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Taylor’s power law (TPL) describes the variance of population abundance as a power-law function of the mean abundance for a single or a group of species. Using consistently sampled long-term (1958–2001) multimesh capture data of Lake Kariba in Africa, we showed that TPL robustly described the relationship between the temporal mean and the temporal variance of the captured fish assemblage abundance (regardless of species), separately when abundance was measured by numbers of individuals and by aggregate weight. The strong correlation between the mean of abundance and the variance of abundance was not altered after adding other abiotic or biotic variables into the TPL model. We analytically connected the parameters of TPL when abundance was measured separately by the aggregate weight and by the aggregate number, using a weight–number scaling relationship. We utilized TPL to find the number of samples required for fixed-precision sampling and compared the number of samples when sampling was performed with a single gillnet mesh size and with multiple mesh sizes. These results facilitate optimizing the sampling design to estimate fish assemblage abundance with specified precision, as needed in stock management and conservation.

Background: Taylor's law (TL) states a linear relationship on logarithmic scales between the variance and the mean of a nonnegative quantity. TL has been observed in spatiotemporal contexts for the population density of hundreds of species including humans. TL also describes temporal variation in human mortality in developed countries, but no explanation has been proposed. Objective: To understand why and to what extent TL describes temporal variation in human mortality, we examine whether the mortality models of Gompertz, Makeham, and Siler are consistent with TL. We also examine how strongly TL differs between observed and modeled mortality, between women and men, and among countries. Methods: We analyze how well each mortality model explains TL fitted to observed occurrence-exposure death rates by comparing three features: the log-log linearity of the temporal variance as a function of the temporal mean, the age profile, and the slope of TL. We support some empirical findings from the Human Mortality Database with mathematical proofs. Results: TL describes modeled mortality better than observed mortality and describes Gompertz mortality best. The age profile of TL is closest between observed and Siler mortality. The slope of TL is closest between observed and Makeham mortality. The Gompertz model predicts TL with a slope of exactly 2 if the modal age at death increases linearly with time and the parameter that specifies the growth rate of mortality with age is constant in time. Observed mortality obeys TL with a slope generally less than 2. An explanation is that, when the parameters of the Gompertz model are estimated from observed mortality year by year, both the modal age at death and the growth rate of mortality with age change over time. Conclusions: TL describes human mortality well in developed countries because their mortality schedules are approximated well by classical mortality models, which we have shown to obey TL. Contribution: We provide the first theoretical linkage between three classical demographic models of mortality and TL

