305(b)(2) Confidence interval for mean line obtained by regression [ETM 3.5.2.2] If n measurements of EPNL (y1, y2,….,yn ) are obtained under significantly varying values of engine-related parameter (x1, x2,….,xn ) respectively, then a polynomial can be fitted to the data by the method of least squares. For determining the mean EPNL, , the following polynomial regression model is assumed to apply:
ߤൌܤ ܤ ଵݔܤ ଶݔଶ ⋯ܤ ݔ .
The estimate of the mean line through the data of the EPNL is given by:
ݕൌܾ ܾ ଵݔܾ ଶݔଶ ⋯ܾ ݔ .
Each regression coefficient (Bi) is estimated by bi from the sample data using the method of least squares in a process summarized as follows.
Each observation (xi, yi) satisfies the equations:
ݕ ൌܤ ܤ ଵݔ ܤ ଶݔଶ ⋯. ܤ ݔ ߝ ൌܾ ܾ ଵݔ ܾ ଶݔଶ ⋯. ܾ ݔ ݁ , Where ߝ and ݁ are, respectively, the random error and residual associated with the EPNL. The random error ߝ is assumed to be a random sample from a normal population with mean zero and standard deviation . The residual (݁) is the difference between the measured value and the estima te of the value using the estimates of the regression coefficients and xi . Its root mean square value (s) is the sample estimate for . These equations are often referred to as the normal equations.
Table 5. Student's t-distribut ion (for 90 percent confidence) for various degrees of freedom Degrees of freedom () t.95, 1 6.314 2 2.920 3 2.353 4 2.132 5 2.015 6 1.943 7 1.895 8 1.860 9 1.833 10 1.812 12 1.782 14 1.761 16 1.746 18 1.734 20 1.725 24 1.711 30 1.697 60 1.671 >60 1.645 The n data points of measurements (xi, yi) are processed as follows:
Each elemental vector ሺݔሻ and its transpose ሺݔᇱሻ are formed such that:
ݔ ൌሺ 1ݔ ݔଶ..ݔ ሻ, a row vector; and ۇݔ 1 ଶ ۊݔݔᇱ ൌ ۈ ۋ ,a column vector. ۈ. ۋ ۈ. ۋ .
ݔۉی A matrix ܺ is formed from all the elemental vectors xi for i = 1, ….., n. ܺᇱ is the transpose of ܺ .A matrix ܣ is defined such that ܣ ൌܺ ᇱܺ and a matrix ܣିଵ is the inverse of ܣ In addition, ݕ ൌሺ ݕଵݕଶ …ݕሻ, and ܾ ൌሺ ܾܾଵ …ܾ ሻ, with ܾ determined as the solution of the normal equations:
ݕ ൌ ܾܺ and ݕ′ܺ ൌܾܺ′ܺ ൌܾܣ , Page 86
to give ܾ ൌܣ ିଵݕ′ܺ .
The 90 percent confidence interval CI 90 for the mean value of the EPNL estima ted with the associated value of the engine-related parameter x0 is then defined as:
ܫܥଽ ൌ ݕሺݔሻേݐ ଽହ,ఋݒݏሺݔሻ, where ݒሺݔሻ ൌ √ݔܣିଵݔᇱ .
,Thusܫܥଽ ൌ ݕሺݔሻേ.ݐ ଽହ,ఋݏ√ݔܣିଵݔᇱ where:
— ݔ ൌሺ 1 ݔݔଶ …ݔ ሻ;
— ݔᇱ is the transpose of ݔ ;
— ݕሺݔሻis the estimate of the mean value of the EPNL at the associated value of the engine-related parameter;
— .ݐଽହ, is obtained for ζ degrees of freedom. For the general case of a multiple regression analysis involving K independent variables (i.e., K + 1 coefficients) ζ is defined as ζ = n-K-1 (for the specific case of a polynomial regression analysis, for which k is the order of curve fit, there are k variables independent of the dependent variable, and so ζ = n-K-1); and ሺ௬ି௬ሺ௫ሻሻమ — ݏൌ √∑ససభ ିିଵ , the estimate of ߪ the true standard deviation.
