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MATHPAC image-analysis library is collection of general-purpose mathematical and statistical routines and special-purpose data-analysis and pattern-recognition routines for image analysis. MATHPAC library consists of Linear Algebra, Optimization, Statistical-Summary, Densities and Distribution, Regression, and Statistical-Test packages.
The Maritime Aerosol Network (MAN) has been collecting data over the oceans since November 2006. The MAN archive provides a valuable resource for aerosol studies in maritime environments. In the current paper we investigate correlations between ship-borne aerosol optical depth (AOD) and near-surface wind speed, either measured (onboard or from satellite) or modeled (NCEP). According to our analysis, wind speed influences columnar aerosol optical depth, although the slope of the linear regression between AOD and wind speed is not steep (approx. 0.004 - 0.005), even for strong winds over 10m/s. The relationships show significant scatter (correlation coefficients typically in the range 0.3 - 0.5); the majority of this scatter can be explained by the uncertainty on the input data. The various wind speed sources considered yield similar patterns. Results are in good agreement with the majority of previously published relationships between surface wind speed and ship-based or satellite-based AOD measurements. The basic relationships are similar for all the wind speed sources considered; however, the gradient of the relationship varies by around a factor of two depending on the wind data used
A method is developed to retrieve surface ground temperature (Tg) and atmospheric moisture using clear sky fluxes (CSF) from CERES-TRMM observations. In general, the clear sky outgoing long-wave radiation (CLR) is sensitive to upper level moisture (q(sub h)) over wet regions and Tg over dry regions The clear sky window flux from 800 to 1200 /cm (RadWn) is sensitive to low level moisture (q(sub j)) and Tg. Combining these two measurements (CLR and RadWn), Tg and q(sub h) can be estimated over land, while q(sub h) and q(sub t) can be estimated over the oceans. The approach capitalizes on the availability of satellite estimates of CLR and RadWn and other auxiliary satellite data. The basic methodology employs off-line forward radiative transfer calculations to generate synthetic CSF data from two different global 4-dimensional data assimilation products. Simple linear regression is used to relate discrepancies in CSF to discrepancies in Tg, q(sub h) and q(sub t). The slopes of the regression lines define sensitivity parameters that can be exploited to help interpret mismatches between satellite observations and model-based estimates of CSF. For illustration, we analyze the discrepancies in the CSF between an early implementation of the Goddard Earth Observing System Data Assimilation System (GEOS-DAS) and a recent operational version of the European Center for Medium-Range Weather Prediction data assimilation system. In particular, our analysis of synthetic total and window region SCF differences (computed from two different assimilated data sets) shows that simple linear regression employing (Delta)Tg and broad layer (Delta)q(sub l) from 500 hPa to surface and (Delta)q(sub h) from 200 to 500 hPa provides a good approximation to the full radiative transfer calculations, typically explaining more than 90% of the 6-hourly variance in the flux differences. These simple regression relations can be inverted to "retrieve" the errors in the geophysical parameters. Uncertainties (normalized by standard deviation) in the monthly mean retrieved parameters range from 7% for (Delta)T to about 20% for (Delta)q(sub t). Our initial application of the methodology employed an early CERES-TRMM data set (CLR and Radwn) to assess the quality of the GEOS2 data. The results showed that over the tropical and subtropical oceans GEOS2 is, in general, too wet in the upper troposphere (mean bias of 0.99 mm) and too dry in the lower troposphere (mean bias of -4.7 mm). We note that these errors, as well as a cold bias in the Tg, have largely been corrected in the current version of GEOS-2 with the introduction of a land surface model, a moist turbulence scheme and the assimilation of SSTM/I total precipitable water.
