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1、英 文 翻 译系 别专 业班 级学生姓名学 号指导教师报告日期Data Curve Fitting Based on MATLAB Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,possibly subject to constraints. Curve fitting can involve eitherinterpolation, where an exact fit to the
2、 data is required, or smoothing, in which a smooth function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random er
3、rors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables.Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to adegr
4、ee of uncertaintysince it may reflect the method used to construct the curve as much as it reflects the observed data. Research and Application of a New Method of Curve Fitting.The technique of curve fitting is used proverbially for the image processing, reverse engineering, test data processing, et
5、c. It is inequitable to process physical parameters by some usual methods of curve fitting. Those methods are performed only by minimizing the fitting error of one physical parameter, but do not take other parameters into account. The new method of curve fitting processes each physical parameter equ
6、ally The simulation also proves that this new curve fitting method is right and effective. In the experiment of sound velocity, the voltammetry to measure the resistance experiment and the volt-ampere characteristic of diode experiment data processing as an example, introduced the experiment data pr
7、ocessing based on MATLAB.With the traditional experimental data processing methods, experimental data is processed by using MATLAB can effectively avoid the error caused by manual processing, but also can reduce the computational workload, obtain accurate curve fitting, thereby increasing the accura
8、cy of data processing and fast,rom graphic display results also can be more intuitive to judge the validity of the experiment.Mathematical expression Given set of discrete data(xk,yk) (k=1,2,m),(1) Where xk is the independent variable x (scalar or vector, i.e., a mono-or polyhydric variable) values;
9、 yk of (scalar) corresponding to the value of the dependent variable y. Curve fitting is to seek to solve the problem of (1) to adapt the laws of the analytical expression of the backgroundy=f(x,b),(2) Making best approximation in some sense or fit (1), (x, b) is called fitting model;? Parameters to
10、 be determined, when b) only appears when the linear, called a linear model? otherwise non-linear.Amount(k=1,2,，m) In xk place called residual or remaining fit,The standard measure of goodness of fit is usually 或 Where k 0 as weight coefficient or weight(Unless otherwise specified, generally taken t
11、o be the average weight,wk(k=1,2,m),At this time without mention weight).When the parameter b) make T (b) or Q (b) to achieve the most hours,Appropriate (2) are referred to (1) the weighted Chebyshev fitting meaning or weighted least squares sense,latter is more simple and most commonly used in the
12、calculation. General linear model to determine the model parameters are parameters b) generalized polynomial coefficients,thatf(x,b)=b0g0(x)+b1g1(x)+bngn(x) (3) Wherein g0, g1, ., gn called basis functions.Gj on various different choices may constitute a variety of typical and commonly used linear m
13、odel.From the point of view of function approximation, equation (3) can be approximated reflect the nature of many of nonlinear models.In the least squares sense (3) fitting a linear model with a discrete set of points (1),parameter b can be obtained by solving equations=0(i=0,n)to determine,that so
14、lution on b0,b1,bn of linear algebraic equations(i=0,1,，n),(4) Formula (i,j0,1,，n), Equations (4) commonly referred to as the normal equation or the normal equation, when m n, generally have a unique solution.As for the case of non-linear model and the principle of least squares, the parameter b) ca
15、n be determined (see numerical solution of nonlinear equations, optimization) nonlinear equations or calculations about the method optimization.Select the model For a given discrete data (1), the need to properly select the general model (2) of the function f (x, b) the type and specific form, which