305(c) Confidence Interval for Static-Test-Derived NPD Curves [ETM 3.5.3] When static test data are used in family certifications, NPD curves are formed by the linear combination of baseline flight regressions, baseline projected static regressions and derivative projected static regressions in the form:
EPNLDF = EPNLBF – EPNLBS + EPNLDS, or using the notation adopted above:
ݕிሺݔሻ ൌ ݕிሺݔሻ െ ݕௌሺݔሻ ݕௌሺݔሻ, where:
DF denotes derivative flight;
BF denotes baseline flight;
BS denotes baseline static; and DS denotes derivative static.
Confidence intervals for the derivative flight NPD curves are obtained by pooling the three data sets, each with their own polynomial regression. The confidence interval for the mean derived EPNL at engine-related parameter x0, i.e., for ߤிሺݔሻ, is given by:
ܫܥଽሺݔሻ ൌ ݕிሺݔሻേ ݒ′ݐ ிሺݔሻ where:
ݒிሺݔሻ ൌ√ ሺ ݏிݒிሺݔሻሻଶ ሺ ݏௌݒௌሺݔሻሻଶ ሺ ݏௌݒௌሺݔሻሻଶ with sBF, sBS, sDS, vBF (x0), vBS (x0), vDS (x0) computed as explained in 3.5.2.2 for the respective data sets indicated by the subscripts BF, BS and DS, and ݐᇱ ൌ ሺݏிݒிሺݔሻሻଶݐி ሺ ݏௌݒௌሺݔሻሻଶݐௌ ሺ ݏௌݒௌሺݔሻሻଶݐௌ ሺݏிݒிሺݔሻሻଶ ሺ ݏௌݒௌሺݔሻሻଶ ሺ ݏௌݒௌሺݔሻሻଶ where tBF, tBS, tDS are the t95, ζ values each evaluated with the respective degrees of freedom ζBF, ζBS, ζDS as they arise in the corresponding regressions.
305(d) Confidence Interval for Analytically Derived NPD Curves [ETM 3.5.4] Analysis may be used to determine the effect of changes in noise source components on certificated levels. This is accomplished by analytically determining the effect of hardware change on the noise component it generates. The resultant delta (Δ) is applied to the original configuration, and new noise levels are computed. The changes may occur on the baseline configuration or on subsequent derivative configurations. The confidence intervals for this case are computed using the appropriate method from 305(b) and 305(c).
If Δ̂ represents the analytically determined change and if it is assumed that it may deviate from the true unknown ∆by some random amount, d, such that:
߂̂ ൌ∆݀ , Where d is assumed to be normally distributed with mean zero and known variance ߬ଶ, then the confidence interval for ߤሺݔሻ ∆ is given by:
ሺݕሺݔሻ ߂ ̂ ᇱݒሻേݐ ᇱሺݔሻ, Where ݒᇱሺݔሻ ൌ√ ݒ ሺ ݔሻଶ ߬ ଶ and t’ is as above without change.
305(e) Adequacy of the Model [ETM 3.5.5] 305(e)(1) Choice of engi ne-related parameter [ETM 3.5.5.1] Every effort should be made to determine th e most appropriate engine-related parameter, x, which may be a combination of various simpler parameters.
305(e)(2) Choice of regression model [ETM 3.5.5.2] Page 88
It is not recommended in any case that polynomials of gr eater complexity than a simple quadratic be used for certification purposes, unless there is a clear basis for using a higher order polynomial.
Standard texts on multiple regression should be consulted, and the data available should be examined to show the adequacy of the model chosen.
305(f) Worked Example of the Determination of 90 percent Confidence Intervals from the Pooling of Three Data Sets [ETM 3.5.6] 305(f)(1) Introduction [ETM 3.5.6.1] This section presents an example of the derivation of the 90 percent confidence intervals arising from the pooling of three data sets. Worked examples and guidance material are presented for the calculation of confidence intervals for a clustered data set as well as for first order (i.e., straight line) and second order (i.e., quadratic) regression curves. In addition this section also shows how the confidence interval must be established for the pooling together of several data sets.
Consider the theoretical evaluation of the certification noise levels for an aircraft retrof itted with silenced engines.
