Simple Regression Estimator Research Articles

A Comparison of Model-Assisted Estimators to Infer Land Cover/Use Class Area Using Satellite Imagery

Remote sensing provides timely, economic, and objective data over a large area and has become the main data source for land cover/use area estimation. However, the classification results cannot be directly used to calculate the area of a given land cover/use type because of classification errors. The main purpose of this study is to explore the performance and stability of several model-assisted estimators in various overall accuracies of classification and sampling fractions. In this study, the confusion matrix calibration direct estimator, confusion matrix calibration inverse estimator, ratio estimator, and simple regression estimator were implemented to infer the areas of several land cover classes using simple random sampling without replacement in two experiments: a case study using simulation data based on RapidEye images and that using actual RapidEye and Huan Jing (HJ)-1A images. In addition, the simple estimator using a simple random sample without replacement was regarded as a basic estimator. The comparison results suggested that the confusion matrix calibration estimators, ratio estimator, and simple regression estimator could provide more accurate and stable estimates than the simple random sampling estimator. In addition, high-quality classification data played a positive role in the estimation, and the confusion matrix inverse estimators were more sensitive to the classification accuracy. In the simulated experiment, the average deviation of a confusion matrix calibration inverse estimator decreased by about 0.195 with the increasing overall accuracy of classification; otherwise, the variation of the other three model-assisted estimators was 0.102. Moreover, the simple regression estimator was slightly superior to the confusion matrix calibration estimators and required fewer sample units under acceptable classification accuracy levels of 70%–90%.

Estimating area from an accuracy assessment error matrix

A map of land cover or land-cover change produced from remotely sensed data is linked to estimation of area of land cover or land-cover change via an accuracy assessment of the map. A variety of area estimators have been proposed based on different approaches to using the estimated error matrix produced from an accuracy assessment along with information available from the map. These estimators include a stratified estimator (where the strata are the map classes), several model-assisted estimators incorporating the map information as auxiliary variables in a variety of different models, and a bias-adjusted estimator that corrects for classification error when area is computed directly from the map. In some cases the same area estimator results from more than one approach. For stratified random sampling with the map classes defining the strata, the model-assisted and bias-adjusted estimators are equivalent to the stratified estimator of area that would typically be used with this sampling design. Thus the commonly used stratified estimator is the lone choice for stratified random sampling. For simple random sampling, the bias-adjusted estimator and a model-assisted difference estimator are equivalent, but other model-assisted options include poststratified (i.e., applying a stratified estimator to data obtained from a simple random sample), ratio, and simple regression estimators. A simulation study demonstrates that for simple random sampling, the poststratified estimator almost always has the smallest variance among these estimators. The only exception to the superior performance of the poststratified estimator occurred when overall accuracy was very high, the true proportion of area was small (i.e., less than 2%), and the accuracy assessment sample size was small (n=100). Because the poststratified estimator for simple random sampling is equivalent to the stratified estimator used with stratified random sampling, the stratified estimator provides a unified, simple approach to area estimation for these two commonly used sampling designs.

Spousal Violence in Southwest Nigeria: Prevalence and Correlates

Spousal violence is increasingly a health issue all over the world, especially in Africa where an unhealthy mix of tradition, inequality and ignorance aggravates the scourge. Despite numerous interventions from human right groups and NGO’s, the problem is still widespread. This study investigates the prevalence of two forms of spousal violencephysical and sexual violence and its correlation among the people of Southwest Nigeria. Data was collected from 300 ever married or cohabiting women through an interview method, and it was processed with SPSS to generate simple percentages and logistic regression estimates. The sample was selected through multistage stratified random sampling technique across all the states of Southwest Nigeria. Spousal violence was measured using a Shorthand and Modified Conflict Tactics Scale. The result indicated spousal violence prevalence rate of 47.3% for ever experience of spousal violence, and 32% for spousal violence prevalence in the 12months preceding the survey. The common forms of physical violence are: kicking/pushing- 31%, slapping-15.5% and arm twisting/throwing things at-14.1%; while the most common form of sexual violence is forced intercourse with 12.7% and 11.5% for ever experience and experience within the 12 months preceding the survey respectively. The experience of spousal violence varied substantially with number of living children, educational levels of women, union status and women’s attitude towards wife beating. The paper concluded that there is a need for massive girl child education and the enlistment of social, political, religious and other leaders in speaking out against spousal violence.