Organizational process improvement often emphasizes reductions in variation; mean performance may remain unchanged if the effect is primarily a reduction in extreme values (high quantiles). Such data may be best analyzed by modeling the temporal regression relationship at the 0.95th quantile or 95th percentile (cases that exceed 95 percent of the distribution) controlling for the independent variables. A conventional linear regression analysis assumes the relationship observed for the mean holds at the different quantiles of the response variable. The estimated temporal mean trend basically determines the shape of temporal trends at individual quantiles. The resulting trend estimates cannot be necessarily true or close to the true temporal trend at a particular quantile of interest. Least-squares analysis is not valid when addressing a performance improvement goal stated as “95 percent of patients have less than 10 days between first contact and first treatment.” Quantile regression, on the other hand, estimates the 95th percentile directly and allows us to investigate the question without constraining the trend at 95 percent to be of a certain shape. System-level studies of the performance measure “wait time to treatment” are rare in the U.S. health care system despite the importance of this measure. The need for systematically monitored and analyzed process measures is particularly pressing within the substance abuse treatment field (Garnick et al. 2002). Patients seeking treatment for alcohol and drug use disorders often encounter waiting lists and experience delays between the day they request care and the day they enter care. The delays reflect a fixed capacity for new patients and limited financing for expanding access to care (Hoffman and McCarty 2012). The organization and delivery of care, however, can also inhibit access to care and discourage patients from attending addiction treatment appointments (Stark, Campbell, and Brinkerhoff 1990; Hser et al. 1998; Ford et al. 2007). NIATx was the first widespread application of process improvement techniques to substance abuse treatment (http://www.niatx.net) (Capoccia et al. 2007). Participating agencies were trained to use a simplified version of the Institute for Healthcare Improvement's hospital improvement support system (http://www.ihi.org). A cross-site evaluation of NIATx participants examined: (1) change in days between first contact and first treatment and (2) the percentage of clients that began treatment and completed the first four units of care. Thirteen agencies that began participation in NIATx in August 2003 reduced days to treatment 37 percent—from 19.6 to 12.4 days across all levels of care. Retention in care also improved; the proportion of clients who completed a first session of care and returned for a second and third session of care increased 18 percent between the first and second session (72–85 percent) and 17 percent between the first and third session of care (62–73 percent) (McCarty et al. 2007). A subsequent analysis of 14 outpatient and intensive outpatient treatment programs within the second NIATx cohort replicated the reduction in wait time and the improvement in retention and noted that the first cohort sustained the gains during a 20-month follow-up (Hoffman et al. 2008). Previous analyses of wait time to treatment within the Network for the Improvement of Addiction Treatment (NIATx) employed a conventional least-squares regression analysis and estimated rates of change in the mean of the monthly averaged outcome variables (McCarty et al. 2007; Hoffman et al. 2008). Analysis of mean function is useful, but other aspects of the response distribution can also be of interest. Patient wait times may have unequal variation due to complex interactions between variables or unobserved exogenous noises that are not accounted for in the regression model. Figure 1 shows distributions of patient wait time in a NIATx program. The distribution of all wait times across the 15-month study period appears in Figure 1a. Figure 1b presents wait time distributions for month 1 (first), month 8 (mid), and month 15 (last) and illustrates change in wait time distributions over time. This variation in the distributions implies that more than one single slope describes the temporal trends in wait time from the first contact to the first treatment. In the presence of such heterogeneity, conventional least-squares regression models may underestimate, overestimate, or fail to detect important changes occurring locally at a certain quantile of data, because it focuses on changes in the means (Terrell et al. 1996; Cade, Terrell, and Schroeder 1999). Figure 2 provides expository guidelines where quantile regression is desirable over the conventional least-squares regression analysis. Figure 1 The Distributions of Wait Time to Treatment in Program 8. (A) Density Plot of All Wait Time. (B) Density Plot of Wait Time in the First (n = 39), Mid (n = 49) and Last Months (n = 33) Figure 2 Guidelines How to Choose between Least-Squares Regression and Quantile Regression Quantile regression (Koenker and Bassett 1978) extends the concept of percentile or quantile in the univariate analysis to regression and estimates regression relationships specific to a certain percentile of the response variable controlling for the independent variables. In its simplest application, quantile regression examines median regression and describes changes in the center of the distribution. As the median is robust to outliers in the univariate analysis, median regression is also robust to outliers. Median, however, is just one of the many quantiles that can be examined. A series of quantile regression for different percentiles estimates multiple rates of change, providing a more robust and detailed description of the regression relationships that can be overlooked by other regression methods. Quantile regression is well established and available in common statistical software (e.g., proc quantreg procedure in SAS; quantreg package in R). Applications include economics (Hendricks and Koenker 1992; Buchinsky 1994; Manning, Blunmberg, and Moutlton 1995; Poterba and Rueben 1995); biology, especially for growth curve and duration models (Cade, Terrell, and Schroeder 1999; Koenker and Geling 2001); medicine, especially for reference charts (Cole 1988; Cole and Green 1992; Wei et al. 2006); environmental modeling (Pandey and Nguyen 1999); and infrastructure studies (He, Simpson, and Wang 2000). More examples are available in Koenker (2005).

