4Misc_Start4Platform@ xHH@Rg(HHdh xHH@Rg(HHdh x HH@Rg(HHdh ^Graph*WDashSettings#  ! - l6Normal@ Monaco<HHHH$$4 4 4 4 4 4 homey=dMacintoshHD:Users:stu:Desktop:< MacintoshHDFUTH+Desktop E stuFU/MacintoshHD:Users:stu:DesktopDesktop MacintoshHDUsers/stu/Desktop/ RecentWindowsAdvanced Topics.ihfAnalysis.ihfCurve Fitting.ihfDATA2FIT:ELLIPSE_X_DATA,ELLIPSE_Y_DATAErrors.ihfGraph1:ELLIPSE_AML_CONFIDENCE;ELLIPSE_Y_DATA vs ELLIPSE_X_DATA;...Igor Reference.ihf 4Misc_EndTXOPState_Start PeakFunctions2-64fAnalysis.ihf4XOPState_End\P•// - Everything is commented and updated as much as I can except fastGuaranteedEllipseFit •// - 40% speed-up overally by speeding the fastLevenbergMarquardtStep by ~5x by threading on an 8-core machine. •// - The fastGuaranteedEllipseFit also has some iterations over the # data points so should be threaded similarly. •// - There's a bug somewhere where the iterations refuse to converge. Cmd+. and run again and it usually works, but should track this error down and figure out why it doesn't happen all the time, or at all. •// - Update: Fixed the bug where iterations would not converge, it had to do with the threading of the normalized covariance list, which sometimes would only do the first group of threads. In both instances. So set to NaN and iterate until NaN is no longer present. •// - Update: There's still a bug somewhere in the confidence calculation where it seems to thread improperly and return a weird confidence interval and sigmas ~2x what they should be. •Make/O/N=100 ELLIPSE_X_DATA, ELLIPSE_Y_DATA, ELLIPSE_X_FIT_DIR1, ELLIPSE_Y_FIT_DIR1, ELLIPSE_X_FIT_DIR2, ELLIPSE_Y_FIT_DIR2, ELLIPSE_X_FIT_AML, ELLIPSE_Y_FIT_AML, ELLIPSE_X_PERFECT, ELLIPSE_Y_PERFECT •Edit ELLIPSE_X_DATA, ELLIPSE_Y_DATA •DoWindow/C DATA2FIT •Make/O/N=(0,0) ELLIPSE_AML_CONFIDENCE •Make/O/N=(1,1) ELLIPSE_AML_CONFIDENCE •Display /W=(251,45,1028,670) ELLIPSE_Y_DATA vs ELLIPSE_X_DATA •AppendToGraph ELLIPSE_Y_FIT_DIR1 vs ELLIPSE_X_FIT_DIR1 •AppendToGraph ELLIPSE_Y_FIT_DIR2 vs ELLIPSE_X_FIT_DIR2 •AppendToGraph ELLIPSE_Y_FIT_AML vs ELLIPSE_X_FIT_AML •AppendToGraph ELLIPSE_Y_PERFECT vs ELLIPSE_X_PERFECT •AppendImage ELLIPSE_AML_CONFIDENCE •ModifyImage ELLIPSE_AML_CONFIDENCE ctab= {0.75,1,PastelsMap20,1} •ModifyGraph width={perUnit,100,bottom},height={perUnit,100,left} •ModifyGraph mode(ELLIPSE_Y_DATA)=3,marker(ELLIPSE_Y_DATA)=1,msize(ELLIPSE_Y_DATA)=4,mrkThick(ELLIPSE_Y_DATA)=2,rgb(ELLIPSE_Y_DATA)=(17476,17476,17476),rgb(ELLIPSE_Y_FIT_DIR2)=(2,39321,1) •ModifyGraph lSize(ELLIPSE_Y_FIT_DIR1)=2,lSize(ELLIPSE_Y_FIT_DIR2)=2,lSize(ELLIPSE_Y_FIT_AML)=2 •ModifyGraph lStyle(ELLIPSE_Y_FIT_DIR1)=2,lStyle(ELLIPSE_Y_FIT_DIR2)=3,lStyle(ELLIPSE_Y_FIT_AML)=8 •ModifyGraph lStyle(ELLIPSE_Y_PERFECT)=1 •ModifyGraph rgb(ELLIPSE_Y_FIT_AML)=(1,4,52428),rgb(ELLIPSE_Y_PERFECT)=(13107,13107,13107) •Legend/C/N=text0/J/B=(65535,65535,65535,32768)/A=MC/X=-21.75/Y=-43.93 "\\s(ELLIPSE_Y_DATA) Data to Fit\n\\s(ELLIPSE_Y_PERFECT) Ideal Ellipse" •AppendText "\\s(ELLIPSE_Y_FIT_DIR1) Direct: Fitzgibbon \\f02et al\\f00. (1997)\r\\s(ELLIPSE_Y_FIT_DIR2) Direct: Halíř & Flusser (1998)" •AppendText "\\s(ELLIPSE_Y_FIT_AML) Approx. M-L: Szpak \\f02et al\\f00. (2015)" •ColorScale/C/N=text1/B=(65535,65535,65535,32768)/A=MC/X=39.75/Y=12.65 •ColorScale/C/N=text1 image=ELLIPSE_AML_CONFIDENCE, axisRange={0.75,1,0} •AppendText "\\f02p\\f00-value or ~confidence (ℙ data were drawn from an ellipse in this band)" •// •//At this point: Everything has updated comments and has been as threaded as I can make it. One commenting/formatting exception is fastGuaranteedEllipseFit. There is a bug SOMEWHERE with an out of index issue. •//At this point: •// - Everything has updated comments and has been as threaded as I can make it. One commenting/formatting exception is fastGuaranteedEllipseFit. •// - There *might* be a bug somewhere having to do with precision relative to Matlab, search for "precision" in the procedure and read the notes. •// - There *is* a bug somewhere having to do with threading, for when threads = # CPU, results are different than when threads = any value < # CPU. •// •//To Run: ellipseforModeling(2, 1, 5, 3, 110, -35, 85, 0.01, ELLIPSE_X_DATA, ELLIPSE_Y_DATA);ellipseforModeling(2, 1, 5, 3, 110, 0, 360, 0.0, ELLIPSE_X_PERFECT, ELLIPSE_Y_PERFECT);EllipseFitDIR1(mean(ELLIPSE_X_DATA),mean(ELLIPSE_Y_DATA), ELLIPSE_X_DATA, ELLIPSE_Y_DATA);EllipseFitDIR2(mean(ELLIPSE_X_DATA),mean(ELLIPSE_Y_DATA), ELLIPSE_X_DATA, ELLIPSE_Y_DATA);EllipseFitAML(mean(ELLIPSE_X_DATA),mean(ELLIPSE_Y_DATA), ELLIPSE_X_DATA, ELLIPSE_Y_DATA);ELLIPSE_AML_CONFIDENCE = ELLIPSE_AML_CONFIDENCE > 0.99999 ? NaN : ELLIPSE_AML_CONFIDENCE;KillWaves/A/Z;KillVariables/A/Z;KillStrings/A/Z •// NOTE: You MUST have at least 6 pairs of {x,y} points, otherwise the ellipse is under-determined. •// •ellipseforModeling(2, 1, 5, 3, 110, -45, 95, 0.015, ELLIPSE_X_DATA, ELLIPSE_Y_DATA);ellipseforModeling(2, 1, 5, 3, 110, 0, 360, 0.0, ELLIPSE_X_PERFECT, ELLIPSE_Y_PERFECT);EllipseFitDIR1(mean(ELLIPSE_X_DATA),mean(ELLIPSE_Y_DATA), ELLIPSE_X_DATA, ELLIPSE_Y_DATA);EllipseFitDIR2(mean(ELLIPSE_X_DATA),mean(ELLIPSE_Y_DATA), ELLIPSE_X_DATA, ELLIPSE_Y_DATA);EllipseFitAML(mean(ELLIPSE_X_DATA),mean(ELLIPSE_Y_DATA), ELLIPSE_X_DATA, ELLIPSE_Y_DATA);ELLIPSE_AML_CONFIDENCE = ELLIPSE_AML_CONFIDENCE > 0.99999 ? NaN : ELLIPSE_AML_CONFIDENCE;KillWaves/A/Z;KillVariables/A/Z;KillStrings/A/Z Ellipse Parameters from Fitzgibbon et al. (1997) "Direct" Method: x0: 5.11772 y0: 2.81997 a÷2: 0.793925 b÷2: 0.446691 theta: 102.72 Ellipse Parameters from Halíř & Flusser (1998) "Direct" Method: x0: 5.11772 y0: 2.81997 a÷2: 0.793925 b÷2: 0.446691 theta: 102.72 Time to fit the ellipse was 2.76715 seconds. Ellipse Parameters from Szpak et al. (2015) "AML" Method: x0: 5.01441 ± 0.0404135 y0: 2.97887 ± 0.0639208 a÷2: 0.972639 ± 0.0733425 b÷2: 0.497692 ± 0.0189362 theta: 109.193 ± 1.75838 ! 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E?WFb?Yb=??ʙT?8?h?^z?:FQ?#](?P??mЫ?jՁ?}W?dxj-?r'N?l0?:`Ű?QZ?*_?Er+7?c?BH?>?L?LWvB{?ۯX?07?*Y?cy?a???Vب??K|?Si?; X?>I?0}c major) Variable switcher = minor minor = major major = switcher f_flipped = 1 endif //print "semi-major",major //print "semi-minor",minor //Calculate the tilt angle of the major axis in terms of counter-clockwise rotation // from the x-axis. NOTE that this requires the inverse cotangent function, but Igor // apparently does not have that. So I used the definition of inverse cotangent (uses // i (as in SQRT(-1)) and natural logs) in order to compute this. NOTE that the if() // statements come from equation 23 from http://mathworld.wolfram.com/Ellipse.html // and do not account for Igor's own idiosyncrasies. Variable angle = 0 Make/O/N=1/C aa; aa=sqrt(-1) if(abs(b) <= 1e-10) if(a < c) angle = 0 else angle = 1/2*pi endif else Variable dummy=(a-c)/(2*b) if(a < c) angle = real( 1/2 * aa[0]/2*(ln((dummy-aa[0])/dummy)-ln((dummy+aa[0])/dummy))) else angle = real(pi/2 + 1/2 * aa[0]/2*(ln((dummy-aa[0])/dummy)-ln((dummy+aa[0])/dummy))) endif endif if(f_flipped == 1) angle -= pi/2 endif if(angle < 0) angle += pi endif //print "angle",angle*180/pi //print "eccentricity",sqrt(1-minor/major) //print "ellipticity",major/minor //***** STORE THE VALUES *****// printf, "Ellipse Parameters from Fitzgibbon et al. (1997) \"Direct\" Method:\r\tx0: %g\r\ty0: %g\r\ta÷2: %g\r\tb÷2: %g\r\ttheta: %g\r", x0, y0, major, minor, angle*180./pi ellipseforModeling(major*2., minor*2., x0, y0, angle*180./pi, 0, 360, 0, ELLIPSE_X_FIT_DIR1, ELLIPSE_Y_FIT_DIR1) //LATITUDE_ELLIPSE_IMAGE[i_counter_crater] = y0/map_projection+y_center_mass_degrees //LONGITUDE_ELLIPSE_IMAGE[i_counter_crater] = x0/(map_projection*cos(y0*pi/180))+x_center_mass_degrees //DIAM_ELLIPSE_MAJOR_IMAGE[i_counter_crater] = major*2. //DIAM_ELLIPSE_MINOR_IMAGE[i_counter_crater] = minor*2. //DIAM_ELLIPSE_ANGLE_IMAGE[i_counter_crater] = angle*180./pi //DIAM_ELLIPSE_ECCEN_IMAGE[i_counter_crater] = sqrt(1-minor^2/major^2) //DIAM_ELLIPSE_ELLIP_IMAGE[i_counter_crater] = major/minor End ThreadSafe Function EllipseFitDIR2(x_center_mass_degrees, y_center_mass_degrees, rim_lon_temp, rim_lat_temp) Variable x_center_mass_degrees, y_center_mass_degrees Wave rim_lon_temp, rim_lat_temp Wave ELLIPSE_X_FIT_DIR2 = root:ELLIPSE_X_FIT_DIR2 Wave ELLIPSE_Y_FIT_DIR2 = root:ELLIPSE_Y_FIT_DIR2 //Variable map_projection = 2*PI*d_planet_radius/360 //scaling factor //***** Perform the Ellipse Fit. *****// //Calculate