I would like to extend my deepest gratitude to Michael Friendly for introducing me to the work of Guerry. I am extremely grateful to Ludovic Lebart, Gilles Palsky, Bernard Ycard, and Michael Friendly for encouraging me to write my notes on Guerry’s Plate XVII. I also want to thank Brian Francis for the information on the British libraries and, of course, the two referees. All errors and false interpretations remain mine.
1Michael Friendly in a recent article (Friendly 2022) where most useful references can be found; his article includes in particular a reproduction of the graphic which stimulated the present interest in librations, the last plate in Guerry’s 1864 book (see below Fig. 4). The book title translates as "Moral statistics of England compared to moral statistics of France » (Guerry 1864). The topic was not new in 19th century. In particular, England had been the subject of many French observers. Among a significant list of authors are two early statisticians Charles Dupin (1784-1873) and Alexandre Moreau de Jonnès (1778-1870), both members of the Academy of Sciences. Lively descriptions of the works of statisticians of this time and of their acceptance by the civil society can be found in (Bru, Bru, and Bienaymé 1997), (Desrosières 1988) and Bernard Ycart (Ycart 2016). Guerry’s publication contains two distinct parts. First, a long introduction on the history of statistics, its developments in new fields (demography, meteorology, criminology, literature), its differentiation from probability calculus, and the specificity of data analytics (statistique analytique). Preceeding the Introduction itself is a reproduction of the reports from the Academy of Sciences by which Gerry was awarded twice a Montyon prize (in Statistics) in 1833 and 1860. It is followed by a "Table of names cited in the Introduction » which shows more than five hundreds entries! The Introduction is followed by an Atlas preceded by a full page dedicated to the "Explanation of the terminology and symbols of statistical analysis ».
2The Atlas contains 12 paired statistical maps (France and England including Wales on separated pages, Plates I to XII) presenting data on 6 broad categories of crime, 2 on education (both countries, Plates XVIII to XIV), 1 on suicide (France only, Plate XV). Plate XVI graphically displays the propensity (penchant) to 11 types of crime in France and England according to age. Plate XVII, the last plate in the Atlas, is a complicated graphical representation of crimes according to types and related matters (religion, sex, education,...) as recorded from various sources in the English counties. Undeniably, Plate XVII singles out. The graphic reflects two main ambitions: i) a description of the compared variability of the distributions of each type of crimes and ii) a systematic representation of the intensity of the links between a fixed list of a-priory explanatory variables and each types of crime. In all likelihood, this plate was for Guerry the masterpiece of his Atlas.
3The Atlas contains 12 paired statistical maps (France and England including Wales on separated pages, Plates I to XII) presenting data on 6 broad categories of crime, 2 on education (both countries, Plates XVIII to XIV), 1 on suicide (France only, Plate XV). Plate XVI graphically displays the propensity (penchant) to 11 types of crime in France and England according to age. Plate XVII, the last plate in the Atlas, is a complicated graphical representation of crimes according to types and related matters (religion, sex, education,...) as recorded from various sources in the English counties. Undeniably, Plate XVII singles out. The graphic reflects two main ambitions: i) a description of the compared variability of the distributions of each type of crimes and ii) a systematic representation of the intensity of the links between a fixed list of a-priory explanatory variables and each types of crime. In all likelihood, this plate was for Guerry the masterpiece of his Atlas.
- 1 Polymath Charles Dupin (1784-1873), astronomer Louis Mathieu (1783-1875), chemist and agronomist Je (...)
- 2 The Société de Statistique de Paris was founded in 1860 and the first issue of its Journal appeared (...)
