The Two-Piece Normal, Binormal, or Double Gaussian Distribution: Its Origin and Rediscoveries
aa r X i v : . [ s t a t . M E ] M a y Statistical Science (cid:13)
Institute of Mathematical Statistics, 2014
The Two-Piece Normal, Binormal, orDouble Gaussian Distribution: Its Originand Rediscoveries
Kenneth F. Wallis
Abstract.
This paper traces the history of the two-piece normal distri-bution from its origin in the posthumous
Kollektivmasslehre (1897) ofGustav Theodor Fechner to its rediscoveries and generalisations. Thedenial of Fechner’s originality by Karl Pearson, reiterated a centurylater by Oscar Sheynin, is shown to be without foundation.
Key words and phrases:
Gustav Theodor Fechner, Gottlob FriedrichLipps, Francis Ysidro Edgeworth, Karl Pearson, Francis Galton, OscarSheynin.
1. INTRODUCTION
The two-piece normal distribution came to pub-lic attention in the late 1990s, when the Bank ofEngland and the Sveriges Riksbank began to pub-lish probability forecasts of future inflation, usingthis distribution to represent the possibility that thebalance of risks around the central forecast mightnot be symmetric. The forecast probabilities thatfuture inflation would fall in given intervals couldbe conveniently calculated by scaling standard nor-mal probabilities, and the resulting density forecastswere visualised in the famous forecast fan charts.In both cases the authors of the supporting tech-nical documentation (Britton, Fisher and Whitley,1998; Blix and Sellin (1998)) refer readers to John-son, Kotz and Balakrishnan (1994) for discussion ofthe distribution. These last authors state (page 173)that “the distribution was originally introduced byGibbons and Mylroie (1973),” a reference that post-dates the first edition of
Distributions in Statis-
Kenneth F. Wallis is Emeritus Professor ofEconometrics, Department of Economics, University ofWarwick, Coventry CV4 7AL, United Kingdom e-mail:[email protected].
This is an electronic reprint of the original articlepublished by the Institute of Mathematical Statistics in
Statistical Science , 2014, Vol. 29, No. 1, 106–112. Thisreprint differs from the original in pagination andtypographic detail. tics (Johnson and Kotz (1970)), in which the two-piece normal distribution made no appearance, un-der this or any other name. On the contrary, thedistribution was originally introduced in Fechner’s
Kollektivmasslehre (1897) as the
Zweispaltiges or Zweiseitige Gauss’sche Gesetz . In his monumentalhistory of statistics, Hald (1998) prefers the lattername, which translates as the “two-sided Gaussianlaw,” and refers to it as “the Fechner distribution”(page 378). However Fechner’s claim to originalityhad been disputed by Pearson (1905), whose denialof Fechner’s originality has recently been repeatedby Sheynin (2004). In this paper we reappraise thesource and nature of the various claims, and recordseveral rediscoveries of the distribution and exten-sions of Fechner’s basic ideas. As a prelude to thediscussion, there follows a brief technical introduc-tion to the distribution.A random variable X has a two-piece normaldistribution with parameters µ, σ and σ if it hasprobability density function (PDF) f ( x ) = (cid:26) A exp[ − ( x − µ ) / σ ] , x ≤ µ,A exp[ − ( x − µ ) / σ ] , x ≥ µ, (1)where A = ( √ π ( σ + σ ) / − . The distribution isformed by taking the left half of a normal distribu-tion with parameters ( µ, σ ) and the right half ofa normal distribution with parameters ( µ, σ ), andscaling them to give the common value f ( µ ) = A atthe mode, µ , as in (1). The scaling factor applied to K. F. WALLIS
Fig. 1.
