I have a graph which appears to be random but there seems to be a pattern hidden in it. How do I analyze it? The below graph seems to be random but I think the peaks seem to repeat after a certain period. Are there any statistical tools I can use to analyze this?

Edit:
I am adding the plot of its autocorrelation:

 A: The plot that you made looks like using the function xcorr which computes an autocorrelation function without normalization.
$$R(l) = \sum_{\forall k:0\leq k \leq 800 \text{ and } 0\leq k+l \leq 800} R(k)\cdot R(k+l)$$
This is why you get this piramid shape. The number of terms in the sum changes/decline when the lag $l$ gets larger. This makes the sum become less.

If you normalize the function then you should get something like

This function can tell you whether there is a periodic behaviour. For instance there might be peaks repeating every 4 or 8/9 time points.
In my plot you see that the correlation is not very significant (the blue dotted lines point the boundary above/below which the correlations are significant). But, I sampled data from your image. If instead you do this with your raw data then the correlation might be stronger and the pattern could be more clear.
A: If you suspect that there are periodic signals in your data, you can analyse it in the frequency domain.  I would recommend you start by plotting the signal intensity using the intensity function in the ts.extend package in R.  Intensity spikes in the frequency domain indicate periodic signals in the data, and harmonics of these spikes occur when those periodic signals have a shape that is not perfectly sinusoidal.
