Frequency Polygon Research Articles

An Approach to Proving Limit Theorems for Dependent Random Variables

Previous article Next article An Approach to Proving Limit Theorems for Dependent Random VariablesB. S. NakhapetyanB. S. Nakhapetyanhttps://doi.org/10.1137/1132080PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– 48:1287 0219.60027 Google Scholar[2] A. V. Bulinskii, Limit Theorems for Random Processes and Fields, Vol. Moscow, Moscow State University Press, 1981, (In Russian.) Google Scholar[3] Hiroshi Takahata, On the central limit theorem for weakly dependent random fields, Yokohama Math. J., 31 (1983), 67–77 85m:60052 0532.60043 Google Scholar[4] E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047–1050 84c:60035 0496.60020 CrossrefGoogle Scholar[5] B. S. Nakhapetyan, Central limit theorem for a random field satisfying a strong mixing condition, Dokl. Armen. SSR, 61 (1075), 210–213, (In Russian.) Google Scholar[6] B. S. Nakhapetyan, The central limit theorem for random fields satisfying a strong mixing conditionMulticomponent Random Systems, Nauka, Moscow, 1978, 276–288, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fieldsThe Annals of Applied Probability, Vol. 32, No. 3 | 1 Jun 2022 Cross Ref Compound Poisson approximation for regularly varying fields with application to sequence alignmentBernoulli, Vol. 27, No. 2 | 1 May 2021 Cross Ref Conditional Hazard Estimate for Functional Random FieldsJournal of Statistical Theory and Practice, Vol. 8, No. 2 | 24 March 2014 Cross Ref Robust nonparametric estimation for spatial regressionJournal of Statistical Planning and Inference, Vol. 140, No. 7 | 1 Jul 2010 Cross Ref Asymptotic normality of frequency polygons for random fieldsJournal of Statistical Planning and Inference, Vol. 140, No. 2 | 1 Feb 2010 Cross Ref Estimation of the trend function for spatio-temporal modelsJournal of Nonparametric Statistics, Vol. 21, No. 5 | 1 Jul 2009 Cross Ref Kernel regression estimation for random fieldsJournal of Statistical Planning and Inference, Vol. 137, No. 3 | 1 Mar 2007 Cross Ref LOCAL LINEAR FITTING UNDER NEAR EPOCH DEPENDENCEEconometric Theory, Vol. 23, No. 01 | 6 December 2006 Cross Ref Local linear spatial regressionThe Annals of Statistics, Vol. 32, No. 6 | 1 Dec 2004 Cross Ref Stochastic Dynamics of Fluctuations in Classical Continuous SystemsJournal of Functional Analysis, Vol. 185, No. 1 | 1 Sep 2001 Cross Ref Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal DistributionLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Kernel density estimation for random fields (density estimation for random fields)Statistics & Probability Letters, Vol. 36, No. 2 | 1 Dec 1997 Cross Ref Kernel density estimation for random fields: The L1 TheoryJournal of Nonparametric Statistics, Vol. 6, No. 2-3 | 1 Jan 1996 Cross Ref Kernel density estimation on random fieldsJournal of Multivariate Analysis, Vol. 34, No. 1 | 1 Jul 1990 Cross Ref Volume 32, Issue 3| 1988Theory of Probability & Its Applications389-571 History Submitted:01 April 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132080Article page range:pp. 535-539ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Basic Statistics for the Behavioral Sciences

Basic Statistics for the Behavioral Sciences

Frequency Polygons: Theory and Application

Abstract In this article I investigate the theoretical properties and applications of the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. For estimating an unknown probability density function using a random sample, the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error, achieving the same rate of convergence to zero of the integrated mean squared error as non-negative kernel estimators. Data-based algorithms for constructing frequency polygons are discussed and illustrated. One is based on a histogram with bin width equal to 2.15sn -1/5, where s is an estimate of the standard deviation from a sample of size n. Another approach is based on the method of generalized crossvalidation. The bivariate frequency polygon is also investigated. Comparisons are made between the frequency polygon and other density estimators. Examples with data in one and two dimensions are presented.