A method is developed to retrieve surface ground temperature (T(sub g)) and atmospheric moisture using clear sky fluxes (CSF) from CERES-TRMM observations. In general, the clear sky outgoing longwave radiation (CLR) is sensitive to upper level moisture (q(sub l)) over wet regions and (T(sub g)) over dry regions The clear sky window flux from 800 to 1200/cm (RadWn) is sensitive to low level moisture (q(sub t)) and T(sub g). Combining these two measurements (CLR and RadWn), Tg and q(sub h) can be estimated over land, while q(sub h) and q(sub l) can be estimated over the oceans. The approach capitalizes on the availability of satellite estimates of CLR and RadWn and other auxiliary satellite data. The basic methodology employs off-line forward radiative transfer calculations to generate synthetic CSF data from two different global 4-dimensional data assimilation products. Simple linear regression is used to relate discrepancies in CSF to discrepancies in T(sub g), q(sub h) and q(sub l). The slopes of the regression lines define sensitivity parameters that can be exploited to help interpret mismatches between satellite observations and model-based estimates of CSF. For illustration, we analyze the discrepancies in the CSF between an early implementation of the Goddard Earth Observing System Data Assimilation System (GEOS-DAS) and a recent operational version of the European Center for Medium-Range Weather Prediction data assimilation system. In particular, our analysis of synthetic total and window region SCF differences (computed from two different assimilated data sets) shows that simple linear regression employing Delta(T(sub g)) and broad layer Delta(q(sub l) from .500 hPa to surface and Delta(q(sub h)) from 200 to .300 hPa provides a good approximation to the full radiative transfer calculations. typically explaining more than 90% of the 6-hourly variance in the flux differences. These simple regression relations can be inverted to "retrieve" the errors in the geophysical parameters. Uncertainties (normalized by standard deviation) in the monthly mean retrieved parameters range from 7% for Delta(T(sub g)) to about 20% for Delta(q(sub l)). Our initial application of the methodology employed an early CERES-TRMM data set (CLR and Radwn) to assess the quality of the GEOS2 data. The results showed that over the tropical and subtropical oceans GEOS2 is, in general, too wet in the upper troposphere (mean bias of 0.99 mm) and too dry in the lower troposphere (mean bias of -4.7 min). We note that these errors, as well as a cold bias in the T(sub g). have largely been corrected in the current version of GEOS-2 with the introduction of a land surface model, a moist turbulence scheme and the assimilation of SSM/I total precipitable water.
This study is a direct result of an on-going project to model the reliability of a large real-time control avionics system. In previous modeling efforts with this system, hardware reliability models were applied in modeling the reliability behavior of this system. In an attempt to enhance the performance of the adapted reliability models, certain software attributes were introduced in these models to control for differences between programs and also sequential executions of the same program. As the basic nature of the software attributes that affect software reliability become better understood in the modeling process, this information begins to have important implications on the software development process. A significant problem arises when raw attribute measures are to be used in statistical models as predictors, for example, of measures of software quality. This is because many of the metrics are highly correlated. Consider the two attributes: lines of code, LOC, and number of program statements, Stmts. In this case, it is quite obvious that a program with a high value of LOC probably will also have a relatively high value of Stmts. In the case of low level languages, such as assembly language programs, there might be a one-to-one relationship between the statement count and the lines of code. When there is a complete absence of linear relationship among the metrics, they are said to be orthogonal or uncorrelated. Usually the lack of orthogonality is not serious enough to affect a statistical analysis. However, for the purposes of some statistical analysis such as multiple regression, the software metrics are so strongly interrelated that the regression results may be ambiguous and possibly even misleading. Typically, it is difficult to estimate the unique effects of individual software metrics in the regression equation. The estimated values of the coefficients are very sensitive to slight changes in the data and to the addition or deletion of variables in the regression equation. Since most of the existing metrics have common elements and are linear combinations of these common elements, it seems reasonable to investigate the structure of the underlying common factors or components that make up the raw metrics. The technique we have chosen to use to explore this structure is a procedure called principal components analysis. Principal components analysis is a decomposition technique that may be used to detect and analyze collinearity in software metrics. When confronted with a large number of metrics measuring a single construct, it may be desirable to represent the set by some smaller number of variables that convey all, or most, of the information in the original set. Principal components are linear transformations of a set of random variables that summarize the information contained in the variables. The transformations are chosen so that the first component accounts for the maximal amount of variation of the measures of any possible linear transform; the second component accounts for the maximal amount of residual variation; and so on. The principal components are constructed so that they represent transformed scores on dimensions that are orthogonal. Through the use of principal components analysis, it is possible to have a set of highly related software attributes mapped into a small number of uncorrelated attribute domains. This definitively solves the problem of multi-collinearity in subsequent regression analysis. There are many software metrics in the literature, but principal component analysis reveals that there are few distinct sources of variation, i.e. dimensions, in this set of metrics. It would appear perfectly reasonable to characterize the measurable attributes of a program with a simple function of a small number of orthogonal metrics each of which represents a distinct software attribute domain.
