Logarithm Of Mean Research Articles

Validating statistical properties of resilience indicators derived from simulated longitudinal performance measures of farmed animals

Resilience is commonly defined as the ability of an individual to be minimally affected or to quickly recover from a challenge. Improvement of animals’ resilience is a vital component of sustainable livestock production but has so far been hampered by the lack of established quantitative resilience measures. Several studies proposed that summary statistics of the deviations of an animal’s observed performance from its target performance trajectory (i.e., performance in the absence of challenge) may constitute suitable quantitative resilience indicators. However, these statistical indicators require further validation. The aim of this study was to obtain a better understanding of these resilience indicators in their ability to discriminate between different response types and their dependence on different response characteristics of animals, and data recording features. To this purpose, milk-yield trajectories of individual dairy cattle differing in resilience, without and when exposed to a short-term challenge, were simulated. Individuals were categorised into three broad response types (with individual variation within each type): Fully Resilient animals, which experience no systematic perturbation in milk yield after challenge, Non-Resilient animals whose milk yield permanently deviates from the target trajectory after challenge and Partially Resilient animals that experience temporary perturbations but recover. The following statistical resilience indicators previously suggested in the literature were validated with respect to their ability to discriminate between response types and their sensitivity to various response features and data characteristics: logarithm of mean of squares (LMS), logarithm of variance (LV), skewness (S), lag-1 autocorrelation (AC1), and area under the curve (AUC) of deviations. Furthermore, different methods for estimating unknown target trajectories were evaluated. All of the considered resilience indicators could distinguish between the Fully Resilient response type and either of the other two types when target trajectories were known or estimated using a parametric method. When the comparison was between Partially Resilient and Non-Resilient, only LMS, LV, and AUC could correctly rank the response types, provided that the observation period was at least twice as long as the perturbation period. Skewness was in general the least reliable indicator, although all indicators showed correct dependency on the amplitude and duration of the perturbations. In addition, all resilience indicators except for AC1 were robust to lower frequency of measurements. In general, parametric methods (quantile or repeated regression) combined with three resilience indicators (LMS, LV and AUC) were found the most reliable techniques for ranking animals in terms of their resilience.

Nuclear magnetic resonance pore radius transformation method and fluid mobility characterization of shale oil reservoirs

The mobility of fluid is important for evaluating shale oil reservoirs, and it is closely related to the pore structure. Nuclear magnetic resonance (NMR) can effectively reveal the pore size distribution from nanometer to micron, providing a way to study the relationship between pore structure and mobility. However, the transverse relaxation time (T2) of NMR also contains bulk relaxation and diffusion relaxation, which will affect the accuracy of pore structure calculation. To characterize the mobility of shale oil by NMR, we proposed a pore radius transformation method with relaxation correction. The NMR simulations were carried out based on different digital core models, which were constructed according to X-CT results of shale cores. And the comparisons were performed to verified the accuracy and efficiency of the presented method. Then, the relation of geometric means of pore radius and mobility was studied through NMR experiments, and the NMR logging data of well J31 were processed by proposed method. The results show that the relative errors between corrected T2 distributions and surface relaxation distributions are smaller, and the geometric means of pore radius after relaxation correction are closer to those of digital core models, which proves the efficiency of the presented method. The logarithm of geometric mean of pore radius obtained from T2 distribution of shale core NMR experiment is positively correlated with S1/TOC, which can represent shale mobility. The processing results of NMR logging data in well J31 show two sections with large geometric means of pore radius, indicating that they have good mobility, which are consistent with the oil test results.

Non-regular Frameworks and the Mean-of-Order p Extreme Value Index Estimation

Most of the estimators of parameters of rare and large events, among which we distinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory, are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarithm of the power mean of order \(p=0\) of a certain set of statistics. Only for heavy tails, i.e. a positive EVI, quite common in many areas of application, and trying to improve the performance of the classical Hill EVI-estimators, instead of the aforementioned geometric mean, we can more generally consider the power mean of order-p (MO\(_p\)) and build associated MO\(_p\) EVI-estimators. The normal asymptotic behaviour of MO\(_p\) EVI-estimators has already been obtained for \(p<1/(2\xi )\), with consistency achieved for \(p<1/\xi \), where \(\xi \) denotes the EVI. We shall now consider the non-regular case, \(p\ge 1/(2\xi )\), a situation in which either normal or non-normal sum-stable laws can be obtained, together with the possibility of an ‘almost degenerate’ EVI-estimation.

