1D-120.18601231666082185D+020.18600812734259759D+020.1D-130.18585133432225120D+020.18600812734259759D+020.1D-140.18596235662471372D+020.18600812734259759D+020.1D-150.20539125955565396D+020.18600812734259759D+020.1D-160.00000000000000000D+000.18600812734259759D+020.1D-170.00000000000000000D+000.18600812734259759D+020.1D-180.00000000000000000D+000.18600812734259759D+020.1D-190.00000000000000000D+000.18600812734259759D+02Taking the imaginary parts of both sides of equation(3)and dividing both sides by h yields(4)Im[F(x0+ih)]/h=F (x0)−h2F(3)(x0)/3!+···.The left-hand side of(4)is an approximation to F (x0)with approximation er-ror O(h2).Im[F(x0+ih)]/h is real although it involves a complex argument,and, importantly,it is not subject to subtractive cancellation.2.Numerical examples.We present two examples which illustrate the power of this approach.The output comes from a FORTRAN program run on a Digital Equipment Corporation VAX6000-620,using standard FORTRAN complex variable features.Thefirst example is in double precision,and the second is in single precision. Both examples show that equation(2)yields an accurate derivative approximation for any realistic value of h whereas equation1quickly succumbs to subtractive can-cellation as h decreases.Example1.Let F(x)=x9/2and x0=1.5;then to seventeen places F (1.5)= 18.600812734259759.Table1shows the approximation obtained from equations(1) and(2)for h ranging from0.01to10−19.Equation(1)approximation improves for a while then deteriorates quickly,finally becoming0.0because of subtractive cancella-tion.Equation(2)approximation experiences no such problems.Example2.Lyness and Sande[4]used the function F(x)=e x/(sin(x)3+cos(x)3) to illustrate a similar type of calculation.Again let x0=1.5;the value of the derivative is3.62203.The output is shown in Table2;here single precision is used.The table has the same behavior as shown in Table1.3.Summary.In this note we have shown that unexpected results are obtained by using a complex variable approach to estimating derivatives of real valued func-tions.112CLASSROOM NOTEST ABLE2F(x)=e x/(sin(x)3+cos(x)3).h Equation1Equation20.100000E-010.362298E+010.362109E+010.100000E-020.362229E+010.362202E+010.100000E-030.362158E+010.362203E+010.100000E-040.360012E+010.362203E+010.100000E-050.357628E+010.362203E+010.100000E-060.476837E+010.362203E+010.100000E-070.000000E+000.362203E+010.100000E-080.000000E+000.362203E+010.100000E-090.000000E+000.362203E+010.100000E-100.000000E+000.362203E+01REFERENCES[1]P.B REZILLON,J-F.S TAUB,A-M.P ERAULT-S TAUB,AND G.M ILHAUD,Numerical estimationof thefirst order derivative:Approximate evaluation of an optimal step,Comput.Math.Appl.,7(1981),pp.333–347.[2]J.N.L YNESS AND C.B.M OLER,Numerical differentiation of analytic functions,SIAM J.Numer.Anal.,4(1967),pp.202–210.[3]J.N.L YNESS,Numerical algorithms based on the theory of complex variables,Proc.ACM22ndNat.Conf.,Thompson Book Co.,Washington,DC,1967,pp.124–134.[4]J.N.L YNESS AND G.S ANDE,Algorithm413-ENTCAF and ENTCRE:Evaluation of normalizedTaylor coefficients of an analytic function,Comm.ACM14,10(1971),pp.669–675.[5]R.S TEPLEMAN AND N.W INARSKY,Adaptive numerical differentiation,p.,33(1979),pp.1257–1264.。

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《金融计量学》习题及习题答案

《金融计量学》习题及习题答案

诚实考试吾心不虚 ，公平竞争方显实力， 考试失败尚有机会 ，考试舞弊前功尽弃。

上海财经大学《 Financial Econometrics 》课程考试卷一课程代码 课程序号姓名 学号 班级Part 1 T erm Explanation （20 marks ）1．White Noise 2．RandomWalk3．Akaike Information Criterion 4．Jarque-Bera Statistic 5．Chow T estImportant Point ：1．White Noise ：White Noise is the special case of stationary stochastic process. We call a stochastic process purely random or white noise if it has zero mean, constant variance and is serially uncorrelated.2．RandomWalk: Random walk means that the stochastic process is nonstationary and value of this period is highly related to the past values. For example, the stock price today may equal the yesterday ’s price plus a random shock. Random walk without drift can be expressed as t t t u y y +=-13．Akaike Information Criterion: AIC provide a way to select the better regression model among several models by comparing their forecast performance. The lower the AIC, the better the forecast performance will be. AIC will also be used to determine the lag length in ARDL approach.4．Jarque-Bera Statistic: The Jarque-Bera test is the test of normality . We first calculate the skewness and the kurtosis, and it is also based on the residual of the regression.The Jarque-Bera S tatistic=)24)3(6(22-+K S n , where S is the skewness and K is the kurtosis,n is sample size, and for normal distribution, S=0, K=3, if JB statistic is not significantly different from zero, p value is quite low , we reject the null hypothesis that the residual is normally distributed.5．Chow T est: The test of structural change of the regression. The estimate of the parameter of the regression may not retain the same through the entire time period; we use the Chow test to test whether the relationship is stable and find the break point. It develop the F statistics=)/(/)(k N RSS mRSS RSS ur ur r --, the null hypothesis is the regression is stable.Part 2 Explain main purpose(s) of constructing following two models and making comments on the empirical results. (25marks)1.Gregory Chow (1966)where M = natural logarithm of total money stock Y p = natural logarithm of permanent income Y = natural logarithm of current income R = natural logarithm of rate of interest2.Taylor and Newhouse (1969)本题答题要点：1。

