MATLAB Examples

ellipseFit4HC_EXAMPLE

(c) Viktor Witkovsky (witkovsky@savba.sk) Ver.: 31-Jul-2014 18:27:32

```clear ```

GENERATE the uncorrelated measurements: x and y

`Feel free to experiment with the setting parameters`
```alpha0true = 0; % x-center (offset x) beta0true = 0; % y-center (offset x) alpha1true = 1; % x-amplitude beta1true = 1; % y-amplitude phi0true = 0; % phase offset sigma = 0.001; % std of x and y independent errors n = 500; % number of measurements points cycles = 1; % measurement interval, # cycles phitrue = cycles*(2*pi)*sort(rand(n,1)); % true phases phi_i % true values: X and Y Xmean = @(t) alpha0true + alpha1true * cos(t); Ymean = @(t) beta0true + beta1true * sin(t + phi0true); % measurements: X + error, Y + error, x = Xmean(phitrue) + sigma * randn(size(phitrue)); y = Ymean(phitrue) + sigma * randn(size(phitrue)); ```

Fit the ellipse based on measured data

```options.alpha = 0.05; options.correlation = 0; options.displconst = 633.3/(4*pi); options.displunit = 'nanometer [nm]'; result = ellipseFit4HC(x,y,options); disp(result) ```
``` ------------------------------------------------- ESTIMATED ELLIPSE PARAMETERS X^2 + B*Y^2 + C*X*Y + D*X + F*Y + G = 0 ------------------------------------------------- ESTIMATE STD DF FACTOR LOWER UPPER ___________ __________ ___ ______ ___________ __________ B 1.0001 0.0002517 495 1.9648 0.99956 1.0005 C -7.2954e-05 0.00025273 495 1.9648 -0.0005695 0.0004236 D -5.1167e-06 0.00012533 495 1.9648 -0.00025136 0.00024113 F -8.4915e-05 0.00012687 495 1.9648 -0.00033418 0.00016435 G -1 0.00015294 495 1.9648 -1.0003 -0.99973 ------------------------------------------------- ESTIMATED ELLIPSE PARAMETERS X(phi) = alpha_0 + alpha_1 * cos(phi) Y(phi) = beta_0 + beta_1 * sin(phi + phi_0 ------------------------------------------------- ESTIMATE STD DF FACTOR LOWER __________ __________ ___ ______ ___________ alpha_0 2.5599e-06 6.2665e-05 495 1.9648 -0.00012056 beta_0 4.2455e-05 6.343e-05 495 1.9648 -8.2169e-05 alpha_1 1 7.6468e-05 495 1.9648 0.99987 beta_1 0.99999 7.7723e-05 495 1.9648 0.99984 phi_0 3.6476e-05 0.00012584 495 1.9648 -0.00021078 UPPER __________ alpha_0 0.00012568 beta_0 0.00016708 alpha_1 1.0002 beta_1 1.0001 phi_0 0.00028373 Description: 'Ellipse Fit by Iterated Locally Best Linea...' ABCDEF_Description: 'A*x^2 + 2*B*x*y + C*y^2 + 2*D*x + 2*E*y + ...' ABCDEF_Names: {'A' 'B' 'C' 'D' 'E' 'F'} ABCDEF_fit: [1x6 double] ABCDEF_std: [1x6 double] BCDFG_Description: 'x^2 + B*y^2 + C*x*y + D*x + F*y + G = 0' BCDFG_Names: {'B' 'C' 'D' 'F' 'G'} BCDFG_fit: [1x5 double] BCDFG_std: [1x5 double] BCDFG_cov: [5x5 double] EllipsePars_Description_X: 'X(phi) = alpha_0 + alpha_1 * cos(phi)' EllipsePars_Description_Y: 'Y(phi) = beta_0 + beta_1 * sin(phi + phi_0)' EllipsePars_Names: {1x5 cell} EllipsePars_fit: [1x5 double] EllipsePars_std: [1x5 double] mu_X_fit: [500x1 double] mu_X_std: [500x1 double] nu_Y_fit: [500x1 double] nu_Y_std: [500x1 double] X_fitfun: @(t)alpha0_fit+alpha1_fit*cos(t) Y_fitfun: @(t)beta0_fit+beta1_fit*sin(t+phi0_fit) x_res_fit: [500x1 double] x_res_std: [500x1 double] y_res_fit: [500x1 double] y_res_std: [500x1 double] sigma_fit: 9.9570e-04 sigma2_fit: 9.9141e-07 phi_fit: [500x1 double] phi_std: [500x1 double] phi_std_max: 0.0010 phi_std_min: 9.9765e-04 Displacement_Description: 'Displacement_fit = Displacement_const * ph...' Displacement_fit: [500x1 double] Displacement_std: [500x1 double] Displacement_std_max: 0.0507 Displacement_std_min: 0.0503 Displacement_const: 50.3964 Displacement_unit: 'nanometer [nm]' TABLE_BCDFG: [5x6 table] TABLE_EllipsePars: [5x6 table] TABLE_Displacements: [500x7 table] x_data: [500x1 double] y_data: [500x1 double] used_Correlation_XY: 0 options: [1x1 struct] criterion: 7.6311e-13 loops: 4 tictoc: 0.0132 ```