Adherence to a lower versus higher intensity physical activity intervention in the Breast Cancer & Physical Activity Level (BC-PAL) Trial.

The first aim is to examine adherence to a lower versus higher intensity physical activity (PA) prescription in breast cancer survivors in the Breast Cancer & Physical Activity Level (BC-PAL) Trial. The second aim is to assess associations between baseline characteristics with mean PA adherence in both intervention groups combined. Forty-five participants were randomized to a 12-week, home-based lower (300 min/week, 40-59% heart rate reserve (HRR)) or higher (150 min/week, 60-80% HRR) intensity PA intervention, or no intervention/control. Both intervention groups received Polar A360® trackers and were included in this analysis (n=30). Study outcomes assessed on a weekly basis with the Polar A360® activity tracker throughout the intervention included relative adherence to the prescribed PA interventions (% of PA prescription goal met), and the absolute amount of PA time ≥40% of HRR. Baseline predictors of adherence included demographic characteristics, cardiorespiratory fitness, habitual PA and sedentary time, quality of life measures, and motivational variables from the Theory of Planned Behavior. For our primary aim, a linear mixed model was used to assess the effects of randomization group, time (intervention weeks 1-12), and the interaction of these factors on the natural logarithm of PA adherence. For our secondary aim, the association between each baseline predictor with the natural logarithm of mean weekly PA adherence was assessed, with randomization group added as a covariate. Higher relative time within the prescribed HRR zone was noted in the lower versus higher intensity PA groups (eβ=3.12, 95% CI=1.97, 4.95). No differences in adherence across time were noted. Social support was inversely associated with relative PA time within the prescribed HRR zone (eβ=0.83, 95% CI=0.72, 0.97) and absolute PA time ≥40% of HRR (eβ= 0.82, 95% CI: 0.71, 0.93). Baseline VO2max was inversely associated with relative PA adherence (eβ=0.98, 95% CI=0.95, 0.99). No other baseline measures were associated with PA adherence. There were no significant changes in absolute PA time ≥40% of HRR across time or between groups. However, the lower intensity PA group averaged over 3 times the relative amount of PA within the prescribed HRR zone compared to the higher intensity PA group. Finally, lower peer support and cardiorespiratory fitness at baseline were associated with higher PA adherence. The recent rise in popularity of commercially available activity trackers provides new opportunities to promote PA participation remotely, and these devices can be used to continuously and objectively measure PA levels as an indicator of intervention adherence. Future studies are needed to explore baseline predictors, facilitators, and barriers to sustained activity tracker use to promote PA behavior change and intervention adherence in cancer survivors. This study was registered at www. gov (No. NCT03564899) on June 21, 2018.

Logarithmic Ratio and Product-type Estimators of Population Mean

Based on the natural logarithm of known population mean of an auxiliaryvariable, x, the study introduces logarithmic ratio and product-type estimatorsof the population mean of the study variable, y, in simple random samplingwithout replacement (SRSWOR) scheme. Part of the eciency conditions forthe proposed logarithmic estimators to be more ecient than the existing ex-ponential ratio and product-type estimators, as well as the customary ratio andproduct-type estimators, is that the natural logarithm of the known populationmean of the auxiliary variable, x, must be greater than 2. Generally, there is ahigh tendency for the proposed logarithmic estimators to be more ecient thanexisting customary and exponential ratio and product-type estimators whenthe natural logarithm of the auxiliary variable population mean is greater than2. The theoretical results are illustrated and conrmed using some numericaldatasets.

Robustness of statistical methods when measure is affected by ceiling and/or floor effect.

