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Hello Scott

I am reading a book on ecology where I have found a number of graphs with a curved best fit line (please see attached photo).

I am familiar with the least squares method of fitting a straight line to scattered points, but I did not know there is a technique to fit a curved line--or are these lines merely freehand estimates?

If there is a mathematical technique for curved best fit lines where may I learn about it?

Thanks.

Pete

Whatever estimate you're trying to do must be done to the data.

You have an x value with a y value you are trying to approximate.

If you are trying to fit an exponential to the curve, as in y = Ax^2 + Bx + C.

Let's take the data

1 2

2 5

3 9

4 15

Number = 4

Sum x = 1+2+3+4 = 10

Sum x^2 = 1+4+9+16 = 30

Sum x^3 = 1+8+27+64 = 100

Sum x^4 = 1+16+81+256 = 354

Sum Y = 2+5+9+16 = 32

Sum XY = 2+10+27+60 = 99

Sum (X^2)Y = 2+20+81+240 = 343

Given the sum of X to powers as 4, 10. 30, 100, 354 and

the sum of (x to a power)Y to be 32, 99, 343, we need to set up a matrix.

The matrix to solve would be

4 10 30 32

10 30 100 99

30 100 354 343

The equation would by y = a + bx + cx^2.

a, b, and c are the elements in the matrix, in order.

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