3chapter3-3.ppt

3.3 Applying the Least-squares Criterion Fitting a Straight Line Suppose a model of the form y = Ax + B is expected and it has been decided to use the m data points (xi, yi), i = 1, 2,…, m, to estimate A and B. Denote the lease-squares estimate of y = Ax + B by y = ax + b. Applying the least-squares criterion to this situation requires the minimization of A necessary condition for optimality yields the equations These equations can be rewritten to give Fitting a Power Curve Now lets use the least-squares criterion to fit a curve of the form y = Axn, where n is fixed, to a given collection of data points. Call the least-squares estimate of the model f(x) = axn. Application of the criterion then requires minimization of A necessary condition for optimality gives the equation For example, lets fit y = Ax2 to the data shown in the table and predict the value of y when x = 2.25. Transformed Least-Squares Fit Although the least-squares criterion appears easy to apply in theory, in practice it may be difficult. For example, consider fitting the model y = AeBx using the least-squares criterion. It will yields a nonlinear equations not easy to solve. Many simple models result in derivatives that are very complex or in system of equations that are difficult to solve. For this reason, we use transformation to approximate the least-squares model. Suppose we wish to fit the power curve y = AxN to a collection of data points. Lets denote the estimate of A by a and the estimate of N by n. Taking the logarithm of both sides of the equation y = axn yields Solving for the slope n and intercept lna with the transformed variables and given m data points, we have Suppose we still wish to fit a quadratic y = Ax2 to the collection of data. Denote the estimate of A by a and taking the logarithm of both sides of the equation y = ax2 yields lny = lna + 2lnx. By minimizing we obtain that lna = 1.1432, a = 3.1368, so the model is y = 3.1368x2. The model predic