Second-degree Models Research Articles

Variation of Grain Height Characteristics of Electroplated cBN Grinding-Wheel Active Surfaces Associated with Their Wear

During the operation of a single-layer grinding wheel (SLGW), irreversible changes occur on its active surface due to wear. The study of grinding-wheel microgeometry changes can be based on the measurement of the surface texture as well as the determination and analysis of its parameters. The article deals with the selection of suitable texture parameters and an appropriate mathematical model carrying information about the SLGW condition. In the study, samples of Pyrowear 53 steel were ground using electroplated cBN single-layer grinding wheels until they were completely worn out or removed assumed volume of the workpiece material. Each SLGW worked with constant process parameters. Among the 144 parameters tested, the highest sensitivity to changes in wheel active surfaces caused by wear was shown by the mean value of the mean island heights Zmean_m. In-depth research was conducted for Zmean_m and reduced peak height Spk. Compared to Spk, Zmean_m has proven to be a better measure of wear, especially when large areas of sticking occur. Moreover, the second-degree models linking Zmean_m and Spk to the process parameters and the specific material loss were better suited to the empirical data than the exponential models.

Ionospheric modeling for short‐ and long‐term predictions of F region parameters over Indian zone

The equatorial and low latitudinal F region ionosphere is highly dynamic and unpredictable because of various geophysical mechanisms operating therein. In the present study, two HF prediction models for short and long term are developed for equatorial and low‐latitude F region ionosphere. In the first approach, multiple regression analysis (MRA) for the dependence of F region parameters, namely, foF2 and M(3000)F2, on solar 2800 MHz flux (F10) and geomagnetic index Ap are generated, and in the second one, second‐degree (SD) coefficients are generated, both by fitting monthly median foF2 and M(3000)F2 with corresponding 12 monthly mean sunspot numbers (R12) using data over three solar cycles. For MRA, daily foF2 and M(3000)F2 values for each hour obtained from Delhi (28.6°N, 77.1°E) digital ionosonde for about half a solar cycle are used. MRA coefficients for foF2 and M(3000)F2 are obtained for every month over 2400 UT times using daily F10 and Ap values separately for quiet (Ap < 25) and disturbed (Ap > 25) periods. Similarly, SD coefficients are obtained each month at all local times for all the 14 stations covering a geographic latitude range from about 0°N to 45°N. In this way, once appropriate coefficients for each hour for all the 12 months are obtained, they are used by the computer‐based MRA and SD models to predict ionospheric hourly foF2 and hmF2 values for given inputs such as month, F10, Ap, and R12, as the case may be. Predicted model values of foF2 and hmF2, calculated on short‐ and long‐term basis, are then compared with the observed data over Delhi and also with those obtained using international reference ionosphere (IRI)‐2001 model. From our comparative studies it is observed that MRA and SD models show better agreement with observations compared to the IRI model for both long‐ and short‐term basis and among the two the MRA model provides best agreement with the observed ones, even during the magnetic storm periods. The SD model, on the other hand, which is based on monthly median values, is useful for providing long‐term predictions for HF communication applications.

Estimation and Prediction for Low Degree Polynomial Models Under Measurement Errors with an Application to Forest Harvesters

Summary In a modern computer-based forest harvester, tree stems are run in sequence through the measuring equipment root end first, and simultaneously the length and diameter are stored in a computer. These measurements may be utilized for example in the determination of the optimal cutting points of the stems. However, a problem that is often passed over is that these variables are usually measured with error. We consider estimation and prediction of stem curves when the length and diameter measurements are subject to errors. It is shown that only in the simplest case of a first-order model can the estimation be carried out unbiasedly by using standard least squares procedures. However, both the first- and the second-degree models are unbiased in prediction. Also a study on real stem is used to illustrate the models that are discussed.

All-Bias Designs for Spline Functions Joined at the Axes

Abstract Often, a sensible approach to a general design problem is to slightly expand the appropriate all-bias designs. This idea is applied to situations involving polynomial spline models in which it is necessary to (i) guard against second-degree models in some part, or all, of a cuboidal region of interest, while fitting first-degree models; (ii) guard against third-degree models, while fitting second-degree models. All joins are made on the axes. For (i), suitable designs include 2 IV k—p factorial designs in each 2 k -tant. For (ii), consideration of symmetrical designs leads to cuboidal designs of the type previously considered in Draper and Lawrence (1965).

All-Bias Designs for Spline Functions Joined at the Axes

Abstract Often, a sensible approach to a general design problem is to slightly expand the appropriate all-bias designs. This idea is applied to situations involving polynomial spline models in which it is necessary to (i) guard against second-degree models in some part, or all, of a cuboidal region of interest, while fitting first-degree models; (ii) guard against third-degree models, while fitting second-degree models. All joins are made on the axes. For (i), suitable designs include 2 IV k—p factorial designs in each 2 k -tant. For (ii), consideration of symmetrical designs leads to cuboidal designs of the type previously considered in Draper and Lawrence (1965).

Some optimal designs for interpolation and extrapolation

Designs are given for minimizing the effect of random error in interpolation or extrapolation. The case most intensively studied is that for first-degree polynomial models in k factors, although some results are given also for second-degree models. Emphasis is put on designs giving minimum variance at a particular point and satisfying additional requirements such as leading to good estimates of parameters.