Frequency Polygons: Theory and Application

Abstract In this article I investigate the theoretical properties and applications of the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. For estimating an unknown probability density function using a random sample, the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error, achieving the same rate of convergence to zero of the integrated mean squared error as non-negative kernel estimators. Data-based algorithms for constructing frequency polygons are discussed and illustrated. One is based on a histogram with bin width equal to 2.15sn -1/5, where s is an estimate of the standard deviation from a sample of size n. Another approach is based on the method of generalized crossvalidation. The bivariate frequency polygon is also investigated. Comparisons are made between the frequency polygon and other density estimators. Examples with data in one and two dimensions are presented.

Oversmoothed Nonparametric Density Estimates

Abstract The optimal histogram for a sample of size n from a density defined on the entire real line requires at least (2n)1/3 bins, under mild smoothness conditions. Similar bounds exist for the frequency polygon and kernel estimators. Values near these bounds give nearly optimal results for a variety of smooth densities, providing good, quick density estimates.

Serum indole-3-acetic acid in control subjects and newly abstinent alcoholics after an oral loading with l-tryptophan: A preliminary study using liquid chromatography with amperometric detection

Described in this paper is a rapid, isocratic assay for serum indole-3-acetic acid (IAA). The sample preparation involves only protein precipitation using sulfosalicylic acid, and the sensitivity of amperometric detection is in the picogram range. The chromatographic analysis time is approximately 4 min. The devised method was used for a longitudinal study of IAA levels in serum samples from control subjects and newly abstinent alcoholics. Dietary variations were eliminated by administering a 2.0-g loading dose of l-Trp to all subjects investigated. The results are presented in the form of cumulative frequency polygons. Preliminary data indicate no differences in IAA levels between newly abstinent alcoholics and control subjects.

Ontogenic development of autoregressive component waves of waking EEG in normal infants and children

Waking EEGs of 150 normal children aged from 20 days to 15 years were subjected to analysis. Discrete time series of an artifact free segment of the EEG record at 50 samples/sec for twenty seconds was generated and autoregressive (AR) and component analyses were carried out with a minicomputer PDP 11/40 (DEC). The results may be summarized as follows: 1) In the group of 1 year old and less, the power increased with the monthly age, whereas the bio-informing activity amount decreased. In those older than 1 year, both parameters showed maximal values at 1 year and then decreased with age, and the decreases were marked from 1 to 3 years. 2) The first- and second-order component activities of 129 and 677 waves, respectively, were obtained by applying component analysis to 152 EEG records. The frequency polygons of natural frequency of second-order component waves verified several modes, each of which was enhanced in the frequency range of the well-known delta 0, delta 1, theta 1, theta 2, alpha 1, alpha 2, beta 1, beta 2 and beta 3 waves, respectively. 3) The average percent-power of the delta wave (delta 0 + delta 1) decreased with age, especially from 1 to 3 years old, whereas those of beta- and alpha-waves increased with advancing age. That of the theta wave tended to increase from 2 to 4 years of age, and thereafter decreased gradually with increasing age. 4) With increasing age, the durations of damped oscillations were significantly lengthened in delta 1, alpha 1 and beta 3 waves, whereas that in the theta 1 wave was significantly shortened. 5) The bio-informing activity amounts of alpha waves increased from 1 to 3 years with increasing age, whereas those of delta and theta waves decreased. No significant developmental change in the parameters, however, was observed in the beta wave. The results indicate that AR-power spectral and component analyses of EEG are sensitive methods for obtaining valuable information regarding the electrical brain maturation in childhood.

The lingual angle of incisors.

A morphologic trait, not previously registered, on the incisor lingual surface: "The lingual angle of incisors", was studied in Norwegians. The lingual angle is the angle between the floor of the lingual fossa and the dental tubercle. Measurements of the lingual angle could be performed with sufficient accuracy. Deviations from normal distribution in skewness and kurtosis values did not affect statistical testing. The general similarity of the frequency polygons for all incisors in the way they were peaked and stepped is possibly due to participation of the same factors on the genetic level in the determination of the trait. The mean lingual angles were (in degrees): I1sup = 142, I2sup equal 156, I1inf = 153 and I2inf = 154. No significant bilateral asymmetry or sex differences were found. Correlations were high for both central to central and lateral to lateral incisors in both jaws, highest for centrals in the maxilla and laterals in the mandibula. Central to lateral correlations were relatively low, especially in the upper jaw. Interjaw correlations were low, but I1sup showed a tendency to be better correlated than I2sup to lower incisors. I2sup and I1inf were the most variable incisors in each jaw. The dental tubercle probably plays an important role in determining the lingual angle. Measurements of the lingual angle give information on the profile of the lingual surface in inciso-cervical direction. Statistical analysis of a phenotypic trait may give information on factors which determine the trait.

