Savitzky-Golay Filter to smooth noisy data

I'm just posting this to record for posterity something I posted in the chatroom a not-so-long time ago. As I noted there, the following routine will only do smoothing; I had a more general routine for generating the differentiation coefficients, but I still have not been able to find it. (For the more general, but less compact version, see below.) As with Virgil's method (the one in Alexey's answer), this is based on Gorry's procedure (though I have traced the spirit of the algorithm going as far back as Hildebrand's book):

GramP[k_Integer, m_Integer, t_Integer] := (-1)^k HypergeometricPFQ[{-k, k + 1, -t - m}, {1, -2 m}, 1] SavitzkyGolay[n_Integer, m_Integer, t_Integer] := Table[Sum[(Binomial[2 m, k]/Binomial[2 m + k + 1, k + 1]) GramP[k, m, i] GramP[k, m, t] (1 + k/(k + 1)), {k, 0, n}, Method -> "Procedural"], {i, -m, m}] SavitzkyGolay[n_Integer, m_Integer] := Table[SavitzkyGolay[n, m, t], {t, -m, m}] 

Usage is pretty straightforward: n is the order of the polynomial smoothing; 2 m + 1 is the window size, and t tells how much to shift the window.

Added 12/17/2015

Here is a faster routine for evaluating the Gram polynomial, using some undocumented functionality:

GramP[k_Integer, m_Integer, t_Integer] := (-1)^k Internal`DCHypergeometricPFQ[k, {-k, k + 1, -m - t}, {1, -2 m}, 1] 

I managed to finally recover the general SG routine I once wrote through the kind assistance of a friend. To share my joy, I now release this to you:

Options[SavitzkyGolay] = {Derivative -> 0, WorkingPrecision -> Infinity}; SavitzkyGolay[n_Integer?Positive, m_Integer?Positive, t_Integer, OptionsPattern[]] /; 1 < n < 2 m + 1 && -m <= t <= m := Module[{o = OptionValue[Derivative], c, s, h, p, q, u, v, w}, u = UnitVector[o + 1, 1]; v = ConstantArray[0, o + 1]; c = 1/(2 m + 1); s = Join[{Boole[o == 0] c}, Table[h = 0; {p, q} = {2 (2 k - 1), (k - 1) (2 m + k)}/(k (2 m - k + 1)); Do[w = u[[j]]; (* evaluate Gram polynomial and derivatives *) u[[j]] = p (t w + (j - 1) h) - q v[[j]]; v[[j]] = h = w, {j, Min[k, o] + 1}]; c *= (2 m - k + 1) (1 + 1/k)/(2 m + k + 1); c (1 + k/(k + 1)) u[[o + 1]], {k, n}]]; Table[h = s[[n]] + 2 (2 n - 1) (p = s[[n + 1]]) j/(n (2 m - n + 1)); Do[q = p; p = h; (* Clenshaw's recurrence *) h = s[[k]] + 2 (2 k - 1) p j/(k (2 m - k + 1)) - k (2 m + k + 1) q/((k + 1) (2 m - k)), {k, n - 1, 1, -1}]; N[h, OptionValue[WorkingPrecision]], {j, -m, m}] // Developer`ToPackedArray]; SavitzkyGolay[n_Integer?Positive, m_Integer?Positive, opts___?OptionQ] /; 1 < n < 2 m + 1 := Developer`ToPackedArray[Table[SavitzkyGolay[n, m, t, opts], {t, -m, m}]] 

As advertised, it uses no matrices, and instead uses the recurrence relation of the Gram polynomial. If need be, the guts of the routine can be embedded within a Compile[].

Added 12/17/2015

Altho SavitzkyGolayMatrix[] is now built-in in version 10, it is only limited to producing the "central" coefficients, as opposed to the routine SavitzkyGolay[] in this answer that can also generate coefficients for the left and right ends.

SavitzkyGolayMatrix[{2}, 3, 1, WorkingPrecision -> ∞] {1/12, -2/3, 0, 2/3, -1/12} SavitzkyGolay[3, 2, 0 (* central *), Derivative -> 1] {1/12, -2/3, 0, 2/3, -1/12} 

In general, the result of SavitzkyGolayMatrix[] is built from appropriate outer products of coefficient lists.

SavitzkyGolayMatrix[{3, 4}, {2, 3}, WorkingPrecision -> ∞] === Outer[Times, SavitzkyGolay[2, 3, 0], SavitzkyGolay[3, 4, 0]] True 

Several years ago in related MathGroups thread Virgil P. Stokes suggested:

A few years back I wrote a Mathematica notebook that shows how one can obtain the SG smoother from Gram polynomials. The code is not very elegant; but, it is a rather general implementation that should be easy to understand. Contact me if you are interested and I will be glad to forward the notebook to you.

