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> ■ r NPS55-83-013 NAVAL POSTGRADUATE SCHOOL Monterey, California THE CEMS IV OAP ALGORITHM by H. J. Larson T. Jayachandran May 1983 Approved for public release; distribution unlimited Prepared for: Kel ly Air Force Base San Antonio, TX 78241 FedDocs D 208.14/2 NPS-55-83-013 a-.ufs-ss-c-oo NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral J. J. Ekelund D. A. Schrady Superintendent Provost The work reported herein was supported with funds provided by the Directorate of Material Management, Kelly Air Force Base. Reproduction of all or part of this report is authorized. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Entered) DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY CA 93943-5101 REPORT DOCUMENTATION PAGE 1. REPORT NUMBER NPS55-83-013 2. GOVT ACCESSION NO. 4. TITLE (and Subtitle) THE CEMS IV OAP ALGORITHM 7. AUTHORr*; H. J. Larson T. Jayachandran READ INSTRUCTIONS BEFORE COMPLETING FORM 1. RECIPIENT'S CATALOG NUMBER S. TYPE OF REPORT * PERIOD COVERED Technical 6 PERFORMING ORG. REPORT NUMBER • • CONTRACT OR GRANT NUMBER(«; 9. PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, CA 93940 10. PROGRAM ELEMENT. PROJECT, TASK AREA * WORK UNIT NUMBERS MME-83 — 004 11. CONTROLLING OFFICE NAME AND ADDRESS Directorate of Material Management Kelly Air Force Base ATTN: Henry Solar, San Antonio, TX 78241 SA-ALC/MMEI 12. REPORT DATE May 1983 I). NUMBER OF PAGES 25 14. MONITORING AGENCY NAME ft ADDRESS*"// dlltatant from Controlling Olllca) IS. SECURITY CLASS, (ol thl, report) Unclassified 19a. OECLASSIFICATION/OOWNORADINO SCHEDULE 16. DISTRIBUTION STATEMENT (ol thla Raport) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (ol the abattact entered In Block 20. II dIHarant from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on ravaraa alda II nacaaaary and Idantlfy by block number) Oil analysis program Comprehensive Engine Management System CEMS IV 20. ABSTRACT (Continue on ravaraa alda II nacaaaary and Idantlfy by block number) The Comprehensive Engine Management System (CEMS) Phase IV, will provide real time data analysis capability for all Air Force oil analysis laboratories. This paper describes the statistical algorithm used by this system to aid the oil analysis technician in making his recommendations. The algorithm incor- porates usage and oil consumption variables, and employs least squares to minimize the effects of the random errors in the spectrometer readings. DD,: FORM AN 73 1473 EDITION OF 1 NOV 68 IS OBSOLETE S/N 0102- LF- 014- 6601 UNCLASSIFIED SECURITY CLASSIFICATION OP THIS PAGE (Whan Data tntarad) THE CEMS IV OAP ALGORITHM by H. J. Larson T. Jayachandran The Air Force Oil Analysis Program (OAP) uses spectrometry' c analyses of used oil to monitor the levels of metallic contaminants in the lubricat- ing systems of aircraft engines and other types of equipment. Oil samples are analyzed at more or less regular intervals (every 10 operating hours, after every flight), allowing observation of the temporal evolution of the contaminants being monitored. These temporal observations are used, in turn, to recommend special maintenance actions when required, generally triggered by "high" levels of one or more contaminants. During fiscal year 1982, the Air Force contracted for the development of the Comprehensive Engine Management System (CEMS), phase IV. This sys- tem is to operate at two different levels: (a) it is to gather maintenance and other data at the level of the individual base, making this data available at the base level to aid in maintenance decisions. (b) a central data bank is to be located at Tinker Air Force Base, bringing together data from all the individual bases. The OAP data generated by an Air Force base is to be one of the data elements in the base level CEMS IV system. Because of this, upon the adop- tion of the CEMS IV system it is planned that every Air Force oil analysis laboratory will be equipped with a computer terminal, linking it to the base's CEMS IV computer, allowing real time processing of the oil analysis data. The use