the "design matrix" for all points. This comes from the quadratic equation for an // ellipse, where the first term is x^2, second is x*y, third is y^2, fourth is x, fifth is y, and // the last is just 1. The coefficients in each (what we're solving for) are just a, b, c, d, e, and f // (those will be explicitly mentioned later on). //This differs from the Fitzgibbon et al. (1997) approach of one matrix: Here, the authors have // decoupled the matrix into its quadratic and its linear portions ("1" and "2" in my code). Make/D/O/N=(numpnts(rim_lat_temp),3) DesignMatrix1, DesignMatrix2 MultiThread DesignMatrix1[][0] = rim_lon_temp[p]*rim_lon_temp[p] MultiThread DesignMatrix1[][1] = rim_lon_temp[p]*rim_lat_temp[p] MultiThread DesignMatrix1[][2] = rim_lat_temp[p]*rim_lat_temp[p] MultiThread DesignMatrix2[][0] = rim_lon_temp[p] MultiThread DesignMatrix2[][1] = rim_lat_temp[p] MultiThread DesignMatrix2[][2] = 1 //Some matrix manipulation. Read the paper. As with the other matricies, the scatter matrix is // split apart from the Fitzgibbon et al. (1997) implementation. Make/O M_Inverse,W_eigenValues //have to declare these when rtGlobals = 3 MatrixOP/O ScatterMatrix1 = DesignMatrix1^t x DesignMatrix1 MatrixOP/O ScatterMatrix2 = DesignMatrix1^t x DesignMatrix2 MatrixOP/O ScatterMatrix3 = DesignMatrix2^t x DesignMatrix2 MatrixOP/O TempMatrix = -inv(ScatterMatrix3) x ScatterMatrix2^t MatrixOP/O TempMatrixM = ScatterMatrix1 + ScatterMatrix2 x TempMatrix //Effectively multiply TempMatrixM by the inverse of the Constraint Matrix. Make/O/N=(3,3)/D M_product M_product[0][] = TempMatrixM[2][q]/2;M_product[1][] = -TempMatrixM[1][q];M_product[2][] = TempMatrixM[0][q]/2 //Calculate the eigenvalues and vectors of the matrix in the last line above. Make/O M_R_eigenVectors //this is required for the steps below once this routine was changed from Macro to Function MatrixEigenV/R M_product // /R here calculates the eigenvectors (Right eigenvectors), too Redimension/R W_eigenValues //the returned eigenvalues are real and imaginary, but we only want the real component //Calculate a "condition" (see paper ... though I can't find it in the paper, but it's line 11 of the Matlab code). Make/D/O/N=(3,3) ConditionMatrix = 0 ConditionMatrix[][0] = 4*M_R_eigenVectors[0][q] * M_R_eigenVectors[2][q] - M_R_eigenVectors[1][q]^2 //***** Determine the values of the fit parameters. *****// //Fitzgibbon et al. (1997) wanted the maximum eigenvector value. What this paper // wants instead is the eigenvector corresponding to the minimum positive eigenvalue W_eigenValues = W_eigenValues <= 0 ? NaN : W_eigenValues WaveStats/Q W_eigenValues //It defines the matrix of a/b/c/d/e/f parameters in two parts. a1 is a/b/c and is: Make/D/O/N=(3) a1 a1[0] = M_R_eigenVectors[0][V_maxRowLoc] a1[1] = M_R_eigenVectors[1][V_maxRowLoc] a1[2] = M_R_eigenVectors[2][V_maxRowLoc] //a2 is based on TempMatrix * a1: Make/D/O/N=(3) a2; MatrixOP/O a2 = TempMatrix x a1 //Snag the ellipse quadratic coefficients from the eigenvectors. Variable a = a1[0] Variable b = a1[1]/2. Variable c = a1[2] Variable d = a2[0]/2. Variable e = a2[1]/2. Variable f = a2[2] //print a, b, c, d, e, f //Calculate the center of the ellipse. Variable x0 = (c*d-b*e)/(b^2-a*c) //print "x center",x0 Variable y0 = (a*e-b*d)/(b^2-a*c) //print "y center",y0 //Calculate the SEMI-major and -minor axes (see equations 21 and 22 from http://mathworld.wolfram.com/Ellipse.html ). Variable major = Sqrt((2*(a*e^2+c*d^2+f*b^2-2*b*d*e-a*c*f)) / ( (b^2-a*c) * ( sqrt((a-c)^2+4*b^2)-(a+c)) ) ) Variable minor = Sqrt((2*(a*e^2+c*d^2+f*b^2-2*b*d*e-a*c*f)) / ( (b^2-a*c) * (-sqrt((a-c)^2+4*b^2)-(a+c)) ) ) Variable f_flipped = 0 if(minor > major) Variable switcher = minor minor = major major = switcher f_flipped = 1 endif //print "semi-major",major //print "semi-minor",minor //Calculate the tilt angle of the major axis in terms of counter-clockwise rotation // from the x-axis. NOTE that this requires the inverse cotangent function, but Igor // apparently does not have that. So I used the definition of inverse cotangent (uses // i (as in SQRT(-1)) and natural logs) in order to compute this. NOTE that the if() // statements come from equation 23 from http://mathworld.wolfram.com/Ellipse.html // and do not account for Igor's own idiosyncrasies. Variable angle = 0 Make/O/N=1/C aa; aa=sqrt(-1) if(abs(b) <= 1e-10) if(a < c) angle = 0 else angle = 1/2*pi endif else Variable dummy=(a-c)/(2*b) if(a < c) angle = real( 1/2 * aa[0]/2*(ln((dummy-aa[0])/dummy)-ln((dummy+aa[0])/dummy))) else angle = real(pi/2 + 1/2 * aa[0]/2*(ln((dummy-aa[0])/dummy)-ln((dummy+aa[0])/dummy))) endif endif if(f_flipped == 1) angle -= pi/2 endif if(angle < 0) angle += pi endif //print "angle",angle*180/pi //print "eccentricity",sqrt(1-minor/major) //print "ellipticity",major/minor //***** STORE THE VALUES *****// printf, "Ellipse Parameters from Halíř & Flusser (1998) \"Direct\" Method:\r\tx0: %g\r\ty0: %g\r\ta÷2: %g\r\tb÷2: %g\r\ttheta: %g\r", x0, y0, major, minor, angle*180./pi ellipseforModeling(major*2., minor*2., x0, y0, angle*180./pi, 0, 360, 0, ELLIPSE_X_FIT_DIR2, ELLIPSE_Y_FIT_DIR2) //LATITUDE_ELLIPSE_IMAGE[i_counter_crater] = y0/map_projection+y_center_mass_degrees //LONGITUDE_ELLIPSE_IMAGE[i_counter_crater] = x0/(map_projection*cos(y0*pi/180))+x_center_mass_degrees //DIAM_ELLIPSE_MAJOR_IMAGE[i_counter_crater] = major*2. //DIAM_ELLIPSE_MINOR_IMAGE[i_counter_crater] = minor*2. //DIAM_ELLIPSE_ANGLE_IMAGE[i_counter_crater] = angle*180./pi //DIAM_ELLIPSE_ECCEN_IMAGE[i_counter_crater] = sqrt(1-minor^2/major^2) //DIAM_ELLIPSE_ELLIP_IMAGE[i_counter_crater] = major/minor End //Szpak likes to have everything normalized into a "unit" box (doesn't actually // get you to a unit cube, but close). He says that this improves the numerical // stability of the calculations. The math seems to be based on Hartley (1997). //Created July 19, 2018. ThreadSafe Function normalizeData(ELLIPSE_X, ELLIPSE_Y, Matrix_Scaling, Matrix_DataNorm) Wave ELLIPSE_X, ELLIPSE_Y, Matrix_Scaling, Matrix_DataNorm //Create a 3 x N matrix where the first row is all X coordinates, second is Y, // and third is 1. Make/D/O/N=(3,numpnts(ELLIPSE_X)) points MultiThread points[0][] = ELLIPSE_X[q] MultiThread points[1][] = ELLIPSE_Y[q] MultiThread points[2][] = 1 //Calculate the average values of X and Y. Variable meanX = mean(ELLIPSE_X) Variable meanY = mean(ELLIPSE_Y) //Calculate an isotropic scaling factor. Duplicate/D/O ELLIPSE_X ELLIPSE_X_TEMP; ELLIPSE_X_TEMP -= meanX; ELLIPSE_X_TEMP *= ELLIPSE_X_TEMP //X_TEMP now equals (x - x_0)^2 Duplicate/D/O ELLIPSE_Y ELLIPSE_Y_TEMP; ELLIPSE_Y_TEMP -= meanY; ELLIPSE_Y_TEMP *= ELLIPSE_Y_TEMP //Y_TEMP now equals (y - y_0)^2 Duplicate/D/O ELLIPSE_X_TEMP ELLIPSE_TEMP; ELLIPSE_TEMP += ELLIPSE_Y_TEMP //ELLIPSE_TEMP now equals (x - x_0)^2 + (y - y_0)^2 Variable scalingFactor = sqrt( 1./(2*numpnts(ELLIPSE_X)) * sum(ELLIPSE_TEMP) ) //Calculate a scaling matrix. Matrix_Scaling[0][0] = 1./scalingFactor Matrix_Scaling[0][1] = 0 Matrix_Scaling[0][2] = -1./scalingFactor * meanX Matrix_Scaling[1][0] = 0 Matrix_Scaling[1][1] = 1./scalingFactor Matrix_Scaling[1][2] = -1./scalingFactor * meanY Matrix_Scaling[2][0] = 0 Matrix_Scaling[2][1] = 0 Matrix_Scaling[2][2] = 1.0 //Scale the points. Make/D/O/N=(3,numpnts(ELLIPSE_X)) M_product MatrixOP/O M_product = Matrix_Scaling x points //Remove homogenous points. MatrixOP/O M_product = M_product^t Redimension/N=(numpnts(ELLIPSE_X),2) M_product Duplicate/O M_product Matrix_DataNorm End //See http://www.users.on.net/~zygmunt.szpak/sourcecode.html and http://www.users.on.net/~zygmunt.szpak/ellipsefitting.html //Created 18-29 Jul 2018 by converting Matlab code to Igor. Optimizations and extra comments 29 Jul 2018. Function EllipseFitAML(x_center_mass_degrees, y_center_mass_degrees, rim_lon_temp, rim_lat_temp) Variable x_center_mass_degrees, y_center_mass_degrees Wave rim_lon_temp, rim_lat_temp Wave ELLIPSE_X_FIT_AML = root:ELLIPSE_X_FIT_AML Wave ELLIPSE_Y_FIT_AML = root:ELLIPSE_Y_FIT_AML Wave ELLIPSE_X_PERFECT = root:ELLIPSE_X_PERFECT Wave ELLIPSE_Y_PERFECT = root:ELLIPSE_Y_PERFECT //Variable map_projection = 2*PI*d_planet_radius/360 //scaling factor Variable numberOfPoints = numpnts(rim_lon_temp) //***** SET AUTOMATIC MUTLI-THREADING MODE & OTHER THREADING *****// //Several of Igor's built-in functions are automatically, but the threading