Logarithmic scale

Abstract: Regression formulas relating dispersivity and field scale have been suggested in the literature by few researchers, among others: Arya (1986) and Neuman (1990). These studies employ conventional statistical techniques that cannot take into account the relative reliability level of a data point. Neuman (1990), using conventional statistics, discarded the low reliability data obtained by calibrating numerical grid models against solute concentrations of large-scale plumes. Although the numerical modeling data are of low reliability, they are important to consider because of the very limited overall availability of the data. Another problem is with the slope reduction of log{sub 10} {alpha}{sub L} versus log{sub 10} L for large values of L, where L is the field scale, and {alpha}{sub L} is the longitudinal dispersivity. Neuman (1990) used a different line of best fit with a reduced slope for L {ge} 100 m. Fitting two straight lines on a logarithmic scale, one for L {le} 100 m and another one for L {ge} 100 m, takes care of the slope change to some extent, but results in loss of data points for both statistical groups. In order to provide a better statistical model using the available data with varying reliabilities, a weighted least-squaresmore » data fitting method was used to define a relationship between dispersivity and field scale. By using this technique, less reliable data points were not simply ignored, but rather assigned a lower weight. The linear model is further improved by fitting a curve through the data drawn on a log{sub 10} {alpha}{sub L} versus log{sub 10} L diagram. The analysis shows a significant difference between the weighted and the unweighted curves. Further analysis indicated that the increase in longitudinal dispersivity is practically negligible when the scale of flow exceeds 1 km.« less

Abstract: Quantal biological assays are traditionally performed on grouped data. However, grouping disguises the fundamental all-or-none nature of the responses, is not necessary and may not be possible. The statistical method relevant to quantal assays is reconsidered from the viewpoint of the individual responses and with a view to getting the apparatus for their explicit analysis. Specifically, the logistic function is used as a model for the distribution of the probability of responding as a function of dose or concentration. The parameters of the logistic function are estimated by the method of maximum likelihood. Since the function is nonlinear in both the scale and location parameters, solution of the normal equations is achieved by an iterative technique base on a Taylor series expansion. Observations are not grouped so the method is applicable to cases in which several observations are not available at the same dose level as well as to grouped data. Three variants on the basic model are considered: the classical model in which the parameters estimated are the intercept and slope of the associated linear regression and two models in which the parameter of prime pharmacological interest—the ED50—is estimated directly. In one of the latter the ED5O is considered to be symmetrically distributed on a logarithmic scale, in the other on a linear scale. Similarly, when two curves are compared, potency ratios or their logarithms are estimated directly rather than indirectly as the difference or ratio of two variables. One advantage of such direct estimation is that the error valiance can then be obtained directly from the covariance matric obtained during the solution of the normal equations. Numerical examples are given and illustrative computer programs are given in an appendix.

Abstract: The authors considers "scale" a physical attribute of a signal and develop its properties. He presents an operator which represents scale and study its characteristics and representation. This allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal. He obtains explicit expressions for the mean scale, scale bandwidth, instantaneous scale, and scale group delay. Furthermore, he derives expressions for mean time, mean frequency, duration, frequency bandwidth in terms of the scale variable. The short-time transform is defined and used to obtain the conditional value of scale for a given time. He shows that as the windows narrows one obtains instantaneous scale. Convolution and correlation theorems for scale are derived. A formulation is devised for studying linear scale-invariant systems. He derives joint representations of time-scale and frequency-scale, General classes for each are presented using the same methodology as for the time-frequency case. As special cases the joint distributions of Marinovich-Altes (1978, 1986) and Bertrand-Bertrand (1984) are recovered. Also, joint representations of the three quantities, time-frequency-scale are devised. A general expression for the local scale autocorrelation function is given. Uncertainty principles for scale and time and scale and frequency are derived. >

Abstract: Whereas most sensory information is coded on a logarithmic scale, linear expansion of a limited range may provide a more efficient coding for the angular variables important to precise motor control. In four experiments, we show that the perceived declination of gaze, like the perceived orientation of surfaces, is coded on a distorted scale. The distortion seems to arise from a nearly linear expansion of the angular range close to horizontal/straight ahead and is evident in explicit verbal and nonverbal measures (Experiments 1 and 2), as well as in implicit measures of perceived gaze direction (Experiment 4). The theory is advanced that this scale expansion (by a factor of about 1.5) may serve a functional goal of coding efficiency for angular perceptual variables. The scale expansion of perceived gaze declination is accompanied by a corresponding expansion of perceived optical slants in the same range (Experiments 3 and 4). These dual distortions can account for the explicit misperception of distance typically obtained by direct report and exocentric matching, while allowing for accurate spatial action to be understood as the result of calibration.