of regression methodology to aid the OAP decision making proc- ess has been suggested several times (see references [1] through [6]). The main drawback to implementing this type of approach has been the data- analytic requirements of such a system. Many of the OAP laboratories main- tain a heavy workload of sample analyses and are not equipped to handle 1 extensive number crunching prior to giving a laboratory recommendation, based on the numbers produced by the spectrometry c analysis of a used oil sample. Availability of real time data processing, and a direct link between the spectrometer and the CEMS IV computer, will remove this drawback. For any aircraft participating in OAP, used engine oil samples are re- moved on a regular basis. The frequency of such samples depends on the aircraft type involved. Some aircraft are sampled after every flight, some every flying day, some every 10 flight hours, etc. In every case the used oil sample is delivered to an oil analysis laboratory where it is analyzed on a spectrometer. The spectrometer produces a digital readout of the part per million content of any of 20 different contaminants; generally the lab- oratory only analyzes for those metals that are in contact with the lubri- cating oil in the engine sampled. In current practice, the reading for each metal of interest is subjected to two comparisons defined by the Tech- nical Order (T.O.) table for the specific engine sampled. The T.O. table gives a trend value and a number of range values, for each contaminant of interest. The range values provide absolute limits for the amount of the contaminant, regardless of how long it has been since the last sample was taken or since the oil was changed. Thus for iron, for example, the table may say the normal range is 0-12 ppm, the marginal range is 13 to 15 ppm, the high range is 16 to 18 ppm and the abnormal range is 19 ppm and over. The spectrometer operator then observes where the current iron reading falls, relative to these values, and notes the appropriate recommendation. The trend value is used to judge the change in contaminant level from the preceding sample to the current reading. For iron, say the T.O. trend value is 4 ppm. The spectrometer operator must then take the iron reading for the current sample, subtract the iron reading for the previous sample, multiply by 10 and then divide by the change in operating hours. This computation gives an estimate of the rate of change of the iron contamina- tion, standardized to a 10 hour operating period. If the computer value is no more than 4 (for this example) the sample has a normal trend; otherwise the T.O. recommendation is to declare the trend abnormal and appropriate recommendations are made, based on the range and trend results. The oil sample received by the spectrometri c laboratory is removed from the aircraft's oil sump while the engine is still hot, hopefully en- suring that the fluid in the sump is homogeneous and that the contamination in the sample removed is representative of that in the sump. The sample is placed in a small bottle, labelled to identify the aircraft, date and cumu- lative hours since the oil was changed, and then is sent to the laboratory. At the laboratory the sample is well mixed and then a small part of the oil in the bottle is burned by the spectrometer to produce the ppm contaminant counts. The counts produced by the spectrometer are in fact observed values of random variables. This apparent randomness in the readings is caused by a number of factors, including the following. If the oil in the same sample bottle is analyzed two or more times, the counts produced are not the same. This variation in readings is caused by voltage fluctuations, temperature variations, actual length of time the spectrometer uses to burn the oil, variations in the actual contamination contents in the small amounts burned, etc. From a broader view, if a second sample were removed from the aircraft sump it is quite likely that the actual contamination levels in the two bottles are not in fact identical and, of course, both may