The probability density function of the two-piece nor-mal distribution. Dashed line: left half of N ( µ, σ ) and righthalf of N ( µ, σ ) distributions with µ = 2 . and σ < σ . Solidline: the two-piece normal distribution. the left half of the N ( µ, σ ) PDF is 2 σ / ( σ + σ )while that applied to the right half of N ( µ, σ ) is2 σ / ( σ + σ ), so the probability mass under theleft or right piece is σ / ( σ + σ ) or σ / ( σ + σ ),respectively. An example with σ < σ , in which thetwo-piece normal distribution is positively skewed,is shown in Figure 1. The skewness becomes extremeas σ → σ → σ , reaching zero when σ = σ and the dis-tribution is again the normal distribution.The mean and variance of the distribution are E ( X ) = µ + r π ( σ − σ ) , (2) var( X ) = (cid:18) − π (cid:19) ( σ − σ ) + σ σ . (3)Expressions for the third and fourth moments aboutthe mean are increasingly complicated and unin-formative. Skewness is more readily interpreted interms of the ratio of the areas under the two piecesof the PDF, which is σ /σ , or a monotone transfor-mation thereof such as ( σ − σ ) / ( σ + σ ), which isthe value taken by the skewness measure of Arnoldand Groeneveld (1995). With only three parametersthere is a one-to-one relation between (the absolutevalue of) skewness and kurtosis. The conventionalmoment-based measure of kurtosis, β , ranges from3 (symmetry) to 3.8692 (the half-normal extremeasymmetry), hence the distribution is leptokurtic.Quantiles of the distribution can be convenientlyobtained by scaling the appropriate standard nor-mal quantiles. For the respective cumulative distri-bution functions (CDFs) F ( x ) and Φ( z ) we define quantiles x p = F − ( p ) and z p = Φ − ( p ). Then in theleft piece of the distribution we have x α = σ z β + µ ,where β = α ( σ + σ ) / σ . And in the right pieceof the distribution, defining quantiles with referenceto their upper tail probabilities, we have x − α = σ z − δ + µ , where δ = α ( σ + σ ) / σ . In particu-lar, with σ < σ , as in Figure 1, the median of thedistribution is x . = σ Φ − (1 − ( σ + σ ) / σ ) + µ .In this case the three central values are orderedmean > median > mode; with negative skewness thisorder is reversed.Although the two-piece normal PDF is continuousat µ , its first derivative is not and the second deriva-tive has a break at µ , as first noted by Ranke andGreiner (1904). This has the disadvantage of makingstandard asymptotic likelihood theory inapplicable,nevertheless standard asymptotic results are avail-able by direct proof for the specific example.The remainder of this paper is organised as fol-lows. In Section 2 we revisit the distribution’s originin Gustav Theodor Fechner’s Kollektivmasslehre ,edited by Gottlob Friedrich Lipps and published in1897, ten years after Fechner’s death. In Section 3 wenote an early rediscovery, two years later, by Fran-cis Ysidro Edgeworth. In Section 4 we turn to thefirst discussion in the English language of Fechner’scontribution, in a characteristically long and argu-mentative article by Karl Pearson (1905). Pearsonderives some properties of “Fechner’s double Gaus-sian curve,” but asserts that it is “historically in-correct to attribute [it] to Fechner.” We re-examinePearson’s evidence in support of this position, inparticular having in mind its reappearance in Os-car Sheynin’s (2004) appraisal of Fechner’s statis-tical work. Pearson also argues that “the curve isnot general enough,” especially in comparison withhis family of curves. The overall result was that theFechner distribution was overlooked for some time,to the extent that there have been several indepen-dent rediscoveries of the distribution in more recentyears; these are noted in Section 5, together withsome extensions.