I contacted him and received the notebook. I find his implementation of the Savitzky-Golay filter quite stable and working pretty well. Here I publish it with his permission:

Clear[m, i]; (* m, i are global variables !! *) Clear[GramPolys, LSCoeffs, SGSmooth]; GramPolys[mm_, nmax_] := Module[{k, m = mm}, (* equations (1a), (1b) *) (* Define recursive equation for Gram polynomials as a function of m,i for degrees 0,1,...,nmax *) p[m, 0, i] = 1; p[m, -1, i] = 0; p[m_, k_, i_] := p[m, k, i] = 2*(2*k - 1)/(k*(2*m - k + 1))*i*p[m, k - 1, i] - (k - 1)*(2*m + k)/(k*(2*m - k + 1))*p[m, k - 2, i]; (* Return coefficients for degrees 0,1,...,nmax in a list *) Table[p[mm, k, i] // FullSimplify, {k, 0, nmax}] ]; LSCoeffs[m_, n_, d_] := Module[{k, j, sum, clist, polynomial, cclist}, polynomial = GramPolys[m, n]; clist = {}; Do[(* points in each sliding window *) sum = 0; Do[ (* degree loop *) num = (2 k + 1) FactorialPower[2 m, k]; den = FactorialPower[2 m + k + 1, k + 1]; t1 = polynomial[[k + 1]] /. {i -> j}; t2 = polynomial[[k + 1]]; sum = sum + (num/den)*t1*t2 // FullSimplify; (*Print["k,polynomial[[k+1]]: ",k,", ",polynomial[[k+1]]];*) , {k, 0, n}]; clist = Append[clist, sum]; , {j, -m, m}]; Table[D[clist, {i, d}] /. {i -> j}, {j, -m, m}] ]; SGSmooth[cc_, data_] := Module[{m, y, datal, datar, k, kk, n, yy}, n = Length[data]; m = (Length[cc] - 1)/2; (* Left end --- first 2*m+1 points used *) datal = Take[data, 2*m + 1]; (* Smooth first m points (1,2,...,m-1,m) *) kk = 0; Table[(kk = kk + 1; y[k] = ListConvolve[Reverse[cc[[kk]]], datal][[1]]), {k, -m, -1}]; (* Smooth central points (m+1,m+2,...n-m-1) *) y[0] = ListConvolve[Reverse[cc[[m + 1]]], data]; (* Right end --- last 2*m+1 points used *) datar = Take[data, {n - (2*m + 1) + 1, n}]; (* Smooth last m points (n-m,n-m+1,...,n) *) kk = m + 1; Table[(kk = kk + 1; y[k] = ListConvolve[Reverse[cc[[kk]]], datar][[1]]), {k, 1, m}]; (* And now we concatenate the front-end, central, and back- end estimated data values *) yy = Join[Table[y[k], {k, -m, -1}], y[0], Table[y[k], {k, 1, m}]] ]; 

Usage: SGOutput = SGSmooth[LSCoeffs[m,n,d], data] Inputs: m = half-width of smoothing window; i.e., 2m+1 points in smoothing kernel n = degree of LS polynomial (n < 2m+1) d = order of derivative (d =0, smoother; d = 1, 1st derivative; ...) data = list of uniformly sampled (spaced) data values to be smoothed (length(data) >=2m+1) Outputs: SGOutput = list of smoothed data values 

I came across the code pasted below a couple of years back and have been using it ever since. It has the advantage that you can apply it also to {x,y} data right away. (BTW, it looks like PH Lundow's code below, last updated back in 2007, is in parts very similar to the one posted by John...)
Best,
Frank

PS
I just noticed, that the built-in SavitzkyGolayMatrix function does not seem to be capable of generating higher order (>2) derivatives. So it makes perfectly sense to use a separate package for the task even if it's slower.