45—STUDIES OF ADHESIVE-BONDED NON-WOVEN FABRICS PART III: THE DETERMINATION OF FIBRE ORIENTATION AND CURL

Experiments are described in which the orientation and curl of fibre segments in some random, parallel, and cross-laid fabrics were examined with a scanning electron microscope (SEM). A technique for the measurement of these parameters that consists in the enlargement of the SEM photomicrographs to obtain traces of fibre paths is outlined. The distributions of measured values are presented in histograms or frequency polygons to provide a basis for a comparison between different structures. It is also shown that these distributions can be expressed in mathematical formulae that yield a more accurate representation, are more suitable for comparison purposes, and also simplify the theoretical computations of the stress–strain behaviour of non-woven fabrics.

On the Strong Mixing Property for Linear Sequences

On the Strong Mixing Property for Linear Sequences

A statistical description of the normal computerized brain scan

A total of 100 CT brain scans performed with an EMI head scanner were analyzed in order to define the statistical limits of normal. The statistical parameters are means and standard deviations. Frequency polygons of the attenuation coefficients for each of five brain slices have been constructed. The concept of the water to brain ratio is introduced.

Computer program for presentation of grain‐size data by the graphic method

ABSTRACTA computer program is described for a Hewlett‐Packard desk‐top Calculator (Model 9820A)‐Plotter (Model 9862A) which plots a histogram, frequency polygon and/or cumulative curve, as well as standard statistical parameters and percentages of gravel, sand, silt, and clay from raw weight data from a sediment grain‐size analysis. The program utilizes the graphic method for calculating statistical parameters in contrast to other commonly used computer programs derived for the method of moments.This program is advantageous because (1) it is rapid, (2) it allows direct comparison with grain‐size data in the literature which have been computed by the graphic method, (3) either a complete presentation of a distribution can be obtained on one sheet of paper or a series of cumulative curves or frequency polygons can be prepared on a single graph which is ready for publication photocopying, (4) automatic extrapolation of data points from a cumulative curve eliminates inconsistencies arising from manual extrapolation, and (5) the desk‐top calculator is more convenient to use, eliminates punching of data cards, and can be operated by someone without a knowledge of computers and programming. Disadvantages of the program are: (1) some authors feel critical percentiles are more accurately read off cumulative curves plotted on a probability ordinate than on an arithmetic ordinate, (2) the cumulative curve is a series of straight‐line segments, (3) the frequency polygon is not as accurate a representation of grain‐size distribution as a frequency curve and (4) size class intervals for the histogram and frequency polygon are standardized at 0.5φ, even though gravel and mud fractions are analysed at whole φ intervals. For most studies, the advantages of the program outweigh the disadvantages.

A new computer program for correcting particle size distribution data obtained from non-equidistant class intervals

A generalized particle size distribution algorithm has been developed to compensate for two intrinsic shortfalls of electronic sizing instruments. The problem of lower detection limit is handled by first determining if the form of the distribution fits a normal or log-normal model. Then, using the Cohen method for estimating the parameters of a truncated distribution, the missing data are calculated and the usual distribution statistics are computed. To compensate for uneven cell intervals in histogram or frequency polygon representations of size distribution, the cells are treated from an area under the curve consideration rather than height. This gives the true shape, although the lack of resolution in the wider cells is still evident. A FORTRAN computer program is utilized to implement these distribution computations.

Frequency and size of muscle fibers in athletic body build

AbstractIn one prototype athletic subject and two persons of more average body build complete mid‐point cross sections of the biceps brachii muscle were made. All three cross‐sections were photomicrographed on overlaping fields, and composite photomicrographs of the complete muscle cross section were made. An individual fiber count on these photomicrographs was carried out.The number of individual muscle fibers in the athlete and the other two persons amounted to 316,243, 227,233 and 199,240 fibers respectively. Sampling methods were also tried on the sections for the evaluation of different estimation procedures. Fiber size distributions were arrayed in frequency polygons in each case. Six thousand fibers, randomly chosen from different parts of the section were measured in each case by means of an automatic counter. The mean fiber diameters are 42.26 ± 7.616 μ in the athletic subject and 24.1 ± 4.583 μ and 22.54 ± 4.919μ in the other two subjects.