2. THE ORIGINATORS: FECHNER ANDLIPPS
Gustav Theodor Fechner (1801–1887) is known asthe founder of psychophysics, the study of the re-lation between psychological sensation and physi-cal stimulus, through his 1860 book
Elemente derPsychophysik . Stigler’s (1986, pages 242–254) as-sessment of this “landmark” contribution concludes
HE TWO-PIECE NORMAL DISTRIBUTION that “at a stroke, Fechner had created a method-ology for a new quantitative psychology.” However,his final work, Kollektivmasslehre , is devoted moregenerally to the study of mass phenomena and thesearch for empirical regularities therein, with ex-amples of frequency distributions taken from manyfields, including aesthetics, anthropology, astron-omy, botany, meteorology and zoology. In his Fore-word, Fechner mentions the long gestation periodof the book, and states its main objective as theestablishment of a generalisation of the Gaussianlaw of random errors, to overcome its limitations ofsymmetric probabilities and relatively small positiveand negative deviations from the arithmetic mean.He also appeals to astronomical and statistical in-stitutes to use their mechanical calculation powersto produce accurate tables of the Gaussian distri-bution, which he had desperately missed during hiswork on the book. But the book had not been com-pleted when Fechner died in November 1887.The eventual publication of
Kollektivmasslehre in 1897 followed extensive work on the incompletemanuscript by Gottlob Friedrich Lipps (1865–1931).In his Editor’s Preface, Lipps says that he receivedthe manuscript in early 1895 and that material hehas worked on is placed in square brackets in thepublished work. It is not clear how much unfin-ished material was left behind by Fechner or to whatextent Lipps had to guess at Fechner’s intentions.It would appear that the overall structure of thebook had already been set out by Fechner, sincemost of the later chapters have early paragraphs byFechner, before square-bracketed paragraphs beginto appear. Also, some earlier chapters by Fechnerhave forward references to later material that ap-pears in square brackets. In general, Lipps’ mate-rial is more mathematical: he was more of a math-ematician than Fechner, who perhaps had set somesections aside for attention later, only to run outof time. Lipps also has a lighter style: for example,Sheynin (2004, page 54) complains about some ear-lier work that “Fechner’s style is troublesome. Veryoften his sentences occupy eight lines, and some-times much more—sentences of up to 16 lines areeasy to find.” The same is true of the present work.The origin of the two-piece normal distributionis in Chapter 5 of
Kollektivmasslehre , titled “TheGaussian law of random deviations and its general-isations.” Here Fechner uses very little mathemat-ics, postponing more analytical treatment to laterchapters. He first presents a numerical example of the use of the Gaussian distribution to calculate theprobability of an observation falling in a given inter-val. The measure of location is the arithmetic mean, A , and the measure of dispersion is the mean abso-lute deviation, ε (related to the standard deviation,in the Gaussian distribution, by ε = σ p /π ). Ta-bles of the standard normal distribution are not yetavailable, and his calculations proceed via the errorfunction (see Stigler (1986), pages 246–248, e.g.),and prove to be remarkably accurate.In previous work Fechner had introduced other“main values” of a frequency distribution, the Zen-tralwert or “central value” C , and the DichtesteWert or “densest value” D , subsequently knownin English as the median and the mode. Arguingthat the equality of A , C and D is the exceptionrather than the rule, he next introduces the Zweis-paltiges Gauss’sche Gesetz to represent this asym-metry. Calculating mean absolute deviations fromthe mode separately for positive and negative devi-ations from D , the “law of proportions” is invoked,that these should be in the same ratio as the num-bers of observations on which they are based. Onconverting from relative frequencies of observationsto probabilities, and from subset mean absolute de-viations to subset standard deviations, it is seen thatthis is exactly the requirement discussed above, thatthe probabilities below and above the mode are inthe ratio σ /σ , to give a curve that is continuousat the mode. Fechner says that he first discoveredthis law empirically, and warns that determinationof the mode from raw data is not straightforward.He goes on to show that, in this distribution, themedian lies between the mean and the mode.The first mathematical expression of the two nor-mal curves with different precision soon appears inwhat is the first square-bracketed paragraph in thebook and the only such paragraph in Chapter 5.More extensive workings by Lipps appear in Chap-ter 19, “The asymmetry laws,” where every para-graph is enclosed in square brackets. Here Lippstraces the development and properties of the dis-tribution more formally, including an expression forthe density function [equation (6), page 297] whichcorresponds to equation (1) on converting betweenmeasures of dispersion. Nevertheless, the key stepsin that development, in Chapter 5, were Fechner’salone.We note that the second “generalisation” pre-sented later in Chapter 5 of Kollektivmasslehre isa form of log-normal distribution, but this receives