only // kicks in under certain thresholds and in certain functions. Importantly, // the automatic threading exists in MatrixOP, which is heavily used in this // ellipse-fitting method. In tests, even with 1000 points mode 1 versus 8 did // nothing, really, to change the speed. However, this code is retained in case // that were to change. //MultiThreadingControl getMode //writes out to V_autoMultiThread //MultiThreadingControl setMode=1 //mode 8 "enables automatic multithreading unconditionally - regardless of thresholds or the type of the calling thread" //Multithreading setup: Determine how many threads we can spawn, equal to the // number of processors (generally). This assumes that I will *ALWAYS* have // ≥data points than threads available. //IMPORTANT NOTE: In tests, if the nthreads is set to the number of CPUs, there // *will*be* slightly different results at the end. If it's set to anything less // than the number of CPUs, it's fine. At this point, I'm going to leave that to // a future debugger (sorry Stuart of the Future). Variable nthreads = min(ThreadProcessorCount-1,numberOfPoints), tgs, ti, dummy_forThreadGroupRelease Variable thread_start_datum, thread_end_datum //Multithreading requires overhead to manage the threads, check for when one is // free, etc., and so if you're not smrt about it, it can manage to take longer // then doing things on a single thread. To mitigate some of that, what we can // do is we can pre-emptively select ranges of points to farm to a single thread // instead of just doing things one at a time and sending the next item to the // thread. Things are easiest if the number of points (since that's the // majority of the threading here) is evenly divisible by the number of threads, // but that's not always possible. Another factor is sometimes the math is // easier on one thread than others, so things don't end evenly. So I try to // set up things such that the data are divided into 2-3x the number of threads // available ... if there's enough data. Variable thread_interval_datum = numberOfPoints > 2*nthreads ? floor(numberOfPoints/(2*nthreads)) : 1 //***** TIMER *****// Variable timer_start = stopMSTimer(-2) //create a timer //***** CALCULATE INITIAL ELLIPSE PARAMETERS FROM DIR2 TO USE AS A SEED *****// //Extra scaling. Make/D/O/N=(3,3) Matrix_Scaling Make/D/O/N=(numberOfPoints,2) Matrix_DataNorm normalizeData(rim_lon_temp, rim_lat_temp, Matrix_Scaling, Matrix_DataNorm) //Compressed DIR2 code. Make/D/O/N=(numberOfPoints,3) DesignMatrix1, DesignMatrix2 MultiThread DesignMatrix1[][0] = Matrix_DataNorm[p][0]*Matrix_DataNorm[p][0] MultiThread DesignMatrix1[][1] = Matrix_DataNorm[p][0]*Matrix_DataNorm[p][1] MultiThread DesignMatrix1[][2] = Matrix_DataNorm[p][1]*Matrix_DataNorm[p][1] MultiThread DesignMatrix2[][0] = Matrix_DataNorm[p][0] MultiThread DesignMatrix2[][1] = Matrix_DataNorm[p][1] MultiThread DesignMatrix2[][2] = 1 Make/D/O M_Inverse,W_eigenValues MatrixOP/O ScatterMatrix1 = DesignMatrix1^t x DesignMatrix1 MatrixOP/O ScatterMatrix2 = DesignMatrix1^t x DesignMatrix2 MatrixOP/O ScatterMatrix3 = DesignMatrix2^t x DesignMatrix2 MatrixOP/O TempMatrix = -inv(ScatterMatrix3) x ScatterMatrix2^t MatrixOP/O TempMatrixM = ScatterMatrix1 + ScatterMatrix2 x TempMatrix Make/O/N=(3,3)/D M_product MultiThread M_product[0][] = TempMatrixM[2][q]/2;MultiThread M_product[1][] = -TempMatrixM[1][q];MultiThread M_product[2][] = TempMatrixM[0][q]/2 Make/D/O M_R_eigenVectors MatrixEigenV/R M_product Redimension/R W_eigenValues Make/D/O/N=(3,3) ConditionMatrix = 0 MultiThread ConditionMatrix[][0] = 4*M_R_eigenVectors[0][q] * M_R_eigenVectors[2][q] - M_R_eigenVectors[1][q]^2 W_eigenValues = W_eigenValues <= 0 ? NaN : W_eigenValues WaveStats/Q W_eigenValues Make/D/O/N=(3) a1 a1[0] = M_R_eigenVectors[0][V_maxRowLoc] a1[1] = M_R_eigenVectors[1][V_maxRowLoc] a1[2] = M_R_eigenVectors[2][V_maxRowLoc] Make/D/O/N=(3) a2; MatrixOP/O a2 = TempMatrix x a1 Make/D/O/N=6 initialEllipseParameters initialEllipseParameters[0,2] = a1[p] initialEllipseParameters[3,5] = a2[p-3] //Extra scaling. Make/D/O/N=(3,3) YetanotherTempMatrix YetanotherTempMatrix[0][0] = initialEllipseParameters[0] YetanotherTempMatrix[0][1] = initialEllipseParameters[1]/2 YetanotherTempMatrix[0][2] = initialEllipseParameters[3]/2 YetanotherTempMatrix[1][0] = initialEllipseParameters[1]/2 YetanotherTempMatrix[1][1] = initialEllipseParameters[2] YetanotherTempMatrix[1][2] = initialEllipseParameters[4]/2 YetanotherTempMatrix[2][0] = initialEllipseParameters[3]/2 YetanotherTempMatrix[2][1] = initialEllipseParameters[4]/2 YetanotherTempMatrix[2][2] = initialEllipseParameters[5] MatrixOP/O YetanotherTempMatrix = Matrix_Scaling^t x YetanotherTempMatrix x Matrix_Scaling initialEllipseParameters[0] = YetanotherTempMatrix[0][0] initialEllipseParameters[1] = YetanotherTempMatrix[0][1]*2. initialEllipseParameters[2] = YetanotherTempMatrix[1][1] initialEllipseParameters[3] = YetanotherTempMatrix[0][2]*2. initialEllipseParameters[4] = YetanotherTempMatrix[1][2]*2. initialEllipseParameters[5] = YetanotherTempMatrix[2][2] Variable myMatrixNorm = norm(initialEllipseParameters) initialEllipseParameters /= myMatrixNorm //the "initialEllipseParameters" are the scaled parameters from the DIR2 method //***** PERFORM THE ELLIPSE FIT: NORMALIZE (§7 of paper) *****// //Create a diagonal matrix. Make/D/O/N=(6,6) Matrix_E; MultiThread Matrix_E = 0 Matrix_E[0][0] = 1 Matrix_E[1][1] = 0.5 Matrix_E[2][2] = 1 Matrix_E[3][3] = 0.5 Matrix_E[4][4] = 0.5 Matrix_E[5][5] = 1 //Create a permutation matrix for interchanging the 3rd and 4th entries of a // length-6 vector, which requires the Kronecker tensor product. Matlab gives // one result, Igor another, and my math, yet another. Since this is a static, // pre-defined matrix, it can be easily hard-coded. Make/D/O/N=(6,6) Matrix_Permutation; MultiThread Matrix_Permutation = 0 Matrix_Permutation[0][0] = 1 Matrix_Permutation[1][1] = 1 Matrix_Permutation[2][3] = 1 Matrix_Permutation[3][2] = 1 Matrix_Permutation[4][4] = 1 Matrix_Permutation[5][5] = 1 //Create a "duplication" matrix. Also hard-coded. Make/D/O/N=(9,6) Matrix_Duplication; MultiThread Matrix_Duplication = 0 Matrix_Duplication[0][0] = 1 Matrix_Duplication[1][1] = 1 Matrix_Duplication[2][2] = 1 Matrix_Duplication[3][1] = 1 Matrix_Duplication[4][3] = 1 Matrix_Duplication[5][4] = 1 Matrix_Duplication[6][2] = 1 Matrix_Duplication[7][4] = 1 Matrix_Duplication[8][5] = 1 //Do some stuff that's not explained. Make/D/O/N=(6,9) initialEllipseNormalizedSpace //The first matrix operation in Matlab is "E \ Matrix_Permutation" which solves // the equation A*x=b for x. Igor can do that, but of course it's complicated. // In this particular case, all the matricies at the beginning of eq. 7.1-7.2 // are pre-defined, so the solution (initialEllipseNormalizedSpace) can be hard- // coded. initialEllipseNormalizedSpace = 0 initialEllipseNormalizedSpace[0][0] = 1 initialEllipseNormalizedSpace[1][1] = 1 initialEllipseNormalizedSpace[1][3] = 1 initialEllipseNormalizedSpace[2][4] = 1 initialEllipseNormalizedSpace[3][2] = 1 initialEllipseNormalizedSpace[3][6] = 1 initialEllipseNormalizedSpace[4][5] = 1 initialEllipseNormalizedSpace[4][7] = 1 initialEllipseNormalizedSpace[5][8] = 1 //The remainder of equation 7.2, plus normalization. MatrixOP/O initialEllipseNormalizedSpace = initialEllipseNormalizedSpace x inv(tensorProduct(Matrix_Scaling,Matrix_Scaling))^t x Matrix_Duplication x Matrix_Permutation x Matrix_E x initialEllipseParameters myMatrixNorm = norm(initialEllipseNormalizedSpace) initialEllipseNormalizedSpace /= myMatrixNorm //***** PERFORM THE ELLIPSE FIT: COVARIANCE (§5 of paper) *****// //Create a covariance list/matrix. Note that a covariance matrix could be, // instead, passed to this function. If it's not, then we assume it's Gaussian // distributed and, basically, a constant of 1 ... I think. // MATLAB CODE: covList = mat2cell(repmat(eye(2),1,nPts),2,2.