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.
306 Table 7. Example of confidence interval calculation Description Reference thrust Function ܨ ߜ ே Datum Hard wall 1 600 Silenced 1 600 90% confidence interval about the mean CI90 0.3183 0.4817 0.2128 Number of data points n 6 23 6 Degree of curve fit k 0 2 1 Number of independent variables K 0 2 1 Number of degrees of freedom N – K - 1 5 20 4 Student’s t-distribution Z Z2 Z2t t ܫܥଽ ݐ ሺܫܥଽ ݐ ሻଶ ሺܫܥଽ ݐ ሻଶݐ
2.015
0.1580
2.4953 × 10–2
5.0280 × 10–2
1.725
0.2792
7.7979 × 10–2
0.1345
2.132
0.09981
9.9625 × 10 –3
2.1240 × 10–2 ∑Z2 0.1129 ∑(Z2t) T √∑ܼଶ CI ∑ሺܼଶݐሻ ∑ ܼଶ ܶ√∑ܼଶ
0.2060
1.8248
0.3360
0.6131
ADJUSTMENT OF AIRCRAFT NOISE LEVELS FOR THE EFFECTS
OF BACKGROUND NOISE [ETM 3.6] 306(a) Introduction [ETM 3.6.1] The following is provided as guidance material on procedures for adjusting measured aircraft noise levels for the effects of background noise.
The presence of background noise during aircraft noise certification tests can influence measured aircraft sound levels and, in some cases, obscure portions of the spectral time history used to obtain EPNL values. Adjustment procedures should include the following components:
a) testing to determine which portions of the spectral time history, if any, are obscured;
b) adjustment of unobscured levels to determine the aircraft sound levels that would have been measured in the absence of background noise; and
c) replacement or reconstruction of obscured levels by frequency extrapolation, time extrapolation or other means.
Definitions of the terms used in this section are provid ed in 306(b). Although some of the terms have generally accepted meanings, the specific meanings as defined apply herein.
A detailed step-by-step procedure is presented in 306(c) including equations and descriptions of time and frequency extrapolation methods (see 306(c)(2)(J)). Other procedures may be used provided that they have been approved by the FAA.
General considerations that apply to any background noise adjustment procedure are listed in 306(d), including limitations and requirements (see 306(d)(1)) and other special considerations (see 306(d)(2) through 306(d)(4)).
306(b) Definitions [ETM 3.6.2] For the purposes of 306, the following definitions apply:
Adjusted level. A valid one-third octave band level which has been adjusted for measurement conditions, including:
a) the energy contribution of pre-detection noise; and
b) frequency-dependent adjustments such as system frequency response, microphone pressure response and free-field response, and windscreen incidence-dependent insertion loss.
Ambient noise. The acoustical noise from sources other than the te st aircraft present at the microphone site during aircraft noise measurements. Ambient noise is one component of background noise.
Background noise. The combined noise present in a measurement system from sources other than the test aircraft, which can influence or obscure the aircraft noise levels being measured. Typical elements of background noise include, but are not limited to, ambient noise from sour ces around the microphone s ite, thermal electrical noise generated by components in the measurement system, magnetic flux noise (“tape hiss”) from analog tape recorders and digitization noise caused by quantization error in digital converters. Some elements of background noise, such as digitization noise, can obscure the aircraft noise signal, while others, such as ambient noise, can also contribute energy to the measured aircraft noise signal.
Energy-subtraction. Subtraction of one sound pressure level from another, on an energy basis, in the form of the following:
10݈݃ଵሾ10ሺଵಲሻ െ1 0ሺଵಳሻሿ where L A and LB are two sound pressure levels in decibels, with LB being the value subtracted from LA.
Frequency extrapolation. A method for reconstruction of high frequency masked data, based on unmasked data in a lower-frequency one-third octave band from the same spectrum.
High frequency bands. The twelve bands from 800 Hz through 10 kHz inclusive (also see “low frequency bands”).
Last good band (LGB). In the adjustment methodology presented in 3.6.3, for any aircraft one-third band spectrum, Page 98
the LGB is the highest frequency unmasked band within the range of 630 Hz to 10 kHz inclusive, below which there are no masked high frequency bands.
Low frequency bands. The twelve bands from 50 Hz through 630 Hz inclusive (also see “high frequency bands”).
Masked band. Within a single spectrum, any one-third octave band containing a masked level.
Masked level. Any one-third octave band level which is less than or equal to the masking criterion for that band. When a level is identified as being masked, the actual level of aircraft noise in that band has been obscured by background noise and cannot be determined. Masked levels can be reconstructed using frequency extrapolation, time extrapolation or other methods.