4(Bru, Bru, and Bienaymé 1997) recall that, in 1860, the five Montyon prize examiners11 did emphasize in their laudatory report to the Académie of Sciences Guerry’s « terror » of being accused of using probability calculus. Regardless of that, Guerry was awarded his second Montyon prize (Statistics). Possibly pleased, Guerry did not hesitate to reproduce this report at the head of his 1864 book and did not contest his « terror » in an addendum. Six years later, in a speech delivered at Guerry’s funeral and published shortly afterward, Alfred Maury (1817-1892) recalled that Guerry’s work will remain as « a model of true statistics... » (see(Maury et al. 1867), p. 7). Unfortunately, Maury did not develop further the concept. It seems that, for Guerry, statistics was simply accurate data uncontaminated by probabilistic speculations (see (Bru, Bru, and Bienaymé 1997) p. 176). It was then a accepted approach that still has its followers. In particular, it was the approach claimed in the series of Atlas de Statistique Graphique (1879-1900) which will appear a few years later. For example, the 1882 Atlas recalls in its preface that « The purpose of this work is to provide materials and not to present opinions » (« Le but de ce travail est de fournir des matériaux et non de présenter des opinions »)! Surprisingly Guerry is very rarely mentioned in the Journal de la Société de Statistique de Paris2.2 As reflected by the full title of his book (see (Guerry 1864)), and regardless of its ideological standpoint, Guerry has done considerable investigative work to gather the data he presents in his Atlas. Are his data fully objective? To a large extent. But it is known that data always present aspects of social constructs, an issue not discussed in this paper. Were all Guerry’s statistical analyzes uncontaminated by probability calculus? It can be objected that most tools of descriptive statistics can be reformulated in a flexible probability context.
- 3 The Société de Statistique de Paris was founded in 1860 and the first issue of its Journal appeared (...)
- 4 The Société de Statistique de Paris was founded in 1860 and the first issue of its Journal appeared (...)
- 5 See http://ark.bnf.fr/ark:/12148/cb39363150x
5Hippolyte Diard (1795-1877), a high magistrate, reports in his notice devoted to the late Guerry (see (Maury et al. 1867), p. 13) that his book published in 1864 had received a very favorable international audience in Germany, England and the United States of America. In the United Kingdom, the Royal Collection Trust3 owns the book and mentions an inventory book measurement of 58cm × 3cm.) (The format of the book, a large in-folio4 , has possibly hindered its circulation.) And in France? His book is currently held in a few French public libraries. But if the book is not freely available for downloading on Gallica’site it is available in some foreign libraries. In particular, the Bibliotheca Patrimonial Digital of the Universitàt de Barcelona offers a downloadable version which I have used in the Figures below to reproduce some original plates. As a matter of fact, the proofs of the Atlas or of its main plates were circulated in Great-Britain before its publication in 1864. Guerry reports that the proofs had been communicated to Charles Babbage (1791-1871) in July 1860 (see (Guerry 1864), p. XVI) and that Babbage deposited them in the library of the London Statistical Society (see (Guerry 1864), note 6, p. LVI). The Catalogue général of the Bibliothèque Nationale de France lists an edition of the Atlas alone dated circa 18605. The proofs of the final publication in 1864? But the Albert Sloman Library which houses the Royal Statistical Society’s historical book collection does not list either edition.
6I report below the structure of the Atlas with emphasis on the occurrences of the word libration.
- 6 Downloaded from pàgina 81 at https://bipadi.ub.edu/digital/collection/atlesmed/id/78859/rec/2
7On page numbered LXV of the Introduction a two-column list gives the terms and symbols used in the Atlas which follows. This table is reproduced in Figure 116 . Its subtitle is « moral statistics or moral statistics analytics » (statistique morale ou statistique morale analytique). As a branch of science, statistics is singular in French as is analytic which sounds more fundamental than just analysis. Surprisingly the outdated term analytics is currently making a massive come back in the area of statistics: data analytics, diagnostic analytics, etc. The table is reproduced in Figure 1. The list of statistics summaries of location and associated symbols is fair: mean, median, libration, midrange. For most indices, Guerry introduces an iterated use on the partial series of observations with lower (resp. larger) values than the overall value and even iterates for the mean. Unfortunately, most definitions concerning the use of libration are evasive as will be seen. The absence of standard statistical indices of dispersion may seem also strange to a contemporary reader. But it will appear that a skilled use of statistical indices of location may other a sound alternative (see below subsection 2.4).