differ from the actual contamination levels of the sump itself, the quantity of interest. Earlier studies ([3], [4]) have shown that the readings produced by the spectrometer appear to be well described by a normal (Gaussian) distribution, Additionally, it has been shown ([2], [3], [4]) that different serial num- bers of the same type of equipment appear to present individual signatures for contaminant buildup in the oil sump, even though they are presumably "identical" in construction. Previously the only useful environmental variables readily available to the technician in making oil analysis recommendations were the number of flight hours since the oil was changed and the number of hours since over- haul. With the advent of CEMS IV a new environmental variable is now available from other maintenance sources: whether or not fresh oil was added to the sump. Our major task for this year was to provide an imple- mentable statistical algorithm to aid the operator in making recommenda- tions. This algorithm was to take into account (minimize the effects of) the random noise of the spectrometer, the number of flight hours since oil change and the oil addition records. For the A-10 aircraft, maintenance procedures call for oil to be added (if needed) after the oil sample has been removed for the oil analysis pro- gram. The maintenance form on which additions are recorded allows entry of the number of whole units (pints, quarts or gallons) added to the sump; for the A-10 aircraft the unit used is pints. No provision is made for the entry of fractional amounts of units being added to the sump. Thus the oil added values consist of O's and l's, indicating whether or not a one pint can of oil was opened and used to top off the A-10 sump. In actual prac- tice, of course, the amount added is generally a fraction of a pint, but this is not reflected in the records available to CEMS IV. Among other topics, reference [6] discusses the use of oil addition records to estimate wear metal production rates for aircraft engines. This approach is based on several assumptions which may or may not be universally acceptable for all aircraft types. A tacit assumption apparently made is that oil is lost through a leaking or burning phenomenon and that the metallic contaminants are also lost in direct proportion to the oil lost. That is, if the iron contamination level (as measured by the spectrometer) is, say, 10 ppm and one pint of oil has been added to an aircraft sump which holds 10 pints, reference [6] suggests that the iron contamination level shoud be "corrected" to read 11 ppm (a 10% upward adjustment to account for the 10% addition of fresh oil). The procedure suggested in [6] also tacitly assumes a relatively accurate record made of the amount of oil added on each occurrence, rather than a simple 0-1 variable indicating whether a one pint can was opened and partially poured into the sump. This suggested procedure then goes on to suggest fitting least squares regression curves with the corrected concentration as the dependent variable and number of flight hours since oil change as the independent variable. The least squares regression approach certainly seems justified, to minimize the effects of the spectrom- eter errors of measurement of the ppm concentration. The use of the "corrected" concentration, though, does not seem wise for the CEMS IV algo- rithm, in part because of the crude indication of how much oil was added (and thus how much the spectrometer reading should be corrected). Of equal importance, it may not be true that the iron contamination is lost at the same rate as the oil itself. If an evaporative mechanism were causing the oil loss, it seems possible the iron contamination may not evaporate at the same rate as the oil, if at all; if this were true the "corrected" concen- trations would then be too high. The algorithm employed in the CEMS IV prototype uses least squares re- gression methodology to minimize the effects of the random spectrometer errors. It utilizes the number of hours since oil change as an independent variable and, if oil