*(ones(1,nPts))) Make/D/O/N=(2,2,numberOfPoints) covList; covList = 0 MultiThread covList[0][0][] = 1 MultiThread covList[1][1][] = 1 //Becuase the data are now in a new normalized coordinate system, the data // covariance matricies also need to be transformed into the new normalized // coordinate system. The transformation of the covariance matricies into the // new coordinate system can be achieved by embedding the covariance matricies in // a 3x3 matrix (by padding the 2x2 covariance matricies by zeros) and by // multiplying the covariance matricies by Matrix_Scaling from the left and // Matrix_Scaling^t from the right. Make/D/O/N=(2,2,numberOfPoints) normalized_CovList; MultiThread normalized_CovList = NaN Variable counter_datum = 0, i_thread = 0, mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum //intialize variables for the new threading do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_NormalizeCovariances(covList, normalized_CovList, Matrix_Scaling, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_NormalizeCovariances(covList, normalized_CovList, Matrix_Scaling, thread_start_datum, min(numberOfPoints,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < numberOfPoints) do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,2.5) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. //***** PERFORM THE ELLIPSE FIT: EXCLUDE HYPERBOLA *****// //Convert our original parameterization to one that excludes hyperbolas. N.b., // it is assumed that the initialParameters that were passed into the function // do NOT represent a hyperbola nor parabola. Eq. 4.3. Make/D/O/N=5 latentParameters latentParameters[0] = initialEllipseNormalizedSpace[1]/(2.*initialEllipseNormalizedSpace[0]) latentParameters[1] = sqrt(initialEllipseNormalizedSpace[2]/initialEllipseNormalizedSpace[0] - (initialEllipseNormalizedSpace[1]/(2.*initialEllipseNormalizedSpace[0]))^2) latentParameters[2] = initialEllipseNormalizedSpace[3] / initialEllipseNormalizedSpace[0] latentParameters[3] = initialEllipseNormalizedSpace[4] / initialEllipseNormalizedSpace[0] latentParameters[4] = initialEllipseNormalizedSpace[5] / initialEllipseNormalizedSpace[0] //***** PERFORM THE ELLIPSE FIT: ITERATE (§4.2.2-4.2.3) *****// MatrixOP/O Matrix_DataNormT = Matrix_DataNorm^t Make/D/O/N=6 ellipseParametersFinal fastGuaranteedEllipseFit(latentParameters, Matrix_DataNormT, normalized_CovList, ellipseParametersFinal, nthreads) //Normalize. --Commented out because theta (ellipseParametersFinal) is already // normalized as the last step in fastGuaranteedEllipseFit(). //myMatrixNorm = norm(ellipseParametersFinal) //Igor seems to have an odd bug where dividing a matrix by norm(matrix) has huge offsets //ellipseParametersFinal /= myMatrixNorm //Reiteration of precision note from fastGuaranteedEllipseFit(): In trying to // VERY carefully compare results in Igor to Matlab, after the first iteration, // the results differ at the ~15th–16th significant figure. I can't find any // error (yes, I looked into "eps"). Because there doesn't appear to be a code // error on my part, I didn't change anything in the code, such that by the 2nd // iteration, errors are now at the ~4th–6th significant figure. Keep that in // mind if ever trying to debug (i.e., I find other errors). //***** PERFORM THE ELLIPSE FIT: UNSCALE *****// //Convert the final ellipse parameters back to the original coordinate system. // This involves the same equations as 7.1-7.2 above, so we can again hard-code // the initial part of the matrix solution and then calculate the rest. Make/D/O/N=(6,9) estimatedParameters MultiThread estimatedParameters = 0 estimatedParameters[0][0] = 1 estimatedParameters[1][1] = 1 estimatedParameters[1][3] = 1 estimatedParameters[2][4] = 1 estimatedParameters[3][2] = 1 estimatedParameters[3][6] = 1 estimatedParameters[4][5] = 1 estimatedParameters[4][7] = 1 estimatedParameters[5][8] = 1 MatrixOP/O estimatedParameters = estimatedParameters x tensorProduct(Matrix_Scaling,Matrix_Scaling)^t x Matrix_Duplication x Matrix_Permutation x Matrix_E x ellipseParametersFinal //Normalize. myMatrixNorm = norm(estimatedParameters) //Igor seems to have an odd bug where dividing a matrix by norm(matrix) has huge offsets estimatedParameters /= myMatrixNorm myMatrixNorm = sign(estimatedParameters[numpnts(estimatedParameters)-1]) estimatedParameters /= myMatrixNorm //***** CALCULATE ELLIPSE UNCERTAINTY MATRICIES THAT ARE USED LATER *****// //Backup the ellipse parameters. Duplicate/O estimatedParameters algebraicEllipseParameters //The first real step is the same as the one above for the initial guesses, but // now for the final estimates. As before, there's a lot of matrix math here // that can be hard-coded because it never changes. Note that this is identical // to the above for "estimatedParameters" except for the inv() component. Make/D/O/N=(6,9) algebraicEllipseParametersNorm MultiThread algebraicEllipseParametersNorm = 0 algebraicEllipseParametersNorm[0][0] = 1 algebraicEllipseParametersNorm[1][1] = 1 algebraicEllipseParametersNorm[1][3] = 1 algebraicEllipseParametersNorm[2][4] = 1 algebraicEllipseParametersNorm[3][2] = 1 algebraicEllipseParametersNorm[3][6] = 1 algebraicEllipseParametersNorm[4][5] = 1 algebraicEllipseParametersNorm[4][7] = 1 algebraicEllipseParametersNorm[5][8] = 1 MatrixOP/O algebraicEllipseParametersNorm = algebraicEllipseParametersNorm x inv(tensorProduct(Matrix_Scaling,Matrix_Scaling))^t x Matrix_Duplication x Matrix_Permutation x Matrix_E x algebraicEllipseParameters myMatrixNorm = norm(algebraicEllipseParametersNorm) //Igor seems to have an odd bug where dividing a matrix by norm(matrix) has huge offsets algebraicEllipseParametersNorm /= myMatrixNorm //Calculate the noise level of the data from the fit. Make/D/O/N=(2,numberOfPoints) Matrix_DataT; Matrix_DataT[0][] = rim_lon_temp[q]; Matrix_DataT[1][] = rim_lat_temp[q] //noise level is relative to raw data, not normalized Make/D/O/N=(numberOfPoints) Wave_aml counter_datum = 0; i_thread = 0; mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum //intialize variables for the new threading do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_NoiseLevelOfData1(Matrix_DataT, covList, algebraicEllipseParameters, Wave_aml, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_NoiseLevelOfData1(Matrix_DataT, covList, algebraicEllipseParameters, Wave_aml, thread_start_datum, min(numberOfPoints,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < numberOfPoints) do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,2.5) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. //Calculate the "sigma_squared" scaling factor, which is the sum of the AML // cost, divided by the degrees of freedom (# points – # fit parameters). // Then, scale the covariance matrix by this scaling factor. Variable sigma_squared = sum(Wave_aml)/(numberOfPoints-5) covList *= sigma_squared //Now that the covariance matrix is scaled properly, need to re-normalize it. // NOTE: This seems like a lot of duplicated code, but the initial ellipse fit // requires the covariance matrix and it doesn't matter how it's scaled, just // how the *relative* scaling is between points in the covariance matrix. In // the above, we require the actual fit parameters to calculate the residual // noise of the data from the fit in order to calculate the scaling, so this // really does have to be done again. normalized_CovList = NaN counter_datum = 0; i_thread = 0; mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum //intialize variables for the new threading do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_NormalizeCovariances(covList, normalized_CovList, Matrix_Scaling, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_NormalizeCovariances(covList, normalized_CovList, Matrix_Scaling, thread_start_datum, min(numberOfPoints,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < numberOfPoints) do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,2.5) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. //And, repeat the loop above, but this time with the normalized and scaled // covariance matrix, and also computing "Matrix_M" that's used below, from // which we almost directly get the fit parameter uncertainties. Make/D/O/N=(6,6,numberOfPoints) Matrix_M_3D; MultiThread Matrix_M_3D = 0 Make/D/O/N=(6,6) Matrix_M; MultiThread Matrix_M = 0 counter_datum = 0; i_thread = 0; mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum //intialize variables for the new threading do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_NoiseLevelOfData2(Matrix_DataNormT, normalized_covList, algebraicEllipseParametersNorm, Matrix_M_3D, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_NoiseLevelOfData2(Matrix_DataNormT, normalized_covList, algebraicEllipseParametersNorm, Matrix_M_3D, thread_start_datum, min(numberOfPoints,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < numberOfPoints) do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,2.5) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. //Igor