Masking criteria. The spectrum of one-third octave band levels below which measured aircraft sound pressure levels are considered to be masked or obscured by background noise. Masking criteria levels are defined as the greater of:
a) pre-detection noise + 3 dB; or
b) post-detection noise + 1 dB.
Post-detection noise. The minimum levels below which measured noise levels are not considered valid. Usually determined by the baseline of an analysis “window” or by the amplitude non-linearity characteristics of components in the measurement and analysis system. Post-detection noise levels are non-additive (i.e., they do not contribute energy to measured aircraft noise levels).
Pre-detection noise. Any noise which can contribute energy to the measured levels of sound produced by the aircraft, including ambient noise present at the microphone site and active instrumentation noise present in the measurement, record/playback and analysis systems.
Reconstructed level. A level, calculated by frequency extrapolation, time extrapolation, or by other means, which replaces the measured value for a masked band.
Sound attenuation coefficient. The reduction in level of sound within a one-third octave band, in dB per 100 meters, due to the effects of atmospheric absorption of sound.
Time extrapolation. A method for reconstruction of high frequency masked data, based on unmasked data in the same one-third octave band, from a different spectrum in the time history.
Valid or unmasked band. Within a single spectrum, any one-third octave band containing a valid level.
Valid or unmasked level. Any one-third octave band level which exceeds the masking criterion for that band.
306(c) Background Noise Adjustment Procedure [ETM 3.6.3] 306(c)(1) Assumptions [ETM 3.6.3.1]
a) A typical aircraft spectru m measured on the ground contains one-third octave band levels which decrease in amplitude with increasing frequency. This characteristic high frequency roll-off is due primarily to the effects of atmospheric absorption.
b) A typical electronic instrumentation floor spectrum contains one-third octave band levels which increase in amplitude with increasing frequency.
c) Due to the assumptions cited in a) and b), as the observed frequency is increased within a one-third octave band aircraft spectrum and once a band becomes masked, all subsequent higher frequency bands will also be masked. This allows the implementation of a Last Good Band (LGB) label to identify the frequency band above which the bands in a spectrum are masked.
d) If, on occasion, a valid level occurs in a band with higher center frequency than the LGB, its presence will most likely be due to small variations in the pre-detection levels and/or due to levels of the measured aircraft one-third octave band spectrum being close to the levels of the background noise in general, so its energy contribution will not be significant. Note that this assumption is valid only in the absence of significant aircraft-generated tones in the region of masking. Th erefore, the possibility of a level being valid in a band with higher center frequency than the LGB may be ignored. Applicants who prefer to implement algorithms for identifying and handling such situations may do so, but no procedure may be used without prior approval by the FAA.
306(c)(2) Step-by step description [ETM 3.6.3.2] 306(c)(2)(A) Determination of pre-detection noise [ETM 3.6.3.2.1] A time-averaged one-third octave band spectrum of pre-detection noise levels for each test run, or group of runs occurring during a short time period, should be obtained by recording and analyzing ambient noise over a representative period of time (30 seconds or more). Care should be taken to ensure that this “ambient” noise sample reasonably represents that which is present during measured aircraft runs. In recording ambient noise, all gain stages and attenuators should be set as they would be during the aircraft runs in order to ensure that the instrumentation noise is also representative. If multiple gain settings are required for aircraft noise measurements, a separate ambient sample should be recorded at each of the settings used.
306(c)(2)(B) Determination of post-detection noise [ETM 3.6.3.2.2] A one-third octave band spectrum of post-detection noise levels should be determined as a result of testing, or from manufacturer’s specifications, for each measurement/analysis configuration used, including different gain and/or sensitivity settings. These minimum valid levels may be determined on the basis of display limitations (e.g., blanking of the displayed indication when levels fall below a cert ain value), amplitude non-linearity or other non-additive limitations. In cases where more than one component or stage of the measurement/analysis system imposes a set of minimum valid levels, the most restrictive in each one-third octave band should be used.
306(c)(2)(C) Testing of pre-detectio n noise versus post-detection noise [ETM 3.5.3.2.3] The validity of pre-detection noise levels must be established before these levels can be used to adjust valid aircraft noise levels. Any pre-detection noise level that is equal to or less than the post-detection noise level in a particular one-third octave band should be identified as invalid and therefore should not be used in the adjustment procedure.
306(c)(2)(D) Determinati on of masking criteria [ETM 3.6.3.2.4] Once the pre-detection noise and post-detection noise spectra are established, the masking criteria can be identified.
For each one-third octave band, compare the valid pre-detection noise level + 3 dB with the post-detection noise level + 1 dB. The highest of these levels is used as the masking criterion for that band. If there is no valid pre-detection noise level for a particular one-third octave band, then th e post-detection noise level + 1 dB is used as the masking criterion for that band. The 3 dB window above pre-detection levels allows for the doubling of energy which could Page 100