(ProQuest: ... denotes formulae omitted.)1. IntroductionTaylor (1961) established a power law that describes a pattern in ecology regarding the (spatial or temporal) variability of populations. For many species, it describes a linear relationship between the logarithms of the variance (Var) of population size or density (P) and its mean (E):... (1)Although the interpretation of the two parameters a and b is controversial, Taylor (1961: 735) calls the constant a a (less relevant) computing factor and suggests that the slope b is a species-specific index of aggregation. Kilpatrick and Ives (2003) report that many empirical analyses identify values between 1 and 2 for the slope b, due to environmental and demographic stochasticity as well as competitive interactions between species. Kendal (2004b) gives a detailed overview of the history of TL. He shows that TL has mostly been found for population densities in ecology, but that power laws have also been identified in other contexts, such as outbreaks of infectious diseases (Anderson and May 1988; Rhodes and Anderson 1996) and for physical distributions of gene structures within chromosomes (Kendal 2004a). In these non-ecological realizations of TL, Taylor's interpretation of the exponent is obviously not viable.In human demography, TL has been applied by Cohen, Xu, and Brunborg (2013), who verify a log-linear variance to mean relationship for Norway's population (disaggregated in 19 counties) from 1978 to 2010; they suggest using TL as an evaluation criterion for population forecasts, i.e., to determine if such a linear relationship can be found in both observed population data and in forecasts.Vaupel, Zhang, and van Raalte (2011) analyzed variation in the age at death by e[dagger], which is the average number of life-years lost in a population (Vaupel and Canudas Romo 2003). They showed that populations with very low variation typically also had the lowest mean, as measured by life expectancy at birth. In this article, we quantify further the relationship between the variation and the mean of human mortality by testing the application of TL to human death rates and to rates of improvement in mortality: We compute variances and means of human mortality (change) for single ages over time for different countries to analyze their temporal variability across ages and across countries.1.1 Hypothesis 1In our so-called cross-age-scenarios, we analyze temporal variance to temporal mean relationships of mortality in multiple countries on the logarithmic scale across ages. We hypothesize that these relationships are almost log-linear for all ages in each country. Such a finding might indicate that TL could be used to evaluate mortality forecasts and to justify linear assumptions in mortality forecasts on a logarithmic scale.If TL were confirmed in observed mortality data, then TL could be tested in mortality forecasts, and the results of this consistency test would be available immediately after generating a forecast. In contrast, forecast errors can typically only be computed after the mortality of forecast years occurs. Of course, in the long run, the empirical usefulness of using TL as a consistency test would have to be evaluated post hoc.Justifying linear assumptions in mortality forecasting with TL would be beneficial, since many models (Lee and Carter 1992; Renshaw and Haberman 2003, 2006) rely on linear predictors for death rates on a logarithmic scale. Tuljapurkar, Li, and Boe (2000) use the observed long-term linear mortality decline in the G7 countries on the logarithmic scale to justify the modeling of log-linear mortality forecasts. If we found support for our hypothesis of (almost) linear variance to mean relationships for death rates and their rates of change, TL might also justify linear assumptions for forecasting models relying on the change of mortality (Mitchell et al. 2013; Haberman and Renshaw 2012; Bohk and Rau 2014). …

Different procedures underlying the determination of the mean of a set of values impart different weights to these values. Thus, different methods result in different weighted means. In particular, when the weights are directly proportional, or in fact equal, to the (weighted) values, the self-weighted mean is obtained, and when they are inversely proportional to the values, the harmonic mean results. Generally, the former is greater and the latter is smaller than the arithmetic (uniformly weighted) mean. We illustrate cases of sampling procedures and of geometric requirements that give rise to the self-weighted and harmonic means. We call attention to some psychological difficulties and to the didactic value of dealing with cases of differential weighting.

An algorithm for computing the temporal mean of a nonlinear function using the temporal means, covariances and higher order statistical moments of the variables involved in the function is revisited. Furthermore, a pyramidal algorithm is derived, which hierarchically stores the statistical moments of a longer interval of a variable from those of its shorter subintervals. The 2 methodologies together presented here show a systematic way of data storage and show that the long-term mean of a nonlinear process can be analyzed by decomposing it into various shorter sub- time scales such as diurnal and seasonal cycles. For example, the long-term mean of horizontal mois- ture flux can be decomposed into the product of the means of wind speed and humidity observations, plus the covariance of daily means of the 2 variables, and plus the mean of the daily covariances of the 2 variables on each day, where the 3 mean values and the covariance are suggested for storage. The results are exactly the same as those directly calculated from their hourly data. Since only 4 statistical moments are needed, significant data reduction for data distribution can be achieved. The error associated with the data archive replica has been discussed for a highly nonlinear function of which the statistical moments of its variables are only available up to a finite order. A case study using 41 yr of data taken on an urbanizing site on a subtropical island is illustrated.