seems to have weird behavior if I just pass the Matrix_M to the threading // function above, where Matrix_M is += for each instance. Weird behavior as in // different results are returned each time, despite this being 100% determin- // istic and should be identical. We can get around that by storing to a 3D // matrix and then summing the layers to flatten it to 2D. It takes longer, but // hopefully the extra time is less than the time gained by threading above, // since that involves matrix manipulation and this is just addition. for(counter_datum=0; counter_datum Vminus) Duplicate/O dVplus dA Duplicate/O dVminus dB else Duplicate/O dVplus dB Duplicate/O dVminus dA endif //Jacobian matrix of the transformation from theta to eta (geometric). Make/D/O/N=(5,6) etaDtheta etaDtheta[0][] = dA[q] etaDtheta[1][] = dB[q] etaDtheta[2][] = dXcenter[q] etaDtheta[3][] = dYcenter[q] etaDtheta[4][] = dTau[q] //Propogate uncertainty from the algebraic parameter space (theta) to the // geometric parameter space (eta) in a normalized coordinate system for maximum // numerical accuracy. Make/D/O/N=(5,5) Matrix_CovarianceNorm_eta MatrixOP/O Matrix_CovarianceNorm_eta = etaDtheta x Matrix_CovarianceNorm_theta x etaDtheta^t //Apply denormalization step to determine geometric parameter covariance matrix // in the original data space. Make/D/O/N=(5,5) Matrix_Denormalization; Matrix_Denormalization = 0 Matrix_Denormalization[0][0] = 1./Matrix_Scaling[1][1] Matrix_Denormalization[1][1] = 1./Matrix_Scaling[1][1] Matrix_Denormalization[2][2] = 1./Matrix_Scaling[1][1] Matrix_Denormalization[3][3] = 1./Matrix_Scaling[1][1] Matrix_Denormalization[4][4] = 1. Make/D/O/N=(5,5) Matrix_Covariance_eta MatrixOP/O Matrix_Covariance_eta = Matrix_Denormalization x Matrix_CovarianceNorm_eta x Matrix_Denormalization^t //Uncertainty in estimatedParameters is the square-root of the diagonal terms of // the covariance matrix. Variable major_sigma = sqrt(Matrix_Covariance_eta[0][0]) Variable minor_sigma = sqrt(Matrix_Covariance_eta[1][1]) Variable x0_sigma = sqrt(Matrix_Covariance_eta[2][2]) Variable y0_sigma = sqrt(Matrix_Covariance_eta[3][3]) Variable angle_sigma = sqrt(Matrix_Covariance_eta[4][4]) //***** CALCULATE CONFIDENCE INTERVAL (OR, CHI-SQUARED → P-VALUE) *****// //Make the uncertainty matrix. Make/D/O/N=6 estimatedParams_CovWeightNorm MatrixOP/O estimatedParams_CovWeightNorm = Matrix_Permutation x Matrix_Duplication_PINV x inv(tensorProduct(Matrix_Scaling,Matrix_Scaling))^t x Matrix_Duplication x Matrix_Permutation x Matrix_E x estimatedParameters MatrixLLS/O Matrix_E estimatedParams_CovWeightNorm myMatrixNorm = norm(estimatedParams_CovWeightNorm) //Igor seems to have an odd bug where dividing a matrix by norm(matrix) has huge offsets estimatedParams_CovWeightNorm /= myMatrixNorm Variable x_min=WaveMin(ELLIPSE_X_PERFECT)-1, x_max=WaveMax(ELLIPSE_X_PERFECT)+1, y_min=WaveMin(ELLIPSE_Y_PERFECT)-1, y_max=WaveMax(ELLIPSE_Y_PERFECT)+1, x_num=500, y_num=500 Make/D/O/N=(x_num,y_num) ELLIPSE_AML_CONFIDENCE SetScale/I x x_min,x_max,"", ELLIPSE_AML_CONFIDENCE SetScale/I y y_min,y_max,"", ELLIPSE_AML_CONFIDENCE Make/D/O/N=6 ux thread_interval_datum = x_num > 2*nthreads ? floor(x_num/(2*nthreads)) : 1 //reset the thread interval counter_datum = 0; i_thread = 0; mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum //intialize variables for the new threading do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_ConfidenceMatrix(estimatedParameters, Matrix_Covariance_theta, ELLIPSE_AML_CONFIDENCE, x_min, x_max, x_num, y_min, y_max, y_num, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_ConfidenceMatrix(estimatedParameters, Matrix_Covariance_theta, ELLIPSE_AML_CONFIDENCE, x_min, x_max, x_num, y_min, y_max, y_num, thread_start_datum, min(y_num,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < y_num) do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,10) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. //Stuart-style stuff. According to the documentation, the above returns the // chi-squared value for each grid point. An ellipse has 5 degrees of freedom. // So, if you want a p-value of 0.05, according to a chi-squared lookup table, // then if the above value is 11.07... then the p-value is 0.05, meaning there // is a 95% chance that the true ellipse is in that region. (Sort of ... it's // actually about accepting/rejecting null hypotheses to a certain confidence, // but this is close enough conceptually at the moment.) MultiThread ELLIPSE_AML_CONFIDENCE = StatsChiCDF(ELLIPSE_AML_CONFIDENCE,5) //***** CALCULATE BEST-FIT ELLIPSE PARAMETERS. *****// //This uses Szpak's definitions, not mine, so the math is a little different. //Snag the ellipse coefficients. a = estimatedParameters[0] b = estimatedParameters[1] c = estimatedParameters[2] d = estimatedParameters[3] e = estimatedParameters[4] f = estimatedParameters[5] //print a, b, c, d, e, f delta = b^2 - 4*a*c lamdaPlus = 0.5*(a + c - (b^2 + (a - c)^2)^0.5) lamdaMinus = 0.5*(a + c + (b^2 + (a - c)^2)^0.5) psi = b*d*e - a*e^2 - b^2*f + c*(4*a*f - d^2) Vplus = (psi/(lamdaPlus *delta))^0.5 Vminus = (psi/(lamdaMinus*delta))^0.5 //Finally, the values. Variable major = max(Vplus,Vminus) //semi-major axis Variable minor = min(Vplus,Vminus) //semi-minor axis //print "semi-major",major //print "semi-minor",minor //Calculate the center of the ellipse. Variable x0 = (2*c*d-b*e)/delta //print "x center",x0 Variable y0 = (2*a*e-b*d)/delta //print "y center",y0 //Calculate the tilt angle of the major axis in terms of counter-clockwise // rotation from the x-axis. Variable angle = 0 if (Vplus >= Vminus) if(b == 0 && a < c) angle = 0; elseif (b == 0 && a >= c) angle = 0.5*pi; elseif (b < 0 && a < c) angle = 0.5*acot((a - c)/b); elseif (b < 0 && a == c) angle = pi/4; elseif (b < 0 && a > c) angle = 0.5*acot((a - c)/b) + pi/2; elseif (b > 0 && a < c) angle = 0.5*acot((a - c)/b) + pi; elseif (b > 0 && a == c) angle = pi*(3/4); elseif (b > 0 && a > c) angle = 0.5*acot((a - c)/b) + pi/2; endif elseif (Vplus < Vminus) if(b == 0 && a < c) angle = pi/2; elseif (b == 0 && a >= c) angle = 0; elseif (b < 0 && a < c) angle = 0.5*acot((a - c)/b) + pi/2; elseif (b < 0 && a == c) angle = pi*(3/4); elseif (b < 0 && a > c) angle = 0.5*acot((a - c)/b) + pi; elseif (b > 0 && a < c) angle = 0.5*acot((a - c)/b) + pi/2; elseif (b > 0 && a == c) angle = pi/4; elseif (b > 0 && a > c) angle = 0.5*acot((a - c)/b); endif endif angle = -angle //to make it consistent with my definition from Szpak's definition. angle = angle < 0 ? angle + pi : angle angle = angle < 0 ? angle + pi : angle //repeat in case we were in a 180°–360° regime //print "angle",angle*180/pi //print "eccentricity",sqrt(1-minor/major) //print "ellipticity",major/minor //***** RE-SET AUTOMATIC MUTLI-THREADING MODE *****// //Set this parameter back to whatever it was before we started. (See notes at // beginning for why this is commented out, but it remains here.) //MultiThreadingControl setMode=V_autoMultiThread //***** TIMER *****// printf "Time to fit the ellipse was %g seconds.\r", (stopMSTimer(-2)-timer_start)*1e-6 //***** STORE THE VALUES *****// printf, "Ellipse Parameters from Szpak et al. (2015) \"AML\" Method:\r\tx0: %g ± %g\r\ty0: %g ± %g\r\ta÷2: %g ± %g\r\tb÷2: %g ± %g\r\ttheta: %g ± %g\r", x0, x0_sigma, y0, y0_sigma, major, major_sigma, minor, minor_sigma, angle*180./pi, angle_sigma*180./pi ellipseforModeling(major*2., minor*2., x0, y0, angle*180./pi, 0, 360, 0, ELLIPSE_X_FIT_AML, ELLIPSE_Y_FIT_AML) //LATITUDE_ELLIPSE_IMAGE[i_counter_crater] = y0/map_projection+y_center_mass_degrees //LONGITUDE_ELLIPSE_IMAGE[i_counter_crater] = x0/(map_projection*cos(y0*pi/180))+x_center_mass_degrees //DIAM_ELLIPSE_MAJOR_IMAGE[i_counter_crater] = major*2. //DIAM_ELLIPSE_MINOR_IMAGE[i_counter_crater] = minor*2. //DIAM_ELLIPSE_ANGLE_IMAGE[i_counter_crater] = angle*180./pi //DIAM_ELLIPSE_ECCEN_IMAGE[i_counter_crater] = sqrt(1-minor^2/major^2) //DIAM_ELLIPSE_ELLIP_IMAGE[i_counter_crater] = major/minor //***** CLEANUP *****// //KillWaves/Z covList, normalized_CovList, covX_i, Matrix_Scaling End //Created 31-Jul-2018. ThreadSafe Function Threading_NormalizeCovariances(covList, normalized_CovList, Matrix_Scaling, thread_start_datum, thread_end_datum) Wave covList, normalized_CovList, Matrix_Scaling Variable thread_start_datum, thread_end_datum Variable counter_datum for(counter_datum=thread_start_datum; counter_datum nthreads ? floor(numberOfPoints/(nthreads)) : 1 Duplicate/O latentParameters parametersLatent_ThisIteration //Convert latent variables into a length-6 vector (called "parametersEllipse_ThisIteration") representing the equation of an ellipse. Make/D/O/N=6 parametersEllipse_ThisIteration parametersEllipse_ThisIteration[0] = 1 parametersEllipse_ThisIteration[1] = 2.