Display statistics of the fitted displacements (!! Table with n rows !!)

```TABLE1 = result.TABLE_EllipsePars; TABLE2 = result.TABLE_Displacements; disp(TABLE2) disp(['Displacement unit: ' result.Displacement_unit]) ```
``` OBSERVATION ESTIMATE STD DF FACTOR LOWER ___________ ________ ________ ___ ______ _________ 1 0.066443 0.050673 495 1.9648 -0.033118 2 0.27813 0.050673 495 1.9648 0.17857 3 0.73274 0.050673 495 1.9648 0.63318 4 1.1799 0.050672 495 1.9648 1.0803 5 1.258 0.050672 495 1.9648 1.1584 6 1.3765 0.050672 495 1.9648 1.277 7 1.9412 0.050671 495 1.9648 1.8417 8 2.7539 0.05067 495 1.9648 2.6544 9 3.5033 0.050668 495 1.9648 3.4037 10 4.3 0.050667 495 1.9648 4.2005 11 4.609 0.050666 495 1.9648 4.5094 12 5.2952 0.050665 495 1.9648 5.1956 13 5.3387 0.050665 495 1.9648 5.2391 14 5.3537 0.050664 495 1.9648 5.2542 15 5.729 0.050664 495 1.9648 5.6294 16 5.681 0.050664 495 1.9648 5.5814 17 7.0522 0.05066 495 1.9648 6.9527 18 7.0983 0.05066 495 1.9648 6.9987 19 7.171 0.05066 495 1.9648 7.0714 20 8.5729 0.050655 495 1.9648 8.4734 21 9.9495 0.05065 495 1.9648 9.85 22 11.529 0.050644 495 1.9648 11.429 23 12.222 0.050641 495 1.9648 12.123 24 12.583 0.05064 495 1.9648 12.484 25 14.562 0.050631 495 1.9648 14.462 26 15.234 0.050627 495 1.9648 15.134 27 15.439 0.050626 495 1.9648 15.339 28 17.866 0.050614 495 1.9648 17.767 29 18.451 0.05061 495 1.9648 18.352 30 18.672 0.050609 495 1.9648 18.572 31 18.941 0.050608 495 1.9648 18.841 32 19.313 0.050605 495 1.9648 19.214 33 21.537 0.050592 495 1.9648 21.438 34 22.029 0.050589 495 1.9648 21.93 35 22.084 0.050589 495 1.9648 21.985 36 23.726 0.050578 495 1.9648 23.626 37 23.898 0.050577 495 1.9648 23.799 38 24.768 0.050571 495 1.9648 24.669 39 25.666 0.050565 495 1.9648 25.567 40 26.742 0.050558 495 1.9648 26.643 41 26.878 0.050557 495 1.9648 26.778 42 27.917 0.05055 495 1.9648 27.818 43 28.236 0.050548 495 1.9648 28.136 44 28.709 0.050544 495 1.9648 28.609 45 29.071 0.050542 495 1.9648 28.972 46 29.147 0.050541 495 1.9648 29.048 47 29.347 0.05054 495 1.9648 29.248 48 30.029 0.050535 495 1.9648 29.929 49 31.394 0.050525 495 1.9648 31.294 50 31.42 0.050525 495 1.9648 31.321 51 31.474 0.050524 495 1.9648 31.375 52 32.61 0.050516 495 1.9648 32.511 53 33.764 0.050507 495 1.9648 33.665 54 34.314 0.050503 495 1.9648 34.215 55 34.963 0.050498 495 1.9648 34.863 56 35.874 0.050492 495 1.9648 35.775 57 36.064 0.05049 495 1.9648 35.964 58 36.146 0.05049 495 1.9648 36.047 59 36.252 0.050489 495 1.9648 36.153 60 37.349 0.050481 495 1.9648 37.25 61 37.806 0.050477 495 1.9648 37.707 62 40.913 0.050454 495 1.9648 40.813 63 41.717 0.050448 495 1.9648 41.618 64 42.302 0.050444 495 1.9648 42.202 65 43.79 0.050433 495 1.9648 43.691 66 44.133 0.05043 495 1.9648 44.034 67 44.428 0.050428 495 1.9648 44.328 68 45.049 0.050424 495 1.9648 44.95 69 45.139 0.050423 495 1.9648 45.04 70 46.319 0.050414 495 1.9648 46.22 71 46.688 0.050412 495 1.9648 46.589 72 47.3 0.050408 495 1.9648 47.201 73 48.416 0.0504 495 1.9648 48.317 74 49.093 0.050395 495 1.9648 48.994 75 49.2 0.050394 495 1.9648 49.101 76 49.185 0.050395 495 1.9648 49.086 77 49.439 0.050393 495 1.9648 49.339 78 50.068 0.050389 495 1.9648 49.969 79 50.64 0.050385 495 1.9648 50.541 80 50.93 0.050383 495 1.9648 50.831 81 51.3 0.05038 495 1.9648 51.201 82 51.589 0.050378 495 1.9648 51.491 83 53.06 0.050369 495 1.9648 52.961 84 54.115 0.050363 495 1.9648 54.016 85 54.781 0.050358 495 1.9648 54.682 86 54.863 0.050358 495 1.9648 54.764 87 55.305 0.050355 495 1.9648 55.207 88 56.876 0.050346 495 1.9648 56.778 89 57.989 0.05034 495 1.9648 57.89 90 59.173 0.050334 495 1.9648 59.074 91 59.263 0.050333 495 1.9648 59.164 92 59.754 0.050331 495 1.9648 59.655 93 60.315 0.050328 495 1.9648 60.216 94 60.651 0.050326 495 1.9648 60.552 95 60.975 0.050325 495 1.9648 60.876 96 61.113 0.050324 495 1.9648 61.014 97 61.651 0.050321 495 1.9648 61.552 98 62.09 0.050319 495 1.9648 61.991 99 62.96 0.050315 495 1.9648 62.861 100 63.166 0.050315 495 1.9648 63.067 101 63.206 0.050314 495 1.9648 63.107 102 64.562 0.050309 495 1.9648 64.463 103 66.405 0.050302 495 1.9648 66.306 104 66.681 0.050301 495 1.9648 66.583 105 67.888 0.050297 495 1.9648 67.789 106 68.159 0.050296 495 1.9648 68.06 107 68.225 0.050296 495 1.9648 68.126 108 68.982 0.050293 495 1.9648 68.883 109 69.668 0.050291 495 1.9648 69.569 110 70.168 0.05029 495 1.9648 70.069 111 71.875 0.050286 495 1.9648 71.776 112 72.061 0.050286 495 1.9648 71.962 113 72.185 0.050285 495 1.9648 72.086 114 72.425 0.050285 495 1.9648 72.326 115 72.342 0.050285 495 1.9648 72.243 116 72.661 0.050284 495 1.9648 72.562 117 72.611 0.050284 495 1.9648 72.512 118 73.123 0.050284 495 1.9648 73.025 119 75.022 0.050281 495 1.9648 74.923 120 77.027 0.050279 495 1.9648 76.929 121 77.711 0.050278 495 1.9648 77.613 122 77.822 0.050278 495 1.9648 77.723 123 77.908 0.050278 495 1.9648 77.809 124 78.01 0.050278 495 1.9648 77.911 125 78.693 0.050278 495 1.9648 78.594 126 80.8 0.050278 495 1.9648 80.701 127 81.399 0.050279 495 1.9648 81.3 128 81.676 0.050279 495 1.9648 81.577 129 81.891 0.050279 495 1.9648 81.792 130 83.573 0.050281 495 1.9648 83.475 131 85.331 0.050283 495 1.9648 85.232 132 86.171 0.050285 495 1.9648 86.073 133 86.226 0.050285 495 1.9648 86.127 134 87.293 0.050287 495 1.9648 87.194 135 87.392 0.050288 495 1.9648 87.293 136 87.833 0.050289 495 1.9648 87.734 137 88.944 0.050292 495 1.9648 88.845 138 90.295 0.050296 495 1.9648 90.196 139 90.601 0.050297 495 1.9648 90.502 140 90.838 0.050298 495 1.9648 90.739 141 91.198 0.050299 495 1.9648 91.099 142 92.155 0.050302 495 1.9648 92.056 143 92.363 0.050303 495 1.9648 92.264 144 92.494 0.050303 495 1.9648 92.395 145 92.706 0.050304 495 1.9648 92.607 146 93.017 0.050306 495 1.9648 92.918 147 93.721 0.050308 495 1.9648 93.622 148 93.716 0.050308 495 1.9648 93.617 149 94.088 0.05031 495 1.9648 93.989 150 94.563 0.050312 495 1.9648 94.464 151 95.296 0.050315 495 1.9648 95.197 152 95.354 0.050316 495 1.9648 95.255 153 95.85 0.050318 495 1.9648 95.751 154 96.017 0.050319 495 1.9648 95.919 155 96.767 0.050322 495 1.9648 96.668 156 96.772 0.050322 495 1.9648 96.673 157 97.024 0.050324 495 1.9648 96.925 158 97.105 0.050324 495 1.9648 97.006 159 97.625 0.050327 495 1.9648 97.527 160 98.389 0.050331 495 1.9648 98.29 161 101.08 0.050346 495 1.9648 100.98 162 101.36 0.050348 495 1.9648 101.26 163 101.96 0.050351 495 1.9648 101.86 164 104.15 0.050365 495 1.9648 104.05 165 107.96 0.050392 495 1.9648 107.86 166 108.09 0.050392 495 1.9648 107.99 167 108.45 0.050395 495 1.9648 108.35 168 108.55 0.050396 495 1.9648 108.45 169 108.94 0.050399 495 1.9648 108.84 170 109.13 0.0504 495 1.9648 109.03 171 109.47 0.050403 495 1.9648 109.37 172 109.86 0.050406 495 1.9648 109.76 173 110.24 0.050408 495 1.9648 110.14 174 111.65 0.050419 495 1.9648 111.55 175 111.72 0.05042 495 1.9648 111.62 176 112.5 0.050426 495 1.9648 112.4 177 113.07 0.05043 495 1.9648 112.97 178 115.26 0.050448 495 1.9648 115.16 179 116.15 0.050455 495 1.9648 116.05 180 118.94 0.050478 495 1.9648 118.85 181 119.15 0.05048 495 1.9648 119.05 182 119.18 0.05048 495 1.9648 119.08 183 119.31 0.050481 495 1.9648 119.21 184 119.45 0.050482 495 1.9648 119.35 185 120.14 0.050488 495 1.9648 120.04 186 120.73 0.050492 495 1.9648 120.63 187 121.04 0.050495 495 1.9648 120.95 188 121.25 0.050497 495 1.9648 121.15 189 122.27 0.050505 495 1.9648 122.17 190 122.37 0.050506 495 1.9648 122.27 191 122.57 0.050508 495 1.9648 122.47 192 123.01 0.050511 495 1.9648 122.91 193 123.96 0.050519 495 1.9648 123.86 194 124.15 0.050521 495 1.9648 124.05 195 125.39 0.050531 495 1.9648 125.29 196 125.86 0.050534 495 1.9648 125.76 197 127.01 0.050544 495 1.9648 126.91 198 128.59 0.050556 495 1.9648 128.49 199 129.3 0.050561 495 1.9648 129.2 200 129.44 0.050563 495 1.9648 129.34 201 130.44 0.05057 