Two computer programs developed according to two general types of exponential models for conducting nonlinear exponential regression analysis are described. Least squares procedure is used in which the nonlinear problem is linearized by expanding in a Taylor series. Program is written in FORTRAN 5 for the Univac 1108 computer.
The number of tornado outbreak cases studied in detail was increased from the original 8. Detailed ground and aerial studies were carried out of two outbreak cases of considerable importance. It was demonstrated that multiple regression was able to predict the tornadic potential of a given thunderstorm cell by its cirrus anvil plume characteristics. It was also shown that the plume outflow intensity and the deviation of the plume alignment from storm relative winds at anvil altitude could account for the variance in tornadic potential for a given cell ranging from 0.37 to 0.82 for linear to values near 0.9 for quadratic regression. Several predictors were used in various discriminant analysis models and in censored regression models to obtain forecasts of whether a cell is tornadic and how strong tornadic it could be potentially. The experiments were performed with the synoptic scale vertical shear in the horizontal wind and with synoptic scale surface vorticity in the proximity of the cell.
A diagnostic analysis of the VVP (volume velocity processing) retrieval method is presented, with emphasis on understanding the technique as a linear, multivariate regression. Similarities and differences to the velocity-azimuth display and extended velocity-azimuth display retrieval techniques are discussed, using this framework. Conventional regression diagnostics are then employed to quantitatively determine situations in which the VVP technique is likely to fail. An algorithm for preparation and analysis of a robust VVP retrieval is developed and applied to synthetic and actual datasets with high temporal and spatial resolution. A fundamental (but quantifiable) limitation to some forms of VVP analysis is inadequate sampling dispersion in the n space of the multivariate regression, manifest as a collinearity between the basis functions of some fitted parameters. Such collinearity may be present either in the definition of these basis functions or in their realization in a given sampling configuration. This nonorthogonality may cause numerical instability, variance inflation (decrease in robustness), and increased sensitivity to bias from neglected wind components. It is shown that these effects prevent the application of VVP to small azimuthal sectors of data. The behavior of the VVP regression is further diagnosed over a wide range of sampling constraints, and reasonable sector limits are established.
At the NASA Glenn Research Center, the NASA engine performance program (NEPP, ref. 1) and the design optimization testbed COMETBOARDS (ref. 2) with regression and neural network analysis-approximators have been coupled to obtain a preliminary engine design methodology. The solution to a high-bypass-ratio subsonic waverotor-topped turbofan engine, which is shown in the preceding figure, was obtained by the simulation depicted in the following figure. This engine is made of 16 components mounted on two shafts with 21 flow stations. The engine is designed for a flight envelope with 47 operating points. The design optimization utilized both neural network and regression approximations, along with the cascade strategy (ref. 3). The cascade used three algorithms in sequence: the method of feasible directions, the sequence of unconstrained minimizations technique, and sequential quadratic programming. The normalized optimum thrusts obtained by the three methods are shown in the following figure: the cascade algorithm with regression approximation is represented by a triangle, a circle is shown for the neural network solution, and a solid line indicates original NEPP results. The solutions obtained from both approximate methods lie within one standard deviation of the benchmark solution for each operating point. The simulation improved the maximum thrust by 5 percent. The performance of the linear regression and neural network methods as alternate engine analyzers was found to be satisfactory for the analysis and operation optimization of air-breathing propulsion engines (ref. 4).
