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… b2), with b determined as the solution of the normal equations: 2104 2105 y = Xb and X’y = X’Xb = Ab 2106 2107 to give 2108 2109 b = A-1X’y. 2110 2111 The 90 percent confidence interval CI90 for the mean value of the SAR estimated with the associated 2112 value of the mass x0 is then defined as: 2113 2114 CI90 = ȳ (x0) ± t.95,ζ s v(x0) 2115 2116 where 𝑣𝑣(𝑥𝑥0) = �𝑥𝑥0 𝑆𝑆−1 𝑥𝑥0′ . 2117 2118 Thus, CI90 = ȳ (𝑥𝑥0) ± 𝑒𝑒.95,𝜁𝜁 𝑒𝑒 �𝑥𝑥0 𝑆𝑆−1 𝑥𝑥0′ , 2119 2120 where: 2121 2122 𝑥𝑥0 = (1 𝑥𝑥0 𝑥𝑥02 ... 𝑥𝑥0𝑘𝑘); 2123 2124 x’ is the transpose of 𝑥𝑥0; 2125 2126 ȳ (𝑥𝑥0) is the estimate of the mean value of the SAR at the associated value of the mass xo; 2127 2128
t.95,ζ is obtained for ζ degrees of freedom. For the general case of a multiple regression analysis involving 2129 K independent variables (i.e. K + 1 coefficients), ζ is defined as ζ = n – K – 1 ; and 2130 2131 𝑒𝑒 = �∑ (𝑦𝑦𝑖𝑖− ȳ(𝑥𝑥𝑖𝑖 ))2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−𝐾𝐾−1 is the estimate of σ, the true standard deviation. 2132 2133 2134 A4-3 Worked examples of the determination of 90 percent confidence intervals 2135 2136 Direct flight testing 2137 2138 Example 1: In which the confidence interval is less than the confidence interval limit. 2139 2140 Consider the following set of 6 independent measurements of SAR obtained by flight test around one of 2141 the three reference masses of the CO2 emissions evaluation metric. After correction to reference 2142 specifications, the following clustered data set of SAR values is obtained: 2143 2144 2145 Table A4-2. 2146 2147 Measurements of SAR — Example 1 2148 2149 Measurement number Corrected SAR (km/kg) 1 0.38152 2 0.38656 3 0.37988
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Measurement number Corrected SAR (km/kg) 4 0.38011 5 0.38567 6 0.37820 2150 2151 The number of data points (n) = 6 2152 2153 The degrees of freedom (n-1) = 5 2154 2155 The Student t-distribution for 90 percent confidence and 5 degrees of freedom (𝑒𝑒(.95 ,5) ) = 2.015. (See 2156 Table A4-1.) 2157 2158 Note.— 6 is the minimum number of test points required, as stated in § A38.5.3.3. 2159 Estimate of the mean SAR ( 𝑆𝑆𝑆𝑆𝑆𝑆 ) for the clustered data set: 2160 2161 𝑆𝑆𝑆𝑆𝑆𝑆 = 1 𝑛𝑛 �∑ 𝑆𝑆𝑆𝑆𝑆𝑆(𝑖𝑖)𝑖𝑖=𝑛𝑛𝑖𝑖=1 � = 0.38282 km/kg. 2162 2163 Estimate of the standard deviation(s): 2164 2165 𝑒𝑒 = �∑ �𝐷𝐷𝐴𝐴𝐷𝐷(𝑖𝑖)−𝐷𝐷𝐴𝐴𝐷𝐷�������2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−1 = 0.00344 km/kg . 2166 2167 2168 Confidence interval determination 2169 2170 The 90 percent confidence interval (CI90) is calculated as follows (see 3.3.2): 2171 2172 𝐶𝐶𝐶𝐶90 = 𝑆𝑆𝑆𝑆𝑆𝑆 ± 𝑒𝑒(.95,𝑛𝑛−1) s √𝑛𝑛 = 0.38282 ± 2.015 x
0.00344 √6 = 0.38282 ± 0.00283 km/kg. 2173 2174 2175 Check of confidence interval limits 2176 2177 The confidence interval extends to ±0.00283 km/kg around the mean SAR value of the clustered data set 2178
(0.38282 km/kg). This represents ±0.74 percent of the mean SAR value, which is below the confidence 2179 interval limit of 1.5 percent, as defined in § A38.5.3.2. 2180 2181 As a result, the SAR value of 0.38282 km/kg associated to one of the reference masses of the FEM can 2182 be used for the metric determination. 2183 2184 Example 2: In which the confidence interval exceeds the confidence interval limit. 2185 2186 Consider the following set of 6 independent measurements of SAR obtained by flight test around one of 2187 the three reference masses of the FEM. After correction to reference specifications, the following 2188 clustered data set of SAR values is obtained: 2189 2190 2191 2192 2193 2194 2195