addition records are available, it lets least squares itself determine the "corrections" to be applied to the spectrometer con- tamination readings. This algorithm wi 1 1 now be described. The contaminants monitored for the TF34 engine in the A-10 aircraft are Fe, Ag, Al, Cr, Cu, Ni, Ti . These 7 different contaminants are treated separately and in the same way. The following discussion refers to only one contaminant; it is understood that the same procedures are applied to each and that the data from each different serial number are treated separately. Define Y. = Spectrometer contaminant reading for the i — sample. T. = Number of hours since oil change when i — sample is taken. a. u a. = Amount of oil added to the sump, after the i — sample is sent to the spectrometer. x x = i X. = I a- lY./H , the accumulated "correction" to the spectrometer i j=2 J-i J reading in the spirit of reference [6]; the TF34 sump is assumed to hold 11 pints, the reason for the divisor of 11. e. = random spectrometer measurement error on the i — sample. Formally, the computations in the algorithm then are consistent with the assumption that (1) Y, - H * » 1 T 1 ♦ 6 2 X i ♦ a, , where the e. 's are independent normal random variables with mean and variance o . Standard formulas for unweighted least squares, with two independent variables (T. and X i ) are employed to estimate the unknown param- 2 eters Bq, Bp B 2 and a . Detailed definition of these formulas is provided in Appendix I. The synthetic variable X. in equation (1) is created from the oil added values (a.) and the spectrometer contaminant readings; indeed X. is the "correction" to be applied to the i — contaminant reading by one of the procedures described in [6], granted the full pint was added to the sump. If we were to assume that B 2 = "1 then equation (1) is equivalent to the wear metal production rate estimation procedure given in [6]. Use of X. in this way gives (1) an autoregressi ve flavor. It is interesting to note that the use of regular unweighted least squares on (1) does in fact yield true least squares estimates of 3q, Bi and B 2 » as "" s proved in Appendix II, With the added assumption of normality of the e. 's , the estimates used are also maximum likelihood. The CEMS IV algorithm applies two statistical tests to the spectrometer reading, in addition to the T.O. limits mentioned earlier. These two tests are meant to be similar in spirit to the T.O. range and T.O. trend compari- sons; they differ from these T.O. comparisons in that they are determined by the historical data base for the serial number being analyzed. These two statistical tests are called the Primary test (similar to the T.O. range comparison) and the Secondary test (similar to the T.O. trend comparison). After the spectrometer analysis has been made for an incoming sample (say from serial number 1111), the computer calls up the prior data base for 2 serial number 1111, and uses this data to estimate By, Bj, B 2 and a from equation (1). This estimated equation then is used to extrapolate forward to the hours since oil change value (T, ) and the X. value for the new sample o ,t ■■ 1 1 o 2 . o -8 UJ o o o z to to o I Ndd NOdl I 2 T I I I O to a. O O o Z> O I 9 ♦ tidd NOdl just analyzed. This extrapolation produces three numbers, labelled N, M, H in Figure 1 (plotted for a case in which no oil has been added); these num- bers are 90%, 95% and 99% prediction limits for what contaminant reading one would expect at this time, based on the data base. The current contaminant reading (just produced by the spectrometer) is labelled C. Table 1 gives the result of the primary test, based on the relationship of C to N, M, H. Table 1 Current value Primary test result C <_ N Normal N < C <_ M Marginal M < C _< H High H < C Abnormal As pictured in Figure 1 the primary test result would be normal. The Secondary test adds the current reading C to the data base and then splits the data base into two parts: the earliest 75% of the data records versus the latest 25% of the data records. Equation (1) is then fit separately to each of