This GIT respository contains all files needed for an adequate analysis of the gait (6MWT) accelerometer data.
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 `function [RealignedAcc,RotationMatrixT,Flags] = RealignSensorSignalHRAmp(Acc, FS)` `% History` `% 2013/05/31 SR solve problem of ML-AP rotation of appr. 90 degrees ("Maximum number of function evaluations has been exceeded") ` `% 2013/08/15 SR use literature HR (i.e. amplitude not power, and harmonic frequencies not sin^4 windows) ` `% 2013/08/16 SR correct: use abs(X) instead of sqrt(X.*X)` `% 2013/09/11 SR use only relevant F (containing odd or even harmonics) and speed up determination of harmonic frequencies ` `%% Define VT direction (RVT) as direction of mean acceleration` `MeanAcc = mean(Acc);` `RVT = MeanAcc'/norm(MeanAcc);` `RMLGuess = cross([0;0;1],RVT); RMLGuess = RMLGuess/norm(RMLGuess);` `RAPGuess = cross(RVT,RMLGuess); RAPGuess = RAPGuess/norm(RAPGuess);` `%% Estimate stride frequency` `StrideFrequency = StrideFrequencyFrom3dAcc(Acc, FS);` `AccDetrend = detrend(Acc,'constant');` `N = size(AccDetrend,1);` `%% calculate complex DFT` `F = FS*(0:(N-1))'/N;` `dF = FS/N;` `FHarmRange = [(1:20)'-0.1, (1:20)'+0.1]*StrideFrequency;` `EvenRanges = FHarmRange(2:2:end,:);` `OddRanges = FHarmRange(1:2:end,:);` `EvenHarmonics = zeros(size(F));` `for j=1:size(EvenRanges,1)` ` IXRange = [EvenRanges(j,1)/dF, EvenRanges(j,2)/dF]+1;` ` IXRangeRound = min(N,round(IXRange));` ` EvenHarmonics(IXRangeRound(1):IXRangeRound(2)) = EvenHarmonics(IXRangeRound(1):IXRangeRound(2)) + 1;` ` EvenHarmonics(IXRangeRound(1)) = EvenHarmonics(IXRangeRound(1)) - (IXRange(1)-IXRangeRound(1)+0.5);` ` EvenHarmonics(IXRangeRound(2)) = EvenHarmonics(IXRangeRound(2)) - (IXRangeRound(2)-IXRange(2)+0.5);` `end` `OddHarmonics = zeros(size(F));` `for j=1:size(OddRanges,1)` ` IXRange = [OddRanges(j,1)/dF, OddRanges(j,2)/dF]+1;` ` IXRangeRound = min(N,round(IXRange));` ` OddHarmonics(IXRangeRound(1):IXRangeRound(2)) = OddHarmonics(IXRangeRound(1):IXRangeRound(2)) + 1;` ` OddHarmonics(IXRangeRound(1)) = OddHarmonics(IXRangeRound(1)) - (IXRange(1)-IXRangeRound(1)+0.5);` ` OddHarmonics(IXRangeRound(2)) = OddHarmonics(IXRangeRound(2)) - (IXRangeRound(2)-IXRange(2)+0.5);` `end` `%% select only frequencies that contain odd or even harmonics` `RelevantF = OddHarmonics~=0 | EvenHarmonics~=0;` `OddHarmonics = OddHarmonics(RelevantF);` `EvenHarmonics = EvenHarmonics(RelevantF);` `DFT = fft(AccDetrend,N,1);` `DFT = DFT(RelevantF,:);` `DFTMLG = DFT*RMLGuess;` `DFTAPG = DFT*RAPGuess;` `%% Initial estimation of ML-AP realignment (educated guess)` `% We start of by finding the directions ML = MLguess + x*APguess for which ` `% the 'harmonic ratio' for the ML directions is maximal or minimal, and we ` `% do this by varying x in still the power variant of HR instead of amplitude:` ` HRML = @(x) real(((OddHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG))...