*parametersLatent_ThisIteration[0] parametersEllipse_ThisIteration[2] = parametersLatent_ThisIteration[0]*parametersLatent_ThisIteration[0] + abs(parametersLatent_ThisIteration[1])^2. parametersEllipse_ThisIteration[3] = parametersLatent_ThisIteration[2] parametersEllipse_ThisIteration[4] = parametersLatent_ThisIteration[3] parametersEllipse_ThisIteration[5] = parametersLatent_ThisIteration[4] Variable myMatrixNorm = norm(parametersEllipse_ThisIteration) //Igor seems to have an odd bug where dividing a matrix by norm(matrix) has huge offsets parametersEllipse_ThisIteration /= myMatrixNorm //Variable and parameter initializations that describe how to loop. Variable keep_going = 1 //"true" Variable eta_updated = 0 //"false" --in some cases, a Levenberg-Marquardt step does not decrease the cost function, so the parameters (parametersLatent_ThisIteration) are not updated Variable counter_k = 0 //loop counter (starts at 0, while Matlab starts at 1) Variable maxIter = 100 //maximum loop iterations Variable tolDelta = 1e-7 //step-size tolerance Variable tolCost = 1e-7 //cost tolerance Variable toleta = 1e-7 //parameter tolerance Variable tolGrad = 1e-7 //gradient tolerance Variable tolBar = 15.5 //barrier tolerance (prevent ellipse from converging on parabola) Variable tolDet = 1e-5 //minimum allowable magnitude of conic determinant (prevent ellipse from convering on degenerate parabola (eg. two parallel lines) ) //Some matrix stuff that is not explained. The Matrix_Identity is the Identity Matrix. Make/D/O/N=(3,3) Fprim; Fprim = 0 Fprim[0][2] = 2 Fprim[1][1] = -1 Fprim[2][0] = 2 Make/D/O/N=(6,6) Matrix_F, Matrix_Identity; Matrix_F = 0; Matrix_Identity = 0 Matrix_F[0,2][0,2] = Fprim[p][q] MatrixOP/O Matrix_Identity = Identity(6) //More initializations. Make/D/O/N=(maxIter) cost; cost = 0 //allocate space for cost of each iteration Make/D/O/N=(5,maxIter) parametersLatent_PerIteration; parametersLatent_PerIteration = 0 //allocate space for the latent parameters of each iteration Make/D/O/N=(6,maxIter) parametersEllipse_PerIteration; parametersEllipse_PerIteration = 0 //and for the parameters representing the ellipse equation Make/D/O/N=(5,maxIter) delta; delta = 0 //allocate space for the parameter direction of each iteration //Store parameters associated with the first iteration (though we've overwritten some of the names). MultiThread parametersEllipse_PerIteration[][counter_k] = parametersEllipse_ThisIteration[p] MultiThread parametersLatent_PerIteration[][counter_k] = parametersLatent_ThisIteration[p] //Start with some random search direction (here we choose all 1) -- we can initialize with // anything we want, so long as the norm of the vector is not smaller than tolDeta. The // initial search direction is not used in any way in the algorithm, but it can't be NaN. delta[][counter_k] = 1 //Main execution loop. Variable covList_valueToFix = covList[0][0][0] //at random, Igor bug: covList temporarily goes to 0, making covX_i 0 and can't refer to other cells in covList to make them bigger, which makes BB 0, which makes myScalar 0, which makes MMatrix Inf do Make/D/O/N=(numberOfPoints) residuals; residuals = 0 //allocate space for residuals Make/D/O/N=(numberOfPoints,5) Matrix_Jacobian; Matrix_Jacobian = 0 //allocate space for the jacobian matrix based on AML component MultiThread parametersLatent_ThisIteration[] = parametersLatent_PerIteration[p][counter_k] //Convert latent variables into length-6 vector (called t) representing the equation of an ellipse. Make/D/O/N=6 parametersEllipse_ThisIteration parametersEllipse_ThisIteration[0] = 1 parametersEllipse_ThisIteration[1] = 2.*parametersLatent_ThisIteration[0] parametersEllipse_ThisIteration[2] = parametersLatent_ThisIteration[0]*parametersLatent_ThisIteration[0] + abs(parametersLatent_ThisIteration[1])^2. parametersEllipse_ThisIteration[3] = parametersLatent_ThisIteration[2] parametersEllipse_ThisIteration[4] = parametersLatent_ThisIteration[3] parametersEllipse_ThisIteration[5] = parametersLatent_ThisIteration[4] //Jacobian matrix of the transformation from parametersLatent_ThisIteration to theta parameters. Make/D/O/N=(6,5) jacob_latentParameters; jacob_latentParameters = 0 jacob_latentParameters[1][0] = 2 jacob_latentParameters[2][0] = 2*parametersLatent_ThisIteration[0] jacob_latentParameters[2][1] = 2*abs(parametersLatent_ThisIteration[1])^(2-1)*sign(parametersLatent_ThisIteration[1]) jacob_latentParameters[3][2] = 1 jacob_latentParameters[4][3] = 1 jacob_latentParameters[5][4] = 1 //We impose the additional constraint that theta will be unit norm, // so we need to modify the jacobian matrix accordingly. Make/D/O/N=(5,6) Pt myMatrixNorm = GenVecNorm(parametersEllipse_ThisIteration, 2)^2 MatrixOP/O Pt = Matrix_Identity - ( (parametersEllipse_ThisIteration x parametersEllipse_ThisIteration^t) / myMatrixNorm ) myMatrixNorm = 1. / GenVecNorm(parametersEllipse_ThisIteration, 2) MatrixOP/O jacob_latentParameters = myMatrixNorm * Pt x jacob_latentParameters //Unit normal constraint. myMatrixNorm = norm(parametersEllipse_ThisIteration) parametersEllipse_ThisIteration /= myMatrixNorm //Calculate the residuals for each datum. The outputs from this are "grad", // "residulals", and "Matrix_Jacobian". The latter two are declared above. Make/D/O/N=6 grad Variable counter_datum = 0, i_thread = 0, mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_ResidualsAndJacobian(dataPts, covList, parametersEllipse_ThisIteration, jacob_latentParameters, grad, residuals, Matrix_Jacobian, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) if(thread_end_datum < numberOfPoints) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_ResidualsAndJacobian(dataPts, covList, parametersEllipse_ThisIteration, jacob_latentParameters, grad, residuals, Matrix_Jacobian, thread_start_datum, min(numberOfPoints,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < numberOfPoints) endif do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,2.5) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. //Approximate Hessian matrix. MatrixOP/O Matrix_Hessian = Matrix_Jacobian^t x Matrix_Jacobian //Sum of squares cost for the current iteration. MatrixOP/O tempResult = residuals^t x residuals cost[counter_k] = tempResult[0][0] //Now, use a Levenberg-Marquadt step to update the parameters. eta_updated = fastLevenbergMarquardtStep(Matrix_Jacobian, residuals, parametersEllipse_PerIteration, delta, cost, dataPts, covList, numberOfPoints, Matrix_Hessian, jacob_latentParameters, parametersLatent_PerIteration, eta_updated, counter_k, 2, nthreads) //Preparations for various stopping criteria tests. //Convert latent variables into length-6 vector (called "parametersEllipse_ThisIteration") representing the equation of an ellipse ... yet again. Make/D/O/N=5 etaThisTime; etaThisTime[] = parametersLatent_PerIteration[p][counter_k] Make/D/O/N=5 etaNextTime; etaNextTime[] = parametersLatent_PerIteration[p][counter_k+1] Make/D/O/N=6 parametersEllipse_ThisIteration parametersEllipse_ThisIteration[0] = 1 parametersEllipse_ThisIteration[1] = 2.