8The Atlas presents 15 statistical maps visualizing the geographic repartition of non-negative quantitative variables. This mode of representation had been publicized in 1827 by Charles Dupin and in 1837 by Adolphe d’Angeville. (Statistical maps for qualitative variable were in use much earlier, a well circulated example being the map of the salt tax published by Jacques Necker (1732-1804) in his 1781 Compte rendu au Roi.)
Figure 1
Table of terms and symbols used in the Atlas.
Courtesy of Biblioteca Patrimonial Digital de la Universitàt de Barcelona (BiPaDi): Creative Commons Public Domain Mark 1.0 .
Figure 2
Plate II of the Atlas
Courtesy of Biblioteca Patrimonial Digital de la Universitàt de Barcelona (BiPaDi): Creative Commons Public Domain Mark1.0 .
9For each count variable, it seems that Guerry used an 8 level grayscale which corresponds to 7 boundary intervals set by the average values m–, m-, m-+, m, m+-, m+, and m++, where m is the mean value of the variable displayed, m+ the mean of values greater than m, m+-, the mean of the values greater than m and lower than m+, etc. (Gerry uses variations on the letter M rather than m.) Note that any measure of central tendency could be used instead of the usual mean, librations for example as discussed below in subsection 3.1.
- 7 Downloaded from pàgina 89 at https://bipadi.ub.edu/digital/collection/atlesmed/id/78859/rec/2
10The maps are marginally annotated by statistical graphics. An example is given by Plate II which is represented here as Figure 27 . It represents the number of crimes against people in France as established by départements in 1825. The graphic at the bottom of the plate describes the data used to produce the statistical map. This graphic is intended to visualize the distribution of the observed counts. It would be considered non standard nowadays: observed values of counts on the vertical axis, ranked départements as jointed intervals on the horizontal axis. The values of m–,..., m++ and of the libration (\(\cup\cap\), a sort of lying S reflecting an oscillation) are reported on the interpolated curve.
11The values of the upper and lower librations are also reported on this graph by their symbolic representation \((\cup \text{ and } \cap)\). The departments presenting values closest to the librations are emphasized by darker columns. This simple device allows to link the départements in the map (they are numbered 1 to n) to the observed values \((y_{1} > y_{2} > . . . > y_{n})\) and to the values taken by the indices of central values above mentioned. Superimposed on that graph are also the values of the librations of some « ordered » series (see below subsection 3.2).
- 8 Downloaded from pàgina 145 at https://bipadi.ub.edu/digital/collection/ atlesmed/id/78859/rec/2
12Plate XVI of the Atlas is reproduced here as Figure 38 . Plate XVI consists in a parallel display of the propensity to 11 types of crime in France and England as a function of classes of age for France and 7 for England. The ingenuity is in the pairing of crimes committed in two different judiciary systems. In the heading of this Plate, the symbol for representing the « libration centre for each age »,\(\cup\cap\) , is given but apparently not clearly used (see also the reproduction of Plate XVI on p. 18 of Friendly [9]). Note on the first row of graphics, two protohistograms hanging upside down; unfortunately, their construction principle is unclear.
Figure 3
Plate XVI of the Atlas: parallel display of the propensity to 11 types of crime in France and England as a function of 7 classes of age.
Courtesy of Biblioteca Patrimonial Digital de la Universitàt de Barcelona (BiPaDi): Creative Commons Public Domain Mark1.0 .
Figure 4
Plate XVII of the Atlas: parallel statistics of type of crimes in England.
Courtesy of Biblioteca Patrimonial Digital de la Universitàt de Barcelona (BiPaDi): Creative Commons Public Domain Mark1.0 .