` ` /((EvenHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG)));` `% or by substititutions` `a = real((OddHarmonics.*DFTAPG)'*DFTAPG);` `b = real((OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG);` `c = real((OddHarmonics.*DFTMLG)'*DFTMLG);` `d = real((EvenHarmonics.*DFTAPG)'*DFTAPG);` `e = real((EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG);` `f = real((EvenHarmonics.*DFTMLG)'*DFTMLG);` `% and by setting d/dx(HRML(x))==0 we get the quadratic formula` `aq = a*e-b*d;` `bq = 2*a*f-2*c*d;` `cq = b*f-c*e;` `% DT = bq^2-4*aq*cq = 4*(a*f-*c*d)*(a*f-c*d) - 4*(a*e-b*d)*(b*f-c*e)` `% = 4aaff +4ccdd -8acdf -4abef -4bcde +4acee + 4bbdf ` `% = 4aaff +4ccdd -4abef -4bcde +4ac(ee-df) +4(bb-ac)df ` `x = (-bq +[1 -1]*sqrt(bq.^2-4*aq*cq))/(2*aq);` `% with hopefully two solutions for x. ` `% from here on use amplitude HR:` `HRML = @(x) real((OddHarmonics'*abs(DFTMLG+x*DFTAPG))...` ` /(EvenHarmonics'*abs(DFTMLG+x*DFTAPG)));` `% now we take the x for which the HR is maximal, and the x rotated 90` `% degrees fo which it is minimal, and take the mean angle between them` `HR1 = HRML(x(1));` `HR2 = HRML(x(2));` `if HR1 > HR2` ` thetas = atan([x(1),-1/x(2)]);` `else` ` thetas = atan([-1/x(1),x(2)]);` `end` `theta = mean(thetas);` `xeg = tan(theta);` `if abs(diff(thetas)) > pi/2 % thetas are more than 90 degrees apart, and the 'mean of the two axes' should be rotated 90 degrees ` ` xeg = -1/xeg;` `end` `%% Numerical search for best harmonic ratios product` `% Now we want to improve the solution, since the two above directions are ` `% not necessarily orthogonal. Thus we want to find the orthogonal ` `% directions ML = MLguess + x*APguess, and AP = APguess - x*MLguess, ` `% for which the product of 'harmonic ratios' for the ML and AP directions ` `% is maximal, and we do this by varying x, starting with the educated guess` `% above using power instead of amplitude:` `% HRML = ((OddHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG))...` `% /((EvenHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG));` `% HRAP = ((EvenHarmonics'.*(-x*DFTMLG'+DFTAPG'))*(-x*DFTMLG+DFTAPG))...` `% /((OddHarmonics'.*(-x*DFTMLG'+DFTAPG'))*(-x*DFTMLG+DFTAPG));` `% and` `% HR_Prod = HRML*HRAP = ...` `% Now use amplitude` `Minus_HR_Prod = @(x) -real(...` ` (OddHarmonics'*abs(DFTMLG+x*DFTAPG))...` ` /(EvenHarmonics'*abs(DFTMLG+x*DFTAPG))...` ` *(EvenHarmonics'*abs(-x*DFTMLG+DFTAPG))...