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2260 Figure A4-5. Measured SAR versus gross mass — Example 3 2261 2262 2263 Estimate of the mean SAR model by polynomial regression 2264 2265 In order to estimate the SAR model (SARav) as a function the airplane gross mass (m), a polynomial 2266 regression of second order is proposed so that: 2267 2268 SARav = B0 + B1 m + B2 m² 2269 2270 Each observation (mi, SARi), for I = 1, …, 12 satisfies the equation: 2271 2272 SAR(i) = b0 + b1 mi + b2 mi² + ei 2273 2274 where ei = residual error (difference between the measured SAR value and its estimate). 2275 2276 Under a matrix form this gives: 2277 2278 ⎝ ⎜ ⎛ 𝑆𝑆𝑆𝑆𝑆𝑆1 𝑆𝑆𝑆𝑆𝑆𝑆2 𝑆𝑆𝑆𝑆𝑆𝑆3 ⋮ 𝑆𝑆𝑆𝑆𝑆𝑆12⎠ ⎟ ⎞ = ⎝ ⎜ ⎛ 1 𝑒𝑒1 𝑒𝑒1² 1 𝑒𝑒2 𝑒𝑒2² 1 𝑒𝑒3 𝑒𝑒3² ⋮ ⋮ ⋮ 1 𝑒𝑒12 ⋯ 𝑒𝑒12²⎠ ⎟ ⎞ �𝑏𝑏0𝑏𝑏1 𝑏𝑏2 � + ⎝ ⎜ ⎛ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹3 ⋮ 𝐹𝐹12⎠ ⎟ ⎞ 2279 2280
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SAR = M b + e 2281 2282 where: 2283 2284 SAR = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛
0.928
0.905
0.908
0.884
0.850
0.845
0.833
0.818
0.792
0.781
0.779
0.771⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ M = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 1 17800 17800² 1 17970 17970² 1 18400 18400² 1 18850 18850² 1 19500 19500² 1 19950 19950² 1 20180 20180² 1 20350 20350² 1 21000 21000² 1 21500 21500² 1 21870 21870² 1 22150 22150²⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ b = � 𝑏𝑏0 𝑏𝑏1 𝑏𝑏2 � e = ⎝ ⎜ ⎛ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹3 ⋮ 𝐹𝐹n⎠ ⎟ ⎞. 2285 2286 The least square principle consists in looking for the parameter values of vector B, minimizing the sum of 2287 the squares of residuals, i.e.: 2288 2289 Min ∑ 𝐹𝐹i²𝑖𝑖=12𝑖𝑖=1 = min ∑ (𝑆𝑆𝑆𝑆𝑆𝑆(i) − 𝑏𝑏0 − 𝑏𝑏1 𝑒𝑒i − 𝑏𝑏2 𝑒𝑒i2)𝑖𝑖=12𝑖𝑖=1 ². 2290 2291 It is equivalent to look for the solutions of 𝜕𝜕(∑ 𝑟𝑟i²) 𝜕𝜕𝑏𝑏j = 0 for j = (0, 1, 2). 2292 2293 The solution B = � 𝐵𝐵0 𝐵𝐵1 𝐵𝐵2 � is given by: 2294 2295 A-1 M’ SAR (see A4-2) 2296 2297 where: 2298 2299 M’ = transpose of M; and 2300 2301 A-1 = (M’ M)-1 = inverse of (M’ M). 2302 2303 Finally, B = �
2.402921963 −0.000120515
2.10695 𝑥𝑥10-09 �; and 2304 2305 SARav = 2.402921963 – 0.000120515 m + 2.10695.10-9 m². 2306 2307 Figure A4-6 provides a representation of the mean SAR model as a function of the airplane gross mass. 2308 2309
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2310 Figure A4-6. SAR model (trend line) versus gross mass — Example 3 2311 2312 2313 The mean SAR values at each of the three reference gross masses of the FEM are as follows: 2314 2315 Table A4-7. 2316 2317 Mean SAR values — Example 3 2318 2319 Reference mass Mass value (kg) Mean SAR value (km/kg) Low mass 17,825 0.92418 Mid mass 19,953 0.83710 High mass 22,080 0.76914 2320 Estimate of the standard deviation(s): 2321 2322 𝑒𝑒 = �∑ (SARi−SARav(i))2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−𝐾𝐾−1 = 0.00765 km/kg 2323 2324
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where: 2325 2326 the number of data points (n) = 12; 2327 2328 K = 2 for a second order polynomial regression (see A4-2); and 2329 2330 the degrees of freedom (n-K-1) = 9. 2331 2332 2333 Confidence interval determination 2334 2335 The 90 percent confidence interval (CI90) at an airplane gross mass m0 is calculated as follows (see A4-2336
2): 2337 2338 𝐶𝐶𝐶𝐶90 = SARav(m0) ± 𝑒𝑒(.95 ,𝑛𝑛−𝐾𝐾−1) 𝑒𝑒 �m0 A-1 m0' 2339 2340 where: 2341 2342 the Student t-distribution for 90 percent confidence and 9 degrees of freedom t(.95, 9) = 1.833. (See Table 2343 A4-1); 2344 2345 m0 = (1 m0 m0²) and m0’ = � 1 𝑒𝑒0 𝑒𝑒0² �; and 2346 2347 A-1 = (M’ M)-1 = inverse of (M’ M). 2348 2349 Figure A4-8 provides a representation of the 90 percent confidence interval as a function of airplane 2350 gross mass. 2351 2352 2353
