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The case is considered of n distinct, normally distributed classes or populations of two dimensional response vectors x = (lambda sub 1, lambda sub 2), where lambda sub i is a measurement of the relative reponse of x along channel i. The problem dealt with is to determine the best channel in the sense of divergence and in the sense of minimizing the probability of misclassification.
The problem dealt with concerns feature selection or reducing the dimension of the data to be processed from n to k. By reducing the dimension of the data from n to k, classification time is generally reduced. Yet the dimension reduction should not be so great that classification accuracy is impaired. Thus, the general problem is considered of classifying an n-dimensional observation vector x into one of m-distinct classes where each class is normally distributed with mean and covariance. It is shown that the probability of misclassification is minimized if a maximum likelihood classification procedure is used to classify the data. The dimension of each observation vector to be processed is conveniently reduced by performing the transformation y = Bx, where B is a K by n matrix of rank k. Thus, the n-dimensional classification problem transforms into a k-dimensional classification problem.
A FORTRAN 4 routine is given which may be used to generate random observations of a continuous real valued random variable. Normal distribution of F(x), X, E(akimas), and E(linear) is presented in tabular form.
The probability with which any given proton fluence level will be exceeded during a space mission is computed for missions to be flown during the active phase of the next solar cycle (1977-1983). This probability is a function of fluence level, proton energy threshold, and mission duration. Data on the major solar proton events of 1966-1972 are given; it is argued that only this data set (and not that of the previous solar cycle) is appropriate for estimating next-cycle fluences. Probable numbers of each of the two types of events are estimated from Burrell's extension of Poisson statistics. Fluences of all future anomalously large events are assumed to have a common spectrum, that given by the August 1972 event. Fluences of the ordinary events are assumed to obey a log normal distribution. It is shown that for much of the confidence-level mission-duration regime of interest, at least one anomalously large event will occur; and given such an occurrence, the ordinary-event contribution to mission fluence is negligible.
A condition for the Gateaux differentiability of the probability of misclassification as a function of a feature selection matrix B, assuming a maximum likelihood classifier and normally distributed populations, is given. It is also shown that if the probability of error has a local minimum at B then it is differentiable at B.
The probability with which any given solar proton fluence level will be exceeded during a space mission is computed for mission to be flown during the active phase of the next solar cycle (1977-1983). This probability is a function of fluence level, proton energy threshold, and mission duration. Calculations are based on 1966-1972 data only. In estimating mission fluences, a distinction is made between ordinary and anomalously large events. Probable numbers of each type of event are estimated from Burrell's extension of Poisson statistics. Fluences of all anomalously large events are assumed to have a spectrum given by the August 1972 event, while fluences of the ordinary events are assumed to obey a log normal distribution.
Common to all nondestructive type testing of hardware that provides the necessary confidence in the design is the question regarding the proper magnitude of the test level. The objective of this investigation was to study the possibility of establishing cost optimized test levels of a rather general nature. This investigation was based on studying the influence of test level on a cost of error cost model that reflects the adverse effects of 'undertesting' and 'overtesting'. For the assumed conditions of normal distribution for the random variables and a protoflight spacecraft case, the results indicated that the limits for an optimum test factor can range between slightly less than 1.0 to slightly more than 1.5.
Monte Carlo studies using population models intended to represent response surface applications are reported. Simulated experiments were generated by adding pseudo random normally distributed errors to population values to generate observations. Model equations were fitted to the observations and the decision procedure was used to delete terms. Comparison of values predicted by the reduced models with the true population values enabled the identification of deletion strategies that are approximately optimal for minimizing prediction errors.
The local linearization method for axisymmetric flow is combined with the transonic equivalence rule to calculate pressure distribution on slender bodies at free-stream Mach numbers from .8 to 1.2. This is an approximate solution to the transonic flow problem which yields results applicable during the preliminary design stages of a configuration development. The method can be used to determine the aerodynamic loads on parabolic arc bodies having either circular or elliptical cross sections. It is particularly useful in predicting pressure distributions and normal force distributions along the body at small angles of attack. The equations discussed may be extended to include wing-body combinations.
Maximum-likelihood estimates are discussed which are based on an unlabeled sample of observations, of unknown parameters in a mixture of normal distributions. Several successive approximation procedures for obtaining such maximum-likelihood estimates are described. These procedures, which are theoretically justified by the local contractibility of certain maps, are designed to take advantage of good initial estimates of the unknown parameters. They can be applied to the signature extension problem, in which good initial estimates of the unknown parameters are obtained from segments which are geographically near the segments from which the unlabeled samples are taken. Additional problems to which these methods are applicable include: estimation of proportions and adaptive classification (estimation of mean signatures and covariances).