Trend analysis of geophysical data

Trend analysis of aeromagnetic data has been treated in a few publications during recent years. In this paper a short review is given of the methods used and results from comparisons between different methods. It is attempted to make a more general use of one of the methods presented—the triangle method—on all sorts of geophysical data, including terrain analysis. By the aid of a computer programme for rectangular grid systems the trend analysis is easily performed. The results are given as frequency polygons and polar diagrams, and a discussion is made over the reliability of the resulting frequencies. The amount of information from one analysis can be increased by using two different lengths of the class intervals and by operating with two grid spacings. The triangle method is tested with random and dependent data. Finally an example related to the treatment of gravimetric, magnetic, and topographic maps over a small survey area is presented.

Notes on the Possible Misuse and Errors of Cumulative Percentage Frequency Graphs for the Comparison of Prehistoric Artefact Assemblages

For many years archaeologists interested in the study of the Palaeolithic in North Africa and Eurasia have been using cumulative percentage frequency graphs for the comparison of prehistoric artefact assemblages. For examples we refer to the references at the foot of this text. However, it is perhaps time to carefully review this technique and its future utility.Statistical techniques and mathematical models are slowly infiltrating and reshaping the discipline of archaeology—increasing the power and depth of both analysis and synthesis. It is perhaps already possible to distinguish the cumulative advance of these techniques in archaeology from the initial role of demonstrative aids and methods of data display towards an increasingly powerful analytical role with a capacity for predictive inference. In the ranks of the first generation we have the early use of graphs, frequency polygons and histograms for mapping severely limited numbers of percentages or attribute ratios. In the second generation of statistical techniques the impact of the computer is felt for the first time and archaeology is developing an array of exploratory attempts to integrate the probing capacity of such methods as factor analysis, matrix analysis and principal component analysis.

A survey of pH1 and ultimate pH values of British progeny‐test pigs

Summary. 1. A total of 2878 progeny‐test pigs from five stations, and of 1532 commercial pigs, were surveyed for the pH1 values, at 45 min after death, and the ultimate pH values, at 24 hr after death, of their LD muscles at the last rib. Values are also available for the heads of the adductor and semimembranosus muscles, where they are exposed in the gammons on splitting the carcase.2. Factors which affect the pH1 values are the method of stunning, the breed of pig and the waiting time between station and factory, before slaughter.(a) CO2 stunning lowers the mean pH1 values and increases the % of these values lying below pH 6.0 about three‐fold, as compared with electrical stunning. The effect is easily detectable. In extreme cases, a distinctly bimodal distribution of the pH1 values appears with CO2‐stunned pigs.(b) Of the progeny‐tested breeds, Large White shows the lowest % of low pH1 values and Welsh and Landrace the highest, but the differences (about two‐fold) are not as marked as those brought about by changing from electrical to CO2‐stunning of any particular breed. Commercial pigs of mixed breed, for instance, show a lower % of low pH1 values even than Large White, but this % can be raised beyond that for electrically stunned Large White, if the commercial pigs are stunned with CO2.(c) Waiting time between leaving the station and slaughter has a marked effect on the shape and position of the frequency polygons for pH1. As the time is increased, so the frequency polygons become broader, the mean pH1 shifts to lower pH values and the % of values below pH 6.0 increases. At the shortest waiting times, the frequency polygons and the other two parameters for progeny‐tested pigs are almost indentical with those for commercial pigs, and may even be an improvement on them, in the sense that the tendency to watery pork is further decreased. This effect of waiting time is independent of ambient temperature, which itself affects the distribution of pH1 values but only to a minor degree. There is, similarly, a tendency for the % of low pH, values to be affected by season, being lower in spring and early summer than in autumn and winter, contrary to expectation.3. Factors which do not significantly affect the distribution of the pH, values, or their mean, are the sex of the pig, and the deep muscle temperature at 45 min after slaughter. In fact, the pH, values are completely randomly distributed with respect to deep muscle temperature which in its turn appears to be quite unrelated to the ambient temperature in the holding pens or the factory.4. There are some differences between the pH, values of gammon and LD muscles, the former tending to be slightly higher on average, but randomly distributed, so that a pH, value of below 6‐0 in the LD muscle may be accompanied, on the same carcase, by one of 6.30 or higher in the gammon or vice‐versa. For the purposes of an overall survey of the likely occurrence of watery pork, the differences between the two muscles are negligible.5. The ultimate pH values of the progeny‐tested pigs are raised as the waiting time before slaughter increases, due no doubt to increased utilization of glycogen reserves during the tension and excitement of waiting. Ambient temperature, on the other hand, has no significant effect on the ultimate pH, even at the longest waiting time of 1 ½ hr, although the pigs may have been exposed during his time to temperatures below 4.5°C. Hence, the wastage of glycogen by shivering, observed by earlier workers, does not seem to occur here.The grand mean of all the ultimate pH values was 5.68, which is higher than that of 5.50 usually observed in large samples of Danish Landrace.

