🚀 go-pugleaf

An excellent and well documented method for forming power spectra from
gappy 
data is to use covariances formed from the existing portions of data.  For
example, the alternative methods (and the fortran code to execute them)
can
be found in the book "Digital Spectral Analysis, with applications" by
S.L.
Marple (Prentice-Hall, 1987).

Covariance-based methods are often referred to as "auto-regressive" models
(or AR-spectra;  I can't say why this name has stuck).  Marple reviews the
relative advantages of AR versus "periodogram" (fft) and a few others
(e.g.
Proni's method).

As for the spikes, your on your own.  A method which has worked less
poorly
than others for me in the past is based on using a threshold applied to
the 
difference between each value and the median value in its neighborhood. 
An
unsatisfactory aspect of it is that it has two adjustable (subjective)
parameters (the size of the neighborhood, and the threshold).  A
redeeming 
feature is that there are enough adjustable parameters to make it reject
most
of the points which you would have chosen by eye, if you had the time.

Good luck.

	-jerry smith