- 9 Downloaded from pàgina 149 at https://bipadi.ub.edu/digital/collection/atlesmed/id/78859/rec/2
13Plate XVII is reproduced here as Figure 419 (see also (Friendly 2022), p. 19). Guerry others there a one sheet summary of numerous statistical analyses, a memento of several hundred calculations! Useful for whom? Certainly for him! For an occasional reader? Not really! Notwithstanding a striking artistic quality, Plate XVII is hard to read. No clear message, except in its title. About 60 small symbols (including a dozen letters typed in 4 fonts) are repeatedly sparkled on most rows!
14Each row represents the distribution of the observed counts of a specific type of crime (bigamy, etc.) or of an aggregation of crimes (violent crime, etc.) over the 52 counties of Great-Britain. There are in all 32 types of crime displayed as rows: 23 elementary types, 9 (3 + 6) aggregations or averages.
15Every type of crime counts is ordered (decreasing values from the left to the right). Each county is anonymously represented by a rectangle of constant size except for the county ranked 26th (median rank) which is twice as large. It must be emphasized that if any two rows are compared, the rectangles in a same column do not necessarily represent the same county. For the first 23 elementary types, four shades of blue correspond to the intervals defined by the 5 number summary (min, m-, m, m+, max) of the corresponding counts (mean and conditional means). For the last 9 aggregations of crime, four shades of grey correspond to the intervals defined by the 5 number summary (min, \(\cup\), \(\cup\cap\) , \(\cap\) , max) of the corresponding counts (lower libration, libration, and upper librations). Guerry’s approach is not too far from that of John Tukey (1915-2000) who will advocate the more robust 5-number summary (min, lower Hinges, Median, upper Hinges, max) where the Hinges are conditional medians (see (Tukey 1977), p. 33).
16As previously mentioned, county names are not reported in their associated rectangle except for the highest observed value (on the left of the most left rectangle) and the lowest observed lower (on the right of the most right rectangle).
17However, for any value in the range of the distribution, an implicit rule allowed to find « the » county with the closest observed value: the incidence point (le point d’incidence) for this value. Thus, for each crime type, the values of any summary statistics describing the distribution of a type of crime (mean, midrange, median, libration, etc.) could be represented by specific symbols printed in the corresponding rectangle.
18How does Guerry describe the fact that two series observed on the same statistical units may relate? His exact wording is « relationships of reciprocal dependence, of coincidence or of opposition » (« rapports de coïncidence coïncidence ou d’opposition », see p. XLVIII of the Introduction) or « relationships of coincidence positive, negative, or neutral » (« rapports de dépendance réciproque, positive, négative ou neutre », see page LXVL of the Introduction). Guerry remains cautious in his formulation of dependence. I risk below an explanation for his circumspection.
19Adolphe d’Angeville (see (d’Angeville 1836), p. 76) had earlier investigated the quantification of dependence between two series based on rank considerations, a premonitory intuition (see (Ycart 2016), pp. 23-24). Unfortunately, d’Angeville reported an association between the taxes on doors and windows (a proxy for household wealth) and the level of education which was found preposterous by members of the Académie Royale des Sciences (see (d’Angeville 1836), p. 76).
20Why libration? The term libration (from the Latin libratio balancing) was certainly known to Guerry. An early French definition of the word can be found in the Dictionnaire de l’Académie Française (1777) (Duplain 1777):
Libration is used in Astronomy and means an apparent or real oscillation of a satellite.
21Investigating the Moon libration, but not under that designation, the German astronomer Johann Tobias Mayer (1723-1762) published a clever solution to a multiple linear regression in 1750 (see (Stigler 1986), Tobias Mayer and the libration of the Moon, pp. 16-25). This was approximately fifty years before the introduction of Least Squares! But Mayer’s problem does not directly relate to Guerry’s introduction of libration indices. However I shall suggest a connection with the moon in my concluding remarks.