16、 is the basis of fitting results.If known, (1) the actual background of the law, that is the dependent variable y dependence of the empirical formula of the independent variable x has an expression determined directly take appropriate empirical formula is fitting model.On the contrary, through the m
17、odel (3) of the basis functions g0, g1, ., gn (number and types) of different choices, each corresponding proposed merger by choosing the good effect.Function g0, g1, ., gn adaptation plays a role the model for testing, it is also known test function.Another way is: the number and types into a suffi
18、cient number of test functions in the model (3), by means of statistical methods in mathematical correlation analysis and test of significance of the test function contains screened individually or sequentially with establish more appropriate model (see regression analysis).Certainly, the above meth
19、od may fit residuals (as a new discrete data) is performed again to compensate for the lack of the first fitting.In conclusion, when the intrinsic link between the variables in the data is not clear, as the choice to adapt the model to fit generally requires repeated testing and analysis to identify
20、.Procedure（一） Draw a scatter plot, select the appropriate type of curve.Generally based on the nature of the information can be combined with the expertise to determine the type of curve data, not really taht can be plotted on graph paper squares scatter plot, according to the distribution of scatte
21、red points, choose close, the appropriate curve type.Can be plotted on graph paper squares scatter plot, according to the distribution of scattered points, choose close, the appropriate curve type.（二） Be variable transformationY=f(Y),X=g(X)(12.37)The two variables transformed linear relationship.（三）
22、 Solving linear equations and variance analysis by the least squares method（四） Convert the linear equations on the original variables X, Y of the function expression基于MATLAB的数据曲线拟合分析 曲线拟合是构建的过程曲线，或数学函数，具有最适合于一系列的数据点，可能受到约束。曲线拟合可涉及无论是插值，其中一个确切的适合的数据是必需的，或平滑，其中一个“平滑”功能构造，大约拟合数据。一个相关的话题是回归分析，它更侧重于问题的统计
23、推断如多少不确定性存在于一条曲线，它是适合与随机误差观测数据。拟合曲线可以作为一种辅助手段进行数据可视化，推断功能在没有数据的情况下，值，并总结两个或多个变量之间的关系。外推法是指使用拟合曲线的超出范围的观测数据，并受程度的不确定性，因为它可能反映了用于构造曲线一样，因为它反映了观测数据的方法。曲线拟合技术在图像处理、逆向工程以及测试数据的处理等领域中的应用越来越广泛。目前常见的一些曲线拟合方法中, 对各个物理量的处理有失公平性原则，通常是在处理中确保某一个物理量的拟合误差达到“最小”, 而没有考虑到其它物理量的拟合误差。本文从这一思路出发, 给出了一种新的曲线拟合方法, 采用这种曲线拟合方法
24、, 对每个物理量的重视程度是相同的。实际的曲线拟合结果表明本文所提出的曲线拟合方法是正确和有效的。以声速测定实验、伏安法测电阻实验和二极管伏安特性实验的数据处理为例，介绍了 Matlab 在实验数据处理中的应用。与传统的实验数据处理方法相比，用 Matlab 处理实验数据能有效地避免手工处理所带来的误差,而且可减少计算工作量，得到准确的拟合曲线,从而增加数据处理的准确性及快捷性，从图形显示结果还可以更加直观地判断实验的正确性。数学表述 设给定离散数据(xk,yk) (k=1,2,m),(1)式中xk为自变量x(标量或向量,即一元或多元变量)的取值；yk为因变量 y（标量）的相应值。曲线拟合要解
25、决的问题是寻求与(1)的背景规律相适应解析表达式y=f(x,b),(2)使它在某种意义下最佳地逼近或拟合(1),?(x,b)称为拟合模型；为待定参数,当b)仅在?中线性地出现时，称模型为线性的，否则为非线性的。量(k=1,2,，m)称为在xk处拟合的残差或剩余，衡量拟合优度的标准通常有 或 式中k0为权系数或权重（如无特别指定，一般取为平均权重，即wk(k=1,2,m),此时无需提到权）。当参数b)使T(b)或Q(b)达到最小时，相应的(2)分别称为在加权切比雪夫意义或加权最小二乘意义下对 (1)的拟合，后者在计算上较简便且最为常用。模型中参数的确定 一般的线性模型是以参数 b)为系数的广义多
26、项式，即f(x,b)=b0g0(x)+b1g1(x)+bngn(x) (3)式中g0,g1,，gn称为基函数。对诸gj的不同选取可构成多种典型的和常用的线性模型。从函数逼近的观点来看，式(3)还能近似地体现许多非线性模型的性质。在最小二乘意义下用线性模型(3)拟合离散点组(1)，参数b可通过解方程组=0(i=0,n)来确定，即解关于b0,b1,bn的线性代数方程组(i=0,1,，n),(4)式中 (i,j0,1,，n),方程组(4)通常称为法方程或正规方程，当mn时一般有惟一解。至于非线性模型以及非最小二乘原则的情形,参数b)可通过解非线性方程组或最优化计算中的有关方法来确定（见非线性方程组数
27、值解法、最优化）。模型的选择对于给定的离散数据(1),需恰当地选取一般模型(2)中函数f(x,b)的类别和具体形式,这是拟合效果的基础。若已知(1)的实际背景规律，即因变量y对自变量 x的依赖关系已有表达式形式确定的经验公式，则直接取相应的经验公式为拟合模型。反之，可通过对模型(3)中基函数g0,g1,gn(个数和种类)的不同选取，分别进行相应的拟合并择其效果佳者。函数g0,g1,gn对模型的适应性起着测试的作用，故又称为测试函数。另一种途径是:在模型(3)中纳入个数和种类足够多的测试函数，借助于数理统计方法中的相关性分析和显著性检验，对所包含的测试函数逐个或依次进行筛选以建立较适合的模型（见回归分析）。当然，上述方法还可对拟合的残差（视为新的离散数据）再次进行，以弥补初次拟合的不足。总之，当数据中变量之间的内在联系不明确时，为选择到相适应的模型，一般需要反复地进行拟合试验和分析鉴别。步骤（一）绘制散点图，选择合适的曲线类型一般根据资料性质结合专业知识便可确定资料的曲线类型，不能确定时，可在方格坐标纸上绘制散点图，根据散点的分布，选择接近的、合适的曲线类型。（二）进行变量变换Y=f(Y),X=g(X)(12.37)使变换后的两个变量呈直线关系。（三）按最小二乘法原理求线性方程和方差分析（四）将直线化方程转换为关于原变量X、Y的函数表达式