these two pieces, giving two estimates of 3-1, the rate of change of the contaminant. These two estimates of B, are then compared (using a Student's t statistic) to see if it appears likely that the rate of change in the later data is larger than that in the earlier part. This test is pictured in Figure 2, again for a case with no oil additions. ■f- h If the computed t statistic exceeds the 95 — quantile of the appropriate t distribution, the secondary test produces an abnormal recommendation; otherwise the recommendation returned is normal. For the two slopes pictured in Figure 2, the Secondary test result is normal. At this point the computer has evaluated four recommendations for each element: the T.O. range and trend values, as well as the Primary and Second- ary statistical test results. It then takes the worst case of the T.O. range and Primary tests, and the worst case of the T.O. trend and Secondary tests and uses these as entries in the T.O. decision making guidance table. The result of this is the computer's recommendation for each element for the cur- rent sample (standard A, B, C, E, F, H, J, P, S, T, or U as used in the JOAP laboratory manual). This computer recommendation may be accepted (used) by the OAP technician or may be overridden and changed by him if he feels that to be appropriate. This discussion of the Primary and Secondary tests has referred to a historical data base for each serial number. For many aircraft types, oil changes are widely separated in time and for some types, the oil is never changed. Thus, if all historical data were maintained, the data base could become quite large for each serial number, requiring a very large, accessible data storage facility for each engine at each base. Of equal importance, it seems intuitively reasonable that older data gets "stale", that ancient his- tory has little bearing on the judgment of the current state of an engine's health. Because of this the statistical data base for each engine consists only of the 20 latest historical records which were accepted as being normal st for the given engine. The current spectrometer reading makes the 21 — record, the largest number used in the statistical algorithm; as each new record is accepted as being normal, it replaces the oldest record in the data base. When a new engine enters the program, or an old engine has an oil change, for the first 7 records only the T.O. range and trend computations 10 4- U are used for the computer recommendation. When the 8 — record becomes avail- able, the Primary test is also applied (but the Secondary is not); this is also the case for records 9, 10, 11 and 12. The 13 — and all subsequent records are subjected to both the Primary and Secondary tests. The statisti- cal algorithm has been programmed in APL at the Naval Postgraduate School and in Fortran by the CEMS IV contractor. Several data sets have been used, giving identical results from both programs. 11 Formulas for CEMS IV OAP H. Larson, T. Jayachandran I Primary Test Model 1 Element values Y, , Y ? , ..., Y N , Time values T, , T ? , ..., T.. , Computed "oil addition" values X,, X ? , ..., X N , Compute sums of squares and cross products ss - iv 2 - M i:> Y l T i N-l »I ■ IT? - 5£ T L l N-l SS X " ^ X i " N-l SP YT ■ I Y 1 T i " N-l SP YX " ^Vi " N-l _ (I^OdX.) SP TX " ^ T i X i " N-l Compute denominator D = (SS T )(SS X ) - (SP-™) Compute coefficients ^ = [(SS X )(SP YT ) - (SP TX )(SP YX )] t D 6 2 = [(SS T )(SP YX ) - (SP TX )(SP yT )] i D Iy. It. £x. and means ? = ^| , T = ^| , X = ^^j 2 12 Compute residual RES = SS y - B 1 (SP YJ ) - 3 2 (SP YX ) Current values for time, "oil addition 1 Current value for element - Y, V X N N Compute increment INC = < /y + E(T N -T) 2 SS X + (X N -X) 2 SS T - 2(T N -f)(X N -X)SP TX ] * D) gff 1/2 Compute predicted element value P = Y+e^-T) + b 2 (x n -x) Compute limits L ] = P + t 9 (N-4)INC L 2 = P + t g5 (N-4)INC L 3 = P + t >gg (N-4)INC where t g(N-4), t gr(N-4), t gg(N-4) are quantiles of the t-distribution, N-4 degrees of freedom Sample reading is normal if Y.. < L, Sample reading is marginal if