Total ozone measurements derived from the TOMS and SBUV backscattered solar UV instrument series cover the period from late 1978 to the present. As the SBUV series of instruments comes to an end, we look to the 10 years of data from the AURA Ozone Monitoring Instrument (OMI) and two years of data from the Ozone Mapping Profiler Suite (OMPS) on board the Suomi National Polar-orbiting Partnership satellite to continue the record. When combining these records to construct a single long-term data set for analysis we must estimate the uncertainty in the record resulting from potential biases and drifts in the individual measurement records. In this study we present a Monte Carlo analysis used to estimate uncertainties in the Merged Ozone Dataset (MOD), constructed from the Version 8.6 SBUV2 series of instruments. We extend this analysis to incorporate OMI and OMPS total ozone data into the record and investigate the impact of multiple overlapping measurements on the estimated error. We also present an updated column ozone trend analysis and compare the size of statistical error (error from variability not explained by our linear regression model) to that from instrument uncertainty.
The author has identified the following significant results. A linear regression between percent nonvegetative land and the time variable was completed for the two sample areas. Sample area no. 1 showed an average vegetation loss of 1.901% per year, while the loss for sample area no. 2 amounted to 5.889% per year. Two basic reasons for the difference were assumed to play a role: the difference in access potential and the amount of already fragmented vegetation complexes in existence during the first year of the comparative analysis - 1970. Sample area no. 2 was located closer to potential access points and was more fragmented initially.
To investigate the ability of column (or partial column) information to represent surface air quality, results of linear regression analyses between surface mixing ratio data and column abundances for O3 and NO2 are presented for the July 2011 Maryland deployment of the DISCOVER-AQ mission. Data collected by the P-3B aircraft, ground-based Pandora spectrometers, Aura/OMI satellite instrument, and simulations for July 2011 from the CMAQ air quality model during this deployment provide a large and varied data set, allowing this problem to be approached from multiple perspectives. O3 columns typically exhibited a statistically significant and high degree of correlation with surface data (R(sup 2) > 0.64) in the P- 3B data set, a moderate degree of correlation (0.16 < R(sup 2) < 0.64) in the CMAQ data set, and a low degree of correlation (R(sup 2) < 0.16) in the Pandora and OMI data sets. NO2 columns typically exhibited a low to moderate degree of correlation with surface data in each data set. The results of linear regression analyses for O3 exhibited smaller errors relative to the observations than NO2 regressions. These results suggest that O3 partial column observations from future satellite instruments with sufficient sensitivity to the lower troposphere can be meaningful for surface air quality analysis.
Landsat MSS data transformed into Kauth-Thomas greenness were averaged over 5 n.mi x 6 n.mi. sample segments from the U.S. Great Plains winter and spring wheat (Triticum aestivum) regions, and related by regression analysis to yields reported by county, crop reporting district (CRD) and state levels. Evidence of a linear relation between winter- and spring-wheat yields and Landsat spectral data at a broad scale is shown for 1978 and 1979. A common slope of about 1.6 (Bu/A)/unit greenness is discerned for the relation between yield and spectral greenness. Tests at both a smaller scale on sets of field-level spectal data and yield and at a large scale on 25 mi. x 25 mi. gridded spectral data from the NOAA-6 AVHRR sensor support the relation. The implications of these results to yield estimation from satellite spectral data are discussed.