K. F. WALLIS less emphasis and is not our present focus of atten-tion.
3. AN EARLY REDISCOVERY: EDGEWORTH
In 1898–1900 Edgeworth contributed a five-partarticle “On the representation of statistics by math-ematical formulae” to the
Journal of the Royal Sta-tistical Society , each part appearing in a different is-sue of the journal. His objective was “to recommendformulae which have some affinity to the normal lawof error, as being specially suited to represent statis-tics of frequency.” The first two parts deal with the“method of translation,” or transformations to nor-mality, and the “method of separation,” or mixturesof normals, using modern terminology.In the third part Edgeworth considers the “methodof composition,” in which he constructs “a compositeprobability-curve , consisting of two half-probabilitycurves of different types, tacked together at the mode , or greatest ordinate, of each, so as to forma continuous whole, as in the accompanying figure”(1899, page 373, emphasis in original; the figure isvery similar to the solid line in Figure 1 above). Hegives expressions for the two appropriately scaledhalf-normal curves, as above, using the modulus ,equal to √
4. THE CRITICS: PEARSON AND SHEYNIN
The first English-language discussion of Fechner’scontribution appears in a 44-page article by KarlPearson, published in 1905 in
Biometrika , the jour- nal he had co-founded four years earlier. The arti-cle is a response to a review of Pearson’s and Fech-ner’s works on skew variation by Ranke and Greiner(1904) in the leading German anthropology journal.Pearson’s title quotes most of the title of the Ger-man article, omitting its reference to anthropology,and adds the words “A rejoinder,” although the run-ning head throughout his article is “Skew variation,a rejoinder.” He explains that the German journalhad provisionally accepted a rejoinder, but when itarrived the editors did not “see fit to publish” hisreply, so he placed it in
Biometrika , of which he was,in effect, managing editor. From a statistical pointof view this seems to have been a more appropriateoutcome, since his article contains much general sta-tistical discussion and is most often cited for its in-troduction of the terms platykurtic, leptokurtic andmesokurtic.However, Pearson’s article also contains extensiveattacks on Ranke and Greiner, who had argued that,for the anthropologist, only the Gaussian law is ofimportance. In this respect the article is a good ex-ample of his well-documented behaviour. For exam-ple, Stigler (1999, Chapter 1) opens by observingthat “Karl Pearson’s long life was punctuated bycontroversies, controversies he often instigated, usu-ally pursued with a zealous energy bordering on ob-session;” he “was a fighter who vigorously reactedagainst opinions that seemed to detract from hisown theories. Instead of giving room for other meth-ods and seeking cooperation, his aggressive styleled to controversy” (Hald (1998), page 651); he wasever “relentless in controversy” (Cox (2001), page 5)and “beyond question a fierce antagonist” (Porter(2004), page 266). Some of this antagonism is di-rected towards Fechner: although Pearson and Fech-ner are on the same side of the debate with Rankeand Greiner about asymmetry, Pearson sees “Fech-ner’s double Gaussian curve” as a rival to his familyof curves, and criticises it on both statistical andhistorical grounds.Using the parameterisation in terms of σ and σ as in equation (1), Pearson presents expressions forthe first four moments of the distribution. Ratherthan “the rough process by which Fechner deter-mines the mode and obtains the constants of thedistribution,” he shows that “fitting by my methodof moments is perfectly straightforward.” To do this,he obtains the cubic equation discussed above, andsays in a footnote (page 197) “This cubic was, Ibelieve, first given by Edgeworth,” but there is noreference. He observes that the skewness and kur- HE TWO-PIECE NORMAL DISTRIBUTION tosis are not independent of one another, so that“we cannot have any form of symmetry but themesokurtic.” He obtains the bounds on β givenabove, but notes that many empirical distributionswith values outside this range have been observed.Hence, Pearson’s overall conclusion is that “the dou-ble Gaussian curve fails us hopelessly.” Curiously,having defined platykurtic as “more flat-topped”and leptokurtic as “less flat-topped” than the nor-mal curve, as has become standard usage, he con-trarily describes Fechner’s double Gaussian curve asplatykurtic, despite having shown its positive excesskurtosis. Similarly, another curve, the symmetricalbinomial, is said to be “essentially leptokurtic, thatis, β <
3” (page 175).Turning to questions of precedence, Pearson’scounter claims appear in a footnote (page 196) at thestart of the statistical discussion summarised above,which reads as follows:Here again it is historically incorrect to at-tribute these curves to Fechner. They hadbeen proposed by De Vries in 1894, andtermed “half-Galton curves,” and Galtonwas certainly using them in 1897. See thediscussion in Yule’s memoir,
R. Statist.Soc. Jour.
Vol. LX, page 45 et seq.