A simulation study investigated how ceiling and floor effect (CFE) affect the performance of Welch's t-test, F-test, Mann-Whitney test, Kruskal-Wallis test, Scheirer-Ray-Hare-test, trimmed t-test, Bayesian t-test, and the "two one-sided tests" equivalence testing procedure. The effect of CFE on the estimate of group difference and on its confidence interval, and on Cohen's d and on its confidence interval was also evaluated. In addition, the parametric methods were applied to data transformed with log or logit function and the performance was evaluated. The notion of essential maximum from abstract measurement theory is used to formally define CFE and the principle of maximum entropy was used to derive probability distributions with essential maximum/minimum. These distributions allow the manipulation of the magnitude of CFE through a parameter. Beta, Gamma, Beta prime and Beta-binomial distributions were obtained in this way with the CFE parameter corresponding to the logarithm of the geometric mean. Wald distribution and ordered logistic regression were also included in the study due to their measure-theoretic connection to CFE, even though these models lack essential minimum/maximum. Performance in two-group, three-group and 2 × 2 factor design scenarios was investigated by fixing the group differences in terms of CFE parameter and by adjusting the base level of CFE. In general, bias and uncertainty increased with CFE. Most problematic were occasional instances of biased inference which became more certain and more biased as the magnitude of CFE increased. The bias affected the estimate of group difference, the estimate of Cohen's d and the decisions of the equivalence testing methods. Statistical methods worked best with transformed data, albeit this depended on the match between the choice of transformation and the type of CFE. Log transform worked well with Gamma and Beta prime distribution while logit transform worked well with Beta distribution. Rank-based tests showed best performance with discrete data, but it was demonstrated that even there a model derived with measurement-theoretic principles may show superior performance. Trimmed t-test showed poor performance. In the factor design, CFE prevented the detection of main effects as well as the detection of interaction. Irrespective of CFE, F-test misidentified main effects and interactions on multiple occasions. Five different constellations of main effect and interactions were investigated for each probability distribution, and weaknesses of each statistical method were identified and reported. As part of the discussion, the use of generalized linear models based on abstract measurement theory is recommended to counter CFE. Furthermore, the necessity of measure validation/calibration studies to obtain the necessary knowledge of CFE to design and select an appropriate statistical tool, is stressed.

SPIHT Algorithm With Adaptive Selection of Compression Ratio Depending on DWT Coefficients

In mobile multimedia devices, the frame memory compression (FMC) technique by embedded compression (EC) is becoming an increasingly important video-processing method for reducing the external data bandwidth requirement, which, in turn, results in power savings. Among various EC schemes, the combination of discrete wavelet transform (DWT) and set partitioning in hierarchical trees (SPIHT) is widely used for FMC because it achieves high compression efficiency with low computational complexity. However, there is room for improvement in the conventional DWT and SPIHT algorithm because it compresses all blocks with the same compression ratio without taking into account the correlation between DWT coefficients and the SPIHT algorithm. This study proposes a novel one-dimensional (1-D) DWT and SPIHT algorithm, which enhances the quality of the compressed video by internally applying an adaptive compression ratio for the SPIHT algorithm based on DWT coefficients while keeping the same bit-stream size. The block complexity is predicted from the distribution of DWT coefficients. Then, simple blocks are aggressively compressed with a low compression ratio, while the complex blocks are compressed with a high ratio. Furthermore, to achieve the best video quality, each compression ratio is decided by an optimization technique based on mathematical formulation. Precisely, the logarithm of mean squared error by the SPIHT algorithm is assumed to be linearly correlated with the logarithm of processed DWT coefficients. Experimental results are provided that support the aforementioned model. Compared to the conventional 1-D DWT and SPIHT algorithm, the proposed scheme remarkably improves the video quality by an average of 2.23 dB in peak signal-to-noise ratio when the target compression ratio for the SPIHT algorithm is 5/16.

Chagas disease vector control and Taylor's law.

BackgroundLarge spatial and temporal fluctuations in the population density of living organisms have profound consequences for biodiversity conservation, food production, pest control and disease control, especially vector-borne disease control. Chagas disease vector control based on insecticide spraying could benefit from improved concepts and methods to deal with spatial variations in vector population density.Methodology/Principal findingsWe show that Taylor's law (TL) of fluctuation scaling describes accurately the mean and variance over space of relative abundance, by habitat, of four insect vectors of Chagas disease (Triatoma infestans, Triatoma guasayana, Triatoma garciabesi and Triatoma sordida) in 33,908 searches of people's dwellings and associated habitats in 79 field surveys in four districts in the Argentine Chaco region, before and after insecticide spraying. As TL predicts, the logarithm of the sample variance of bug relative abundance closely approximates a linear function of the logarithm of the sample mean of abundance in different habitats. Slopes of TL indicate spatial aggregation or variation in habitat suitability. Predictions of new mathematical models of the effect of vector control measures on TL agree overall with field data before and after community-wide spraying of insecticide.Conclusions/SignificanceA spatial Taylor's law identifies key habitats with high average infestation and spatially highly variable infestation, providing a new instrument for the control and elimination of the vectors of a major human disease.