The confidence limit is a standard measure of the accuracy of the result in any statistical analysis. Most of the confidence limits are derived as follows. The data are first divided into subsections and then, under the ergodic assumption, the temporal mean is substituted for the ensemble mean. Next, the confidence limit is defined as a range of standard deviations from this mean. However, such a confidence limit is valid only for linear and stationary processes. Furthermore, in order for the ergodic assumption to be valid, the subsections have to be statistically independent. For non‐stationary and nonlinear processes, such an analysis is no longer valid. The confidence limit of the method here termed EMD/HSA (for empirical mode decomposition/Hilbert spectral analysis) is introduced by using various adjustable stopping criteria in the sifting processes of the EMD step to generate a sample set of intrinsic mode functions (IMFs). The EMD technique acts as a pre‐processor for HSA on the original data, producing a set of components (IMFs) from the original data that equal the original data when added back together. Each IMF represents a scale in the data, from smallest to largest. The ensemble mean and standard deviation of the IMF sample sets obtained with different stopping criteria are calculated, and these form a simple random sample set. The confidence limit for EMD/HSA is then defined as a range of standard deviations from the ensemble mean. Without evoking the ergodic assumption, subdivision of the data stream into short sections is unnecessary; hence, the results and the confidence limit retain the full‐frequency resolution of the full dataset. This new confidence limit can be applied to the analysis of nonlinear and non‐stationary processes by these new techniques. Data from length‐of‐day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied. By providing a confidence limit for this new approach, a stable range of stopping criteria for the decomposition or sifting phase (EMD) has been established, making the results of the final processing with HSA, and the entire EMD/HSA method, more definitive.

Two measures of tracking performance for the LMS (least mean square) algorithm are compared and contrasted. These are the conventional time-average or temporal mean of the nonstationary mean-squared error (MSE) in excess of the minimum attainable MSE, and the novel temporal root-mean-squared value of the excess MSE, which takes into account the temporal variance as well as the temporal mean of the nonstationary MSE. These measures are evaluated for the LMS algorithm applied to two time-variant system identification problems, one involving a random Markov system and the other a periodic system. Optimal step-size parameters and minimum misadjustments are evaluated. It is shown that the conventional time-average performance measure is adequate only when the degree of nonstationarity is sufficiently low. For higher degrees of nonstationarity, the time-average performance measure can be misleading in studies of the tracking behavior of the LMS algorithm. >

An important aspect in road safety research concerns the development of analytical tools to identify road sites with high risk. Within a context of optimization subject to financial constraints, decisions have to be taken as to which sites should be considered for treatment or safety improvement. The most economically reasonable selection criterion is to select those sites which had the highest accident rate in the preceding year. This is a bad procedure because of the well known regression to the mean problem. Even if no remedial treatment is made, the number of accidents recorded at the same site in the following year will naturally decrease toward its temporal mean. In other word, very high accident rates should be viewed as outliers.

There are several measurements of central tendency for a distribution of values. The most frequently used is the “average,” often called the mean , but more accurately described as the arithmetic mean . This is calculated as the sum of all values divided by the total number of values. But many data sets are skewed, with a disproportionate …

The three-dimensional structure of temporal mean, variance and covariance fields of the Northern Hemisphere wintertime circulation is documented by making use of twice-daily hemispheric analyses at 10 pressure levels (1000–100 mb) for 11 winters (1965–76). The Northern Hemisphere is partitioned into seven longitudinal sectors which correspond, respectively, to 1) the entrance regions of the two principal jet streams; 2) the two major oceanic storm tracks (areas with pronounced geopotential height variability in the 2.5–6 day period range); 3) the western portions of the North American and Eurasian land masses; and 4) the central part of the Asian continent. Meridional cross sections of various seasonal mean statistics averaged over meridians comprising these individual sectors are displayed. The fields presented in this manner include 1) time-averaged zonal wind and temperature; 2) temporal variance statistics of zonal and meridional wind components, temperature, and geopotential height; 3) merid...

