A statistical test for distinguishing between two possible processes I am researching a physical domain where I know one of two processes is taking place. I have variables X,Y which should display a different correlation depending on the process taking place. X is set, Y is observed. I want to test my observations to know which of the two processes is taking place. 
Process A should yield $Y(X) = K (constant)$
Process B should yield $Y(X) = [A*sin(4*X+\phi+22.5)+B]/[C*sin(2*X+\phi)+D]$ (pseudo-sinusoidal) 

A,B,C,D,$\phi$ are unknown constants. $\phi$ represents a possible shift of the horizontal axis
Here is my data which I think looks much more like process B than A but I want to give it a statistical significance. 

 A: You can handle this problem as a nonlinear regression analysis, where you posit the regression function in process $B$ along with some appropriate error distribution.  For example, if you assume homoskedastic errors then you might use the model:
$$Y_i = \frac{a \cdot \sin(4x_i + \phi + 22.5) + b}{c \cdot \sin(2x_i + \phi) + d} + \varepsilon_i
\quad \quad \quad \quad \quad
\varepsilon_1, ..., \varepsilon_n \sim \text{IID N}(0, \sigma^2).$$
Process $A$ is a special case of this process when $a = c = 0$.  Thus, you can test whether the additional complexity of process $B$ is required by testing the hypotheses:
$$H_0: a = c = 0
\quad \quad \quad \quad \quad 
H_A: a \neq 0 \text{ or } c \neq 0.$$
If there is significant evidence in favour of the alternative hypothesis (at your chosen significance level) then this means that there is significant evidence in favour of process $B$.  You can use the nls function in R to fit nonlinear regression models, and then you will be able to conduct the required hypothesis test.
