We are given the data set (0,0),(1,2),(2,16),(3,52),(4,118),(5,223) and we are asked to fit these in a power model y=ax^b :
(1) The easiest way is to input the data in Excel or a graphing utility and perform a power regression yielding y=2.04x^2.93 with a,b rounded to two decimal places.
(2) To determine if the data set will fit a power model well we can plot (lnx,lny) and determine if the plot is approximately a straight line.
Taking the natural logarithm of the coordinates yields the points (0,.6931),(.6931,2.773),(1.0986,3.9512),(1.3862,4.7707),(1.6094,5.4072). These points appear to lie on a line. Performing linear regression on these points yields the line y=.7146+2.929x
So we have lny=lna+blnx Exponentiating both sides with base e gives us:
y=e^(lna)*e^(blnx)=a*(e^(lnx))^b=ax^b where a=e^(.7146) and b=2.929
So the power model is y~~2.04x^(2.93)
http://mathworld.wolfram.com/LeastSquaresFittingPowerLaw.html
Friday, June 2, 2017
Your data set is [(0,0),(1,2),(2,16),(3,52),(4,118),(5,223)]. Determine the values of a and b rounded to 2 places for the power model, model(x)=ax^b.
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