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Overview of Curve Fitting Models and Methods in Lab. VIEW1. Curve Fitting in Lab. VIEWWhat is Curve Fitting? The purpose of curve fitting is to find a function f(x) in a function class . The function f(x) minimizes the residual under the weight W. The residual is the distance between the data samples and f(x). A smaller residual means a better fit. In geometry, curve fitting is a curve y=f(x) that fits the data (xi, yi) where i=0, 1, 2. Refer to the Lab. VIEW Help for information about using these VIs. The following graphs show the different types of fitting models you can create with Lab. VIEW. Figure 1. Curve Fitting Models in Lab. VIEWBefore fitting the data set, you must decide which fitting model to use. An improper choice, for example, using a linear model to fit logarithmic data, leads to an incorrect fitting result or a result that inaccurately determines the characteristics of the data set. Therefore, you first must choose an appropriate fitting model based on the data distribution shape, and then judge if the model is suitable according to the result. Every fitting model VI in Lab. VIEW has a Weight input. The Weight input default is 1, which means all data samples have the same influence on the fitting result. In some cases, outliers exist in the data set due to external factors such as noise. If you calculate the outliers at the same weight as the data samples, you risk a negative effect on the fitting result. Therefore, you can adjust the weight of the outliers, even set the weight to 0, to eliminate the negative influence. You also can use the Curve Fitting Express VI in Lab. VIEW to develop a curve fitting application. Curve Fitting Methods. Different fitting methods can evaluate the input data to find the curve fitting model parameters. Each method has its own criteria for evaluating the fitting residual in finding the fitted curve. By understanding the criteria for each method, you can choose the most appropriate method to apply to the data set and fit the curve. Practical strain gage measurements, are strain s. In this paper we begin by describing two algorithms that operate on the Web graph. The Web as a Graph: Measurements, Models, and Methods Book Title. To view the rest of this content please follow the download PDF link. Estimation Method Based on Heart 1Rate Measurement Published: Feb 2005. MODEL VALIDATION Methods. Graph structure in the web Andrei Broder 1. Keywords: graph structure, diameter, web measurement 1. Risk Analysis and Quantification 1 What is Risk Analysis? The methods can be: Qualitative Methods. Why develop a Risk Model? For the measurement of the likelihood of occurrence of the risk and the impact. In Lab. VIEW, you can apply the Least Square (LS), Least Absolute Residual (LAR), or Bisquare fitting method to the Linear Fit, Exponential Fit, Power Fit, Gaussian Peak Fit, or Logarithm Fit VI to find the function f(x). The LS method finds f(x) by minimizing the residual according to the following formula: where n is the number of data sampleswi is the ith element of the array of weights for the data samplesf(xi) is the ith element of the array of y- values of the fitted modelyi is the ith element of the data set (xi, yi)The LAR method finds f(x) by minimizing the residual according to the following formula: The Bisquare method finds f(x) by using an iterative process, as shown in the following flowchart, and calculates the residual by using the same formula as in the LS method. The Bisquare method calculates the data starting from iteration k. Figure 2. Bisquare Method Flowchart. Because the LS, LAR, and Bisquare methods calculate f(x) differently, you want to choose the curve fitting method depending on the data set. For example, the LAR and Bisquare fitting methods are robust fitting methods. Use these methods if outliers exist in the data set. The following sections describe the LS, LAR, and Bisquare calculation methods in detail. LS Method. The least square method begins with a linear equations solution. Ax = b. A is a matrix and x and b are vectors. Ax–b represents the error of the equations. The following equation represents the square of the error of the previous equation. E(x) = (Ax- b)T(Ax- b) = x. TATAx- 2b. TAx+b. Tb. To minimize the square error E(x), calculate the derivative of the previous function and set the result to zero: E’(x) = 0. ATAx- 2. ATb = 0. ATAx =ATbx = (ATA)- 1. ATb. From the algorithm flow, you can see the efficiency of the calculation process, because the process is not iterative. Applications demanding efficiency can use this calculation process. The LS method calculates x by minimizing the square error and processing data that has Gaussian- distributed noise. If the noise is not Gaussian- distributed, for example, if the data contains outliers, the LS method is not suitable. You can use another method, such as the LAR or Bisquare method, to process data containing non- Gaussian- distributed noise. LAR Method. The LAR method minimizes the residual according to the following formula: From the formula, you can see that the LAR method is an LS method with changing weights. If the data sample is far from f(x), the weight is set relatively lower after each iteration so that this data sample has less negative influence on the fitting result. Therefore, the LAR method is suitable for data with outliers. Bisquare Method. Like the LAR method, the Bisquare method also uses iteration to modify the weights of data samples. In most cases, the Bisquare method is less sensitive to outliers than the LAR method. Comparing the