The approach noise level for the “flight datum” aircraft was derived from a clustered data set of noise levels measured at nominally reference conditions, to which were added source noise corrections derived from a quadratic least-squares curve fit through a series of data points made at different e ngine thrusts. In order to evaluate the noise levels for the aircraft fitted with acoustically treated engines, a further source noise curve, assumed to be a straight least-squares regression line, was established from a series of measurements of the silenced aircraft. Each of the three databases is assumed to be made up of data unique to each base.
The clustered data set consists of six EPNL levels for the nominal datum hard wall condition. These levels have been derived from measurements which have been fully corrected to the hard wall approach reference condition.
The two curves which determine the ac oustical changes are the regression cu rves (in this example quadratic and straight-line least-squares fit curves) for the plots of EPNL against normalized thrust for the hard wall and silenced conditions. These are presented in Figure 6 where the dotted lines plotted about each line represent the boundaries of 90 percent confidence.
Each of the two curves is made up of the full set of data points obtained for each condition during a series of back-toback tests. The least-squares fits therefore have associated with them all the uncertainties contained within each data set. It is maintained that the number of data points in each of the three sets is large enough to constitute a statistical sample.
Corrected net thrust per engine
90.0
91.0
92.0
93.0
94.0
95.0
96.0
97.0
98.0
99.0
100.0 1 000 1 200 1 400 1 600 1 800 2 000 2 200 Hardwall Silenced ( = 89.93 + 0.001843 )y x EPNL Figure 6. Regression curves for plots of EPNL against normalized thrust for hard wall and silenced conditions 305(f)(2) Confidence interval for a clustered data set [ETM 3.5.6.2] The confidence interval for the clustered data set is defined as follows:
Let EPNLi be the individual values of EPNL, n = number of data points; and t = Student’s t-distribution for ( n-1) degrees of freedom (i.e., the number of degrees of freedom associated with a clustered data set).
Then the confidence interval ܫܥ ൌ ܮܰܲܧ േݐ √ ௦ , where s, the estimate of the standard deviation, is defined as:
ሺாேିாேሻమݏൌ √∑ససభ , andିଵ సభ ாேܮܰܲܧ ൌ ∑స .
Suppose that the clustered set of EPNL values consists of the following:
Run number EPNL 1 95.8 2 94.8 3 95.7 4 95.1 5 95.6 6 95.3 Then the number of data points (n) = 6, the degrees of freedom (n-1) = 5, and the Student’s t-distribution for 5 degrees of freedom = 2.015 (see Table 5 ), and so:
స ாே ܮܰܲܧ ൌ ∑సభ ൌ9 5 . 3 8, ሺாேିாேሻమݏൌ √∑ససభ ൌ 0.3869,ିଵ and the confidence interval (CI) is calculated as follows:
ܫܥ ൌ ܮܰܲܧ േݐ √ ௦ ൌ 95.38 േ 2.015 .ଷ଼ଽ ൌ 95.38 േ 0.3183√ 305(f)(3) Confidence in terval for a first order regression curve [ETM 3.5.6.3] Suppose that the regression curve for one of the source noise data sets, the silenced case, can best be represented by a straight line least-squares fit curve (i.e., a first order polynomial).
The equation for this regression line is of the general form:
Y = a + bX where Y represents the dependent variable EPNL, and X represents the independent variable, in this case normalized thrust ிಿ ఋ .
Although for higher order polynomial least-squares curves a re gression line’s coefficients (i.e., the solutions to the “normal equations”) are best established through computer matrix solutions, the two coefficients for a straight-line fit, a and b, can be determined from the following two simple formulae for the measured values of X and Y, Xi and ∆ :
స స ܽൌ ∑సభ ି ∑సభ ; and మ ܾൌ ௩ మ , where: ൌ ௌೣ ௌೣ స స ∑స ݕݔܵଶ ൌ ∑సభ െ ∑సభ మ సభ ;and Page 91
స మ ష ܵ௫ଶ ൌ ∑సభ సభ ሻଶ െሺ ∑ .