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Table A4-3. 2196 2197 Measurements of SAR — Example 2 2198 2199 Measurement number Corrected SAR (km/kg) 1 0.15208 2 0.15795 3 0.15114 4 0.15225 5 0.15697 6 0.15834 2200 The number of data points (n) = 6 2201 2202 The degrees of freedom (n-1) = 5 2203 The Student t-distribution for 90 percent confidence and 5 degrees of freedom (𝑒𝑒(.95 ,5) ) = 2.015. (See 2204 Table A4-1) 2205 2206 Estimate of the mean SAR ( 𝑆𝑆𝑆𝑆𝑆𝑆 ) for the clustered data set: 2207 2208 𝑆𝑆𝑆𝑆𝑆𝑆 = 1 𝑛𝑛 �∑ 𝑆𝑆𝑆𝑆𝑆𝑆(𝑖𝑖)𝑖𝑖=𝑛𝑛𝑖𝑖=1 � = 0.15479 km/kg. 2209 2210 Estimate of the standard deviation(s): 2211 2212 𝑒𝑒 = �∑ �𝐷𝐷𝐴𝐴𝐷𝐷(𝑖𝑖)−𝐷𝐷𝐴𝐴𝐷𝐷�������2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−1 = 0.0033 km/kg . 2213 2214 2215 Confidence interval determination 2216 2217 The 90 percent confidence interval (CI90) is calculated as follows (see 3.3.2): 2218 2219 𝐶𝐶𝐶𝐶90 = 𝑆𝑆𝑆𝑆𝑆𝑆 ± 𝑒𝑒(.95,𝑛𝑛−1) s √𝑛𝑛 = 0.15479 ± 2.015 x
0.0033 √6 = 0.15479 ± 0.00271 km/kg. 2220 2221 2222 Check of confidence interval limits 2223 2224 The confidence interval extends to ±0.00271 km/kg around the mean SAR value of the clustered data set 2225
(0.15479 km/kg). This represents ±1.75 percent of the mean SAR value, which is above the confidence 2226 interval limit of 1.5 percent, as defined in § A38.5.3.2. 2227 2228 In such case, a penalty equal to the amount that the 90 percent confidence interval exceeds ±1.5 percent 2229 is applied to the mean SAR value, i.e. (1.75-1.50) = 0.25 percent. The mean SAR value is therefore be 2230 penalized by an amount of 0.25 percent as follows: 2231 2232 𝑆𝑆𝑆𝑆𝑆𝑆 = (1 −
0.25
100) 𝑥𝑥 0.15479 = 0.15440 km/kg. 2233 2234 As a result, the SAR value of 0.15440 km/kg associated to one of the reference masses of the FEM can 2235 be used for the metric determination. 2236
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2237 Regression model 2238 2239 Example 3: In which the confidence interval at each of the three reference masses of FEM is less than 2240 the confidence interval limit. 2241 2242 Consider the following set of 12 measurements of SAR obtained by flight test at optimum speed and 2243 optimum altitude as a function of the airplane gross mass. After SAR correction to reference 2244 specifications, the following data set is obtained: 2245 2246 Table A4-4. 2247 2248 Measurements of SAR — Example 3 2249 2250 Measurement number and reference mass Gross mass (mi) (kg) Corrected SAR (SARi) (km/kg) 1 17,800 0.928 Low mass (*) 17,825 2 17,970 0.905 3 18,400 0.908 4 18,850 0.884 5 19,500 0.850 6 19,950 0.845 Mid mass (*) 19,953 7 20,180 0.833 8 20,350 0.818 9 21,000 0.792 10 21,500 0.781 11 21,870 0.779 High mass (*) 22,080 12 22,150 0.771 2251 (*) Low, mid and high mass represent the reference masses of the FEM, as defined in §38.13(b). 2252 2253 The number of data points (n) = 12 2254 2255 Note.— 12 is the minimum number of test points requested, as stated in § A38.5.3.4. 2256 2257 A representation of the above measurement points is proposed in Figure A4-5. 2258 2259
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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
0.95
0.90
0.85
0.80
0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 3 — Measured SAR versus gross mass SAR (km/kg) Measured data points High mass Mid mass Low mass
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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
0.95
0.90
0.85
0.80
0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 3 — SAR model (trend line) versus gross mass SAR (km/kg) Measured data points High mass Low mass Mid mass Trend line
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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
0.90
0.85
0.80
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
0.95
0.90
0.85
0.80
0.75 17 000 18 000 19 000 20 000 21 000 22 000 23 000 Gross mass - m (kg) Example 4 — Measured SAR versus gross mass SAR (km/kg) Measured data points High mass Mid mass Low mass