multivariate statistical analysis

multivariate statistical analysis

Multivariate Statistical AnalysisMultivariate statistical analysis is a powerful tool used to analyze complex data sets that involve multiple variables. It provides insights into the relationship between variables and allows us to understating the underlying structure and patterns within the data.IntroductionIn a univariate analysis, we typically examine a single variable at a time, such as analyzing the distribution of income or the average temperature over time. However, real-world problems are often more complex, involving multiple variables that interact with each other.Multivariate statistical analysis aims to address this complexity by considering multiple variables simultaneously. It allows us to identify how different variables are related to each other and to understand the joint behavior of these variables.Types of Multivariate Statistical AnalysisThere are various types of multivariate statistical analysis techniques, each designed to answer different research questions and address different types of data. Some of the commonly used techniques include:Principal Component Analysis (PCA)PCA is a dimensionality reduction technique that aims to identify the most important variables or combinations of variables that explain the majority of the variance in the data. It transforms the original variables into a smaller set of uncorrelated variables called principal components.1.Identify the variables and their correlations.2.Determine the eigenvectors and eigenvalues of the correlationmatrix.3.Select the principal components based on their eigenvalues.4.Interpret the results and analyze the contribution of eachprincipal component.Factor AnalysisFactor analysis is another dimensionality reduction technique that aims to identify the underlying latent variables or factors that explain the covariance structure in the data. It is often used in psychology and social sciences to understand the factors that influence human behavior.1.Define the variables and their relationships.2.Estimate the factor loadings, which represent the strength of therelationship between each variable and each factor.3.Determine the number of factors to retain based on criteria suchas eigenvalues or explained variance.4.Interpret the factors and analyze their impact on the variables.Cluster AnalysisCluster analysis is a technique used to group similar observations or variables into clusters based on their characteristics. It helps to identify patterns and similarities within the data and can be used for segmenting customers, grouping genes, or classifying objects.1.Select the variables to be included in the cluster analysis.2.Choose a distance metric to measure the similarity betweenobservations or variables.3.Apply a clustering algorithm such as k-means or hierarchicalclustering.4.Evaluate the clusters and interpret the results.Discriminant AnalysisDiscriminant analysis aims to identify the variables that best discriminate between two or more groups. It is often used in marketing and biomedical research to predict group membership based on a set of predictors.1.Define the groups and the predictor variables.2.Estimate the discriminant function coefficients.3.Evaluate the model’s performance using measures such as accuracyor confusion matrix.4.Interpret the discriminant function and analyze the impact of eachpredictor variable.Advantages of Multivariate Statistical AnalysisMultivariate statistical analysis offers several advantages over univariate analysis:1.Identifying hidden patterns: By considering multiple variablessimultaneously, we can uncover hidden relationships and patternsthat are not visible in univariate analysis.2.Reducing dimensionality: Multivariate techniques such as PCA andfactor analysis help to reduce the dimensionality of the data byidentifying the most important variables or factors.3.Better understanding of complex systems: Multivariate analysisallows for a more comprehensive understanding of complex systemsby considering the interdependencies between variables.4.Improved predictive modeling: By incorporating multiplepredictors, multivariate techniques can improve the accuracy androbustness of predictive models.5.More efficient data exploration: Multivariate techniques providea systematic and structured approach to data exploration, enablingresearchers to efficiently explore and analyze large and complexdata sets.ConclusionMultivariate statistical analysis is a valuable tool for researchers and analysts working with complex data sets. It allows for a deeper understanding of the relationships between variables, uncovering hidden patterns, and providing insights into complex systems. By employing techniques such as PCA, factor analysis, cluster analysis, and discriminant analysis, researchers can make more informed decisions and develop robust models. It is essential to choose the appropriate technique based on the research question and the characteristics of the data.。

