Performing a moving linear fit to 1D data in Python

I have a 1D array of data and wish to extract the spatial variation. The standard way to do this which I wish to pythonize is to perform a moving linear regression to the data and save the gradient...

def nssl_kdp(phidp, distance, fitlen):
    kdp=zeros(phidp.shape, dtype=float)
    myshape=kdp.shape
    for swn in range(myshape[0]):
        print "Sweep ", swn+1
        for rayn in range(myshape[1]):
            print "ray ", rayn+1
            small=[polyfit(distance[a:a+2*fitlen], phidp[swn, rayn, a:a+2*fitlen],1)[0] for a in xrange(myshape[2]-2*fitlen)]
            kdp[swn, rayn, :]=array((list(itertools.chain(*[fitlen*[small[0]], small, fitlen*[small[-1]]]))))
    return kdp

This works well but is SLOW... I need to do this 17*360 times...

I imagine the overhead is in the iterator in the [ for in arange] line... Is there an implimentation of a moving fit in numpy/sci开发者_JS百科py?

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the calculation for linear regression is based on the sum of various values. so you could write a more efficient routine that modifies the sum as the window moves (adding one point and subtracting an earlier one).

this will be much more efficient than repeating the process every time the window shifts, but is open to rounding errors. so you would need to restart occasionally.

you can probably do better than this for equally spaced points by pre-calculating all the x dependencies, but i don't understand your example in detail so am unsure whether it's relevant.

so i guess i'll just assume that it is.

the slope is (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2) where the "2" is "squared" - http://easycalculation.com/statistics/learn-regression.php

for evenly spaced data the denominator is fixed (since you can shift the x axis to the start of the window without changing the gradient). the (ΣX) in the numerator is also fixed (for the same reason). so you only need to be concerned with ΣXY and ΣY. the latter is trivial - just add and subtract a value. the former decreases by ΣY (each X weighting decreases by 1) and increases by (N-1)Y (assuming x_0 is 0 and x_N is N-1) each step.

i suspect that's not clear. what i am saying is that the formula for the slope does not need to be completely recalculated each step. particularly because, at each step, you can rename the X values as 0,1,...N-1 without changing the slope. so almost everything in the formula is the same. all that changes are two terms, which depend on Y as Y_0 "drops out" of the window and Y_N "moves in".

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I've used these moving window functions from the somewhat old scikits.timeseries module with some success. They are implemented in C, but I haven't managed to use them in a situation where the moving window varies in size (not sure if you need that functionality).

http://pytseries.sourceforge.net/lib.moving_funcs.html

Head here for downloads (if using Python 2.7+, you'll probably need to compile the extension itself -- I did this for 2.7 and it works fine):

http://sourceforge.net/projects/pytseries/files/scikits.timeseries/0.91.3/

I/we might be able to help you more if you clean up your example code a bit. I'd consider defining some of the arguments/objects in lines 7 and 8 (where you're defining 'small') as variables, so that you don't end row 8 with so many hard-to-follow parentheses.

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Ok.. I have what seems to be a solution.. not an answer persay, but a way of doing a moving, multi-point differential... I have tested this and the result looks very very similar to a moving regression... I used a 1D sobel filter (ramp from -1 to 1 convolved with the data):

def KDP(phidp, dx, fitlen):
    kdp=np.zeros(phidp.shape, dtype=float)
    myshape=kdp.shape
    for swn in range(myshape[0]):
        #print "Sweep ", swn+1
        for rayn in range(myshape[1]):
            #print "ray ", rayn+1
            kdp[swn, rayn, :]=sobel(phidp[swn, rayn,:], window_len=fitlen)/dx
    return kdp

def sobel(x,window_len=11):
    """Sobel differential filter for calculating KDP
    output:
        differential signal (Unscaled for gate spacing
        example:
    """
    s=np.r_[x[window_len-1:0:-1],x,x[-1:-window_len:-1]]
    #print(len(s))
    w=2.0*np.arange(window_len)/(window_len-1.0) -1.0
    #print w
    w=w/(abs(w).sum())
    y=np.convolve(w,s,mode='valid')
    return -1.0*y[window_len/2:len(x)+window_len/2]/(window_len/3.0)

this runs QUICK!