` ` /(OddHarmonics'*abs(-x*DFTMLG+DFTAPG)));` `[xopt1,Minus_HR_Prod_opt1,exitflag1]=fminsearch(Minus_HR_Prod,xeg);` `if exitflag1 == 0 && abs(xopt1) > 100 % optimum for theta beyond +- pi/2` ` [xopt1a,Minus_HR_Prod_opt1a,exitflag1a]=fminsearch(Minus_HR_Prod,-xopt1);` ` if ~(exitflag1a == 0 && abs(xopt1a) > 100)` ` xopt1org = xopt1;` ` xopt1 = xopt1a;` ` Minus_HR_Prod_opt1 = Minus_HR_Prod_opt1a;` ` end` `end` `options = optimset('LargeScale','off','GradObj','off','Display','notify');` `[xopt2,Minus_HR_Prod_opt2,exitflag2]=fminunc(Minus_HR_Prod,xopt1,options);` `if exitflag2 == 0 && abs(xopt2) > 100 % optimum for theta beyond +- pi/2` ` [xopt2a,Minus_HR_Prod_opt2a,exitflag2a]=fminsearch(Minus_HR_Prod,-xopt2);` ` if ~(exitflag2a == 0 && abs(xopt2a) > 100)` ` xopt2 = xopt2a;` ` Minus_HR_Prod_opt2 = Minus_HR_Prod_opt2a;` ` end` `end` `RML = (RMLGuess+xopt2*RAPGuess)/sqrt(1+xopt2^2);` `RAP = cross(RVT,RML); RAP = RAP/norm(RAP);` `RT = [RVT,RML,RAP];` `if nargout > 1` ` RotationMatrixT = RT;` `end` `RealignedAcc = Acc*RT;` `if nargout > 2` ` Flags = [exitflag1,exitflag2];` `end` `if nargin > 2 && plotje` ` Thetas = pi*(-0.49:0.01:0.49)';` ` Xs = tan(Thetas);` ` HR_Prod = nan(size(Xs));` ` HRMLs = nan(size(Xs));` ` for i=1:numel(Xs),` ` HR_Prod(i,1) = -Minus_HR_Prod(Xs(i));` ` HRMLs(i,1) = HRML(Xs(i));` ` end` ` figure(); ` ` semilogy(Thetas,[HR_Prod,HRMLs.^2]);` ` hold on;` ` semilogy(atan(x),[HR1,HR2].^2,'r.');` ` semilogy(atan([xopt1;xopt2]),-[Minus_HR_Prod_opt1;Minus_HR_Prod_opt2],'g.');` ` semilogy(atan(xeg),-Minus_HR_Prod(xeg),'kx');` `end` `%% attempt to solve it analytically:` `% % We can rewrite HR_Prod as:` `% % HR_Prod = ` `% % ( (x.^2)*((OddHarmonics.*DFTAPG)'*DFTAPG) ...` `% % + x*((OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG)...` `% % + ((OddHarmonics.*DFTMLG)'*DFTMLG) )...` `% % /` `% % ( (x.^2)*((EvenHarmonics.*DFTAPG)'*DFTAPG) ...` `% % + x*((EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG)...` `% % + ((EvenHarmonics.*DFTMLG)'*DFTMLG) )` `% % *` `% % ( (x.^2)*((EvenHarmonics.*DFTMLG)'*DFTMLG) ...` `% % - x*((EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG)...` `% % + ((EvenHarmonics.*DFTAPG)'*DFTAPG) )...` `% % /` `% % ( (x.^2)*((OddHarmonics.*DFTMLG)'*DFTMLG) ...` `% % - x*((OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG)...` `% % + ((OddHarmonics.*DFTAPG)'*DFTAPG) )` `% % ` `% % or by making these substitutions:` `% a2 = (OddHarmonics.*DFTAPG)'*DFTAPG;` `% a1 = (OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG;` `% a0 = (OddHarmonics.*DFTMLG)'*DFTMLG;` `% c2 = (EvenHarmonics.*DFTAPG)'*DFTAPG;` `% c1 = (EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG;` `% c0 = (EvenHarmonics.*DFTMLG)'*DFTMLG;` `% b2 = c0;` `% b1 = c1;` `% b0 = c2;` `% d2 = a0;` `% d1 = a1;` `% d0 = a2;` `% % we get:` `% % HR_Prod = (x.^2*a2 + x*a1 + a0) * (x.^2*b2 + x*b1 + b0) / (x.^2*c2 + x*c1 + c0) * (x.^2*d2 + x*d1 + d0)` `% % or:` `% % HR_Prod = N/D ` `% % with` `% % N = ( x.^4*(a2*b2)...` `% % +x.^3*(a2*b1+a1*b2)...` `% % +x.^2*(a2*b0+a1*b1+a0*b2)...` `% % +x *(a1*b0+a0*b1)...` `% % + a0*b0 )...