Pearson was familiar with De Vries (1894), havingused two of his J-shaped botanical frequency distri-butions as Examples XI and XII in his 1895 arti-cle on skew variation. De Vries said that these de-served the name half-Galton (i.e., half-normal) sim-ply on the basis of the appearance of the empiri-cal distributions, and no fitting was attempted, nordid he make any proposal to place two such curvestogether to give a more general asymmetric distri-bution. Fechner’s curve had not been proposed byDe Vries. [Edgeworth knew that his composite curvehad not either, noting at the outset (1899, page 373)that “It will be observed that the following construc-tion is not much indebted to the “half-Galtonian”curve employed by Professor De Vries.”]Galton comes a little closer, but Pearson is againincorrect. His citation is inaccurate, since he clearlyhas in mind Yule’s paper read at the Royal Statis-tical Society in January 1896, published with dis-cussion later that year (Yule (1896a)). Galton hadopened the discussion at the meeting and mentionedhis method of percentiles as an alternative to themethod of frequency curves developed by Pearsonand applied by Yule. In response to a request at themeeting, he provided a memorandum giving fuller information on his method, which was publishedin the same issue of the Society’s journal (Galton(1896)), together with a reply by Yule (1896b). Gal-ton explains how his method of percentiles, in thisexample method of deciles, smooths the original fre-quency table or “frequency polygon” of Yule by in-terpolating deciles and plotting them. He then men-tions another approach, namely . . . the extremely rude and scarcely defen-sible method, but still a sometimes ser-viceable one, of looking upon skew-curvesas made up of the halves of two differ-ent normal curves pieced together at themode. . . .
On trying it, again for curios-ity’s sake, with the present series for allthe five years, there was of course no er-ror for the 2nd, 5th, and 8th deciles, . . . because he had inferred the spread or standard devi-ation of the lower half-normal distribution from thelower 20% point of the standard normal distribu-tion, and similarly for the upper part; he goes on todiscuss the errors of fit at the other deciles. But no“law of proportions” or scaling is applied, and theresulting curve is discontinuous, like the initial twohalves of normal curves in Figure 1. Yule (1896b)recognises this in his response, noting that, in con-trast, his skew-curve “presents a continuous distri-bution round the mode.” Galton was certainly not using Fechner’s curves.The erroneous assertions in Pearson’s footnotemay be due to his combativeness. Several authorsalso discuss the tremendous volume of work he un-dertook. For example, Cox (2001, page 6) observesthat he “wrote more than 90 papers in
Biometrika in the period up to 1915, few of them brief, and ap-pears to have been the moving spirit behind manymore.” He founded not only the journal but also theBiometrics Laboratory at University College Lon-don at this time. His son Egon remarks that thevolume of work “led inevitably to a certain hurry inexecution” (E. S. Pearson (1936), page 222). Thisremark is made during discussion of one of Pear-son’s two well-known errors, recently reappraised byStigler (2008), but it perhaps also applies to themistakes discussed above, which are of a smaller or-der of magnitude. Nevertheless, Pearson’s assertionsin the quoted footnote are mistaken, and his chal-lenge to Fechner’s claim to priority is unjustified,and thereby unjust.Sheynin (2004), in his review of Fechner’s statis-tical work, has a very brief discussion of the double-
K. F. WALLIS sided Gaussian law, quoting from sections of
Kollek-tivmasslehre that had been worked on by Lipps, andhence underestimating the role of Fechner’s law ofproportions. In his discussion (page 68) he statesthat the double-sided Gaussian law was not origi-nal to Fechner, this having been pointed out, force-fully, by Pearson (1905). As if quoting from Pearson,and giving no citation for De Vries (1894), Sheyninstates “De Vries, in 1894, had applied the double-sided law.” In this statement “applied” is somewhatstronger than Pearson’s “proposed,” hence is furtherfrom the truth, and Sheynin’s denial of Fechner’soriginality is similarly inaccurate and unjust.