495 1.9648 130.34 202 131.02 0.050575 495 1.9648 130.92 203 131.03 0.050575 495 1.9648 130.93 204 131.39 0.050577 495 1.9648 131.29 205 131.42 0.050578 495 1.9648 131.32 206 132 0.050582 495 1.9648 131.9 207 133.86 0.050595 495 1.9648 133.76 208 134.77 0.050602 495 1.9648 134.67 209 135.81 0.050609 495 1.9648 135.71 210 137.6 0.05062 495 1.9648 137.5 211 138.32 0.050625 495 1.9648 138.22 212 138.83 0.050628 495 1.9648 138.73 213 139.4 0.050631 495 1.9648 139.31 214 140.42 0.050637 495 1.9648 140.32 215 140.81 0.050639 495 1.9648 140.71 216 140.82 0.05064 495 1.9648 140.72 217 140.89 0.05064 495 1.9648 140.79 218 140.86 0.05064 495 1.9648 140.76 219 142.32 0.050648 495 1.9648 142.22 220 142.88 0.05065 495 1.9648 142.78 221 143.34 0.050653 495 1.9648 143.24 222 143.36 0.050653 495 1.9648 143.26 223 144.71 0.050659 495 1.9648 144.61 224 146.81 0.050668 495 1.9648 146.72 225 147.27 0.05067 495 1.9648 147.17 226 147.59 0.050671 495 1.9648 147.49 227 148.18 0.050673 495 1.9648 148.08 228 148.69 0.050675 495 1.9648 148.59 229 149.24 0.050677 495 1.9648 149.14 230 149.38 0.050677 495 1.9648 149.28 231 149.84 0.050679 495 1.9648 149.74 232 151.06 0.050682 495 1.9648 150.96 233 151.29 0.050683 495 1.9648 151.19 234 151.37 0.050683 495 1.9648 151.27 235 151.57 0.050684 495 1.9648 151.47 236 153.05 0.050687 495 1.9648 152.95 237 153.34 0.050688 495 1.9648 153.24 238 154.51 0.05069 495 1.9648 154.41 239 154.46 0.05069 495 1.9648 154.36 240 154.93 0.05069 495 1.9648 154.83 241 155.09 0.050691 495 1.9648 154.99 242 155.27 0.050691 495 1.9648 155.17 243 155.32 0.050691 495 1.9648 155.22 244 156.26 0.050692 495 1.9648 156.16 245 157.73 0.050693 495 1.9648 157.63 246 157.9 0.050693 495 1.9648 157.8 247 -158.2 0.050693 495 1.9648 -158.3 248 -157.1 0.050693 495 1.9648 -157.2 249 -156.82 0.050693 495 1.9648 -156.92 250 -156.26 0.050693 495 1.9648 -156.36 251 -155.46 0.050693 495 1.9648 -155.56 252 -155.01 0.050692 495 1.9648 -155.11 253 -154.71 0.050692 495 1.9648 -154.81 254 -154.51 0.050692 495 1.9648 -154.61 255 -154.33 0.050692 495 1.9648 -154.43 256 -153.62 0.050691 495 1.9648 -153.72 257 -152.81 0.05069 495 1.9648 -152.91 258 -152.73 0.05069 495 1.9648 -152.83 259 -150.77 0.050686 495 1.9648 -150.87 260 -150.11 0.050684 495 1.9648 -150.21 261 -149.84 0.050684 495 1.9648 -149.94 262 -148.88 0.050681 495 1.9648 -148.98 263 -147.97 0.050678 495 1.9648 -148.07 264 -147.78 0.050678 495 1.9648 -147.88 265 -146.57 0.050674 495 1.9648 -146.67 266 -146.58 0.050674 495 1.9648 -146.67 267 -145.09 0.050668 495 1.9648 -145.19 268 -143.16 0.05066 495 1.9648 -143.26 269 -143.05 0.05066 495 1.9648 -143.15 270 -142.93 0.050659 495 1.9648 -143.03 271 -141.68 0.050653 495 1.9648 -141.78 272 -141.55 0.050652 495 1.9648 -141.65 273 -141.21 0.050651 495 1.9648 -141.31 274 -140.84 0.050649 495 1.9648 -140.94 275 -140.7 0.050648 495 1.9648 -140.8 276 -139.8 0.050643 495 1.9648 -139.9 277 -138.69 0.050637 495 1.9648 -138.79 278 -138.65 0.050637 495 1.9648 -138.75 279 -137.09 0.050628 495 1.9648 -137.19 280 -136.74 0.050626 495 1.9648 -136.84 281 -136.42 0.050624 495 1.9648 -136.52 282 -136.16 0.050622 495 1.9648 -136.26 283 -135.59 0.050618 495 1.9648 -135.69 284 -135.2 0.050616 495 1.9648 -135.3 285 -134.35 0.05061 495 1.9648 -134.45 286 -134.18 0.050609 495 1.9648 -134.28 287 -134.03 0.050608 495 1.9648 -134.13 288 -132.55 0.050598 495 1.9648 -132.65 289 -132.6 0.050598 495 1.9648 -132.7 290 -131.06 0.050587 495 1.9648 -131.16 291 -130.6 0.050584 495 1.9648 -130.69 292 -130.28 0.050582 495 1.9648 -130.38 293 -129.91 0.050579 495 1.9648 -130.01 294 -126.83 0.050555 495 1.9648 -126.93 295 -126.84 0.050555 495 1.9648 -126.94 296 -126.73 0.050554 495 1.9648 -126.82 297 -126.29 0.050551 495 1.9648 -126.39 298 -125.73 0.050547 495 1.9648 -125.83 299 -125.34 0.050543 495 1.9648 -125.44 300 -125.26 0.050543 495 1.9648 -125.36 301 -123.3 0.050527 495 1.9648 -123.4 302 -123.13 0.050526 495 1.9648 -123.23 303 -122.88 0.050523 495 1.9648 -122.98 304 -121.26 0.05051 495 1.9648 -121.36 305 -120.7 0.050505 495 1.9648 -120.8 306 -119.75 0.050498 495 1.9648 -119.85 307 -119.04 0.050492 495 1.9648 -119.14 308 -118.83 0.05049 495 1.9648 -118.93 309 -118.07 0.050484 495 1.9648 -118.17 310 -117.81 0.050481 495 1.9648 -117.91 311 -117.76 0.050481 495 1.9648 -117.86 312 -117.69 0.05048 495 1.9648 -117.79 313 -116.4 0.05047 495 1.9648 -116.49 314 -116.2 0.050468 495 1.9648 -116.3 315 -116.04 0.050467 495 1.9648 -116.14 316 -115.79 0.050465 495 1.9648 -115.89 317 -115.16 0.050459 495 1.9648 -115.26 318 -115.02 0.050458 495 1.9648 -115.12 319 -114.4 0.050453 495 1.9648 -114.5 320 -113.68 0.050447 495 1.9648 -113.78 321 -113.27 0.050444 495 1.9648 -113.37 322 -112.5 0.050437 495 1.9648 -112.6 323 -112.45 0.050437 495 1.9648 -112.55 324 -111.84 0.050432 495 1.9648 -111.94 325 -110.54 0.050422 495 1.9648 -110.64 326 -110.39 0.05042 495 1.9648 -110.49 327 -109.83 0.050416 495 1.9648 -109.93 328 -106.52 0.050391 495 1.9648 -106.62 329 -105.39 0.050382 495 1.9648 -105.49 330 -105.42 0.050383 495 1.9648 -105.52 331 -105.19 0.050381 495 1.9648 -105.29 332 -105.15 0.050381 495 1.9648 -105.25 333 -102.96 0.050365 495 1.9648 -103.06 334 -102.34 0.050361 495 1.9648 -102.44 335 -100.06 0.050346 495 1.9648 -100.16 336 -100.06 0.050346 495 1.9648 -100.15 337 -100.08 0.050347 495 1.9648 -100.18 338 -99.652 0.050344 495 1.9648 -99.751 339 -98.955 0.05034 495 1.9648 -99.054 340 -97.781 0.050333 495 1.9648 -97.88 341 -97.701 0.050332 495 1.9648 -97.8 342 -96.387 0.050325 495 1.9648 -96.486 343 -96.305 0.050325 495 1.9648 -96.404 344 -95.575 0.050321 495 1.9648 -95.674 345 -95.055 0.050318 495 1.9648 -95.153 346 -95.002 0.050318 495 1.9648 -95.101 347 -94.945 0.050318 495 1.9648 -95.044 348 -93.623 0.050311 495 1.9648 -93.722 349 -93.442 0.050311 495 1.9648 -93.541 350 -93.099 0.050309 495 1.9648 -93.198 351 -92.918 0.050308 495 1.9648 -93.017 352 -92.823 0.050308 495 1.9648 -92.922 353 -92.358 0.050306 495 1.9648 -92.456 354 -91.867 0.050304 495 1.9648 -91.965 355 -90.742 0.0503 495 1.9648 -90.84 356 -90.466 0.050298 495 1.9648 -90.565 357 -90.335 0.050298 495 1.9648 -90.434 358 -89.406 0.050295 495 1.9648 -89.505 359 -87.913 0.05029 495 1.9648 -88.012 360 -87.013 0.050288 495 1.9648 -87.112 361 -86.623 0.050287 495 1.9648 -86.722 362 -85.302 0.050284 495 1.9648 -85.401 363 -85.052 0.050283 495 1.9648 -85.151 364 -83.246 0.05028 495 1.9648 -83.344 365 -82.833 0.05028 495 1.9648 -82.932 366 -80.619 0.050278 495 1.9648 -80.718 367 -80.548 0.050278 495 1.9648 -80.646 368 -79.963 0.050278 495 1.9648 -80.062 369 -78.842 0.050278 495 1.9648 -78.94 370 -78.542 0.050278 495 1.9648 -78.641 371 -77.545 0.050279 495 1.9648 -77.643 372 -77.292 0.050279 495 1.9648 -77.391 373 -76.798 0.050279 495 1.9648 -76.897 374 -73.831 0.050283 495 1.9648 -73.93 375 -73.175 0.050284 495 1.9648 -73.274 376 -72.934 0.050285 495 1.9648 -73.033 377 -72.77 0.050285 495 1.9648 -72.868 378 -72.312 0.050286 495 1.9648 -72.411 379 -72.253 0.050287 495 1.9648 -72.351 380 -72.058 0.050287 495 1.9648 -72.156 381 -71.549 0.050288 495 1.9648 -71.648 382 -70.909 0.05029 495 1.9648 -71.008 383 -70.36 0.050292 495 1.9648 -70.458 384 -69.997 0.050293 495 1.9648 -70.096 385 -69.691 0.050294 495 1.9648 -69.789 386 -68.277 0.050299 495 1.9648 -68.376 387 -68.051 0.050299 495 1.9648 -68.15 388 -65.831 0.050308 495 1.9648 -65.93 389 -65.255 0.050311 495 1.9648 -65.353 390 -65.192 0.050311 495 1.9648 -65.291 391 -64.898 0.050313 495 1.9648 -64.996 392 -64.877 0.050313 495 1.9648 -64.976 393 -64.253 0.050316 495 1.9648 -64.352 394 -62.638 0.050324 495 1.9648 -62.737 395 -61.842 0.050328 495 1.9648 -61.941 396 -61.414 0.05033 495 1.9648 -61.513 397 -61.336 0.050331 495 1.9648 -61.435 398 -60.632 0.050335 495 1.9648 -60.731 399 -59.93 0.050339 495 1.9648 -60.029 400 -59.757 0.05034 495 1.9648 -59.856 401 -59.022 0.050345 495 1.9648 -59.121 402 -58.54 0.050347 495 1.9648 -58.639 403 -58.097 0.05035 495 1.9648 -58.196 404 -56.92 0.050358 495 1.9648 -57.019 405 -56.354 0.050362 495 1.9648 -56.453 406 -54.924 0.050371 495 1.9648 -55.023 407 -54.77 0.050373 495 1.9648 -54.869 408 -54.567 0.050374 495 1.9648 -54.666 409 -54.078 0.050377 495 1.9648 -54.177 410 -53.902 0.050379 495 1.9648 -54.001 411 -53.472 0.050382 495 1.9648 -53.571 412 -53.381 0.050382 495 1.9648 -53.48 413 -52.22 0.050391 495 1.9648 -52.319 414 -51.825 0.050394 495 1.9648 -51.924 415 -50.928 0.050401 495 1.9648 -51.027 416 -49.55 0.050411 495 1.9648 -49.649 417 -49.071 0.050415 495 1.9648 -49.17 418 -47.365 0.050428 495 1.9648 -47.464 419 -46.347 0.050437 495 1.9648 -46.446 420 -45.955 0.05044 495 1.9648 -46.054 421 -45.782 0.050441 495 1.9648 -45.881 422 -45.119 0.050446 495 1.9648 -45.218 423 -44.62 0.05045 495 1.9648 -44.719 424 -43.995 0.050456 495 1.9648 -44.094 425 -43.988 0.050456 495 1.9648 -44.088 426 -43.977 0.050456 495 1.9648 -44.076 427 -43.771 0.050457 495 1.9648 -43.87 428 -43.3 0.050461 495 1.9648 -43.399 429 -42.549 0.050467 495 1.9648 -42.648 430 -42.213 0.05047 495 1.9648 -42.312 431 -41.992 0.050472 495 1.9648 -42.091 432 -40.018 0.050488 495 1.9648 -40.117 433 -39.206 0.050495 495 1.9648 -39.305 434 -38.77 0.050498 495 1.9648 -38.87 435 -38.001 0.050504 495 1.9648 -38.1 436 -37.058 0.050512 495 1.9648 -37.157 437 -36.708 0.050515 495 1.9648 -36.808 438 -36.607 0.050516 495 1.9648 -36.706 439 -36.629 0.050515 495 1.9648 -36.728 440 -36.251 0.050518 495 1.9648 -36.351 441 -36.266 0.050518 495 1.9648 -36.366 442 -36.099 0.05052 495 1.9648 -36.198 443 -35.435 0.050525 495 1.9648 -35.535 444 -34.056 0.050536 495 1.9648 -34.156 445 -33.746 0.050538 495 1.9648 -33.845 446 -33.288 0.050542 495 1.9648 -33.387 447 -31.231 0.050557 495 1.9648 -31.33 448 -30.744 0.050561 495 1.9648 -30.843 449 -28.851 0.050574 495 1.9648 -28.951 450 -28.67 0.050576 495 1.9648 -28.769 451 -28.161 0.050579 495 1.9648 -28.261 452 -27.516 0.050584 495 1.9648 -27.615 453 -27.422 0.050584 495 1.9648 -27.521 454 -27.018 0.050587 495 1.9648 -27.117 455 -26.858 0.050588 495 1.9648 -26.957 456 -26.467 0.050591 495 1.9648 -26.567 457 -26.485 0.050591 495 1.9648 -26.584 458 -25.779 0.050595 495 1.9648 -25.878 459 -25.579 0.050597 495 1.9648 -25.678 460 -24.753 0.050602 495 1.9648 -24.853 461 -24.674 0.050602 495 1.9648 -24.774 462 -23.737 0.050608 495 1.9648 -23.836 463 -21.611 0.05062 495 1.9648 -21.71 464 -21.278 0.050622 495 1.9648 -21.378 465 -20.421 0.050627 495 1.9648 -20.52 466 -20.433 0.050627 495 1.9648 -20.532 467 -20.459 0.050626 495 1.9648 -20.558 468 -19.97 0.050629 495 1.9648 -20.069 469 -19.171 0.050633 495 1.9648 -19.27 470 -18.893 0.050634 495 1.9648 -18.992 471 -18.755 0.050635 495 1.9648 -18.855 472 -17.862 0.050639 495 1.9648 -17.961 473 -17.942 0.050639 495 1.9648 -18.042 474 -17.511 0.050641 495 1.9648 -17.611 475 -17.511 0.050641 495 1.9648 -17.611 476 -16.941 0.050643 495 1.9648 -17.041 477 -16.598 0.050645 495 1.9648 -16.697 478 -16.493 0.050645 495 1.9648 -16.592 479 -16.414 0.050646 495 1.9648 -16.513 480 -14.21 0.050654 495 1.9648 -14.309 481 -14.021 0.050655 495 1.9648 -14.121 482 -13.045 0.050658 495 1.9648 -13.144 483 -12.452 0.05066 495 1.9648 -12.552 484 -11.314 0.050663 495 1.9648 -11.414 485 -10.775 0.050664 495 1.9648 -10.875 486 -10.202 0.050666 495 1.9648 -10.301 487 -9.9277 0.050666 495 1.9648 -10.027 488 -9.8317 0.050667 495 1.9648 -9.9312 489 -9.3933 0.050668 495 1.9648 -9.4928 490 -9.2314 0.050668 495 1.9648 -9.331 491 -8.2837 0.05067 495 1.9648 -8.3832 492 -7.0638 0.050672 495 1.9648 -7.1633 493 -6.4429 0.050672 495 1.9648 -6.5424 494 -4.8267 0.050674 495 1.9648 -4.9262 495 -4.5049 0.050674 495 1.9648 -4.6045 496 -3.7284 0.050674 495 1.9648 -3.828 497 -2.5452 0.050674 495 1.9648 -2.6448 498 -2.0393 0.050674 495 1.9648 -2.1389 499 -1.3021 0.050674 495 1.9648 -1.4017 500 -0.93418 0.050674 495 1.9648 -1.0337 UPPER ________ 0.166 0.37769 0.8323 1.2794 1.3576 1.4761 2.0408 2.8535 3.6029 4.3996 4.7085 5.3947 5.4382 5.4533 5.8285 5.7805 7.1518 7.1978 7.2705 8.6724 10.049 11.628 12.322 12.683 14.661 15.333 15.538 17.966 18.551 18.771 19.04 19.413 21.637 22.129 22.184 23.825 23.998 24.868 25.765 26.842 26.977 28.017 28.335 28.808 29.17 29.247 29.446 30.128 31.493 31.519 31.574 32.709 33.864 34.413 35.062 35.973 36.163 36.245 36.351 37.448 37.905 41.012 41.816 42.401 43.889 44.233 44.527 45.149 45.238 46.418 46.787 47.399 48.515 49.192 49.299 49.284 49.538 50.167 50.739 51.029 51.399 51.688 53.159 54.214 54.88 54.962 55.404 56.975 58.088 59.271 59.362 59.853 60.414 60.75 61.074 61.212 61.75 62.189 63.059 63.264 63.305 64.661 66.504 66.78 67.987 68.258 68.323 69.081 69.767 70.266 71.974 72.16 72.284 72.524 72.441 72.76 72.71 73.222 75.121 77.126 77.81 77.92 78.007 78.109 78.792 80.899 81.498 81.774 81.99 83.672 85.43 86.27 86.325 87.391 87.49 87.932 89.043 90.394 90.7 90.937 91.297 92.254 92.462 92.593 92.805 93.116 93.819 93.815 94.187 94.662 95.395 95.453 95.949 96.116 96.866 96.871 97.123 97.204 97.724 98.488 101.17 101.46 102.06 104.25 108.06 108.19 108.55 108.65 109.03 109.23 109.57 109.96 110.34 111.74 111.81 112.6 113.17 115.36 116.25 119.04 119.25 119.27 119.41 119.55 120.24 120.83 121.14 121.35 122.36 122.46 122.67 123.11 124.06 124.25 125.49 125.96 127.11 128.69 129.39 129.54 130.54 131.12 131.13 131.49 131.52 132.1 133.96 134.87 135.9 137.7 138.42 138.92 139.5 140.52 140.91 140.92 140.99 140.96 142.42 142.97 143.44 143.46 144.81 146.91 147.37 147.69 148.28 148.79 149.33 149.48 149.94 151.16 151.39 151.47 151.67 153.15 153.44 154.61 154.56 155.03 155.19 155.37 155.42 156.36 157.83 158 -158.11 -157 -156.72 -156.16 -155.36 -154.92 -154.61 -154.41 -154.23 -153.52 -152.71 -152.63 -150.67 -150.01 -149.74 -148.78 -147.87 -147.68 -146.47 -146.48 -144.99 -143.06 -142.95 -142.83 -141.58 -141.45 -141.11 -140.74 -140.6 -139.7 -138.59 -138.55 -136.99 -136.64 -136.32 -136.06 -135.49 -135.1 -134.25 -134.08 -133.93 -132.45 -132.5 -130.96 -130.5 -130.18 -129.81 -126.73 -126.74 -126.63 -126.19 -125.63 -125.24 -125.16 -123.2 -123.03 -122.78 -121.16 -120.6 -119.65 -118.94 -118.73 -117.98 -117.71 -117.66 -117.59 -116.3 -116.1 -115.94 -115.69 -115.06 -114.92 -114.3 -113.58 -113.17 -112.4 -112.36 -111.74 -110.44 -110.29 -109.73 -106.42 -105.29 -105.33 -105.09 -105.05 -102.86 -102.24 -99.965 -99.957 -99.981 -99.553 -98.856 -97.682 -97.602 -96.288 -96.206 -95.476 -94.956 -94.903 -94.846 -93.524 -93.343 -93.001 -92.819 -92.724 -92.259 -91.768 -90.643 -90.367 -90.236 -89.307 -87.814 -86.915 -86.524 -85.203 -84.953 -83.147 -82.734 -80.52 -80.449 -79.864 -78.743 -78.443 -77.446 -77.193 -76.7 -73.732 -73.076 -72.835 -72.671 -72.213 -72.154 -71.959 -71.45 -70.81 -70.261 -69.898 -69.592 -68.179 -67.952 -65.732 -65.156 -65.094 -64.799 -64.778 -64.155 -62.539 -61.743 -61.315 -61.237 -60.533 -59.831 -59.658 -58.923 -58.441 -57.998 -56.821 -56.255 -54.825 -54.671 -54.468 -53.979 -53.803 -53.373 -53.282 -52.121 -51.726 -50.829 -49.451 -48.972 -47.266 -46.248 -45.856 -45.683 -45.02 -44.521 -43.896 -43.889 -43.878 -43.672 -43.201 -42.449 -42.114 -41.893 -39.918 -39.106 -38.671 -37.902 -36.958 -36.609 -36.508 -36.53 -36.152 -36.167 -36 -35.336 -33.957 -33.646 -33.188 -31.132 -30.644 -28.752 -28.57 -28.062 -27.416 -27.322 -26.918 -26.758 -26.368 -26.386 -25.679 -25.48 -24.654 -24.575 -23.637 -21.512 -21.179 -20.321 -20.333 -20.359 -19.87 -19.071 -18.793 -18.656 -17.762 -17.843 -17.412 -17.412 -16.842 -16.498 -16.393 -16.314 -14.11 -13.922 -12.945 -12.353 -11.215 -10.676 -10.102 -9.8281 -9.7321 -9.2937 -9.1319 -8.1841 -6.9642 -6.3433 -4.7271 -4.4053 -3.6289 -2.4457 -1.9398 -1.2025 -0.83462 Displacement unit: nanometer [nm] ```