The Sea Garfish, Reporhamphus melanochir (Cuvier & Valenciennes) (Hemi-ramphidae), in South Australia: Breeding, Age Determination, and Growth Rate

There are seven macroscopic gonad stages recognizable in male sea garfish. and eight in females with which ova diameter measurements have been correlated . In South Australia the spawning period lasts from late September to early March. and activity is most intense during October and November . A second period of breeding activity may occur in February . A few females are sexually mature at 2 years. but the majority do not spawn for the first time until the end of 3 years .This applies also to males. A latent maturation cycle is demonstrated in immature female garfish. The use of otoliths for age determination is described and the method validated by Petersen length frequency polygons. The rate of growth has been deduced from calculated intermediate lengths of younger garfish, based on an otolith lengthlfish length relation and the mean lengths of older age groups. The weight of the garfish varies as the cube of the length. Females attain a larger average size than males after the third pear. Gulf St. Vincent females grow faster than those in other regions; males, however, are variable. Application of these results to a management policy for the commercial fishery is severely restricted by the inadequate official catch statistics. However, it is recommended that netting of garfish in Coffin's Bay be prohibited, but that no change in the minimum legal length should be made for the present.

Studies of the Kwanto Loam

The patterns of sedimentation, grain size, and roundness, and mineralogical composition of the Kwanto loam in Shimosa terrace land, Chiba prefecture, Japan were studied., The Kwatno loam is stratified upon the sandy gravel bed which lies on the Narita formation, and mainly consists of rock fragments of weathered basic tuff., The Kwanto loam is classified into two groups, the upper and the lower A pumice bed lies about 20 cm above the bottom of the upper Kwanto loam (Fig., 1)., The mean grain size of the upper Kwanto loam by weight percentage is bout 3., 6∼3., 8 Phi in diameter while the lower Kwanto loam is 2., 8∼3., 2 Phi., The variation in grain size in the upper Kwanto loam is far less than in the lower (Table 2)., The frequency polygons in Fig., 3 show the difference between the upper and lower., The roundness in the upper Kwanto loam varies from 0., 20 to 0., 22 while in the lower from 0., 33 to 0., 37 (Fig., 5)., The two loams can be distinguished fairly well by their mineralogical composition., The upper Kwanto loam rich in mafic minerals, such as olivine and pyroxene, while the lower is rich in felsic minerals (Table 3)., From the evidence just mentioned, it may be inferred that the upper Kwanto loam is aeolian while the lower is aqueous, and there is a time interval of sedimentation between these two loams.,

Note on the frequency distribution of geothermal gradients

Tables of temperature gradients observed in the Earth's crust have been compiled by H. C. Spicer [see “References” at end of paper, Spicer, 1942], These data cover several pages, and it was considered convenient to summarize them for teaching purposes.This was done by obtaining the frequency of various geothermal gradients for intervals of 2°C/km for the three intervals of depths given in the original tables, that is, 30.5 to 305 m, 305 to 610 m, and more than 610 m. Three frequency polygons were obtained, shown in Figure 1. Eight odd values over 62°C/km were omitted. The histograms for the two upper levels show distinctly bimodal distributions. In the 30.5–305–m group (198 observations) the maximal frequencies are the steps 18 to 20°C/km and 34 to 36°C/km. In the 305–610 m group (175 observations) the second mode is in the same interval as in the first group, but the first maximum lies in the somewhat lower interval of 14 to 16°C/km. The third group of data (138 observations), taken at depths of more than 610 m shows a main mode in approximately the same range as the other two groups with several minor modes. The irregularities in this last histogram might be attributed, in part, to the smaller number of observations in the third group.