(1976). The Interpretation of R 2 in Autoregressive-Moving Average Time Series Models. The American Statistician: Vol. 30, No. 4, pp. 175-180.

The positive probability that an estimated moving average process is noninvertible is studied for maximum likelihood estimation of a university process. Upper and lower bounds for the probability in the first-order case are obtained as well as limits when the sample size tends to infinity. Higher order moving average models and autoregressive moving average models are also treated.

Four estimates of the current process average are compared in the situation when measurements are made at equal intervals of time and two types of variance components, σ2 2, the variance of a random-shift, cumulative component, and σ2 1, a transitory measurement component, are present. The four estimators are the current measurement, the m.v.u.e. (minimum variance unbiased estimate), the geometric moving average and the arithmetic moving average. The coefficients of the m.v.u.e. depend on c = σ2 2/σ2 1 and n, the number of periods of time at which measurements have been made. The inefficiencies caused by using the geometric moving average when c is small and n is small and by using the arithmetic moving average when c is large and n is large are studied and emphasized because of their practical importance.

Reference is first made to the fact that the “mean age” derived from the arithmetic mean of assurance functions for the same term (e.g. annuity values, premiums) is always higher than the arithmetic mean of the agesx. The “mean term” derived from the arithmetic mean of assurance functions for the same age at entry, on the other hand, is always smaller than the arithmetic mean of the termsn. In the case of premiums and assurance values in particular, the harmonic mean of then’s comes very close to the “mean term” derived from the arithmetic mean of the respective functions. A valuation method involving grouping, based on mean age and mean unexpired term, can never be fully satisfactory, since the determination of the mean age requires equal terms, whereas the determination of the mean unexpired term requires equal ages. These two requirements are not compatible, apart from the trivial case of identical assurance functions.

Events overtake the condition of the landscape, and change the reactions of consumers to that landscape. Values calculated by the expert as valid for today will not always be a correct assessment.

For the analysis of temporal cross-section situations random coefficient regression models have been proposed in the literature. In such an RCR-model with homoscedastic disturbances several estimation procedures for the mean value parameter are discussed.

Publisher Summary This chapter provides an overview of time series. A time series is a set of observations of a variable made at different points of time and arranged in chronological order, each observation representing the value of the variable either at a given moment or during the interval of time between this observation and the preceding one. In general, the observations forming a time-series as made at equidistant intervals of time are considered. The factors affecting time-series may be recurring or nonrecurring, or evolutionary, periodic, or random. The method of moving averages consists in determining the average value for a certain number of terms of a time series and taking this average as the trend normal value for the middle of the period covered in the calculation of the average, that is, the period extent of the moving average.

(1978). Extended Mean Values. The American Mathematical Monthly: Vol. 85, No. 2, pp. 84-90.

I HAVE long been dissatisfied with the method of taking the arithmetic mean as the most probable value of a comparatively few direct observations of a quantity. This is certainly the legitimate result of the theory of probability, or “method of least squares,” when one knows nothing to guide one in giving more weight to one than to another observation.

STATISTICIANS are familiar with the moving average. For example, when monthly prices are given, it may be desirable to eliminate the seasonal variation. The average of the monthly prices for a calendar year may be found, then the average of the prices from the February of this year to the January of the next, then from March to February, and so on. Each average thus formed involves just once each of the twelve months; and such averaging would seem a good method for eliminating the effects of the seasonal cycle. A question, however, arises: When we take out one cycle, such as the twelve-month cycle, are we likely to put in another cycle, with substantial waves? Under certain conditions, the answer is: Yes. But to understand why such an unwelcome cycle intrudes itself, some explanation is required. As a basis for studying cycles, it is often assumed that data contain an additive chance constituent. That is, it is assumed that the rth measurement Ur may be analyzed thus