Plot the stat. uncertainty of displacement

```displ_std = result.Displacement_std; figure plot(phitrue, displ_std) xlabel('phi'); ylabel(['Statistical uncertainty of displacement, ',result.Displacement_unit]); title(['Statistical Uncertainty: Min: ', ... num2str(result.Displacement_std_min), ', Max: ', ... num2str(result.Displacement_std_max) ]); ```

Plot the displacements with expanded uncertainties

```displ_fit = result.Displacement_fit; const = result.Displacement_const; displ_true = const*phitrue; displ_res = displ_fit - displ_true; ix = (result.phi_fit-phitrue) < -1; displ_res(ix) = displ_res(ix) + 2*pi*const; figure plot(displ_true, displ_res ,'o') hold plot(displ_true, 2*displ_std ,'r-') plot(displ_true, -2*displ_std ,'r-') grid xlabel('True displacement'); ylabel('Displacement Residuals'); title('Expanded Uncertainty of Fitted Displacement'); ```
```Current plot held ```

Plot the X residuals with the expanded uncertainties

```figure plot(phitrue,result.x_res_fit,'.') hold plot(phitrue,2*result.x_res_std,'r-') plot(phitrue,-2*result.x_res_std,'r-') grid xlabel('True phi'); ylabel('x residuals'); title('Expanded Uncertainty of Fitted X Residuals'); ```
```Current plot held ```

Plot the Y residuals with the expanded uncertainties

```figure plot(phitrue,result.y_res_fit,'.') hold plot(phitrue,2*result.y_res_std,'r-') plot(phitrue,-2*result.y_res_std,'r-') grid xlabel('True phi'); ylabel('y residuals'); title('Expanded Uncertainty of Fitted Y Residuals'); ```
```Current plot held ```

CORRELATED MEASUREMENTS

```clear ```

GENERATE the correlated measurements: x and y

```alpha0true = 0; % x-center (offset x) beta0true = 0; % y-center (offset x) alpha1true = 1; % x-amplitude beta1true = 0.98; % y-amplitude phi0true = pi/10; % phase offset sigma = 0.005; % std of x and y independent errors n = 1000; % number of measurements points cycles = 1; % measurement interval, # cycles phitrue = cycles*(2*pi)*sort(rand(n,1)); % true phases phi_i % true values: X and Y Xmean = @(t) alpha0true + alpha1true * cos(t); Ymean = @(t) beta0true + beta1true * sin(t + phi0true); rho = 0.9; err = mvnrnd([0 0],sigma^2*[1 rho; rho 1],n); % AR errors + correlated x and y % rhoAR = -.99995; % err = sigma*filter(1,[1 rhoAR],mvnrnd([0 0],[1 rho; rho 1],n)); x = Xmean(phitrue) + err(:,1); y = Ymean(phitrue) + err(:,2); ```

Fit the ellipse based on measured data

```options.alpha = 0.05; options.correlation = rho; % Set the true correlation (assumed to be known) options.displconst = 633.3/(4*pi); options.displunit = 'nanometer [nm]'; result = ellipseFit4HC(x,y,options); disp(result) ```
``` ------------------------------------------------- ESTIMATED ELLIPSE PARAMETERS X^2 + B*Y^2 + C*X*Y + D*X + F*Y + G = 0 ------------------------------------------------- ESTIMATE STD DF FACTOR LOWER UPPER __________ __________ ___ ______ ___________ __________ B 1.0403 0.00071011 995 1.9624 1.039 1.0417 C -0.63017 0.00071325 995 1.9624 -0.63157 -0.62877 D 8.6817e-05 0.00031468 995 1.9624 -0.00053069 0.00070433 F 0.00015771 0.00032069 995 1.9624 -0.00047159 0.00078702 G -0.90401 0.00037431 995 1.9624 -0.90475 -0.90328 ------------------------------------------------- ESTIMATED ELLIPSE PARAMETERS X(phi) = alpha_0 + alpha_1 * cos(phi) Y(phi) = beta_0 + beta_1 * sin(phi + phi_0 ------------------------------------------------- ESTIMATE STD DF FACTOR LOWER ___________ __________ ___ ______ ___________ alpha_0 -7.4391e-05 0.00019574 995 1.9624 -0.0004585 beta_0 -9.8329e-05 0.00019179 995 1.9624 -0.00047468 alpha_1 0.99969 0.00025686 995 1.9624 0.99919 beta_1 0.98011 0.00024759 995 1.9624 0.97963 phi_0 0.31405 0.00034675 995 1.9624 0.31337 UPPER __________ alpha_0 0.00030972 beta_0 0.00027803 alpha_1 1.0002 beta_1 0.9806 phi_0 0.31473 Description: 'Ellipse Fit by Iterated Locally Best Linea...' ABCDEF_Description: 'A*x^2 + 2*B*x*y + C*y^2 + 2*D*x + 2*E*y + ...' ABCDEF_Names: {'A' 'B' 'C' 'D' 'E' 'F'} ABCDEF_fit: [1x6 double] ABCDEF_std: [1x6 double] BCDFG_Description: 'x^2 + B*y^2 + C*x*y + D*x + F*y + G = 0' BCDFG_Names: {'B' 'C' 'D' 'F' 'G'} BCDFG_fit: [1.0403 -0.6302 8.6817e-05 1.5771e-04 -0.9040] BCDFG_std: [1x5 double] BCDFG_cov: [5x5 double] EllipsePars_Description_X: 'X(phi) = alpha_0 + alpha_1 * cos(phi)' EllipsePars_Description_Y: 'Y(phi) = beta_0 + beta_1 * sin(phi + phi_0)' EllipsePars_Names: {1x5 cell} EllipsePars_fit: [-7.4391e-05 -9.8329e-05 0.9997 0.9801 0.3140] EllipsePars_std: [1x5 double] mu_X_fit: [1000x1 double] mu_X_std: [1000x1 double] nu_Y_fit: [1000x1 double] nu_Y_std: [1000x1 double] X_fitfun: @(t)alpha0_fit+alpha1_fit*cos(t) Y_fitfun: @(t)beta0_fit+beta1_fit*sin(t+phi0_fit) x_res_fit: [1000x1 double] x_res_std: [1000x1 double] y_res_fit: [1000x1 double] y_res_std: [1000x1 double] sigma_fit: 0.0051 sigma2_fit: 2.5536e-05 phi_fit: [1000x1 double] phi_std: [1000x1 double] phi_std_max: 0.0062 phi_std_min: 0.0020 Displacement_Description: 'Displacement_fit = Displacement_const * ph...' Displacement_fit: [1000x1 double] Displacement_std: [1000x1 double] Displacement_std_max: 0.3104 Displacement_std_min: 0.0991 Displacement_const: 50.3964 Displacement_unit: 'nanometer [nm]' TABLE_BCDFG: [5x6 table] TABLE_EllipsePars: [5x6 table] TABLE_Displacements: [1000x7 table] x_data: [1000x1 double] y_data: [1000x1 double] used_Correlation_XY: 0.9000 options: [1x1 struct] criterion: 2.1270e-12 loops: 7 tictoc: 0.0255 ```

Display statistics of the fitted displacements (!! Table with n rows !!)

