You can have a go at using simple function minimisation: best fit
Start with a vector that represents all your shape characteristics: (amplitude, frequency, phase, offset)
x0 = [1, 10, 0, 0];
Make a function that generates a sine wave over a bunch of time values using those characteristics:
waveFn = @(x,t) x(1) * sin(t * 2*pi*x(2) - x(3)) + x(4);
Now you need to make that function into an objective function for minimisation. That is, a sum of square errors. Let's assume you have your motion values in the vector data and the corresponding time values in t. For this example, I'm just going to invent t.
t = 1:length(data);
objFn = @(x) sum( (data - waveFn(x,t)) .^ 2 );
Then you take a 'reasonable' initial guess, which can be quite unreasonable if you like because this is a simple sine-wave with presumably an obvious primary frequency... And you put that into fminsearch() - the bare-bones swiss army knife function solver.
options = optimset( 'MaxFunEvals', 10000, 'MaxIter', 500 );
x = fminsearch( objFn, x0, options );
Out pops your solved wave parameters, and you can generate the wave to test how good it looks. Hopefully it looks good.
dataFit = waveFn( x, t );
plot( [data(:) dataFit(:)] );
legend( 'Original', 'Fitted' );
You can play with your objective function. Declare a real function instead of an anonymous lazy-function, and you can do whatever you like to compare the data with the fit... You might want to do calculations on variance, or maximum error, or any number of things. Whatever number you spit out, remember that smaller means better.
Hope this is a help, and is actually relevant to what you are trying to achieve. =)