Curvelet transform with learning-based tiling

Compact signal and image representations are of crucial importance in a variety of application areas. Wavelet and wavelet-like transforms typically divide the frequency plane in a systematic non-adaptive approach. In this paper, we propose a learning-based method for adapting frequency domain tiling using the curvelet transform as the basis algorithm. The optimal tiling that better represents a single image or a given class of images is computed using denoising performance as the cost function. Simulated additive white Gaussian noise is removed from a given image using a thresholding algorithm. The curvelet tiling that generates maximal denoising performance as measured by PSNR or the logarithm of mean squared error (MSE) is considered optimal. The major adaptations considered are the number of scale decompositions, angular decompositions per scale/quadrant, and scale locations. A global optimization algorithm combining the three adaptations is proposed. Signal representations by adaptive curvelets are shown to outperform default curvelets in partial reconstruction error. Furthermore, adaptive curvelets are used in compressed sensing recovery of incomplete seismic datasets and face images. Visual and numerical improvements across a variety of images and different subsampling ratios are reported. Finally, adaptive curvelets denoising performance is tested on seismic datasets. Our results establish clear numerical and visual performance advantages over the default curvelet transform and the non-local means algorithm (NLM).

Pricing Asian options of discretely monitored geometric average in the regime‐switching model

This paper provides analytic pricing formulas of discretely monitored geometric Asian options under the regime‐switching model. We derive the joint Laplace transform of the discount factor, the log return of the underlying asset price at maturity, and the logarithm of the geometric mean of the asset price. Then using the change of measures and the inversion of the transform, the prices and deltas of a fixed‐strike and a floating‐strike geometric Asian option are obtained. As the numerical results, we calculate the price of a fixed‐strike and a floating‐strike discrete geometric Asian call option using our formulas and compare with the results of the Monte Carlo simulation. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Taylor’s power law and the stability of crop yields

Taylor’s power law (TPL) describes the empirical relationship σ2=aμb where σ2 are sample variances and μ are sample means in subsets of data in a data set. Equivalently, TPL states that the logarithm of the sample variance is a linear function of the logarithm of the sample mean across different subsets of data. Here we show that crop yields follow this relationship in several published data sets from varied situations. We show that TPL is frequently, but not always, valid for various factors structuring the data including varieties, crop species, trial environments or countries. We propose that the residuals from the linear regression of log(σ2) against log(μ) can be used as a measure of stability, called POLAR (Power Law Residuals). We compare POLAR stability with other commonly used measures of stability, and show that POLAR stability offers an advantage over some frequently used stability measures.

Effect of sauna-based heat acclimation on plasma volume and heart rate variability.

We investigated the effect of post-exercise sauna bathing on plasma volume (PV) expansion and whether such responses can be tracked by changes in heart rate (HR)-based measures. Seven, well-trained male cyclists were monitored for 35 consecutive days (17 days baseline training, 10 days training plus sauna, 8 days training). Sauna exposure consisted of 30 min (87 °C, 11 % relative humidity) immediately following normal training. Capillary blood samples were collected while resting seated to assess PV changes. HR (HRwake) and vagal-related HR variability (natural logarithm of square root mean squared differences of successive R-R intervals, ln rMSSDwake) were assessed daily upon waking. A sub-maximal cycle test (5 min at 125 W) was performed on days 1, 8, 15, 22, 25, 29, and 35 and HR recovery (HRR60s) and ln rMSSDpostex were assessed post-exercise. Effects were examined using magnitude-based inferences. Compared with baseline, sauna resulted in: (1) peak PV expansion after four exposures with a likely large increase [+17.8 % (90 % confidence limits, 7.4; 29.2)]; (2) reduction of HRwake by a trivial-to-moderate amount [-10.2 % (-15.9; -4.0)]; (3) trivial-to-small changes for ln rMSSDwake [4.3 % (1.9; 6.8)] and ln rMSSDpostex [-2.4 % (-9.1; 4.9)]; and (4) a likely moderate decrease in HRR60s [-15.6 % (-30.9; 3.0)]. Correlations between individual changes in PV and HR measures were all unclear. Sauna bathing following normal training largely expanded PV in well-trained cyclists after just four exposures. The utility of HR and HRV indices for tracking changes in PV was uncertain. Future studies will clarify mechanisms and performance benefits of post-training sauna bathing.