```TABLE1 = result.TABLE_EllipsePars; TABLE2 = result.TABLE_Displacements; disp(TABLE2) disp(['Displacement unit: ' result.Displacement_unit]) ```
``` OBSERVATION ESTIMATE STD DF FACTOR LOWER UPPER ___________ ________ ________ ___ ______ _______ _______ 1 1.2196 0.11976 995 1.9624 0.98454 1.4546 2 2.0558 0.11858 995 1.9624 1.8231 2.2885 3 2.3752 0.11815 995 1.9624 2.1433 2.607 4 4.1426 0.11584 995 1.9624 3.9153 4.3699 5 4.4604 0.11545 995 1.9624 4.2339 4.687 6 4.8572 0.11497 995 1.9624 4.6316 5.0828 7 4.6004 0.11528 995 1.9624 4.3742 4.8266 8 4.59 0.11529 995 1.9624 4.3638 4.8163 9 4.6821 0.11518 995 1.9624 4.4561 4.9081 10 5.0404 0.11475 995 1.9624 4.8153 5.2656 11 5.0176 0.11477 995 1.9624 4.7923 5.2428 12 5.553 0.11414 995 1.9624 5.329 5.777 13 6.2602 0.11333 995 1.9624 6.0379 6.4826 14 6.6402 0.11291 995 1.9624 6.4186 6.8617 15 6.8591 0.11267 995 1.9624 6.638 7.0802 16 7.2358 0.11226 995 1.9624 7.0155 7.456 17 7.7238 0.11174 995 1.9624 7.5046 7.9431 18 7.9199 0.11154 995 1.9624 7.7011 8.1388 19 8.087 0.11137 995 1.9624 7.8684 8.3055 20 7.9215 0.11154 995 1.9624 7.7026 8.1404 21 8.4959 0.11095 995 1.9624 8.2781 8.7136 22 8.5153 0.11093 995 1.9624 8.2976 8.733 23 8.4272 0.11102 995 1.9624 8.2094 8.6451 24 8.5857 0.11086 995 1.9624 8.3681 8.8032 25 8.6625 0.11078 995 1.9624 8.4451 8.8799 26 8.6875 0.11076 995 1.9624 8.4701 8.9048 27 8.8788 0.11057 995 1.9624 8.6619 9.0958 28 9.021 0.11043 995 1.9624 8.8043 9.2377 29 8.8546 0.11059 995 1.9624 8.6376 9.0717 30 8.8666 0.11058 995 1.9624 8.6496 9.0836 31 8.9623 0.11049 995 1.9624 8.7455 9.1791 32 9.2986 0.11016 995 1.9624 9.0824 9.5147 33 9.9527 0.10954 995 1.9624 9.7378 10.168 34 10.265 0.10925 995 1.9624 10.051 10.48 35 10.04 0.10946 995 1.9624 9.8253 10.255 36 10.654 0.1089 995 1.9624 10.441 10.868 37 10.796 0.10877 995 1.9624 10.582 11.009 38 10.87 0.10871 995 1.9624 10.657 11.084 39 11.139 0.10847 995 1.9624 10.926 11.352 40 11.322 0.10831 995 1.9624 11.109 11.534 41 11.445 0.1082 995 1.9624 11.232 11.657 42 11.825 0.10788 995 1.9624 11.614 12.037 43 11.762 0.10793 995 1.9624 11.55 11.974 44 11.864 0.10785 995 1.9624 11.652 12.075 45 12.458 0.10736 995 1.9624 12.247 12.669 46 13.059 0.10688 995 1.9624 12.849 13.269 47 13.807 0.1063 995 1.9624 13.599 14.016 48 14.243 0.10598 995 1.9624 14.035 14.451 49 14.43 0.10584 995 1.9624 14.222 14.638 50 14.949 0.10547 995 1.9624 14.742 15.156 51 15.135 0.10534 995 1.9624 14.928 15.342 52 15.07 0.10539 995 1.9624 14.863 15.277 53 15.215 0.10528 995 1.9624 15.009 15.422 54 15.175 0.10531 995 1.9624 14.968 15.381 55 16.517 0.10442 995 1.9624 16.312 16.721 56 16.642 0.10434 995 1.9624 16.437 16.847 57 16.764 0.10426 995 1.9624 16.56 16.969 58 17.022 0.1041 995 1.9624 16.818 17.226 59 17.339 0.10391 995 1.9624 17.135 17.542 60 17.462 0.10384 995 1.9624 17.258 17.666 61 18.761 0.10309 995 1.9624 18.559 18.963 62 19.044 0.10294 995 1.9624 18.842 19.246 63 19.476 0.10271 995 1.9624 19.275 19.678 64 19.516 0.10269 995 1.9624 19.314 19.717 65 20.402 0.10225 995 1.9624 20.201 20.603 66 21.67 0.10167 995 1.9624 21.47 21.869 67 21.734 0.10164 995 1.9624 21.535 21.934 68 22.057 0.10151 995 1.9624 21.858 22.257 69 22.146 0.10147 995 1.9624 21.947 22.345 70 21.972 0.10154 995 1.9624 21.773 22.171 71 22.543 0.10131 995 1.9624 22.345 22.742 72 22.548 0.1013 995 1.9624 22.349 22.747 73 22.738 0.10123 995 1.9624 22.54 22.937 74 23.023 0.10112 995 1.9624 22.824 23.221 75 23.351 0.101 995 1.9624 23.152 23.549 76 23.358 0.10099 995 1.9624 23.16 23.556 77 23.445 0.10096 995 1.9624 23.247 23.644 78 23.469 0.10095 995 1.9624 23.271 23.667 79 24.499 0.10059 995 1.9624 24.302 24.697 80 24.652 0.10054 995 1.9624 24.455 24.85 81 25.009 0.10043 995 1.9624 24.812 25.206 82 25.581 0.10026 995 1.9624 25.384 25.778 83 25.683 0.10023 995 1.9624 25.487 25.88 84 25.987 0.10015 995 1.9624 25.79 26.183 85 26.619 0.09998 995 1.9624 26.423 26.816 86 26.514 0.10001 995 1.9624 26.318 26.71 87 26.969 0.099895 995 1.9624 26.772 27.165 88 27.506 0.099772 995 1.9624 27.31 27.702 89 27.394 0.099797 995 1.9624 27.198 27.59 90 27.625 0.099746 995 1.9624 27.429 27.82 91 28.148 0.099639 995 1.9624 27.953 28.344 92 28.444 0.099582 995 1.9624 28.249 28.639 93 28.688 0.099538 995 1.9624 28.493 28.884 94 29.274 0.099441 995 1.9624 29.078 29.469 95 29.551 0.099399 995 1.9624 29.356 29.746 96 29.613 0.09939 995 1.9624 29.418 29.808 97 29.774 0.099368 995 1.9624 29.579 29.969 98 30.768 0.099248 995 1.9624 30.573 30.963 99 30.998 0.099226 995 1.9624 30.804 31.193 100 31.214 0.099207 995 1.9624 31.019 31.408 101 32.519 0.099124 995 1.9624 32.325 32.714 102 32.874 0.099113 995 1.9624 32.679 33.068 103 32.928 0.099111 995 1.9624 32.734 33.123 104 32.988 0.09911 995 1.9624 32.793 33.182 105 33.344 0.099104 995 1.9624 33.149 33.538 106 33.292 0.099105 995 1.9624 33.097 33.486 107 33.745 0.099103 995 1.9624 33.551 33.94 108 33.914 0.099105 995 1.9624 33.719 34.108 109 33.792 0.099103 995 1.9624 33.598 33.987 110 33.953 0.099105 995 1.9624 33.758 34.147 111 34.331 0.099112 995 1.9624 34.137 34.526 112 34.504 0.099117 995 1.9624 34.31 34.699 113 34.545 0.099118 995 1.9624 34.35 34.739 114 35.386 0.09916 995 1.9624 35.191 35.58 115 35.372 0.099159 995 1.9624 35.177 35.566 116 35.735 0.099184 995 1.9624 35.541 35.93 117 36.524 0.099256 995 1.9624 36.329 36.719 118 36.337 0.099237 995 1.9624 36.142 36.531 119 36.776 0.099284 995 1.9624 36.582 36.971 120 36.598 0.099264 995 1.9624 36.403 36.793 121 36.863 0.099294 995 1.9624 36.668 37.058 122 37.138 0.099328 995 1.9624 36.943 37.333 123 37.759 0.099415 995 1.9624 37.564 37.955 124 37.906 0.099438 995 1.9624 37.711 38.101 125 37.936 0.099442 995 1.9624 37.741 38.131 126 39.061 0.099644 995 1.9624 38.866 39.257 127 39.192 0.09967 995 1.9624 38.997 39.388 128 39.622 0.099762 995 1.9624 39.427 39.818 129 40.121 0.099877 995 1.9624 39.925 40.317 130 40.189 0.099893 995 1.9624 39.993 40.385 131 40.4 0.099945 995 1.9624 40.204 40.596 132 40.494 0.099969 995 1.9624 40.297 40.69 133 41.147 0.10014 995 1.9624 40.95 41.343 134 41.227 0.10017 995 1.9624 41.031 41.424 135 41.924 0.10037 995 1.9624 41.727 42.121 136 42.178 0.10045 995 1.9624 41.98 42.375 137 42.333 0.1005 995 1.9624 42.136 42.53 138 42.579 0.10058 995 1.9624 42.382 42.776 139 42.982 0.10072 995 1.9624 42.785 43.18 140 42.943 0.10071 995 1.9624 42.745 43.14 141 43.667 0.10097 995 1.9624 43.469 43.866 142 43.818 0.10103 995 1.9624 43.62 44.016 143 44.311 0.10123 995 1.9624 44.113 44.51 144 44.285 0.10122 995 1.9624 44.087 44.484 145 44.692 0.10138 995 1.9624 44.494 44.891 146 44.627 0.10136 995 1.9624 44.428 44.826 147 45.364 0.10167 995 1.9624 45.164 45.563 148 45.349 0.10167 995 1.9624 45.15 45.549 149 46.555 0.10224 995 1.9624 46.354 46.755 150 46.439 0.10218 995 1.9624 46.239 46.64 151 46.675 0.1023 995 1.9624 46.475 46.876 152 46.712 0.10232 995 1.9624 46.512 46.913 153 47.46 0.10271 995 1.9624 47.258 47.661 154 47.487 0.10272 995 1.9624 47.285 47.689 155 48.045 0.10303 995 1.9624 47.843 48.247 156 49.29 0.10377 995 1.9624 49.087 49.494 157 49.509 0.1039 995 1.9624 49.305 49.713 158 49.911 0.10416 995 1.9624 49.707 50.116 159 50.534 0.10457 995 1.9624 50.329 50.74 160 50.561 0.10459 995 1.9624 50.355 50.766 161 50.745 0.10471 995 1.9624 50.54 50.951 162 50.702 0.10468 995 1.9624 50.496 50.907 163 50.586 0.10461 995 1.9624 50.381 50.791 164 50.815 0.10476 995 1.9624 50.609 51.021 165 51.675 0.10537 995 1.9624 51.468 51.882 166 51.55 0.10528 995 1.9624 51.343 51.756 167 51.76 0.10543 995 1.9624 51.553 51.967 168 52.055 0.10565 995 1.9624 51.848 52.262 169 53.97 0.10718 995 1.9624 53.76 54.18 170 54.039 0.10724 995 1.9624 53.829 54.25 171 54.192 0.10737 995 1.9624 53.981 54.403 172 54.442 0.10758 995 1.9624 54.23 54.653 173 54.227 0.1074 995 1.9624 54.016 54.438 174 55.79 0.1088 995 1.9624 55.577 56.004 175 57.603 0.1106 995 1.9624 57.386 57.82 176 57.686 0.11068 995 1.9624 57.469 57.903 177 58.061 0.11108 995 1.9624 57.843 58.279 178 58.389 0.11143 995 1.9624 58.17 58.607 179 59.558 0.11275 995 1.9624 59.336 59.779 180 59.989 0.11325 995 1.9624 59.767 60.211 181 60.367 0.11371 995 1.9624 60.144 60.591 182 60.572 0.11396 995 1.9624 60.348 60.796 183 60.746 0.11417 995 1.9624 60.522 60.97 184 60.869 0.11432 995 1.9624 60.645 61.094 185 61.028 0.11452 995 1.9624 60.803 61.253 186 61.583 0.11522 995 1.9624 61.357 61.809 187 61.902 0.11564 995 1.9624 61.675 62.129 188 61.996 0.11576 995 1.9624 61.769 62.223 189 63.154 0.11733 995 1.9624 62.924 63.385 190 63.631 0.11801 995 1.9624 63.399 63.862 191 63.58 0.11793 995 1.9624 63.348 63.811 192 63.992 0.11853 995 1.9624 63.76 64.225 193 64.172 0.11879 995 1.9624 63.939 64.405 194 64.57 0.11938 995 1.9624 64.336 64.804 195 64.797 0.11972 995 1.9624 64.562 65.031 196 64.924 0.11992 995 1.9624 64.689 65.159 197 65.624 0.121 995 1.9624 65.386 65.861 198 65.694 0.12111 995 1.9624 65.456 65.932 199 66.204 0.12193 995 1.9624 65.965 66.443 200 66.256 0.12202 995 1.9624 66.017 66.496 201 66.429 0.1223 995 1.9624 66.189 66.669 202 67.056 0.12335 995 1.9624 66.814 67.298 203 68.002 0.12499 995 1.9624 67.756 68.247 204 68.373 0.12566 995 1.9624 68.127 68.62 205 68.409 0.12572 995 1.9624 68.162 68.656 206 68.697 0.12625 995 1.9624 68.449 68.945 207 68.794 0.12642 995 1.9624 68.546 69.042 208 69.029 0.12686 995 