*etaNextTime[0] parametersEllipse_ThisIteration[2] = etaNextTime[0]*etaNextTime[0] + abs(etaNextTime[1])^2. parametersEllipse_ThisIteration[3] = etaNextTime[2] parametersEllipse_ThisIteration[4] = etaNextTime[3] parametersEllipse_ThisIteration[5] = etaNextTime[4] myMatrixNorm = norm(parametersEllipse_ThisIteration) parametersEllipse_ThisIteration /= myMatrixNorm //First criterion checks to see if discriminant approaches zero by using a barrier. MatrixOP/O tIt = parametersEllipse_ThisIteration^t x Matrix_Identity x parametersEllipse_ThisIteration MatrixOP/O tFt = parametersEllipse_ThisIteration^t x Matrix_F x parametersEllipse_ThisIteration Variable barrier = tIt[0][0] / tFt[0][0] //Second criterion checks to see if the determinant of conic approaches zero. Make/D/O/N=(3,3) myMMatrix myMMatrix[0][0] = parametersEllipse_ThisIteration[0] myMMatrix[0][1] = parametersEllipse_ThisIteration[1]/2. myMMatrix[0][2] = parametersEllipse_ThisIteration[3]/2. myMMatrix[1][0] = parametersEllipse_ThisIteration[1]/2. myMMatrix[1][1] = parametersEllipse_ThisIteration[2] myMMatrix[1][2] = parametersEllipse_ThisIteration[4]/2. myMMatrix[2][0] = parametersEllipse_ThisIteration[3]/2. myMMatrix[2][1] = parametersEllipse_ThisIteration[4]/2. myMMatrix[2][2] = parametersEllipse_ThisIteration[5] Variable DeterminantConic = MatrixDet(myMMatrix) //Check for various stopping criteria to end the main loop. Duplicate/O etaNextTime etaMinus; etaMinus = etaNextTime-etaThisTime Duplicate/O etaThisTime etaPlus ; etaPlus = etaNextTime+etaThisTime Make/D/O/N=5 deltaNextTime; deltaNextTime[] = delta[p][counter_k+1] Variable myCrazyCriterion = min( norm(etaMinus), norm(etaPlus) ) if ( (myCrazyCriterion < toleta) && (eta_updated == 1) ) keep_going = 0 elseif ( (abs(cost[counter_k]-cost[counter_k+1]) < tolCost) && (eta_updated == 1) ) keep_going = 0 elseif ( (norm(deltaNextTime) < tolDelta) && (eta_updated == 1) ) keep_going = 0 elseif (norm(grad) < tolGrad) keep_going = 0 elseif ( (log(barrier) > tolBar) || (abs(DeterminantConic) < tolDet) ) keep_going = 0 endif //Increment the counter. counter_k += 1 while( (keep_going == 1) && (counter_k < maxIter) ) //Final stuff. Variable iterations = counter_k theta[] = parametersEllipse_PerIteration[p][counter_k] myMatrixNorm = norm(theta) theta /= myMatrixNorm End ThreadSafe Function Threading_ResidualsAndJacobian(dataPts, covList, parametersEllipse_ThisIteration, jacob_latentParameters, grad, residuals, Matrix_Jacobian, thread_start_datum, thread_end_datum) Wave dataPts, covList, parametersEllipse_ThisIteration, jacob_latentParameters, grad, residuals, Matrix_Jacobian Variable thread_start_datum, thread_end_datum Variable counter_datum Duplicate/O grad grad_temp //can't use MatrixOP on "grad" since it's passed to this function for(counter_datum = thread_start_datum; counter_datum < thread_end_datum; counter_datum += 1) //Get the current data pair. Make/D/O/N=2 mm; mm[] = dataPts[p][counter_datum] //Calculate the transformed data pair. Make/D/O/N=6 ux_i ux_i[0] = mm[0]^2 ux_i[1] = mm[0]*mm[1] ux_i[2] = mm[1]^2 ux_i[3] = mm[0] ux_i[4] = mm[1] ux_i[5] = 1 //Calculate the derivative of the transformed data point. Make/D/O/N=(2,6) dux_i; dux_i = 0 dux_i[0][0] = 2*mm[0] dux_i[0][1] = mm[1] dux_i[0][3] = 1 dux_i[1][1] = mm[0] dux_i[1][2] = 2*mm[1] dux_i[1][4] = 1 MatrixOP/O dux_i = dux_i^t //Calculate the outer product. MatrixOP/O AA = ux_i x ux_i^t //Extract the covariance matrix of the ith data pair. Make/D/O/N=(2,2) covX_i; covX_i[][] = covList[p][q][counter_datum] //Some matrix stuff that's not explained. MatrixOP/O BB = dux_i x covX_i x dux_i^t // Igor code seems IDENTICAL to Matlab code *UNTIL* this step, in terms of // results. At this point, in the first iteration, stuff starts to differ at // the 15th–16th decimal point. That shouldn't be a big issue, but it seems // like it can snowball from there. The parametersEllipse_ThisIteration // seems identical to as many decimal points as I can see, as do AA and BB, // but the result tBt and tAt start to deviate. MatrixOP/O tBt = parametersEllipse_ThisIteration^t x BB x parametersEllipse_ThisIteration MatrixOP/O tAt = parametersEllipse_ThisIteration^t x AA x parametersEllipse_ThisIteration //Approximate Maximum-Likelihood cost for the ith data pair. residuals[counter_datum] = sqrt(abs(tAt[0][0]/tBt[0][0])) //at this point, both matricies are 1x1, but Igor still treats them as matricies so must specify which cell //Derivative of Approximate Maximum-Likelihood component. See above comment on the [0][0] Variable myScalar = tBt[0][0] AA /= myScalar myScalar = tAt[0][0] / tBt[0][0]^2 BB *= myScalar AA -= BB //Gradient for Approximate Maximum-Likelihood cost function (row vector). Variable eps = 2^-52 //Matlab definition: eps returns the distance from 1.0 to the next larger double-precision number, that is, 2^-52 myScalar = sqrt((abs(tAt[0][0]/tBt[0][0])+eps)) MatrixOP/O grad_temp = AA x parametersEllipse_ThisIteration grad_temp /= myScalar MatrixOP/O grad_temp = grad_temp^t //Build the Jacobian matrix. MatrixOP/O tempResult = grad_temp x jacob_latentParameters Matrix_Jacobian[counter_datum][] = tempResult[q] endfor //Store back. grad = grad_temp //Reiteration of precision note from above: The results, Matrix_Jacobian and // grad, are slightly off in the first iteration, at the 15th to 16th decimal of // some of the values in the arrays. Some are perfect with respect to Matlab, // but this means that the fastLevenbergMarquardtStep() will also be slightly // off as a result. End Function fastLevenbergMarquardtStep(Matrix_Jacobian, residuals, parametersEllipse_PerIteration, delta, cost, data_points, covList, numberOfPoints, Matrix_Hessian, jacob_latentParameters, parametersLatent_PerIteration, eta_updated, counter_k, rho, nthreads) Wave Matrix_Jacobian Wave residuals Wave parametersEllipse_PerIteration Wave delta Wave cost Wave data_points Wave covList Variable numberOfPoints Wave Matrix_Hessian Wave jacob_latentParameters Wave parametersLatent_PerIteration Variable eta_updated Variable counter_k Variable rho Variable nthreads //***** TIMER *****// //Variable timer_start = stopMSTimer(-2) //create a timer //***** SET AUTOMATIC MUTLI-THREADING MODE & OTHER THREADING *****// //Multithreading setup: Determine how many threads we can spawn, equal to the // number of processors (generally). This assumes that I will *ALWAYS* have // ≥data points than threads available. Variable tgs, ti, dummy_forThreadGroupRelease Variable thread_start_datum, thread_end_datum //Multithreading requires overhead to manage the threads, check for when one is // free, etc., and so if you're not smrt about it, it can manage to take longer // then doing things on a single thread. To mitigate some of that, what we can // do is we can pre-emptively select ranges of points to farm to a single thread // instead of just doing things one at a time and sending the next item to the // thread. Things are easiest if the number of points (since that's the // majority of the threading here) is evenly divisible by the number of threads, // but that's not always possible. Another factor is sometimes the math is // easier on one thread than others, so things don't end evenly ... but that's // not really the case here. Variable thread_interval_datum = numberOfPoints > nthreads ? floor(numberOfPoints/(nthreads)) : 1 //***** SETUP VARIABLES, WAVES, ETC. FOR THIS ITERATION *****// //Some variables needed in this function. Note: Szpak had some of these in an // earlier function but they weren't used there, so we can avoid needing to pass // them by just declaring them here. Variable lamda = 0.01 //damping parameter in Levenberg-Marquardt step Variable damping_multiplier = 15 //used to modify the tradeoff between gradient descent and Hessian based descent in Levenberg-Marquadt step Variable damping_divisor = 1.2 //used to modify the tradeoff between gradient descent and Hessian based descent in Levenberg-Marquadt step //Extract some stuff for just this iteration. Make/D/O/N=5 delta_ThisIteration; delta_ThisIteration[] = delta[p][counter_k] Make/D/O/N=5 parametersLatent_ThisIteration; parametersLatent_ThisIteration[] = parametersLatent_PerIteration[p][counter_k] Variable cost_ThisIteration = cost[counter_k] //Convert latent variables into length-6 vector representing the equation of an // ellipse. Then normalize. Make/D/O/N=6 parametersEllipse_ThisIteration parametersEllipse_ThisIteration[0] = 1 parametersEllipse_ThisIteration[1] = 2.