Event-adjusted evaluation of weather and climate extremes

Abstract. After an overview of existing methods, we present a novel method of "event-adjusted" evaluation of extremeness of weather and climate events. It is based on optimization of both the considered area and the time duration for every event. The method consists of three steps: (i) estimation of return periods of a representative variable at individual sites, performed separately for various time windows; (ii) spatial interpolation of the point return period data; and (iii) searching the area and the time window in which the extremeness of the event was maximum. The extremeness is quantified as the common logarithm of the spatial geometric mean of the return periods multiplied by the radius of a circle of the same area as the one over which the geometric mean is taken. The maximum product is referred to as the weather extremity index (WEI). Two precipitation events, which affected the Czech Republic in May and in August 2010, were evaluated by the WEI for illustration. Validation of the method on sufficiently long data series is still needed. Moreover, the WEI is generally applicable regardless of the studied phenomenon (heavy rains, heat waves, windstorms, etc.). This makes it possible to study various weather and climate extremes from the viewpoint of possible recent and future changes in their frequency, seasonal distribution, and circulation conditions accompanying them.

Taylor's law applies to spatial variation in a human population

Taylor's law (TL), which originated in ecology, states that, in many nonhuman populations, the logarithm of the variance of the population density in a set of populations is linearly related to the logarithm of the mean population density of those populations. TL has not been carefully tested by demographers of human populations. We found that TL described well the time course of spatial variation in the population density of the counties, municipalities, and regions of Norway. Plots of the logarithm of the variance of population density by county as a function of the logarithm of the mean of population density by county, with one point per year from 1978 through 2010, were remarkably close to straight lines. Because of the exceptional situation of Oslo, we conducted two analyses, including and omitting Oslo as a county. When Oslo was included, the relationship of log variance to log mean was non-linear for roughly a decade beginning in 1978, and linear regressions yielded different slopes for TL up to 1984 and after 1985, corresponding to the decline and growth of population in Oslo. When Oslo was excluded from the analysis, the curvature disappeared and the linear relationship of log variance to log mean was remarkably precise throughout 1978-2010 ( R 2 >0.99). This example suggests that non-demographic factors may influence the relationship between the spatial log variance and log mean of population density, but also that, in a stable environment (since the early 1980s), changes in the spatial distribution of population density can follow TL with remarkable precision. Joel E. COHEN, Professor of Populations, The Rockefeller University and Columbia University, 1230 York Avenue, Box 20, New York, New York 10065 USA. E-mail: cohen@rockefeller.edu . Meng XU, Laboratory of Populations, The Rockefeller University, 1230 York Avenue, Box 20, New York, 10065 USA. E-mail: mxu@rockefeller.edu . Helge BRUNBORG, Statistics Norway, Pb 8131 Dep, 0033 Oslo, Norway. E-mail: hbr@ssb.no . Joel E. COHEN, Professor of Populations, The Rockefeller University and Columbia University, 1230 York Avenue, Box 20, New York, New York 10065 USA. E-mail: cohen@rockefeller.edu . Meng XU, Laboratory of Populations, The Rockefeller University, 1230 York Avenue, Box 20, New York, 10065 USA. E-mail: mxu@rockefeller.edu . Helge BRUNBORG, Statistics Norway, Pb 8131 Dep, 0033 Oslo, Norway. E-mail: hbr@ssb.no .

A simple generalisation of the Hill estimator

The classical Hill estimator of a positive extreme value index (EVI) can be regarded as the logarithm of the geometric mean, or equivalently the logarithm of the mean of order p=0, of a set of adequate statistics. A simple generalisation of the Hill estimator is now proposed, considering a more general mean of order p≥0 of the same statistics. Apart from the derivation of the asymptotic behaviour of this new class of EVI-estimators, an asymptotic comparison, at optimal levels, of the members of such class and other known EVI-estimators is undertaken. An algorithm for an adaptive estimation of the tuning parameters under play is also provided. A large-scale simulation study and an application to simulated and real data are developed.