1.9624 68.78 69.278 209 69.515 0.12778 995 1.9624 69.264 69.766 210 69.538 0.12782 995 1.9624 69.287 69.789 211 69.818 0.12836 995 1.9624 69.566 70.069 212 69.792 0.12831 995 1.9624 69.54 70.043 213 70.066 0.12885 995 1.9624 69.813 70.319 214 70.23 0.12917 995 1.9624 69.977 70.483 215 70.246 0.1292 995 1.9624 69.993 70.5 216 70.584 0.12988 995 1.9624 70.329 70.839 217 70.944 0.13061 995 1.9624 70.688 71.201 218 71.532 0.13183 995 1.9624 71.273 71.79 219 71.376 0.1315 995 1.9624 71.118 71.634 220 71.209 0.13115 995 1.9624 70.951 71.466 221 72.012 0.13285 995 1.9624 71.751 72.272 222 71.779 0.13235 995 1.9624 71.519 72.038 223 71.868 0.13255 995 1.9624 71.608 72.128 224 72.136 0.13312 995 1.9624 71.875 72.397 225 72.48 0.13388 995 1.9624 72.217 72.742 226 73.017 0.13508 995 1.9624 72.752 73.282 227 73.071 0.1352 995 1.9624 72.806 73.337 228 73.165 0.13541 995 1.9624 72.899 73.431 229 73.271 0.13566 995 1.9624 73.005 73.537 230 73.719 0.13669 995 1.9624 73.451 73.987 231 73.647 0.13653 995 1.9624 73.379 73.915 232 74.187 0.1378 995 1.9624 73.917 74.458 233 74.773 0.13923 995 1.9624 74.5 75.047 234 74.824 0.13936 995 1.9624 74.551 75.098 235 75.151 0.14017 995 1.9624 74.876 75.426 236 74.784 0.13926 995 1.9624 74.511 75.058 237 74.945 0.13966 995 1.9624 74.671 75.219 238 75.52 0.1411 995 1.9624 75.243 75.797 239 75.488 0.14102 995 1.9624 75.211 75.765 240 75.537 0.14115 995 1.9624 75.26 75.814 241 75.607 0.14133 995 1.9624 75.329 75.884 242 76.147 0.14273 995 1.9624 75.867 76.427 243 76.454 0.14355 995 1.9624 76.172 76.736 244 78.286 0.14867 995 1.9624 77.995 78.578 245 78.538 0.14941 995 1.9624 78.245 78.831 246 79.021 0.15086 995 1.9624 78.725 79.317 247 78.906 0.15051 995 1.9624 78.61 79.201 248 79.872 0.15349 995 1.9624 79.571 80.174 249 79.956 0.15376 995 1.9624 79.654 80.257 250 80.692 0.15613 995 1.9624 80.386 80.999 251 81.706 0.15955 995 1.9624 81.393 82.019 252 82.542 0.1625 995 1.9624 82.223 82.861 253 82.753 0.16326 995 1.9624 82.432 83.073 254 82.66 0.16292 995 1.9624 82.34 82.98 255 83.365 0.16551 995 1.9624 83.04 83.69 256 83.487 0.16597 995 1.9624 83.161 83.812 257 83.895 0.16752 995 1.9624 83.566 84.224 258 84.362 0.16934 995 1.9624 84.03 84.695 259 84.217 0.16877 995 1.9624 83.886 84.548 260 84.413 0.16953 995 1.9624 84.08 84.746 261 84.138 0.16846 995 1.9624 83.807 84.469 262 84.62 0.17035 995 1.9624 84.286 84.954 263 84.952 0.17168 995 1.9624 84.615 85.289 264 85.074 0.17218 995 1.9624 84.736 85.412 265 84.681 0.1706 995 1.9624 84.346 85.016 266 85.495 0.17391 995 1.9624 85.154 85.837 267 85.617 0.17441 995 1.9624 85.275 85.96 268 85.792 0.17515 995 1.9624 85.449 86.136 269 86.301 0.17731 995 1.9624 85.953 86.649 270 86.419 0.17782 995 1.9624 86.07 86.768 271 87.032 0.18052 995 1.9624 86.678 87.386 272 87.531 0.18277 995 1.9624 87.172 87.889 273 87.784 0.18393 995 1.9624 87.423 88.145 274 87.529 0.18276 995 1.9624 87.17 87.887 275 89.015 0.18978 995 1.9624 88.642 89.387 276 89.302 0.1912 995 1.9624 88.927 89.677 277 89.73 0.19334 995 1.9624 89.35 90.109 278 89.476 0.19206 995 1.9624 89.099 89.853 279 89.753 0.19345 995 1.9624 89.374 90.133 280 89.89 0.19415 995 1.9624 89.509 90.271 281 90.481 0.1972 995 1.9624 90.095 90.868 282 90.543 0.19752 995 1.9624 90.155 90.931 283 90.731 0.19851 995 1.9624 90.341 91.12 284 91.115 0.20055 995 1.9624 90.722 91.509 285 90.935 0.19959 995 1.9624 90.543 91.327 286 91.102 0.20049 995 1.9624 90.709 91.496 287 90.996 0.19992 995 1.9624 90.604 91.389 288 91.479 0.20252 995 1.9624 91.082 91.877 289 91.918 0.20494 995 1.9624 91.516 92.32 290 91.807 0.20432 995 1.9624 91.406 92.207 291 91.865 0.20464 995 1.9624 91.463 92.267 292 92.155 0.20626 995 1.9624 91.75 92.559 293 92.136 0.20615 995 1.9624 91.731 92.54 294 92.351 0.20736 995 1.9624 91.944 92.757 295 92.549 0.20849 995 1.9624 92.14 92.958 296 93.21 0.21231 995 1.9624 92.794 93.627 297 93.381 0.21331 995 1.9624 92.962 93.799 298 93.282 0.21273 995 1.9624 92.865 93.699 299 93.379 0.2133 995 1.9624 92.961 93.798 300 93.357 0.21317 995 1.9624 92.939 93.775 301 93.967 0.2168 995 1.9624 93.542 94.393 302 94.609 0.22071 995 1.9624 94.176 95.042 303 94.932 0.2227 995 1.9624 94.495 95.369 304 95.023 0.22327 995 1.9624 94.585 95.461 305 96.471 0.23252 995 1.9624 96.015 96.928 306 96.716 0.23412 995 1.9624 96.257 97.176 307 97.171 0.23712 995 1.9624 96.705 97.636 308 97.286 0.23788 995 1.9624 96.819 97.752 309 97.312 0.23805 995 1.9624 96.845 97.779 310 97.812 0.2414 995 1.9624 97.338 98.286 311 98.18 0.24388 995 1.9624 97.701 98.658 312 98.376 0.24521 995 1.9624 97.895 98.858 313 98.805 0.24813 995 1.9624 98.318 99.292 314 99.075 0.24998 995 1.9624 98.584 99.565 315 99.569 0.25336 995 1.9624 99.071 100.07 316 99.43 0.25241 995 1.9624 98.935 99.925 317 100.06 0.25671 995 1.9624 99.552 100.56 318 99.83 0.25516 995 1.9624 99.329 100.33 319 100.23 0.2579 995 1.9624 99.723 100.73 320 101.12 0.26406 995 1.9624 100.61 101.64 321 102.36 0.27242 995 1.9624 101.83 102.9 322 102.1 0.2707 995 1.9624 101.57 102.64 323 103.24 0.27819 995 1.9624 102.69 103.78 324 103.15 0.27763 995 1.9624 102.61 103.7 325 103.37 0.27904 995 1.9624 102.82 103.92 326 103.8 0.28175 995 1.9624 103.24 104.35 327 103.63 0.2807 995 1.9624 103.08 104.18 328 103.52 0.27997 995 1.9624 102.97 104.07 329 104.07 0.28345 995 1.9624 103.51 104.63 330 103.76 0.28149 995 1.9624 103.2 104.31 331 103.8 0.2818 995 1.9624 103.25 104.36 332 103.99 0.28294 995 1.9624 103.43 104.54 333 104.47 0.28592 995 1.9624 103.91 105.03 334 104.23 0.28444 995 1.9624 103.67 104.79 335 105.03 0.28917 995 1.9624 104.46 105.59 336 104.82 0.28798 995 1.9624 104.26 105.39 337 106.77 0.29833 995 1.9624 106.19 107.36 338 106.97 0.29923 995 1.9624 106.38 107.55 339 107.09 0.29976 995 1.9624 106.5 107.67 340 108.75 0.30601 995 1.9624 108.15 109.35 341 107.67 0.30224 995 1.9624 107.08 108.27 342 108.25 0.30438 995 1.9624 107.65 108.84 343 108.53 0.30533 995 1.9624 107.93 109.13 344 108.56 0.3054 995 1.9624 107.96 109.16 345 108.78 0.30608 995 1.9624 108.18 109.38 346 109.28 0.30746 995 1.9624 108.68 109.88 347 109.96 0.30891 995 1.9624 109.35 110.56 348 110.7 0.30996 995 1.9624 110.09 111.31 349 110.4 0.3096 995 1.9624 109.79 111 350 111.8 0.31042 995 1.9624 111.19 112.41 351 112.46 0.31006 995 1.9624 111.85 113.07 352 112.34 0.31016 995 1.9624 111.73 112.95 353 112.28 0.3102 995 1.9624 111.67 112.89 354 113.23 0.30904 995 1.9624 112.63 113.84 355 114.28 0.30669 995 1.9624 113.67 114.88 356 113.82 0.30784 995 1.9624 113.22 114.43 357 114.13 0.30708 995 1.9624 113.53 114.73 358 115.04 0.30428 995 1.9624 114.45 115.64 359 115.57 0.30232 995 1.9624 114.98 116.16 360 115.84 0.30123 995 1.9624 115.25 116.43 361 115.99 0.30061 995 1.9624 115.4 116.58 362 115.83 0.30128 995 1.9624 115.23 116.42 363 116.4 0.29878 995 1.9624 115.81 116.98 364 116.23 0.29954 995 1.9624 115.64 116.82 365 116.89 0.29641 995 1.9624 116.31 117.47 366 117.28 0.29446 995 1.9624 116.7 117.85 367 117.88 0.2912 995 1.9624 117.31 118.45 368 119.39 0.28226 995 1.9624 118.84 119.94 369 118.91 0.2852 995 1.9624 118.35 119.47 370 119.75 0.28002 995 1.9624 119.2 120.3 371 120.02 0.27828 995 1.9624 119.47 120.56 372 120.19 0.27719 995 1.9624 119.64 120.73 373 119.98 0.27855 995 1.9624 119.43 120.52 374 120 0.27839 995 1.9624 119.45 120.55 375 119.8 0.27965 995 1.9624 119.26 120.35 376 120.81 0.27309 995 1.9624 120.27 121.35 377 121.01 0.27179 995 1.9624 120.47 121.54 378 121.44 0.2689 995 1.9624 120.91 121.96 379 122.16 0.26397 995 1.9624 121.65 122.68 380 121.74 0.26683 995 1.9624 121.22 122.27 381 122.45 0.26202 995 1.9624 121.94 122.96 382 121.91 0.26571 995 1.9624 121.39 122.43 383 122.16 0.26402 995 1.9624 121.64 122.67 384 122.57 0.26119 995 1.9624 122.06 123.08 385 122.55 0.26134 995 1.9624 122.04 123.06 386 122.83 0.25944 995 1.9624 122.32 123.33 387 122.88 0.25907 995 1.9624 122.37 123.39 388 123.46 0.2551 995 1.9624 122.96 123.96 389 123.27 0.2564 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-10.269 -9.7156 975 -9.0426 0.13875 995 1.9624 -9.3149 -8.7704 976 -8.2233 0.13687 995 1.9624 -8.4919 -7.9547 977 -7.51 0.13529 995 1.9624 -7.7755 -7.2445 978 -7.4686 0.1352 995 1.9624 -7.7339 -7.2033 979 -7.3769 0.13501 995 1.9624 -7.6419 -7.112 980 -7.1057 0.13442 995 1.9624 -7.3694 -6.8419 981 -6.9835 0.13416 995 1.9624 -7.2468 -6.7202 982 -7.3527 0.13495 995 1.9624 -7.6175 -7.0879 983 -6.6626 0.13349 995 1.9624 -6.9246 -6.4007 984 -6.2257 0.13259 995 1.9624 -6.4858 -5.9655 985 -6.7355 0.13364 995 1.9624 -6.9977 -6.4732 986 -6.2836 0.13271 995 1.9624 -6.544 -6.0232 987 -5.6518 0.13143 995 1.9624 -5.9097 -5.3939 988 -5.6853 0.1315 995 1.9624 -5.9434 -5.4273 989 -5.5815 0.13129 995 1.9624 -5.8391 -5.3239 990 -5.5543 0.13124 995 1.9624 -5.8118 -5.2967 991 -5.0433 0.13024 995 1.9624 -5.2988 -4.7877 992 -4.0276 0.12832 995 1.9624 -4.2794 -3.7758 993 -4.0132 0.12829 995 1.9624 -4.265 -3.7615 994 -3.8504 0.128 995 1.9624 -4.1016 -3.5993 995 -3.744 0.1278 995 1.9624 -3.9948 -3.4932 996 -3.2357 0.12689 995 1.9624 -3.4847 -2.9867 997 -2.6995 0.12595 995 1.9624 -2.9467 -2.4524 998 -2.4706 0.12556 995 1.9624 -2.717 -2.2242 999 -2.2276 0.12514 995 1.9624 -2.4732 -1.982 1000 -1.3045 0.12361 995 1.9624 -1.5471 -1.0619 Displacement unit: nanometer [nm] ```

Plot the stat. uncertainty of displacement