` `% % and with ` `% % D = ( x.^4*(c2*d2)...` `% % +x.^3*(c2*d1+c1*d2)...` `% % +x.^2*(c2*d0+c1*d1+c0*d2)...` `% % +x *(c1*d0+c0*d1)...` `% % + c0*d0 )` `% % or D = ( x.^4*(a0*b0)...` `% % +x.^3*(a1*b0+a0*b1)...` `% % +x.^2*(a2*b0+a1*b1+a0*b2)...` `% % +x *(a2*b1+a1*b2)...` `% % + a2*b2 )` `% % or by substition of` `% n4 = a2*b2;` `% n3 = a2*b1+a1*b2;` `% n2 = a2*b0+a1*b1+a0*b2;` `% n1 = a1*b0+a0*b1;` `% n0 = a0*b0;` `% % d4 = n0;` `% % d3 = n1;` `% % d2 = n2;` `% % d1 = n3;` `% % d0 = n4;` `% % we get ` `% % N = sum(ni*x^i) and D=sum(di*x^i)` `% % ` `% % Then HR_Prod reaches a maximum or minimum when d/dx (N/D) = 0 ` `% % or (dN/dx)/D - N/(D^2)*(dD/dx) = 0` `% % or (dN/dx)*D - (dD/dx)*N = 0` `% % or` `% % x^7*(4*n4*d4 - 4*n4*d4)` `% % + x^6*(4*n4*d3 - 4*n3*d4 + 3*n3*d4 - 3*n4*d3)` `% % + x^5*(4*n4*d2 - 4*n2*d4 + 3*n3*d3 - 3*n3*d3 + 2*n2*d4 - 2*n4*d2)` `% % + x^4*(4*n4*d1 - 4*n1*d4 + 3*n3*d2 - 3*n2*d3 + 2*n2*d3 - 2*n3*d2 + 1*n1*d4 - 1*n4*d1)` `% % + x^3*(4*n4*d0 - 4*n0*d4 + 3*n3*d1 - 3*n1*d3 + 2*n2*d2 - 2*n2*d2 + 1*n1*d3 - 1*n3*d1)` `% % + x^2*( 3*n3*d0 - 3*n0*d3 + 2*n2*d1 - 2*n1*d2 + 1*n1*d2 - 1*n2*d1)` `% % + x^1*( 2*n2*d0 - 2*n0*d2 + 1*n1*d1 - 1*n1*d1)` `% % + x^0*( 1*n1*d0 - 1*n0*d1)` `% % = 0 ` `% % or` `% % x^6*(1*n4*d3 - 1*n3*d4)` `% % + x^5*(2*n4*d2 - 2*n2*d4)` `% % + x^4*(3*n4*d1 - 3*n1*d4 + 1*n3*d2 - 1*n2*d3)` `% % + x^3*(4*n4*d0 - 4*n0*d4 + 2*n3*d1 - 2*n1*d3)` `% % + x^2*( 3*n3*d0 - 3*n0*d3 + 1*n2*d1 - 1*n1*d2)` `% % + x^1*( 2*n2*d0 - 2*n0*d2)` `% % + x^0*( 1*n1*d0 - 1*n0*d1)` `% % = 0 ` `% % or` `% % (x^6+x^0) * (1*n4*n1 - 1*n3*n0)` `% % + (x^5+x^1) * (2*n4*n2 - 2*n2*n0)` `% % + (x^4+x^2) * (3*n4*n3 - 3*n1*n0 + 1*n3*n2 - 1*n2*n1)` `% % + x^3 * (4*n4*n4 - 4*n0*n0 + 2*n3*n3 - 2*n1*n1)` `% % = 0 ` `% % or` `% % (x^6+x^0) * (1*(a2*b2)*(a1*b0+a0*b1) - 1*(a2*b1+a1*b2)*(a0*b0))` `% % + (x^5+x^1) * (2*(a2*b2)*(a2*b0+a1*b1+a0*b2) - 2*(a2*b0+a1*b1+a0*b2)*(a0*b0))` `% % + (x^4+x^2) * (3*(a2*b2)*(a2*b1+a1*b2) - 3*(a1*b0+a0*b1)*(a0*b0) + 1*(a2*b1+a1*b2)*(a2*b0+a1*b1+a0*b2) - 1*(a2*b0+a1*b1+a0*b2)*(a1*b0+a0*b1))` `% % + x^3 * (4*(a2*b2)*(a2*b2) - 4*(a0*b0)*(a0*b0) + 2*(a2*b1+a1*b2)*(a2*b1+a1*b2) - 2*(a1*b0+a0*b1)*(a1*b0+a0*b1))` `% % = 0 ` `% % or` `% % (x^6+x^0) * (a2*b2*a1*b0 + a2*b2*b1*a0 - a2*b1*a0*b0 - b2*a1*a0*b0)` `% % + (x^5+x^1) * (2*(a2*a2*b2*b0+a2*b2*b2*a0+a2*b2*a1*b1) - 2*(a2*a0*b0*b0+b2*a0*a0*b0+a1*b1*a0*b0))` `% % + (x^4+x^2) * (3*a2*a2*b2*b1 +a2*a2*b1*b0 +3*a2*b2*b2*a1 +a2*b2*a1*b0 +a2*b2*b1*a0 +a2*a1*b1*b1 -a2*a1*b0*b0 -a2*a0*b1*b0 +b2*b2*a1*a0 +b2*a1*a1*b1 -b2*a1*a0*b0 -b2*b1*a0*a0 -a1*a1*b1*b0 -a1*b1*b1*a0 -3*a1*a0*b0*b0 -3*b1*a0*a0*b0)` `% % + x^3 * (4*a2*a2*b2*b2 +2*a2*a2*b1*b1 +4*a2*b2*a1*b1 +2*b2*b2*a1*a1 -2*a1*a1*b0*b0 -4*a1*b1*a0*b0 -2*b1*b1*a0*a0 -4*a0*a0*b0*b0)` `% % = 0 ` `% % which can probably not be solved analytically?.` ``` ``` ``` ```