L < Y.. < L~ Sample reading is high if Sample reading is abnormal if l_ 3 • Y.. 4 < Y N 1 L 3 13 II Primary test Model 2 Element values Y, , Y~, ..., Y N , Time values T-. , T 2 , ..., T N , Compute sums of squares and cross product Y L l N-l SS T ■ IT? - iff L i N-l (IV,.) (IT.) SP YT " I Y i T i " N-l Compute coefficient 3 = SP YT /SS T and means Y = 2>. I T - N-l N-l Compute residual RES = SS„ - (sp yt )' Y SS. Current values for element, time Y.,, T,. Compute increment INC = + (T N" T) N-l SS- 1/2 N-3 Compute predicted element value P = Y + B(T N -T) 14 Compute limits L ] = P + t g (N-3)INC L 2 = P + t g5 (N-3)INC L 3 = P + t gg (N-3)INC where t g (N-3), t >g5 (N-3) t gg (N-3) are quantiles of the t-distribution, N-3 degrees of freedom. Sample reading is normal if Y N < l_. Sample reading is marginal if L, < Y N <_ L ? Sample reading is high if L ? < Y N < L^ Sample reading is abnormal if L~ < Y.. III Secondary test model 1 Element values Y, , Y ? , ..., Y.. Time values T, , T ? , ..., T N Computed "oil addition" values X-,, X 2 , ..., X. Data is split into 2 segments Segment 1 Earliest N, = [.75N] values Segment 2 Remaining N ? = N - N, values 15 For each segment separately, j = 1,2 Compute sums of squares and cross products SS SS T . L i SS Xj pf- K- (IV, > 2 N. >- ' C[T, > 2 N : ' (IX, > 2 N (ZV|)(Tr,) SP YTj * ^Vi " N. SP YX.i * ^i X i (IV ^V SP TXj " ^ T i X i " (IV'IV Compute denominators D. = (SS T .)(SS,. .) - (SP T „.)' Compute coefficients 8, j ■ [(ss xj )(sp yTj ) - (sp txj )(sp yxj )] . D S 2j ■ [(SS Tj )(SP yxj ) - (SP Txj )(SP YTj )] . D. Iy. It. Ix. and means 7j = -+ , Tj » -^ . Xj = -jg 1 J J J Compute residuals RE Sj - SS yj - B l0 .(SP YTj ) - B 2j (SP Yxj ) 16 Compute test value , A A TEST = (3 12 -3 n ) * (RES 1 +RES 2 ) N-6 SS XI SS X2 1/2 Sample trend is normal if TEST < t g g (N-6) Sample trend is abnormal if t qg (N-6) < TEST where t g9 (N-6) is a t-distribution quantile, N-6 degrees of freedom. IV Secondary test Model 2 Element values Y, , Y„, ..., Y N Time values T T T 2' •••' T N Data is split into 2 segments Segment 1 Earliest N, = [.75N] values Segment 2 Remaining N~ = N-N, values For each segment separately, j = 1,2 Compute sums of squares and cross products SS ^Yj 1 i N. SS Tj z 'i N. SP YTj yv.T. - (1*,)^,) 17 Compute coefficients $. = SP j ss T . J = 1,2 Iy. It. and means Y = -^ , T = -^- , j = 1 ,2 Compute residuals RES. = SS V . - 6-SP VT . J Yj p j YTj j = 1 ,2 Compute test value /\ /N TEST = CB 2 -8 1 ) * "RES 1 + RES 2 N-4 SS T1 SS T2 1/2 Sample trend is normal if TEST < t qq(N-4) Sample trend is abnormal if t gg(N-4) < TEST V. Residual SS = 0? test Residual SS is RES in II VI. Residual SS = 0? test Residual SS is RES in I VII. Residua^ = Residual = 0? test Values are RES, and RES ? in IV VIII. Residual -j = Residual „ = 0? test Values are RES, and RES ? in III IX. T.,X. correlation = 1? test With definitions in I the correlation is SP TX r = - , compare with 1. /ss x ss T When the data is split into 2 segments, compute the correlation separately for each SP r. ■ U { . J - 1.2. 19 X Oil addition computation The values for X,, X 2 , ..., X.. are computed using only the data in the current window and do not depend on earlier values. Formula used: X-, = x i + i = x i + a i Y i + i /D where Y. is the ith element reading, a. is the ith "oil" value (generally or 1), D is the sump capacity (unrelated to D used in I). Example 1 2 3 4 5 6 7 8 9 10 OIL = a 1 1 1 1 _FE_=_y. 4 3 3 2 4 5 4 3 3 4 X. i .3 .3 .7 .7 .7 .7 1.0 1.4 D = 10 20 Appendix II For any fixed number, n , of data points the CEMS IV algorithm is consistent with (1) Y. = 3 + B 1 T i + e 2 X. + e . , i = 1,2,. ...n where Y. = spectrometer contaminant reading, sample i T. = time since oil change, sample i X- = i ^i = I a n- i Y .: / 1 1 , where a. = amount of oil added, sample's 1 -j =2 J J J e. = random spectrometer additive noise, sample i . In standard least squares notation, define Y to be the nxl vector of contaminant readings, define X to be the nx3 matrix whose first column is all l's , whose second column contains the n - T. values and