(* :Title: Smooth *) (* :Context: "Smooth`" *) (* :Author: P.H. Lundow Bug reports to [email protected] *) (* :Summary: Functions for smoothing equidistant data with a Savitsky-Golay filter. *) (* :History: 020507 Created. 030417 Smooth extended to lists of y-values. 030425 Added SmoothAll. 030619 Added MinimumDistance and EquidistantData. 030923 Small improvement in SGKernel. 070618 Clean-up. *) (* :Mathematica Version: 4.0 *) (* :Keywords: *) (* :Limitations: *) (* :Discussion: There is room for improvement here. Returned list of smoothed data is shorter than original list. Find a better way to produce many kernels that takes care of the margins. *) BeginPackage["Smooth`"] SGKernel::usage = "SGKernel[left, right, degree, derivative] returns the Savitsky-Golay kernel used by the function Smooth when convoluting the data. The kernel has length left+right+1. Left is the number of leftward data points and right is the number of rightward points. Degree refers to the degree of the polynomial and derivative is the order of the desired derivative.\nRef: Numerical Recipes, chapter 14.8." Smooth::usage = "Smooth[list, window, degree, derivative] returns the smoothed form of equally spaced and sorted data. This is done by fitting polynomials of a given degree to a moving window of the list. Argument list is either a list of pairs {{x1,y1},{x2,y2},...} or a list {y1,y2,...} where the x-values are taken to be 1,2,3... If a derivative of the data is desired then give the order of the derivative, default is 0. \nExample:\n a=Table[{x,Sin[x]+Random[Real,{-1,1}],{x,0,2*Pi,0.001}]; \n b=Smooth[a,15,2]; \n c=Smooth[a,15,2,1]; \n This fits 2:nd degree polynomials to moving windows of width 15 to the data. List b is the smoothed data and list c is the derivative of the data." SmoothAll::usage = "SmoothAll[{vector,matrix}, window, degree, derivative] works like Smooth except that vector is a list of x-values and matrix is a list of lists of y-values at the corresponding x-values. The result is returned on the same form.\nExample:\n xold={1,2,3,4,5}; \n yold={{1,3,5,4,4},{2,3,3,2,1},{3,4,6,4,3}}; \n {xnew,ynew}=SmoothAll[{xold,yold},2,1,0];" MinimumDistance::usage = "MinimumDistance[data] returns the minimum distance between two x-values in the data which is a list of lists. The x-values are the first positions of each sublist." EquidistantData::usage = "EquidistantData[data] extracts the largest chunk of equidistant data from a list of lists.\nExample:\n EquidistantData[{{0,1},{2,8},{3,7},{4,9},{6,3}}] returns the list\n {{2,8},{3,7},{4,9}}." Begin["`Private`"] definitions = {MinimumDistance, EquidistantData, SGKernel, Smooth, SmoothAll} Scan[Unprotect, definitions] MinimumDistance[data:{__List}] := Module[{x}, x = Map[First, data]; Min[Drop[x, 1] - Drop[x, -1]] ] EquidistantData[data:{__List}] := Module[{min, len, pos, tmp}, min = MinimumDistance[data]; tmp = Split[data, (#1[[1]] + min == #2[[1]])&]; len = Map[Length, tmp]; pos = Flatten[Position[len, Max[len], Infinity, 1]][[1]]; tmp[[pos]] ] SGKernel[left_?NonNegative, right_?NonNegative, degree_?NonNegative, derivative_?NonNegative] := Module[{i, j, k, l, matrix, vector}, matrix = Table[ (* matrix is symmetric *) l = i + j; If[l == 0, left + right + 1, (*Else*) Sum[k^l, {k, -left, right}] ], {i, 0, degree}, {j, 0, degree} ]; vector = LinearSolve[ matrix, MapAt[1&, Table[0, {degree+1}], derivative+1] ]; (* vector = Inverse[matrix][[derivative + 1]]; *) Table[ vector.Table[If[i == 0, 1, k^i], {i, 0, degree}], {k, -left, right} ] ] /; derivative <= degree <= left+right Smooth[list_?MatrixQ, window_, degree_, derivative_:0]:= Module[{kernel, list1, list2, margin, space}, margin = Floor[window/2]; list1 = Take[ Map[First, list], {margin + 1, Length[list] - margin} ]; list2 = Map[Last, list]; kernel = SGKernel[margin, margin, degree, derivative]; list2 = ListCorrelate[kernel, list2]; (* Data _should_ be equally spaced, but... *) space = Min[Drop[list1, 1] - Drop[list1, -1]]; list2 = list2*(derivative!/space^derivative); Transpose[{list1, list2}] ] /; derivative <= degree <= 2*Floor[window/2] && $VersionNumber >= 4.0 Smooth[list_?VectorQ, window_, degree_, derivative_:0]:= Module[{pairs}, pairs = MapThread[List, {Range[Length[list]], list}]; Map[Last, Smooth[pairs, window, degree, derivative]] ] SmoothAll[{x_?VectorQ, y_?MatrixQ}, window_, degree_, derivative_:0]:= Module[{old, new, tmp}, tmp = Map[( old = Transpose[{x, #}]; new = Smooth[old, window, degree, derivative]; Map[Last, new])&, y ]; {Map[First, new], tmp} ] Scan[Protect, definitions] End[] EndPackage[]