## Lines, Sines, and Curve Fittings 4 – walk and chew gum

In the three fitting exercises to date, I have first fit one line to the 20th Century data, and then a second to the residuals. In this exercise, I am going to fit two equations simultaneously.

First, the line and sine wave together. I loop around the parameter space of *m* (slope) and *a* (intercept) to fit *y1 = mx + a *while also looping around *A* (amplitude), *b *(phase shift) and *T* (period) to fit *y2 = A * sin ((x-b)/T * (2*pi))*. I add *y1+y2* together and subtract from the data points and looking for the minimum of the residuals squared for the best fit.

In a second case, I look for the best of two sines simultaneously. Similar to the first, I loop around two sets of *A*, *b*, and *T* looking for the least residual squares.

How good is the fit?

The line and sine fit is 0.89

The sine and sine fit is 0.89

In the first set of equations:

m = 0.0046 C/yr

a = 9.0 C

A = 0.09 C

b = 33 yrs

T = 61 yrs

In the second set of equations:

A1 = 0.15 C

b1 = 2 yrs

T1 = 71 yrs

A2 = 0.23 C

b2 = 123 yrs

T2 = 141 yrs

Multiple sines series … oh! oh! … I sense some Fourier analysis in my future. Haven’t tried my hand at that for 20+ years!

The script is here.

I’ve done some Fourier analysis on that same temperature time series. Seems like a nicer way of smoothing (no missing data at the ends).

Hi, Ron. Very nice post.

I have to admit I “peeked ahead” — ran your two models (linear + sine, sine + sine) out through 2020.

If you want to try a different flavor of gum, consider simultaneously fitting exponential + sine models:

y1(t) = y1(0) * exp(kt)

y2(t) = A * sin(((t-b)/T) * 2 * pi)

y(t) = y1(t) + y2(t)

You’ve probably already thought of this, of course.