The 90 percent confidence interval about this regression line for X = x0 is then defined by:
ܫܥଽ ൌ ܻ േݏݐ √ ݔܣିଵݔᇱ where:
t = Student’s t-distribution for 90 percent confidence corresponding to (n – k - 1) degrees of freedom, where k is the order of the polynomial regression line and n is the number of data points;
ݔ ൌሺ 1 ݔሻ;
ᇱݔ ൌሺ ݔ1ሻ;
ܣିଵ is the inverse of ܣ where ܣ ൌܺ′ܺ , with ܺ and ′ܺ defined as in 305(b)(2) from the elemental vectors formed from the measured values of the independent variable ܺ; and ݏൌ√ ݅ൌ1݊െ݇െ1 Where ሺ∆ܻሻ ൌ the difference between the measured value of Y i at its associated value of X i, and the value of Y derived from the straight line least-squares fit curve for X = Xi, and n and k are defined as the number of data points and the order of the polynomial regression line, respectively.
Suppose that the data set consists of the following set of six EPNL values, together with their associated values of engine-related parameter (see Table 6). Note that it would be usual to have more than six data points making up a source noise curve, but in order to limit the size of the matr ices in this example, the number of data points has been restricted.
By plotting this data (see Figure 6) it can be seen by examination that a linear relationship between EPNL (the dependent variable Y) and ி ఋ ಿ (the independent variable X) is suggested with the following general form:
Y = a + bX.
The coefficients a and b of the linear equation are defined as above and may be calculated as follows:
X 1 395 1 505 1 655 1 730 1 810 1 850 Y
92.3
92.9
93.2
92.9
93.4
93.2 XY 128 759 139 815 154 246 160 717 169 054 172 420 X2 1 946 025 2 265 025 2 739 025 2 992 900 3 276 100 3 422 500 ∑X ∑Y ∑XY ∑X2 9 945 557.9 925 010 16 641 575 ହହ.ଽିሺ స ି ∑స ଵ଼ସଷሻሺଽଽସହሻܽൌ ∑సభ సభ ൌ ൌ8 9 . 9 3and మ ܾൌ ௩ మ , where: ൌ ௌೣ ௌೣ స స ∑స ൌ ଽଶହଵ െ ሺଽଽସହሻሺହହ.ଽሻ ݕݔܵଶ ൌ ∑సభ െ ∑సభ మ సభ ൌ 48.46; and ଷ స మ ష సభ సభ ሻଶ ൌ ଵସଵହହ െሺ ଽଽସହ ܵ௫ଶ ൌ ∑ െሺ ∑ ሻଶ ൌ 26.289.6, to give: ܾൌ ସ଼.ସ ଶଶ଼ଽ. ൌ 0.001843.
The 90 percent confidence interval about this regression line is defined as:
ܫܥଽ ൌ ܻ േݏݐ √ ݔܣିଵݔᇱ and is calculated as follows. From the single set of measured independent variables tabulated in Table 6, the matrix, ܺ ,is formed from the elemental row vectors such that:
1 1395 1 1505 ܺ ൌ ۇ 1 1655ۊ1ۈۋ ۈ 1730ۋ 1 1810 ۉ1 1850ی and ′ܺ ,the transpose of ܺ ,where 1 1 1 1 1′ܺ ൌሺ 11395 1505 1655 1730 1810 1850ሻ.
The matrix ܣ is now formed, defined such that ܣ ൌ ܺ′ܺ and so:
ܣ ൌሺ 6 99459945 16641575ሻ, and its inverse ܣିଵ such that:
ܣିଵ ൌሺ 17.5836 െ0.01051െ0.01051 6.3396 ∗ 10ିሻ.
Note.— The manipulation of matrices (i.e., their multiplication and inversion) is best performed by computers via standard routines. Such routines are possible using stan dard functions contained within many commonly used spreadsheets.
Table 6. Values of sample data set Run number ܨ ߜ ே EPNL 1 1 395 92.3 2 1 505 92.9 3 1 655 93.2 4 1 730 92.9 5 1 810 93.4 6 1 850 93.2 To find the 90 percent confidence interval about the regression line for a value of ிಿ (i.e., x0) of 1 600, the row vector ሺݔሻ and its transpose ሺ′ݔሻ, a column vector, are formed such that: ఋ ݔ ൌ ሺ1 1600ሻ, and ′ݔ ൌሺ 11600ሻ.
.