Combining geostatistical models and remotely sensed data to improve tropical tree richness mapping

Information on the spatial distribution and composition of biological communities is essential in designing effective strategies for biodiversity conservation and management. Reliable maps of species richness across the landscape can be useful tools for these purposes. Acquiring such information through traditional survey techniques is costly and logistically difficult. The kriging interpolation method has been widely used as an alternative to predict spatial distributions of species richness, as long as the data are spatially dependent. However, even when this requirement is met, researchers often have few sampled sites in relation to the area to be mapped. Remote sensing provides an inexpensive means to derive complete spatial coverage for large areas and can be extremely useful for estimating biodiversity. The aim of this study was to combine remotely sensed data with kriging estimates (hybrid procedures) to evaluate the possibility of improving the accuracy of tree species richness maps. We did this through the comparison of the predictive performance of three hybrid geostatistical procedures, based on tree species density recorded in 141 sampling quadrats: co-kriging (COK), kriging with external drift (KED), and regression kriging (RK). Reflectance values of spectral bands, computed NDVI and texture measurements of Landsat 7 TM imagery were used as ancillary variables in all methods. The R 2 values of the models increased from 0.35 for ordinary kriging to 0.41 for COK, and from 0.39 for simple regression estimates to 0.52 and 0.53 when using simple KED and RK, respectively. The R 2 values of the models also increased from 0.60 for multiple regression estimates to 0.62 and 0.66 when using multiple KED and RK, respectively. Overall, our results demonstrate that these procedures are capable of greatly improving estimation accuracy, with multivariate RK being clearly superior, because it produces the most accurate predictions, and because of its flexibility in modeling multivariate relationships between tree richness and remotely sensed data. We conclude that this is a valuable tool for guiding future efforts aimed at conservation and management of highly diverse tropical forests.

Consistency and Pythagoras

Pythagorean win share has been one of the fundamental contributions to Sabermetrics. Several hundred articles, both academic and non-academic, have explored variations on Bill James' original formula and its fit to empirical data. This paper considers a variation that is previously unexplored on any systematic level, consistency. After discussing several important contributions to the line of literature, we demonstrate the strong correlation between Pythagorean residuals and several notions of run distribution consistency. Finally, we select the correct form of consistency and use it to construct a simple regression estimator, which improves RMSE by 11%.

An adjusted maximum likelihood method for solving small area estimation problems

For the well-known Fay–Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., a profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate.

Estimating relative rank correlation: Theory and application

In this note we introduce a new novel correlation estimator termed the relative rank correlation (RRC), along with a new descriptive RRC plot. The goal is to estimate the RRC per subject for the purpose of examining “within subject” correlations. The interesting feature of this correlation estimator is that it can be calculated given only a single paired observation per subject. Specific applications of the RRC estimator include model diagnostics, examination of influential observations, graphical exploration and simple description. This approach extends to “within subject” simple linear regression estimation as well.

Pool-based active learning with optimal sampling distribution and its information geometrical interpretation

We propose a pool-based active learning algorithm with approximately optimal sampling distributions. An intuitive understanding of the effectiveness of active learning is also illustrated from the viewpoint of the information geometry. In active learning, one can choose informative input points or input distributions. Appropriate choice of data points is expected in order to make prediction performance more accurate than random data selection. Conventional active learning methods, however, yield serious estimation bias, when parametric statistical models do not include the true probability distribution. To correct the bias, we apply the maximum weighted log-likelihood estimator with approximately optimal input distribution. Optimal input distribution for active learning can be obtained by simple regression estimation. Numerical studies show the effectiveness of the proposed learning algorithm.

Volatility Forecasts for Option Valuations

We document several problems with GARCH type model predictions over the multi-day horizons common to option valuations and value-at-risk models. One, GARCH model forecasts of the return standard deviation - the most common volatility measure and the most appropriate for option valuation and value-at-risk models - are positively biased. Two, the bias is especially severe following high volatility days. Three, in forecasting volatility over longer horizons, the GARCH model puts too little weight on older observations relative to the more recent observations. That is older observations are more important in forecasting volatility next month than in forecasting volatility tomorrow while the GARCH procedure treats them equally at both horizons. We present a simple unbiased regression estimator of the standard deviation of returns which avoids these problems. We find it forecasts better out-of-sample than GARCH, EGARCH, and historical volatility across a wide variety of markets and forecast horizons.