The static response of sea level to the forcing of atmospheric pressure, the so-called inverted barometer (IB) effect, is investigated using TOPEX/POSEIDON data. This response, characterized by the rise and fall of sea level to compensate for the change of atmospheric pressure at a rate of -1 cm/mbar, is not associated with any ocean currents and hence is normally treated as an error to be removed from sea level observation. Linear regression and spectral transfer function analyses are applied to sea level and pressure to examine the validity of the IB effect. In regions outside the tropics, the regression coefficient is found to be consistently close to the theoretical value except for the regions of western boundary currents, where the mesoscale variability interferes with the IB effect. The spectral transfer function shows near IB response at periods of 30 degrees is -0.84 +/- 0.29 cm/mbar (1 standard deviation). The deviation from = 1 cm /mbar is shown to be caused primarily by the effect of wind forcing on sea level, based on multivariate linear regression model involving both pressure and wind forcing. The regression coefficient for pressure resulting from the multivariate analysis is -0.96 +/- 0.32 cm/mbar. In the tropics the multivariate analysis fails because sea level in the tropics is primarily responding to remote wind forcing. However, after removing from the data the wind-forced sea level estimated by a dynamic model of the tropical Pacific, the pressure regression coefficient improves from -1.22 +/- 0.69 cm/mbar to -0.99 +/- 0.46 cm/mbar, clearly revealing an IB response. The result of the study suggests that with a proper removal of the effect of wind forcing the IB effect is valid in most of the open ocean at periods longer than 20 days and spatial scales larger than 500 km.
Repeated measurement will be a feature of the survey data collected during the Quesst missionX-59 community response tests (CRT). Since each participant will report his or her categorical level of annoyance in response to multiple events, the responses from any single individual may be correlated with one another. Several models within the class of generalized linear mixed models (GLMM) are pertinent to the analysis of correlated categorical outcomes; the random intercept logistic regression model is one example. Both Bayesian and frequentist methods for fitting these models are available, with frequentist methods relying on some form of approximation (of either an integral or the integrand) that appears in the marginal likelihood function. Given several anticipated similarities of the X-59 CRT data to data collected during a past risk reduction, Quiet Supersonic Flights 2018 (QSF18), this short note is intended to create awareness. It documents an instance in which a reported population average dose-response relationship derived from QSF18 single event data was distorted by the integral approximation applied in likelihood-based methods. We review some of the available literature on the topic, compare the outputs of several different computational approaches implemented in available statistical software, and present simple corrective actions that may be useful during the Quesst mission.
A candidate math model search algorithm was developed at Ames Research Center that determines a recommended math model for the multivariate regression analysis of experimental data. The search algorithm is applicable to classical regression analysis problems as well as wind tunnel strain gage balance calibration analysis applications. The algorithm compares the predictive capability of different regression models using the standard deviation of the PRESS residuals of the responses as a search metric. This search metric is minimized during the search. Singular value decomposition is used during the search to reject math models that lead to a singular solution of the regression analysis problem. Two threshold dependent constraints are also applied. The first constraint rejects math models with insignificant terms. The second constraint rejects math models with near-linear dependencies between terms. The math term hierarchy rule may also be applied as an optional constraint during or after the candidate math model search. The final term selection of the recommended math model depends on the regressor and response values of the data set, the user s function class combination choice, the user s constraint selections, and the result of the search metric minimization. A frequently used regression analysis example from the literature is used to illustrate the application of the search algorithm to experimental data.
Unlimited simultaneous discrimination intervals in linear regressions
The adequacy of the effective perceived noise level (EPNL) procedure for rating helicopter noise annoyance was investigated. Recordings of 89 helicopters and 30 fixed wing aircraft (CTOL) flyover sounds were rated with respect to annoyance by groups of approximately 40 subjects. The average annoyance scores were transformed to annoyance levels defined as the equally annoying sound levels of a fixed reference sound. The sound levels of the test sounds were measured on various scales, with and without corrections for duration, tones, and impulsiveness. On average, the helicopter sounds were judged equally annoying to CTOL sounds when their duration corrected levels are approximately 2 dB higher. Multiple regression analysis indicated that, provided the helicopter/CTOL difference of about 2 dB is taken into account, the particular linear combination of level, duration, and tone corrections inherent in EPNL is close to optimum. The results reveal no general requirement for special EPNL correction terms to penalize helicopter sounds which are particularly impulsive; impulsiveness causes spectral and temporal changes which themselves adequately amplify conventionally measured sound levels.