From the calculation of ܣିଵ one obtains:
ݔܣିଵ ൌ ሺ1 1600ሻሺ 17.5836 െ0.01051െ0.01051 6.3396 ∗ 10ିሻ ൌ ሺ0.7709 െ 3.6453 ∗ 10ିସሻ, and so:
ݔܣିଵ′ݔ ൌ ሺ0.7709 െ 3.6453 ∗ 10ିସሻሺ 11600ሻ ൌ 0.1876.
The equation for confidence interval also requires the value of standard deviation for the measured data set to be evaluated. From Table 6 and the regression equation for the least-squares best fit straight line, from which is calculated the predicted value of EPNL at each of the six measured values of ிಿ one proceeds as follows: ఋ Page 94
Run number ܨ ߜ ே EPNL (Measured) 1 1 395 92.3 2 1 505 92.9 3 1 655 93.2 4 1 730 92.9 5 1 810 93.4 6 1 850 93.2 Run number EPNL (Predicted) ሺ∆ܮܰܲܧሻଶ 1 92.50 0.03979 2 92.70 0.03911 3 92.98 0.04896 4 93.12 0.04708 5 93.26 0.01838 6 93.34 0.01909 ଶୀ ୀଵ ሺ∆ܮܰܲܧሻ ൌ√ 0.21241ݏൌ √∑ ݊െ݇െ1 6െ1െ1 ൌ 0.2304 for n = 6 and k = 1.
Taking the value of Student’s t-distribution from Table 5 for ( n – k - 1) degrees of freedom (i.e., 4) to be 2.132, the confidence interval about the regression line at ி ఋ ಿ = 1600 is defined as follows:
ܫܥଽ ൌ ܮܰܲܧ ݏݐ √ ݔܣିଵ′ݔ ൌ 92.88 േ ሺ2.132ሻሺ0.2304ሻ√0.1876 ൌ 92.88 േ 0.2128 In order to establish the lines of 90 percent confidence intervals about a regression line, the values of CI90 for a range of values of independent variable(s) should be calculated, through which a line can be drawn. These lines are shown as the dotted lines in Figure 6.
305(f)(4) Confidence int erval for a second order regression curve [ETM 3.5.6.4] The confidence intervals for a second order regression curve are derived in a similar manner to those for a straight line detailed in 305(f)(3). A detailed example of their calculation is not discussed here. However the following points should be borne in mind.
The coefficients of the least-squares regression quadratic line are best dete rmined via computer matrix solutions.
Regression analysis functions are a common feature of many proprietary software packages.
The matrices ݔ, ′ݔ, ݀݊ܽ,ܺ ′ܺ formed during the computation of the confidence interval according to the formula:
ܫܥଽ ൌ ܻ േݏݐ √ ݔܣିଵ′ݔ are formed from 1 × 3 and 3 × 1 row and column vectors respectively, made up from the values of independent variable X according to the following general form:
ᇱݔ ൌ ሺ1 ݔ ݔଶሻ݀݊ܽ ݔ ൌሺ ݔ 1 ݔ ଶ ሻ.
The number of degrees of freedom associated with a multiple regression analysis involving K variables independent of the dependent variable (i.e., with ( K+1) coefficients, including the constant term) is defined as ( n – K - 1) . For a second order regression curve, there are two independent variables and so the number of degrees of freedom is (n-3) .
305(f)(5) Confidence interval for the pooled data set [ETM 3.5.6.5] The confidence interval associated with the pooling of three data sets is defined as follows:
ܫܥ ൌ ܻ േܶ √∑ୀୀଵଷ ܼଶ , where:
ൌ ூܼ ,௧ With CIi = confidence interval for the i-th data set, ti = the value of Student’s t-distribution for the i-th data set, and సయ ܶൌ ∑సభ మ௧ .∑సయ మసభ The different stages in the calculation of the confidence interval at the reference thrust of ி ఋ ಿ ൌ 1600 for the pooling of the three data sets are summarized in Table 7.
305(g) Student’s t-distribution (for 90 percent Confidence) for Various Degrees of Freedom [ETM 3.5.7] The values in the Student’s t-distribution to give a probability of 0.95 that the population mean value (µ) is such that:
ߤݏ ݕ .ݐ ଽହ,ఋ √݊ and thus a probability of 90 percent that ݕ െ.ݐ ଽହ,ఋ √ ݏ ݊ ߤݕ .ݐ ଽହ,ఋ ݏ √݊ are tabulated in Table 5.