Two simple resistant regression estimators

Two simple resistant regression estimators with O P ( n - 1 / 2 ) convergence rate are presented. Ellipsoidal trimming can be used to trim the cases corresponding to predictor variables x with large Mahalanobis distances, and the forward response plot of the residuals versus the fitted values can be used to detect outliers. The first estimator uses 10 forward response plots corresponding to 10 different trimming proportions, and the final estimator corresponds to the “best” forward response plot. The second estimator is similar to the elemental resampling algorithm, but sets of O ( n ) cases are used instead of randomly selected elemental sets. These two estimators should be regarded as new tools for outlier detection rather than as replacements for existing methods. Outliers should always be examined to see if they follow a pattern, are recording errors, or if they could be explained adequately by an alternative model. Using scatterplot matrices of fitted values and residuals from several resistant estimators is a very useful method for comparing the different estimators and for checking the assumptions of the regression model.

A method to combine non-probability sample data with probability sample data in estimating spatial means of environmental variables.

In estimating spatial means of environmental variables of a region from data collected by convenience or purposive sampling, validity of the results can be ensured by collecting additional data through probability sampling. The precision of the pi estimator that uses the probability sample can be increased by interpolating the values at the nonprobability sample points to the probability sample points, and using these interpolated values as an auxiliary variable in the difference or regression estimator. These estimators are (approximately) unbiased, even when the nonprobability sample is severely biased such as in preferential samples. The gain in precision compared to the pi estimator in combination with Simple Random Sampling is controlled by the correlation between the target variable and interpolated variable. This correlation is determined by the size (density) and spatial coverage of the nonprobability sample, and the spatial continuity of the target variable. In a case study the average ratio of the variances of the simple regression estimator and pi estimator was 0.68 for preferential samples of size 150 with moderate spatial clustering, and 0.80 for preferential samples of similar size with strong spatial clustering. In the latter case the simple regression estimator was substantially more precise than the simple difference estimator.

Design-based regression estimators for spatial means of soil properties: the use of two-phase sampling when the means of the auxiliary variables are unknown

The accuracy of design-based sampling strategies can be increased by using regression models at the estimation stage. The usual regression estimator requires that the means of the auxiliary variables are known. If these means are unknown, then it sometimes pays to estimate these from a large preliminary sample that is subsampled in the second phase. Two types of regression estimator, the simple regression estimator and the ratio estimator are compared with the π* estimator in two case studies. In one study, a simulated, bivariate field (correlation coefficient 0.841) was repeatedly sampled by Simple Random Sampling in both phases. The standard error of the simple regression estimator was 61% to 81% of the standard error of the π* estimator; for the ratio estimator, this percentage varied from 78% to 90%. Given the size of the subsample, the gain in precision depends on the size of the first phase sample and the quality of the model. For a sample size ratio of four, the gain was 70% to 75% of the gain that would be achieved if the mean of the auxiliary variable were known. The gain in precision achieved by the ratio estimator, assuming incorrectly that the intercept is 0, was 50% to 60% of the gain achieved by the regression estimator. Despite the incorrect model assumptions, the ratio estimator was approximately design-unbiased, and the confidence intervals were valid. For small sample sizes in the second phase, the ratio estimator performs even slightly better than the regression estimator with respect to validity. This stresses the importance of using simple models. In a second study, the estimated spatial means of the Mean Highest Water table and Mean Lowest Water table of six map units were estimated by Stratified Simple Random Sampling in both phases in combination with the regression estimator that uses the starting depth of hydromorphic properties in the soil profile as an auxiliary variable. The gain in precision did not outweigh the extra costs of collecting the auxiliary data.

