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Expert: - 10/29/2008


Question
QUESTION: The following data represents (hypothetical) energy consumption normalized to the year 1900.  Plot the data.  Test the model Q = aebx by plotting the transformed data.  Estimate the parameters of the model graphically.

x     Year        Consumption
0     1900        1.00
10    1910        2.01
20    1920        4.06
30    1930        8.17
40    1940        16.44
50    1950        33.12
60    1960        66.69
70    1970        134.29
80    1980        270.43
90    1990        544.57
100   2000        1096.63

ANSWER: The problem really depends on whatQ=aebx means.  Is that ae^(bx)?

You need to find what some values are to solve this problem.
N is the number of items (count away and see if there aren't 11 of them).
A=Σx (0 + 10 + 20 + ... 90 + 100) ( it's easier to use x than year).
B=Σx² = 0² + 10² + 20² + .. + 90² + 100²
C=Σy = 1.00 + 2.01 + 4.06 + 8.17 + ... + 544.57 + 1096.63
D=Σ(xy) = 0*1.00 + 10*2.01 + 20*4.06 + ... + 90*544.7 + 100*1096.63.

To get a linear approximation, solve the matrix
|N A||x|
|A B||y|
=
|C|
|D|
where the first one is a 2x2 matrix, the second is a vector with 2 componets and on the other side of the equation (below the = sign) is another vector.

This comes down to solving the two equations
Nx + Ay = C, Ax + By = D

Note that the consumption looks very close to powers of 2
(which are 1, 2, 4, 8, 16, 32, 64,128, 256, 1024; using this it's more like y^n where y is x/10 and the consumption is the result; is the Q suppose to be ae^(bx)? )
---------------------------------------------------------------
That changes things a lot.
Compute the ln() of the consumption column and see what the average number is such that x^n = nth componet of this arrary (starting with element 0).  The equation would then be to find the best fit straight line through the ln() of these points using 0,1,2,...,9,10 for the x values and thr third column as the y values.

Instead of getting a1 + a2*x, you will get a1*e^(a2*x).


---------- FOLLOW-UP ----------

QUESTION: In the question Q = aebx -- I didn't know how to input it on the computer.  The bx should up in the air next to the ae.  How would I plot this data?

Thanks,

Sharon

Answer
YOu need to find a and b, and get the use Excel to put down the x's in one column and the Y's in the other.
Choose Insert Chart XY(Scatter), and then click on the chart shown that is desired.
Note that the top one plots the points, the middle two plot the best fit curve that passes through those points, and the bottom two plot the points as a dot to dot.

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