```displ_std = result.Displacement_std; figure plot(phitrue, displ_std) xlabel('phi'); ylabel(['Statistical uncertainty of displacement, ',result.Displacement_unit]); title(['Statistical Uncertainty: Min: ', ... num2str(result.Displacement_std_min), ', Max: ', ... num2str(result.Displacement_std_max) ]); ```

Plot the displacements with expanded uncertainties

```displ_fit = result.Displacement_fit; const = result.Displacement_const; displ_true = const*phitrue; displ_res = displ_fit - displ_true; ix = (result.phi_fit-phitrue) < -1; displ_res(ix) = displ_res(ix) + 2*pi*const; figure plot(displ_true, displ_res ,'o') hold plot(displ_true, 2*displ_std ,'r-') plot(displ_true, -2*displ_std ,'r-') grid xlabel('True displacement'); ylabel('Displacement Residuals'); title('Expanded Uncertainty of Fitted Displacement'); ```
```Current plot held ```

Plot the X residuals with the expanded uncertainties

```figure plot(phitrue,result.x_res_fit,'.') hold plot(phitrue,2*result.x_res_std,'r-') plot(phitrue,-2*result.x_res_std,'r-') grid xlabel('True phi'); ylabel('x residuals'); title('Expanded Uncertainty of Fitted X Residuals'); ```
```Current plot held ```

Plot the Y residuals with the expanded uncertainties

```figure plot(phitrue,result.y_res_fit,'.') hold plot(phitrue,2*result.y_res_std,'r-') plot(phitrue,-2*result.y_res_std,'r-') grid xlabel('True phi'); ylabel('y residuals'); title('Expanded Uncertainty of Fitted Y Residuals'); ```
```Current plot held ```