Variations in Posttonsillectomy Hemorrhage Rates Are Scale Invariant

Scale invariance is a property of scientific laws or objects that change in a prescribed fashion if measurements are scaled, and is often represented by a power-law relationship. Power laws suggest that events of a large magnitude will be rare, while small events will be much more common, and that a simple mathematical law relates "severity" with frequency. Scale invariance has been demonstrated in scientific fields including physics, social science, and economics. The authors use the complication of a posttonsillectomy hemorrhage to test whether this property is a feature of surgical complications. Non-identifiable data were obtained regarding posttonsillectomy hemorrhage and subcategorized by calendar month, and the percentage rate of posttonsillectomy hemorrhage was calculated. The data were then transformed using a logarithmic function. This transformed data were plotted and a linear regression analysis was performed. The 13-year period studied included 6,381 tonsillectomy procedures. The logarithm of the frequency of a given rate range of posttonsillectomy hemorrhage (y) was linearly related to the logarithm of the geometric mean of the rate range (x). The best-fit straight line was y = -1.3996x + 2.0624 with R2 = 0.851, n = 10, r = 0.922, and P < .001. The authors found that the incidence of posttonsillectomy hemorrhage is scale invariant. The practical implication is that the observation of rare incidences of large hemorrhage rates may not be due to a unique circumstance or a particular operative fault. To reduce the incidence of extreme rates of postoperative hemorrhage, a review of the entire process of tonsillectomy would be required. Scale-invariance analysis may represent a novel tool that should be considered when reviewing surgical complications.

Developmental changes in escape response performance of five species of marine larval fish

We measured the escape response performance by four species of larval fish: Atlantic cod (Gadus morhua), Atlantic herring (Clupea harengus harengus), radiated shanny (Ulvaria subbifurcata), and capelin (Mallotus villosus). The resultant data were combined with information from larval winter flounder (Pleuronectes americanus) to produce general models relating escape response performance to larval size. Three components of the escape response produced models with reasonable predictive ability, including mean and maximum speed during the response, and the distance travelled by larvae during the first 100 ms of the response (the latter representing the contribution of the C-shaped fast start that initiates the response). The logarithm of mean and maximum escape speed increased linearly with increasing larval length. The distance after 100 ms was linearly related to larval length. Our general model for mean escape speed produced significantly higher length-specific speed values than did the general model proposed by Miller et al. (T.J. Miller, L.B. Crowder, J.A. Rice, and E.A. Marschall. 1988. Can. J. Fish. Aquat. Sci. 45: 1657n1670). Resume : Nous avons mesurO liefficacitO de la rOaction de fuite chez les larves de quatre poissons : la morue (Gadus morhua), le hareng (Clupea harengus harengus), liulvaire deux-lignes (Ulvaria subbifurcata) et le capelan (Mallotus

Internal constraints induce localization in an isolated polymer molecule.

We investigate a model of an isolated polymer molecule with the constraint that certain pairs of monomers are in close proximity. A combination of a variational theory and scaling arguments shows that the polymer becomes localized in space, i.e., collapsed, when the product of the density of constraints and the logarithm of the geometric mean of the separation between constraints exceeds a critical value. Applications to the understanding of vulcanization, of protein structure determination, and of protein folding are noted.

River Quality Modeling: Frequency Domain Approach

Complete and equally spaced river quality variables specific conductance (SPC, μScm-1), total dissolved solids (TDS, mgL-1), and the logarithm of the daily discharge mean (DDM, m3s-1) are used in frequency domain modeling. Significant autocorrelation values at lags 1 and 12 indicate that the observations are not random and have a yearly periodicity. Cross‐correlations also indicate the yearly periodicity as well as a negative relationship between TDS and DDM, and SPC and TDS. The cross‐correlation is positive between SPC and TDS. The spectral and cross‐spectral analyses show a significant periodicity at a frequency of νk=0.078 cycles per month, or a period of one year. Because these variables are linearly correlated, an impulse response function can be used to determine the lagging, leading, and/or immediate response of the input(s) on the output. It is found that the best predictor of SPC is DDM. The small contribution from TDS is ignored. Significant coherence and F statistic at a frequency of 0.093 confirms the strong linear relationship between these two variables.

Khintchine's constant for Lüroth representation

Galambos has conjectured that the logarithm of the geometric mean of partial denominators in a Lüroth representation of a number between zero and one is a constant. We here prove that this constant is ≈0.7884 and plays the same role as Khintchine's constant for usual continued fraction expansions.