5. LATER REDISCOVERIES ANDEXTENSIONS
The result of Pearson’s critique appears to havebeen that, with two exceptions discussed below, theFechner distribution, with this attribution, disap-peared from the statistical literature until its reap-pearance in Hald’s (1998) history. Meanwhile, threeindependent rediscoveries occurred.First, in the physics literature, is Gibbons andMylroie’s (1973) “joined half-Gaussian” distribu-tion, cited by Johnson, Kotz and Balakrishnan(1994), as noted above; the distribution is fitted bywhat statisticians recognise as the method of mo-ments. Second, in the statistics literature, is the“three-parameter two-piece normal” distribution ofJohn (1982), also cited by Johnson, Kotz and Bal-akrishnan (1994); John compares estimation by themethod of moments and maximum likelihood. In thesame journal Kimber (1985) notes that John (1982)is a rediscovery, with reference to Gibbons and Myl-roie (1973); he proves the asymptotic normality ofML estimators and provides a likelihood ratio testof symmetry. Finally, in the meteorology literature,Toth and Szentimrey (1990) introduce the “binor-mal” distribution, again fitted by ML, with a testof symmetry. The same name is used by Garvin andMcClean (1997), who nevertheless again attributethe distribution to Gibbons and Mylroie. In all thesearticles the distribution is parameterised in terms ofthe mode, using various symbols, and the standarddeviations σ and σ , as in (1) above. An alternativeparameterisation, with a single explicit skewness pa-rameter, is given by Mudholkar and Hutson (2000),who do acknowledge Fechner’s priority.A modern, but pre-Hald (1998) attribution to Kollektivmasslehre occurs at the start of an explo-ration by Runnenburg (1978) of the mean, median, mode ordering. He notes that Fechner had shownthis
Lagegesetz der Mittelwerte for the two-piecenormal distribution, and investigates more generalconditions in which it holds. The second excep-tional appearance of the Fechner distribution in thestatistical literature pre-1998 is more substantial.Barnard (1989), seeking a family of distributions“which may be expected to represent most of thetypes of skewness liable to arise in practice,” intro-duces the distribution f ( x ) = K exp (cid:20) − (cid:18) − M ( x − µ ) σ (cid:19) a (cid:21) , x ≤ µ,K exp (cid:20) − (cid:18) x − µσ (cid:19) a (cid:21) , x ≥ µ, which reparameterises and generalises the two-piecenormal distribution in equation (1). He calls it theFechner family, because by allowing the skewnessparameter M ( M >
0) to differ from 1 it embod-ies Fechner’s idea in
Kollektivmasslehre of havingdifferent scales for positive and negative deviationsfrom the mode, µ . It also allows for nonnormal kur-tosis by allowing a (1 ≤ a < ∞ ) to differ from 2. Thescale parameter σ is equal to the standard deviationif ( M, a ) = (1 ,
2) but not otherwise, in general. This“Fechner family of unimodal densities” also appearsin a later article (Barnard (1995)), which is cited byHald (1998, page 380). We note that the case a = 1,the asymmetric Laplace distribution, has a consider-able life of its own, beginning before Barnard’s work:see, for example, Kotz, Kozubowski and Podgorski(2001, Chapter 3) and the references therein.Two further extensions of note, independent ofFechner, can be found in Bayesian statistics. Forthe application of Monte Carlo integration with im-portance sampling to Bayesian inference, Geweke(1989) uses “split” (i.e., two-piece) multivariate nor-mal and Student- t distributions as importance sam-pling densities. The generalisation by Fernandez andSteel (1998) is also cast in a Bayesian setting: as inBarnard’s Fechner family, there is a single skewnessparameter, which is convenient whenever it is de-sired to assign priors to skewness; nevertheless, ithas general applicability. For any univariate PDF f ( x ) which is unimodal and symmetric around 0,Fernandez and Steel’s class of two-piece or split dis-tributions p ( x | γ ), indexed by a skewness parameter γ ( γ > p ( x | γ ) = Kf ( γx ) , x ≤ ,Kf (cid:18) xγ (cid:19) , x ≥ HE TWO-PIECE NORMAL DISTRIBUTION where K = 2( γ + γ − ) − . If γ > γ produces the mirror imageof the density function around 0. Unlike Barnard’sFechner family there is no explicit kurtosis parame-ter; kurtosis is introduced, if desired, by the choiceof f ( x ), most commonly as Student- t . An extensionwith two tail parameters to allow different tail be-haviour in an asymmetric two-piece t -distribution isdeveloped by Zhu and Galbraith (2010). ACKNOWLEDGMENTS
I am grateful to Sascha Becker, David Cox, ChrisJones, Malte Kn¨uppel, Kevin McConway, MaryMorgan, Oscar Sheynin, Mark Steel, Stephen Stiglerand two referees for comments and suggestions atvarious stages of this work. I am especially indebtedto Karl-Heinz T¨odter for his expert assistance withthe German-language works cited. Thanks also tothe Resource Delivery team at the University ofWarwick Library. REFERENCES
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