Curve fitting for oscillating data

Question

This is my first question. I have the following data that I'd like to approximate as a parametric function:

\begin{align} y = a + (bx_1 + cx_2 + dx_3 + ex_1x_2 + fx_1x_3 + gx_2x_3 + hx_1x_2x_3 + i)*(j*\sin(kx_1x_2x_3) + l\cos(mx_1x_2x_3)) \end{align}

testdata <- read.csv('..path_to_file/gistfile1.txt', sep = "")
plot(1:35,testdata$X0, col = 'blue', pch = 19, ylim = c(0,8),type = "l",lwd = 2, xlab = "x", ylab = "y_i",cex.lab = 2)
lines(1:35,testdata$X1, col = 'red', pch = 19, ylim = c(0,1),lwd = 2)
lines(1:35,testdata$X1.1, col = 'green', pch = 19, ylim = c(0,1),lwd = 2)
lines(1:35,testdata$X8, col = 'magenta', pch = 19, ylim = c(0,1),lwd = 2)
leg.txt <- (c("y","x1","x2","x3"))
legend(18,8,leg.txt, col  =  c('blue','red','green','magenta'), lty = 1,lwd= 2,border = 'white')

:

Any suggestions how to modify the function shown below so as to get a closer fit?

Answer 1

Numerical judgement of model choice:

You model 36 observations with a model consisting of 12 or 13 predictor variables. This is most likely not a good model. Even if you reach a high $R^2_{adj}$, you most likely model a random pattern. Try to compare a computed $AIC$ (Akaike information criterion) or $BIC$ (Bayesian information criterion) of this model to the one of a simple sinus function with 2 parameters. You might see that your model gets rejected.

Explain-ability of model choice:

In general in Statistics you want to be able to explain the interaction of variables in your model. For example you analyze Death Rates $DR_i$ of a population $P_i$. You find out that $DR_i$ increases exponentially with extreme outside temperatures $T$. So you want to fit a model:

$DR_i = T_{Deviation}*(T -T_{mean})²$

Here you would most likely find biological reasons, for example $T_{mean}$ reflects the average temperature of the natural surroundings of $P_i$. The other fitted model parameter $t_{Deviation}$ reflects $P_i$'s resistance (warm or cold blooded). $P_i$ is cold blooded an so its fitted $t_{Deviation}$ is larger than the one of warm blooded species $P_j$.

Re-usability of your model:

Not very often the best looking fit the best statistical fit. Extrapolate your prediction line and see if it acts like you would expect it to do or not. Even if you do not have to predict extrapolated values you can take this as a measurement how well the model fits the natural circumstances. If no extrapolation is possible, most likely a second data set of measurements would not work on your model. Such a model is not transferable.

In a limited number of fields you can ignore such stuff of course. 3-D reconstruction of surfaces in CAD for example.

Online resource to generate models from 2d data sets:

You will find a regression tool for nonlinear fits with up to three variables here. You type in your $x,y$ matrix and it predicts a huge variety of different models. I think earlier versions had AIC included but maybe it is also somewhere else on this page.