Optimal estimation of suspended‐sediment concentrations in streams

AbstractOptimal estimators are developed for computation of suspended‐sediment concentrations in streams. The estimators are a function of parameters, computed by use of generalized least squares, which simultaneously account for effects of streamflow, seasonal variations in average sediment concentrations, a dynamic error component, and the uncertainty in concentration measurements. The parameters are used in a Kalman filter for on‐line estimation and an associated smoother for off‐line estimation of suspended‐sediment concentrations. The accuracies of the optimal estimators are compared with alternative time‐averaging interpolators and flow‐weighting regression estimators by use of long‐term daily‐mean suspended‐sediment concentration and streamflow data from 10 sites within the United States. For sampling intervals from 3 to 48 days, the standard errors of on‐line and off‐line optimal estimators ranged from 52·7 to 107%, and from 39·5 to 93·0%, respectively. The corresponding standard errors of linear and cubic‐spline interpolators ranged from 48·8 to 158%, and from 50·6 to 176%, respectively. The standard errors of simple and multiple regression estimators, which did not vary with the sampling interval, were 124 and 105%, respectively. Thus, the optimal off‐line estimator (Kalman smoother) had the lowest error characteristics of those evaluated. Because suspended‐sediment concentrations are typically measured at less than 3‐day intervals, use of optimal estimators will likely result in significant improvements in the accuracy of continuous suspended‐sediment concentration records. Additional research on the integration of direct suspended‐sediment concentration measurements and optimal estimators applied at hourly or shorter intervals is needed. Published in 2001 by John Wiley & Sons, Ltd.

Evaluation of sampling methods to quantify discarded fish using data collected during discards project EC 95/094 by Northern Ireland, England and Spain

Two sampling methods, equal probability and probability proportional to x (ppx), were investigated in order to recommend an optimum sampling strategy for carrying out a discards survey in the Irish Sea. The objective of the survey, which took place from April 1997 to August 1998, was to estimate the quantity of fish discarded at fleet level by Northern Ireland trawlers. Additional data collected from English and Spanish trawlers were used to test the generality of the results. Simple random, regression, ratio and ppx estimators were evaluated. The regression, ratio and ppx estimators use additional information on a variable x in order to improve the precision of the estimate of the variable of interest y and offer a potential improvement over the simple random estimator if there is a significant correlation between x and y. In order to compare the various estimators, 1000 independent data sets were generated using a bootstrapping technique for both equal probability and ppx with replacement selection. When comparing the estimators three criteria were examined: (i) bias, if any, of the estimator; (ii) relative precision of the estimators; (iii) accuracy of the formula for the variance of the estimator. Linear relationships between potential x– y combinations were assessed for significance and the residuals about the fitted line examined. This was carried out for each country and gear type for the main commercial species individually and all species combined. Different estimators proved optimal according to the x– y combination examined. In some instances the improvement in precision when using the ppx sampling method was marginal over the equal probability method. For the data sets used in this study the ppx scheme offered insufficient advantage over the simpler equal probability method to justify the greater complexity in implementing ppx.

Using regression models in design‐based estimation of spatial means of soil properties

SummaryThe precision of design‐based sampling strategies can be increased by using regression models at the estimation stage. A general regression estimator is given that can be used for a wide variety of models and any well‐defined sampling design. It equals the π estimator plus an adjustment term that accounts for the differences between the π estimators for the spatial means of the auxiliary variables and the true spatial means of these variables. The regression estimator and ratio estimator follow from certain assumptions on the model and the sampling design. These are compared with the π estimator in two case studies.In one study a bivariate field of linearly related variables was simulated and repeatedly sampled by Simple Random Sampling without replacement and sample sizes 10, 25, 50, 100 and 200. For all sample sizes the ratio of the standard error of the simple regression estimator to that of the π estimator was approximately 55%. The bias of the simple regression estimator was negligibly small. The confidence interval estimators were valid for all sample sizes except for n = 10. Also the ratio estimator was approximately unbiased, and the confidence interval estimators were valid for all sample sizes, even for n = 10. This is remarkable because the ratio estimator assumes that the intercept of the regression line is 0 which was incorrect for the simulated field. On the other hand, only approximately 55% of the potential gain was achieved because the model was inappropriate.In a second study the spatial means of the Mean Highest Watertable of map units were estimated by Stratified Simple Random Sampling and the combined (multiple) regression estimator. The NAP elevation, the local elevation, the Easting and the Northing were used as auxiliary variables. For all map units except one the combined (multiple) regression estimator was more precise than the π estimator. The ratio of the standard errors varied from 0.36 to 1.04. The domain for which the regression estimator was less precise than the π estimator showed strong variation between strata. For this domain it was more efficient to group the strata into two groups and to fit simple models for these groups separately.

