Asymmetric least squares smoothing


// by tony.withers@uwo.ca, using method of Eilers, PHC and Boelens, HFM 
// (2005) Baseline correction with asymmetric least squares smoothing.

// Creates (and overwrites) w_base, a baseline estimate for w_data. The 
// asymmetry parameter (Eilers and Boelens' p) generally takes values 
// between 0.001 and 0.1. Try varying lambda in orders of magnitude 
// between 10^2 and 10^9. Not efficient for large N, try it for w_data 
// with fewer than 1000 points.
function ALS(w_data, lambda, asymmetry)
	wave w_data
	variable lambda, asymmetry
	
	variable i, N=numpnts(w_data), rms=inf
	variable maxIts=20
	
	matrixOp /free  D = identity(N)
	differentiate /EP=1/METH=2/DIM=0 D
	differentiate /EP=1/METH=2/DIM=0 D

	// this step (specifically the matrix multiplication) is slow:
	matrixOp /free H = lambda * (D^t x D)

	duplicate /o/free w_data w, w_new
	w=1

	for (i=0;i<maxIts;i+=1)
		matrixOp /o/free  C = chol(diagRC(w, N, N)+H)
		matrixOp /o w_base = backwardSub(C,(forwardSub(C^t, w * w_data)))
		w_new = asymmetry * (w_data>w_base) + (1-asymmetry) * (w_data<w_base)
		
		// convergence test
		w-=w_new
		wavestats /Q w
		if (v_rms>=rms)		
			return i+1
		else
			rms=v_rms
			w=w_new
		endif
	endfor
	return 0
end

I changed  the original code from tony a little bit to be able to use it on larger datasets:

 

// Originally developed by tony withers:
//
// by tony.withers@uwo.ca, using method of Eilers, PHC and Boelens, HFM
// (2005) Baseline correction with asymmetric least squares smoothing.

// Creates (and overwrites) w_base, a baseline estimate for w_data. The
// asymmetry parameter (Eilers and Boelens' p) generally takes values
// between 0.001 and 0.1. Try varying lambda in orders of magnitude
// between 10^2 and 10^9. Not efficient for large N, try it for w_data
// with fewer than 1000 points.
//
//
// I just changed the code to avoid the slow matrix multiplication.
// The H-matrix is now constructed "manually". This saves time and memory
// allows larger datasets.
// (kmichel@wzw.tum.de)

function ALS(w_data, lambda, asymmetry)
	wave w_data
	variable lambda, asymmetry
	variable i, N=numpnts(w_data), rms=inf
	variable maxIts=20
   
	//    matrixOp /free  D = identity(N)
	//    differentiate /EP=1/METH=2/DIM=0 D
	//    differentiate /EP=1/METH=2/DIM=0 D

	// 		this step (specifically the matrix multiplication) is slow:
	// 		matrixOp /o H = lambda * (D^t x D)
	
	Make /O/N=(N) diag0 = 6
	wave diag0
	diag0[0]=1
	diag0[1]=5
	diag0[N-2]=5
	diag0[N-1]=1

	Make /O/N=(N-1) diag1 = -4
	wave diag1
	diag1[0]=-2
	diag1[N-2]=-2

	Make /O/N=(N-2) diag2 = 1
	Make /O/N=(N,N) H
	wave  H
	matrixoP/o H = setoffdiag(H,0,diag0)
	matrixop/o H = setoffdiag(H,-1,diag1)
	matrixop/o H = setoffdiag(H,1,diag1)
	matrixop/o H = setoffdiag(H,-2,diag2)
	matrixop/o H = setoffdiag(H,2,diag2)
	matrixop/o H = lambda * H
	killwaves diag0, diag1, diag2
	duplicate /o/free w_data w, w_new
	w=1

	for (i=0;i<maxIts;i+=1)
		matrixOp /o /free  C = chol(diagRC(w, N, N)+H)
		matrixOp /o w_base = backwardSub(C,(forwardSub(C^t, w * w_data)))
		w_new = asymmetry * (w_data>w_base) + (1-asymmetry) * (w_data<w_base)
       
		// convergence test
		w-=w_new
		wavestats /Q w
		if (v_rms>=rms)
			killwaves H    
			return i+1
		else
			rms=v_rms
			w=w_new
		endif
	
	endfor

	return 0
end

 

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