*parametersLatent_ThisIteration[0] parametersEllipse_ThisIteration[2] = parametersLatent_ThisIteration[0]*parametersLatent_ThisIteration[0] + abs(parametersLatent_ThisIteration[1])^2. parametersEllipse_ThisIteration[3] = parametersLatent_ThisIteration[2] parametersEllipse_ThisIteration[4] = parametersLatent_ThisIteration[3] parametersEllipse_ThisIteration[5] = parametersLatent_ThisIteration[4] Variable myMatrixNorm = norm(parametersEllipse_ThisIteration) parametersEllipse_ThisIteration /= myMatrixNorm //***** CALCULATE POTENTIAL UPDATES ON THE FIT PARAMETERS *****// //Compute two potential updates for theta based on different weightings of the // identity matrix. MatrixOP/O jacob = Matrix_Jacobian^t x residuals MatrixOP/O Matrix_DMP = jacob_latentParameters^t x jacob_latentParameters Matrix_DMP *= lamda Duplicate/O Matrix_Hessian update_a; update_a = -(Matrix_Hessian + Matrix_DMP) Make/D/O/N=0 M_x; MatrixSolve LU, update_a, jacob Duplicate/O M_x update_a //In a similar fashion, the second potential search direction is computed. MatrixOP/O Matrix_DMP = jacob_latentParameters^t x jacob_latentParameters Matrix_DMP *= lamda/damping_divisor Duplicate/O Matrix_Hessian update_b; update_b = -(Matrix_Hessian + Matrix_DMP) Make/D/O/N=0 M_x; MatrixSolve LU, update_b, jacob Duplicate/O M_x update_b //The potential new parameters are then: Duplicate/O parametersLatent_ThisIteration eta_potential_a; eta_potential_a += update_a Duplicate/O parametersLatent_ThisIteration eta_potential_b; eta_potential_b += update_b //We need to convert from parametersLatent_ThisIteration to theta and impose unit norm constraints. Make/D/O/N=6 t_potential_a t_potential_a[0] = 1 t_potential_a[1] = 2*eta_potential_a[0] t_potential_a[2] = eta_potential_a[0]^2 + abs(eta_potential_a[1])^rho t_potential_a[3] = eta_potential_a[2] t_potential_a[4] = eta_potential_a[3] t_potential_a[5] = eta_potential_a[4] myMatrixNorm = norm(t_potential_a) t_potential_a /= myMatrixNorm Make/D/O/N=6 t_potential_b t_potential_b[0] = 1 t_potential_b[1] = 2*eta_potential_b[0] t_potential_b[2] = eta_potential_b[0]^2 + abs(eta_potential_b[1])^rho t_potential_b[3] = eta_potential_b[2] t_potential_b[4] = eta_potential_b[3] t_potential_b[5] = eta_potential_b[4] myMatrixNorm = norm(t_potential_b) t_potential_b /= myMatrixNorm //Compute new residuals and costs based on these updates. Make/D/O/N=(numberOfPoints) Wave_cost_a, Wave_cost_b; MultiThread Wave_cost_a = 0; MultiThread Wave_cost_b = 0 Variable counter_datum = 0, i_thread = 0, mt = ThreadGroupCreate(nthreads); thread_start_datum = 0; thread_end_datum = thread_start_datum+thread_interval_datum do //Run the first batch of threads. ThreadStart mt, i_thread, Threading_ResidualsAndCosts(Wave_cost_a, Wave_cost_b, data_points, covList, t_potential_a, t_potential_b, thread_start_datum, thread_end_datum) i_thread += 1 thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(i_thread < nthreads) if(thread_end_datum < numberOfPoints) do //Run more as they become available. //Get the index of the first free thread i_thread = 0 if(ThreadGroupWait(mt,0) > 0) //to reduce thread overhead (check every (,X) milliseconds) i_thread = ThreadGroupWait(mt,-2)-1 //store index of first available thread (or <0 if there aren't any) if(i_thread < 0) //if there's nothing available ... continue //... and go back to the beginning of this if-endif loop endif endif //We're only here if we're not still looping above. ThreadStart mt, i_thread, Threading_ResidualsAndCosts(Wave_cost_a, Wave_cost_b, data_points, covList, t_potential_a, t_potential_b, thread_start_datum, min(numberOfPoints,thread_end_datum)) thread_start_datum = thread_end_datum thread_end_datum += thread_interval_datum while(thread_start_datum < numberOfPoints) endif do //the last few threads will keep going outside the do-while loop and result in incomplete work. tgs = ThreadGroupWait(mt,2.5) while(tgs != 0) dummy_forThreadGroupRelease = ThreadGroupRelease(mt) //Release the threads. Variable cost_a = sum(Wave_cost_a) Variable cost_b = sum(Wave_cost_b) //***** For next iteration, calculate appropriate damping and update. *****// //Neither update cost_a nor cost_b reduced the cost. if ( (cost_a >= cost_ThisIteration) && (cost_b >= cost_ThisIteration) ) eta_updated = 0 //since nothing reduced the cost, we are NOT updating eta cost[counter_k+1] = cost_ThisIteration //since nothing reduced the cost, no change in the cost parametersLatent_PerIteration[][counter_k+1] = parametersLatent_ThisIteration[p] //since nothing reduced the cost, no change in parameters parametersEllipse_PerIteration[][counter_k+1] = parametersEllipse_ThisIteration[p] //since nothing reduced the cost, no change in parameters delta[][counter_k+1] = delta_ThisIteration[p] //since nothing reduced the cost, no changes in step direction lamda *= damping_multiplier //next iteration, add more damping factor (to the Matrix_DMP) //Update "b" reduced the cost function, so use that. elseif (cost_b < cost_ThisIteration) eta_updated = 1 //set the flag that eta IS updated cost[counter_k+1] = cost_b //store the new cost parametersLatent_PerIteration[][counter_k+1] = eta_potential_b[p] //update the parameters using "b" parametersEllipse_PerIteration[][counter_k+1] = t_potential_b[p] //update the parameters using "b" MatrixOP/O tempResult = update_b^t //calculate the step direction for the next iteration delta[][counter_k+1] = tempResult[p] //store the step direction lamda /= damping_multiplier //next iteration, add less Identity matrix //Update "a" reduced the cost function, so use that. else eta_updated = 1 //set the flag that eta IS updated cost[counter_k+1] = cost_a //store the new cost parametersLatent_PerIteration[][counter_k+1] = eta_potential_a[p] //update the parameters using "a" parametersEllipse_PerIteration[][counter_k+1] = t_potential_a[p] //update the parameters using "a" MatrixOP/O tempResult = update_a^t //calculate the step direction for the next iteration delta[][counter_k+1] = tempResult[p] //store the step direction //lamda = lamda //next iteration, use the same damping (commented out to save time, but here so you don't think it's an error. endif //***** TIMER *****// //printf "Time for L-M iteration step %d took %g seconds.\r", counter_k, (stopMSTimer(-2)-timer_start)*1e-6 //***** RETURN *****// //What we care about for the next iteration is whether eta is updated. Return- // ing this value lets us use it as a flag to decide whether we are iterating // again or if we're stopping. (Everything else is passed to/from this function // so we don't need to return it.) return(eta_updated) End ThreadSafe Function Threading_ResidualsAndCosts(Wave_cost_a, Wave_cost_b, data_points, covList, t_potential_a, t_potential_b, thread_start_datum, thread_end_datum) Wave Wave_cost_a, Wave_cost_b, data_points, covList, t_potential_a, t_potential_b Variable thread_start_datum, thread_end_datum Variable counter_datum for(counter_datum = thread_start_datum; counter_datum < thread_end_datum; counter_datum += 1) //Get the current data pair. Make/D/O/N=2 mm; mm[] = data_points[p][counter_datum] //Calculate the transformed data pair. Make/D/O/N=6 ux_i ux_i[0] = mm[0]^2 ux_i[1] = mm[0]*mm[1] ux_i[2] = mm[1]^2 ux_i[3] = mm[0] ux_i[4] = mm[1] ux_i[5] = 1 //Calculate the derivative of the transformed data point. Make/D/O/N=(2,6) dux_i; dux_i = 0 dux_i[0][0] = 2*mm[0] dux_i[0][1] = mm[1] dux_i[0][3] = 1 dux_i[1][1] = mm[0] dux_i[1][2] = 2*mm[1] dux_i[1][4] = 1 MatrixOP/O dux_i = dux_i^t //Calculate the outer product. MatrixOP/O AA = ux_i x ux_i^t //Extract the covariance matrix of the ith data pair. Make/D/O/N=(2,2) covX_i; covX_i[][] = covList[p][q][counter_datum] //Some matrix stuff that's not explained. MatrixOP/O BB = dux_i x covX_i x dux_i^t MatrixOP/O t_aBt_a = t_potential_a^t x BB x t_potential_a MatrixOP/O t_aAt_a = t_potential_a^t x AA x t_potential_a MatrixOP/O t_bBt_b = t_potential_b^t x BB x t_potential_b MatrixOP/O t_bAt_b = t_potential_b^t x AA x t_potential_b //Approximate Maximum-Likelihood cost for the ith data pair. Wave_cost_a[counter_datum] += abs(t_aAt_a[0][0]/t_aBt_a[0][0]) Wave_cost_b[counter_datum] += abs(t_bAt_b[0][0]/t_bBt_b[0][0]) endfor End //Converting Matlab to Igor, Igor does not have an option for the general vector // norm, so this function returns that. Note that if myP = 2, it is the // Euclidean Norm (norm() in Igor). //Created 19 Jul 2018. ThreadSafe Function GenVecNorm(myWave, myP) Wave myWave Variable myP Duplicate/O myWave myWaveT //create a temporary copy MultiThread myWaveT[] = abs(myWave[p])^myP //calculate the abs(value)^power return(sum(myWaveT)^(1/myP)) //return the sum of those ^1/power End //Igor does not have acot (inverse cotangent). //Created 19 Jul 2018. ThreadSafe Function acot(myValue) Variable myValue //Use the inverse sine definition from Wolfram. Variable toReturn = myValue > 0 ? -asin(1/sqrt(myValue^2+1)) : asin(1/sqrt(myValue^2+1)) return(toReturn) End