Accelerations in the lateral and vertical directions were measured at two locations on the floor of a three-jet-engine aircraft and at two locations on the floor of a two-jet-engine aircraft during a total of 13 flights, each of which included taxiing, takeoff, ascent, cruise, descent, and landing. Accelerations over the frequency range 0 to 25 Hz were recorded continuously on magnetic tape and were synchronized with the VGH recorders in the aircraft so that vibratory accelerations could be correlated with the operating conditions of the aircraft. From the results it was indicated that the methodology used in segmenting the data, which were obtained in a continuous and repetitive manner, contributes to establishing baseline data representative of the flight characteristics of aircraft. Significant differences among flight conductions were found to occur. The lateral accelerations were approximately 15 percent of the vertical accelerations during flight but as much as 50 to 100 percent of the vertical accelerations during ground operations. The variation between the responses of the two aircraft was not statistically significant. The results also showed that more than 90 percent of the vibratory energy measured during flight occurred in the 0- to 3.0-Hz frequency range. Generally, the vibration amplitudes were normally distributed.
The results are presented of a study which was undertaken to determine if data from a single near-noon pass of Skylab could be used to detect geothermal areas. The size and temperature requirements for a geothermally heated area to be seen by Skylab S-192 MSS X-5 thermal sensor were calculated. This sensor obtained thermal data with the highest spatial resolution of any nonmilitary satellite system. Only very large hot areas could be expected to be unambiguously recognized with a single data set from this instrument. The study area chosen was The Geysers geothermal field in Sonoma County, California, the only geothermal area of significant size scanned by Skylab. Unfortunately, 95% of the Skylab thermal channel data was acquired within 3 hours of local noon. For The Geysers area only daytime X-5 data were available. An analysis of the thermal channel data (10.2 to 12.5 um) revealed that ground temperatures determined by Skylab were normally distributed. No anomalous hot spots were apparent. Computer enhancement techniques were used to delineate the hottest 100 and 300 ground areas (pixel, 75 m by 75 m) within the study region. It was found that the Skylab MSS with the X-5 thermal detector does not have sufficient spatial resolution to locate unambiguously from daytime data any but the largest and hottest convectively created geothermal features, which in general are prominent enough to have been previously recognized.
A Robbins-Monro type multidimensional stochastic approximation algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions.
A technique is presented that uses both cluster analysis and a Monte Carlo significance test of clusters to discover associations between variables in multidimensional data. The method is applied to an example of a noisy function in three-dimensional space, to a sample from a mixture of three bivariate normal distributions, and to the well-known Fisher's Iris data.
A 1- by 3-meter semispan wing of taper ratio 1.0 with NACA 0012 airfoil section contours was tested in the Langley V/STOL tunnel to measure the pressure distribution at five sweep angles, 0 deg, 10 deg, 20 deg, 30 deg, and 40 deg, through an angle-of-attack range from -6 deg to 20 deg. The pressure data are presented as plots of pressure coefficients at each static-pressure tap location on the wing. Flow visualization wing-tuft photographs are also presented for a wing of 40 deg sweep. A comparison between theory and experiment using two inviscid theories and a viscous theory shows good agreement for pressure distributions, normal forces, and pitching moments for the wing at 0 deg sweep.
Compressor face dynamic total pressures from four F-111 flights were analyzed. Statistics of the nonstationary data were investigated by analyzing the data in a quasi-stationary manner. Changes in the character of the dynamic signal are investigated as functions of flight conditions, time in flight, and location at the compressor face. The results, which are presented in the form of rms values, histograms, and power spectrum plots, show that the shape of the power spectra remains relatively flat while the histograms have an approximate normal distribution.
The author has identified the following significant results. The overall spectral response of the strata measured by a mean vector and covariance matrix for each stratum did not show differences among the LACIE phase 3 strata using the machine clustering procedures. This was expected since the large strata gave rise to broad normal distributions with a great deal of overlap. The static stratification of Kansas contains strata which are small in size. The distributions for these strata are not as broad as those based on the LACIE phase 3 partitions, but there is still some confusion since strata from different categories are not spectrally distinct.
A review is presented of the current state of knowledge concerning the emitted flux distributions of Ap stars and their ultraviolet photometric variability especially as it relates to photometric and spectrum variability observed in the visible. A table of Ap stars observed in the ultraviolet from space vehicles is presented along with a graph which shows the observed flux distributions, normalized to unity flux at 3320 A, for the Si star theta Aur and for the normal stars 134 Tau and gamma UMa. Attention is given to ultraviolet color indices and ultraviolet light curves for variable Ap stars.