公交客流预测的英语

公交客流预测的英语

公交客流预测的英语Bus Passenger Flow Prediction.Introduction.Bus passenger flow prediction is a crucial task in transportation planning and management. Accuratepredictions can help transportation agencies optimize their routes, schedules, and fleet management, thus improving the efficiency and effectiveness of public transportation systems. This article delves into the intricacies of bus passenger flow prediction, discussing its importance, methods, challenges, and future prospects.Importance of Bus Passenger Flow Prediction.Bus passenger flow prediction is essential for several reasons. Firstly, it aids in the efficient allocation of resources. By anticipating the passenger flow, agencies can ensure that they have enough buses to meet demand whilealso avoiding unnecessary over-capacity. This helps in reducing costs and waste.Secondly, passenger flow prediction enables agencies to optimize their routing and scheduling decisions. By understanding when and where demand is highest, agencies can design routes that better serve passengers, reduce travel time, and enhance the overall passenger experience.Lastly, accurate predictions can help agencies prepare for突发事件 such as special events, weather changes, or emergencies. By anticipating changes in passenger flow, agencies can plan accordingly, ensuring that their transportation systems remain robust and responsive.Methods of Bus Passenger Flow Prediction.There are several methods used for bus passenger flow prediction, each with its own strengths and limitations.1. Historical Data Analysis: This approach relies on analyzing historical passenger flow data to identifypatterns and trends. It can be used to forecast future passenger flows based on past performance. However, it may not account for sudden changes in demand caused by external factors such as weather or special events.2. Statistical Models: Statistical models, such as regression analysis or time series analysis, are often used to predict passenger flow. These models rely on historical data and various explanatory variables to estimate future passenger numbers. While they can provide valuable insights, they may not capture the complex interactions between various factors that influence passenger flow.3. Machine Learning Algorithms: Machine learning algorithms, such as neural networks, random forests, or support vector machines, have been increasingly used for passenger flow prediction. These algorithms can learn from historical data and identify complex patterns and relationships. They can handle nonlinearities and interactions between various factors, making them more suitable for dynamic and uncertain environments. However, they require large amounts of data for training and may beprone to overfitting if not used carefully.4. Real-Time Monitoring and Prediction: Real-time monitoring systems can provide up-to-date information on passenger flow, enabling agencies to make timelyadjustments to their operations. By combining real-timedata with predictive models, agencies can achieve more accurate predictions and respond more quickly to changes in demand.Challenges in Bus Passenger Flow Prediction.Despite the importance of bus passenger flow prediction, there are several challenges that need to be addressed.1. Data Availability and Quality: Accurate predictions rely on high-quality data. However, obtaining comprehensive and reliable data on passenger flow can be challenging, especially in developing countries or regions with limited infrastructure. Additionally, data quality issues such as missing values, outliers, or inconsistencies can affect the accuracy of predictions.2. Dynamic and Uncertain Environments: Passenger flow is influenced by various dynamic and uncertain factors such as weather, traffic conditions, special events, or public holidays. Modeling these factors accurately can be difficult, and their impact on passenger flow may vary depending on the context and location.3. Complex Interdependencies: Passenger flow is influenced by multiple factors that interact in complex ways. Understanding and capturing these interdependencies can be challenging, especially when dealing with large-scale transportation systems.Future Prospects.Despite these challenges, there are several promising avenues for further research and development in bus passenger flow prediction.1. Advanced Data Collection Techniques: The development of new data collection techniques such as sensor networks,mobile phone data, or social media analytics can provide richer and more granular data on passenger flow. This will enable more accurate predictions and better understanding of passenger behavior.2. Integration of Multiple Data Sources: Integrating multiple data sources such as historical data, real-time monitoring data, and external factors can provide a more comprehensive picture of passenger flow. This integration can help improve the accuracy and reliability of predictions.3. Advancements in Machine Learning and Artificial Intelligence: The ongoing advancements in machine learning and artificial intelligence (AI) provide new opportunities for bus passenger flow prediction. Techniques such as deep learning or reinforcement learning can handle complex patterns and interactions better, enabling more accurate and robust predictions.4. Collaborative and Multi-Stakeholder Approaches: Collaboration between transportation agencies, privatesector partners, and academic institutions can foster innovation and improve the overall quality of passengerflow prediction. Multi-stakeholder approaches can also help ensure that predictions align with the needs and objectives of all parties involved.In conclusion, bus passenger flow prediction is acrucial task that requires a combination of data analysis, modeling techniques, and technological advancements. By addressing the challenges and leveraging the opportunities discussed in this article, transportation agencies can improve their understanding of passenger flow and make more informed decisions about their transportation systems. This, in turn, can lead to more efficient, responsive, and sustainable public transportation services that betterserve the needs of the community.。

高二英语数学建模方法单选题20题

高二英语数学建模方法单选题20题

高二英语数学建模方法单选题20题1.In the process of mathematical modeling, the factor that determines the outcome is called_____.A.independent variableB.dependent variableC.control variableD.extraneous variable答案：B。

本题考查数学建模中的基本术语。

独立变量（independent variable）是指在实验或研究中被研究者主动操纵的变量；因变量dependent variable）是指随着独立变量的变化而变化的变量，在数学建模中决定结果的因素通常是因变量；控制变量（control variable）是指在实验中保持不变的变量；无关变量（extraneous variable）是指与研究目的无关，但可能会影响研究结果的变量。

2.The statement “The value of y depends on the value of x” can be represented by a mathematical model where y is the_____.A.independent variableB.dependent variableC.control variableD.extraneous variable答案：B。

在“y 的值取决于x 的值”这句话中，y 是随着x 的变化而变化的变量，所以y 是因变量。

3.In a mathematical model, the variable that is held constant toobserve the effect on other variables is_____.A.independent variableB.dependent variableC.control variableD.extraneous variable答案：C。