whose third column contains the synthetic X. values, and define 3 to be the 3x1 vector with components 3 Q , 6,, 3 2 • Then it is well known that the simple unweighted least squares estimate for 3 is given by 3 = (X'X)" 1 X'Y where X 1 denotes the transpose of X . To see the form of the least squares estimates of 3q, 3^ and 3 2 for equation (1), let A represent the lower triangular n x n matrix whose first row is all O's, whose i— diagonal and all elements below it in that column are a. , if a. , = , and whose remaining elements are all . A 21 little reflection shows that the elements of the n x 1 vector V = AY then are the values for X,, X 2 ,..., X , the synthetic oil added variable. Addi- tionally, let U represent the n x 2 matrix consisting of the first 2 columns of the matrix X , let y be the 2x1 vector with components 3q and 3i (the first two components of 3) and let e be the n x 1 vector with components e,, e 2 ,... e . Then (1) can be written Y = Uy + 3 2 V + e = Uy + 3 2 AY + e ; rearranging terms, this also can be written Y - 3 2 AY = Uy + e or (I - 3 2 A)Y = Uy + e . The unweighted least squares estimates are by definition the parameter values which minimize Q = e'e = ((I - 3 2 A)Y - Uy)'((I - 3 2 A)Y - Uy) = Y'(I - 3 2 A)'(I - 3 2 A)Y - 2y'U'(I - 3 2 A)Y + y'U'Uy . 90 90 90 Letting ^- represent the vector whose components are jg- and y^- , respectively, it is easy to see that 22 |£ = - 2U'(I - 6 2 A)Y + 2U'U Y , from which it is evident that t = (u'urki'd - i 2 A)Y . Remembering that B 2 is a scalar, a little algebra shows that 3Q . 30, Y'(A' + A)Y + 2 6 2 Y'A'AY + 2 y'U'AY ; setting this equal to zero, and substituting the above solution for y then easily gives Y'(y(A'+A))Y - Y'UCU'uTVaY 2 Y'A'(I - U(U'U)" 1 U')AY as the least squares estimate for 2 • That these solutions for Q , 0, (y) and i-L are in fact the same as the earlier quoted simple unweighted least squares values is easily seen by realizing that e = Y - Uy - 6 2 V ■ v - < u i Av >(-y = Y - X0 since the X matrix is (U | AY ) and the 8 vector is (— |— ) . The well known solution 6 = (X'X)~ X'Y then must consist of the subvectors y and L defined above. If we also assume that e is multivariate normal with mean 2 vector and covariance matrix a I , it is immediately apparent that y 23 and 3 2 are also the maximum likelihood estimates and the maximum likelihood estimate for a is -2 = Y'Y - p'X'Y The CEMS IV algorithm actually uses 2 no s = n - 3 as the estimate for a . 24 References [1] Barr, D. R. and Larson, H. J. "Objective Identification Procedures for the Naval Oil Analysis Program", Technical Report NPS-Bn-La 9091A, September 1969. [2] Barr, D. R. and Larson, H. J. "Identification of Failing Mechanical Systems Through Spectrometri c Oil Analysis", Applied Spectros copy, Vol. 26, Number 1, 1972. [3] Carty, J. J. "An Analysis of Oil Sample Data Obtained from Aircraft Engines by Spectrometry", Naval Postgraduate School Thesis, MS in Operations Research, October 1969. [4] Riceman, J. P. "A Statistical Study of Spectrometri c Oil Analysis Data from the Naval Oil Analysis Program", Naval Postgraduate School Thesis, MS in Operations Research, October 1969. [5] Scheller, K. "Examination of Errors in the Joint Oil Analysis Program Trending Procedures - Suggested Improved Techniques for Trend Analysis of Wear Metal Measurements", Technical Report AFML-TR-79-4194, December 1979. [6] Scheller, K. "Statistical Trend Analysis of Wear Metal Concentration Measurements - Calculation of Significant Wear Metal Production Rates", Technical Report AFML-TR-79-4195, December 1979. 25 DISTRIBUTION LIST No. of Copies SA-ALC/MMEI 20 Kelly Air Force Base San Antonio, TX 78241 Library, Code 0142 4 Naval Postgraduate School Monterey, CA 93940 Dean of Research 1 Code 01 2A Naval Postgraduate School Monterey, CA 93940 Library, Code 55 1 Naval Postgraduate School Monterey, CA 93940 Professor H. J. Larson 45 Code 55La Naval Postgraduate School Monterey, CA 93940 Associate Professor T. Jayachandran 5 Code 53Jy Naval Postgraduate School Monterey, CA 93940 26 DUDLEY KNOX LIBRARY 3 2768 00329690 6