Curve Fitting Methods. If you compare the three curve fitting methods, the LAR and Bisquare methods decrease the influence of outliers by adjusting the weight of each data sample using an iterative process. Unfortunately, adjusting the weight of each data sample also decreases the efficiency of the LAR and Bisquare methods. To better compare the three methods, examine the following experiment. Use the three methods to fit the same data set: a linear model containing 5. The following table shows the computation times for each method: Table 1. Processing Times for Three Fitting Methods. Fitting method. LSLARBisquare. Time. Comparison among Three Fitting Methods. The data samples far from the fitted curves are outliers. In the previous figure, you can regard the data samples at (2, 1. The results indicate the outliers have a greater influence on the LS method than on the LAR and Bisquare methods. You can see that the zeroes occur at approximately (0. Figure 4. General Polynomial Model. This VI calculates the mean square error (MSE) using the following equation: When you use the General Polynomial Fit VI, you first need to set the Polynomial Order input. A high Polynomial Order does not guarantee a better fitting result and can cause oscillation. A tenth order polynomial or lower can satisfy most applications. The Polynomial Order default is 2. This VI has a Coefficient Constraint input. You can set this input if you know the exact values of the polynomial coefficients. By setting this input, the VI calculates a result closer to the true value. General Linear Fit. The General Linear Fit VI fits the data set according to the following equation: y = a. For the General Linear Fit VI, y also can be a linear combination of several coefficients. Each coefficient has a multiplier of some function of x. Therefore, you can use the General Linear Fit VI to calculate and represent the coefficients of the functional models as linear combinations of the coefficients. For example, the following equation defines a model using data from a transducer. The following equation defines the observation matrix H for a data set containing 1. If the data set contains n data points and k coefficients for the coefficient a. Therefore, the number of rows in H equals the number of data points, n. The number of columns in H equals the number of coefficients, k. To obtain the coefficients, a. In curve fitting, splines approximate complex shapes. The Cubic Spline Fit VI fits the data set (xi, yi) by minimizing the following function: where p is the balance parameterwi is the ith element of the array of weights for the data setyi is the ith element of the data set (xi, yi)xi is the ith element of the data set (xi, yi)f. If the Balance Parameter input p is 1, the fitting method is equivalent to cubic spline interpolation. The closer p is to 0, the smoother the fitted curve. The closer p is to 1, the closer the fitted curve is to the observations. The following figure shows the fitting results when p takes different values. Figure 5. Cubic Spline Model. You can see from the previous figure that when p equals 1. When p equals 0. 0, the fitted curve is the smoothest, but the curve does not intercept at any data points. Nonlinear Curve Fit. The Nonlinear Curve Fit VI fits data to the curve using the nonlinear Levenberg- Marquardt method according to the following equation: y = f(x; a. You can use the nonlinear Levenberg- Marquardt method to fit linear or nonlinear curves. However, the most common application of the method is to fit a nonlinear curve, because the general linear fit method is better for linear curve fitting. Lab. VIEW also provides the Constrained Nonlinear Curve Fit VI to fit a nonlinear curve with constraints. You can set the upper and lower limits of each fitting parameter based on prior knowledge about the data set to obtain a better fitting result. The following figure shows the use of the Nonlinear Curve Fit VI on a data set. The nonlinear nature of the data set is appropriate for applying the Levenberg- Marquardt method. Figure 6. Nonlinear Curve Model. Preprocessing. The Remove Outliers VI preprocesses the data set by removing data points that fall outside of a range. The VI eliminates the influence of outliers on the objective function. The following figure shows a data set before and after the application of the Remove Outliers VI. Figure 7. Remove Outliers VIIn the previous figure, the graph on the left shows the original data set with the existence of outliers. The graph on the right shows the preprocessed data after removing the outliers. You also can remove the outliers that fall within the array indices you specify. Some data sets demand a higher degree of preprocessing. A median filter preprocessing tool is useful for both removing the outliers and smoothing out data. Postprocessing. Lab. VIEW offers VIs to evaluate the data results after performing curve fitting. These VIs can determine the accuracy of the curve fitting results and calculate the confidence and prediction intervals in a series of measurements. Robot Check. Enter the characters you see below. Sorry, we just need to make sure you're not a robot. For best results, please make sure your browser is accepting cookies. EDITED BY DAVID ADLERMETRICHANDBOOKPLANNINGANDDESIGNDATASECOND EDITION ArchitecturalPressThis C DMetric Handbook CD. Machinery's Handbook CD-ROM including hundreds. Other subjects in the Handbook that are new or have been. The metric content of the Handbook has been greatly. The Metric Handbook - Architecture must have. 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