The logistic curve is given by L y = 1+ B exp (kx)' It is used...

 

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The logistic curve is given by L y = 1+ B exp (kx)' It is used to model populations that should have growth that is close to exponential (when x is small) but that can only grow to a limiting value L. Ideally-for a given set of data points (xi,yi), i=1..N, we would like to determine the best-fit parameters L, B and k via least squares, but this is a nonlinear problem. It is more difficult than the ones we have encountered in class. So Let's assume for the moment that we can get a good estimate of the value for L, (the largest the population can grow) and we can set this value as known constant. We still want to determine B and k via least squares. 1) [15 pts] Setting the parameter L as a known constant, turn this into a linear least-squares problem by using the right change of variables i.e. Y'=a0+a1*X' where the new variables Y' and X' have to be defined. 2) [5pts] Give an analytic formula for parameters B and k as a function of the data points (X,Yi), i=1..N. HINT : use the linear fit expressions for a0 and al you have obtained in 1)- (see Section 17.1 of the textbook). Do NOT use the Generalized Least square Matrix Formulation with the Z matrix. The logistic curve is given by L y = 1+ B exp (kx)' It is used to model populations that should have growth that is close to exponential (when x is small) but that can only grow to a limiting value L. Ideally-for a given set of data points (xi,yi), i=1..N, we would like to determine the best-fit parameters L, B and k via least squares, but this is a nonlinear problem. It is more difficult than the ones we have encountered in class. So Let's assume for the moment that we can get a good estimate of the value for L, (the largest the population can grow) and we can set this value as known constant. We still want to determine B and k via least squares. 1) [15 pts] Setting the parameter L as a known constant, turn this into a linear least-squares problem by using the right change of variables i.e. Y'=a0+a1*X' where the new variables Y' and X' have to be defined. 2) [5pts] Give an analytic formula for parameters B and k as a function of the data points (X,Yi), i=1..N. HINT : use the linear fit expressions for a0 and al you have obtained in 1)- (see Section 17.1 of the textbook). Do NOT use the Generalized Least square Matrix Formulation with the Z matrix.