Theory & Methods: Calibration of the estimators of variance

This investigation suggests new techniques to calibrate estimators of variance. Estimators of the variance of simple mean, ratio and regression estimators under different sampling schemes are shown to be special cases of the proposed calibration techniques. The approach has more practical use due to recent advances in programming techniques and computational speed. An empirical study has been carried out to address the properties of these proposed strategies.

On estimation of parameters in the bivariate linear errors-in-variables model

We discuss some methods of estimation in bivariate errors-in- variables linear models. We also suggest a method of constructing consistent estimators in the case when the error disturbances have the normal distri- bution with unknown parameters. It is based on the theory of estimating variance components in linear models. A simulation study is presented which compares this estimator with the maximum likelihood one. 1. Introduction. Simple linear regression models describe linear func- tional relationships when there is an observation error in only the depen- dent variable Y. The X (independent variable) is assumed to be mea- sured precisely. The parameters in this model are estimated by the clas- sical ordinary least squares (OLS) method, which gives unbiased estima- tors. In situations where both variables are subject to error the errors-in- variables models are applied. We assume that the true linear relationship is given by y = as+b. The actual observed values are X = s+ and Y = y+ , where the symbols and denote the corresponding error disturbances. The measurement error in X sometimes happens to be overlooked and the OLS estimation of the parameters is chosen because of its familiarity and ease of use. It is well known that in the errors-in-variables model, the simple regression estimator (OLS) is inconsistent. Cochran (1968) has given a general discussion of the consequences of using the OLS estimator in errors-in-variables models. However, there exist cases when this estimator has mean squared error smaller than other esti-

Best linear unbiased estimators for the simple linear regression model using ranked set sampling

When sample observations are expensive or difficult to obtain, ranked set sampling is known to be an efficient method for estimating the population mean, and in particular to improve on the sample mean estimator. Using best linear unbiased estimators, this paper considers the simple linear regression model with replicated observations. Use of a form of ranked set sampling is shown to be markedly more efficient for normal data when compared with the traditional simple linear regression estimators.

Multiple regression for molecular-marker, quantitative trait data from large F2 populations

Molecular marker-quantitative trait associations are important for breeders to recognize and understand to allow application in selection. This work was done to provide simple, intuitive explanations of trait-marker regression for large samples from an F2 and to examine the properties of the regression estimators. Beginning with a(- 1,0,1) coding of marker classes and expected frequencies in the F2, expected values, variances, and covariances of marker variables were calculated. Simple linear regression and regression of trait values on two markers were computed. The sum of coefficient estimates for the flanking-marker regression is asymptotically unbiased for an included additive effect with complete interference, and is only slightly biased with no interference and moderately close (15 cM) marker spacing. The variance of the sum of regression coefficients is much more stable for small recombination distances than variances of individual coefficients. Multiple regression of trait variables on coded marker variables can be interpreted as the product of the inverse of the marker correlation matrix R and the vector a of simple linear regression estimators for each marker. For no interference, elements of the correlation matrix R can be written as products of correlations between adjacent markers. The inverse of R is displayed and used to illustrate the solution vector. Only markers immediately flanking trait loci are expected to have non-zero values and, with at least two marker loci between each trait locus, the solution vector is expected to be the sum of solutions for each trait locus. Results of this work should allow breeders to test for intervals in which trait loci are located and to better interpret results of the trait-marker regression.

Using temperature profile for product quality estimation on a distillation column

A composition estimator based on eight temperature measurements has been implemented on a deethanizer column at the Ethylene plant at Statoil in Stenungsund, Sweden. The column separates a mixture of four components. The estimator is a simple static regression estimator and the regression method is the Partial Least Square Regression, which is extensively employed in chemometrics. The estimator was first tested off-line with very good results, and then also on-line in closed loop for composition control, where it now is implemented. Very accurate estimates have been obtained using this estimator. The need for updating seems limited, and the method should be applicable on a wide range of distillation columns provided the temperature sensors are well distributed and the number is not too small.