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2354 Figure A4-8. 90 percent confidence interval versus gross mass — Example 3 2355 2356 2357 The 90 percent confidence intervals at each of the three reference gross masses of the FEM are as 2358 follows: 2359 2360 Table A4-9. 2361 2362 Confidence intervals — Example 3 2363 2364 Reference mass Mass value (kg) 90% confidence interval (kg/km) Low mass 17 825 CI90 = 0.92418 ± 0.00915 Mid mass 19 953 CI90 = 0.83710 ± 0.00619 High mass 22 080 CI90 = 0.76914 ± 0.00925 2365 Check of confidence interval limits 2366 2367 For each of the three reference gross masses of the FEM, the confidence interval extends around the 2368 mean SAR value by a percentage, as shown in Table A4-10. 2369
0.95
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0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 3 — 90% confidence interval versus gross mass SAR (km/kg) Measured data points High mass 90% CI line - upper 90% CI line - lower Low mass Mid mass Trend line
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2370 Table A4-10. 2371 2372 Check of confidence intervals — Example 3 2373 2374 Reference mass Mass value (kg) 90% confidence interval (percentage of mean SAR) Low mass 17,825 (0.00915/0.92418) x 100 = 0.99% Mid mass 19,953 (0.00619/0.83710) x 100 = 0.74% High mass 22,080 (0.00925/0.76914) x 100 = 1.2% 2375 2376 The 90 percent confidence intervals at each of the three reference gross masses of the FEM are all below 2377 the confidence interval limit of 1.5 percent, as defined in § A38.5.3.2. 2378 2379 As a result, the following mean SAR values associated to each of the three reference masses of the FEM 2380 can be used for the metric determination: 2381 2382 Table A4-11. 2383 2384 Mean SAR values — Example 3 2385 2386 Reference mass Mass value (kg) Mean SAR value (km/kg) Low mass 17,825 0.92418 Mid mass 19,953 0.83710 High mass 22,080 0.76914 2387 2388 Example 4: In which the confidence interval of at least one of the three reference masses of the FEM 2389 exceeds the confidence interval limit. 2390 2391 Consider the following set of 12 measurements of SAR obtained by flight test at optimum speed and 2392 optimum altitude as a function of the airplane gross mass. After SAR correction to reference 2393 specifications, the following data set is obtained: 2394 2395 2396
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Table A4-12. 2397 2398 Measurements of SAR — Example 4 2399 2400 Measurement number and reference mass Gross mass (mi) (kg) Corrected SAR (SARi) (km/kg) 1 17,800 0.932 Low mass (*) 17,825 2 18,200 0.925 3 18,620 0.913 4 18,890 0.889 5 19,350 0.868 6 19,610 0.848 Mid mass (*) 19,953 7 19,920 0.838 8 20,510 0.830 9 20,790 0.806 10 21,220 0.815 11 21,480 0.779 High mass (*) 22,080 12 22,100 0.788 2401 (*) Low, mid and high mass represent the reference masses of the CO2 emissions evaluation metric, as 2402 defined in §38.13(b). 2403 2404 The number of data points (n) = 12 (minimum required, as stated in § A38.5.3.4). 2405 2406 A representation of the above measurement points is proposed in Figure A4-13. 2407 2408 2409
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2410 Figure A4-13. Measured SAR versus gross mass — Example 4 2411 2412 Estimate of the mean SAR model by polynomial regression 2413 2414 In order to estimate the SAR model (SARav) as a function the airplane gross mass (m), a polynomial 2415 regression of second order is proposed, so that: 2416 2417 SARav = B0 + B1 m + B2 m². 2418 2419 Each observation (mi, SARi), for i = 1, …, 12 satisfies the equation: 2420 2421 SAR(i) = b0 + b1 mi + b2 mi² + ei 2422 2423 with ei = residual error (difference between the measured SAR value and its estimate). 2424 2425
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Under a matrix form this gives: 2426 2427 ⎝ ⎜ ⎛ 𝑆𝑆𝑆𝑆𝑆𝑆1 𝑆𝑆𝑆𝑆𝑆𝑆2 𝑆𝑆𝑆𝑆𝑆𝑆3 ⋮ 𝑆𝑆𝑆𝑆𝑆𝑆12⎠ ⎟ ⎞ = ⎝ ⎜ ⎛ 1 𝑒𝑒1 𝑒𝑒1² 1 𝑒𝑒2 𝑒𝑒2² 1 𝑒𝑒3 𝑒𝑒3² ⋮ ⋮ ⋮ 1 𝑒𝑒12 ⋯ 𝑒𝑒12²⎠ ⎟ ⎞ �𝑏𝑏0𝑏𝑏1 𝑏𝑏2 � + ⎝ ⎜ ⎛ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹3 ⋮ 𝐹𝐹12⎠ ⎟ ⎞ 2428 2429 SAR = M b + e 2430 2431 where: 2432 2433 SAR = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛
0.932
0.925
0.913
0.889
0.868
0.848
0.838
0.830
0.806
0.815
0.779
0.788⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ M = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 1 17800 17800² 1 18200 18200² 1 18620 18620² 1 18890 18890² 1 19350 19350² 1 19610 19610² 1 19920 19920² 1 20510 20510² 1 20790 20790² 1 21220 21220² 1 21480 21480² 1 22100 22100²⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ b = � 𝑏𝑏0 𝑏𝑏1 𝑏𝑏2 � e = ⎝ ⎜ ⎛ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹3 ⋮ 𝐹𝐹n⎠ ⎟ ⎞. 2434 2435 The least square principle consists in looking for the parameter values of vector B, minimizing the sum of 2436 the squares of residuals, i.e.: 2437 2438 Min ∑ 𝐹𝐹i²𝑖𝑖=12𝑖𝑖=1 = min ∑ (𝑆𝑆𝑆𝑆𝑆𝑆(i) − 𝑏𝑏0 − 𝑏𝑏1 𝑒𝑒i − 𝑏𝑏2 𝑒𝑒i2)𝑖𝑖=12𝑖𝑖=1 ². 2439 2440 It is equivalent to look for the solutions of 𝜕𝜕(∑ 𝑟𝑟i²) 𝜕𝜕𝑏𝑏j = 0 for j = (0, 1, 2). 2441 2442 The solution B = � 𝐵𝐵0 𝐵𝐵1 𝐵𝐵2 � is given by: 2443 A-1 M’ SAR (see A4-2) 2444 2445 where: 2446 2447 M’ = transpose of M; and 2448 2449 A-1 = (M’ M)-1 = inverse of (M’ M). 2450 2451 Finally, B = �
3.26727172 −0.000205692
4.21798 𝑥𝑥10-09 �; and 2452 2453 SARav = 3.26727172 – 0.000205692 m + 4.21798.10-9 m². 2454 2455 Figure A4-14 provides a representation of the mean SAR model as a function of the airplane gross mass. 2456 2457 2458
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2459 Figure A4-14. SAR model (trend line) versus gross mass — Example 4 2460 2461 2462 The mean SAR values at each of the three reference gross masses of the FEM are as follows: 2463 2464 Table A4-15. 2465 2466 Mean SAR value — Example 4 2467 2468 Reference mass Mass value (kg) Mean SAR value (km/kg) Low mass 17,825 0.94100 Mid mass 19,953 0.84238 High mass 22,080 0.78198 2469 Estimate of the standard deviation(s): 2470 2471 𝑒𝑒 = �∑ (SARi−SARav(i))2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−𝐾𝐾−1 = 0.01050 km/kg 2472 2473
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where: 2474 2475 the number of data points (n) = 12; 2476 2477 K = 2 for a second order polynomial regression (see A4-2); and 2478 2479 the degrees of freedom (n-K-1) = 9. 2480 2481 2482 Confidence interval determination 2483 2484 The 90 percent confidence interval (CI90) at an airplane gross mass m0 is calculated as follows (see A4-2485
2): 2486 2487 𝐶𝐶𝐶𝐶90 = SARav(m0) ± 𝑒𝑒(.95 ,𝑛𝑛−𝐾𝐾−1) 𝑒𝑒 �m0 A-1 m0' 2488 2489 where: 2490 2491 the Student t-distribution for 90 percent confidence and 9 degrees of freedom t(.95, 9) = 1.833 (See Table 2492 A4-1); 2493 2494 m0 = (1 m0 m0²) and m0’ = � 1 𝑒𝑒0 𝑒𝑒0² �; and 2495 2496 A-1 = (M’ M)-1 = inverse of (M’ M). 2497 2498 Figure A4-16 provides a representation of the 90 percent confidence interval as a function of airplane 2499 gross mass. 2500 2501
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2502 Figure A4-16. 90 percent confidence interval versus gross mass — Example 4 2503 2504 The 90 percent confidence intervals at each of the three reference gross masses of the CO2 emissions 2505 evaluation metric are as follows: 2506 2507 Table A4-17. 2508 2509 Confidence intervals — Example 4 2510 2511 Reference mass Mass value (kg) 90% confidence interval (kg/km) Low mass 17,825 CI90 = 0.94100 ± 0.01399 Mid mass 19,953 CI90 = 0.84238 ± 0.00823 High mass 22,080 CI90 = 0.78198 ± 0.01505 2512 Check of confidence interval limits 2513 2514 For each of the three reference gross masses of the FEM, the confidence interval extends around the 2515 mean SAR value by an amount provided in Table A4-18. 2516 2517
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0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 4 — 90% confidence interval versus gross mass SAR (km/kg) Measured data points High mass 90% CI line - upper 90% CI line - lower Low mass Mid mass Trend line