22In Guerry’s book, the term libration appears in several expressions: libration, normal libration, absolute libration, coincidence libration and intersection libration. Most are not clearly defined and a partial clarification is attempted below. My conjecture is that Guerry uses the word libration to name both a central tendency index of an observed non-negative statistical variable and a quantification index of the directed link between a non-negative exogenous variable and a specific type of crime.
23Guerry introduces a normal libration and an absolute libration. Both indices have the same definition but are derived on two series of observation S and series \(\mathbb{S}\) as denoted by Guerry (see p. LXV of the Atlas or Figure 1 above). But the derivation of S from S is not detailed. Assuming that \(\mathbb{S}\) is the sorted version of S, I will risk below an interpretation.
24Let \(X = {x_{i}, i = 1, . . . , n}\) be a series of observed non-negative values listed in a given order (Guerry’s S), its total sum \(\sum_{i=1}^{i=n}x_{i}\), and its mean \(m_{X} = \frac{1}{n} \sum_{i=1}^{i=n} x_{i}\). The value for determining the libration is \(\frac{1}{2} \sum_{i=1}^{i=n} x_{i}\). The normal libration point of the series X is the index \(i^{\star}\) such that \(\sum_{i=1}^{i=n} x_{i} \approx \frac{1}{2}nm_{X}\). Note that if started from n \(\left(\sum_{i=1}^{i=i^{\star}}x_{n+1-i}\right) \approx \frac{1}{2}nm_{X}\) the procedure may return a different value for \(i ^{\star}\).
25Now the series X might have been sorted by decreasing values as is the case for the 32 types of crime paralleled in Plate XVII: \(x_{1} > x_{2} > . . . > x_{n}\) (Guerry’s S?). In that case « the » value and observation obtained as above is called absolute libration and can be introduced in Guerry’s graphics: for each type of crime in Plate XVII, the value of the associated libration is materialized in the corresponding rectangle by its specific symbol (\(\cup\cap\)). (It is also the case for the midrange which has its own specific symbol)
26The label of Plate XVII in the Atlas translates as « General causes of crimes » and its subtitle as « compared librations of type of crimes with linked variables observed on the same geographic units ». Obviously, the problem brings forth statistical concepts such as regression (known at that time), correlation or rank correlation (unknown at that time). The approach chosen by Guerry seems to be intrinsically linked to the system of visualisation adopted in Plate XVII: a label representing X is positioned onto the distribution of Y in a way which reflects the negative, neutral or positive influence of X on Y.
27How a libration index can be used for quantifying an oriented link between two series simultaneously observed X and Y ? Guerry, in the page devoted to the symbols and terminology inserted before Plates I to XVII (see (Guerry 1864), p. LXV), introduces the notions of « ordering » and « ordered » series:
\(\Omega_{IJT...}\) : Order of decreasing progression of the terms T of the « ordering » series \(\Omega_{IJT...}\) Fixed order (see plate XVII.).
(\(\Omega_{IJT...}\) : Ordre de progression décroissante des termes T de la série ordonnatrice \(\Omega_{IJT...}\) Ordre fixe. (Voy. pl. XVII.).)
\(O_{ab...}\): Order of superposition of each of the terms t, of the « ordered » series \(O_{ab...}\) on each of the terms corresponding in the « ordering » series \(\Omega_{IJT...}\) Variable order. (See Plate XVII.)
(\(O_{ab...}\): Ordre de superposition de chacun des termes t, de la série ordonnée \(O_{ab...}\) sur chacun des termes respectivement correspondants T, de la série ordonnatrice \(\Omega_{IJT...}\) Ordre variable. (Voy. planche XVII.))
28To be more specific, let Y be the statistical variable of the counts of a given type of crime observed on n counties (the « ordering » series \(\Omega_{IJT...}\)) and X an accompanying non-negative variable observed on the same counties (the « ordered » series \(O_{ab...}\)) which may be linked to Y . The data at hand are then \(\{(y_{i}, x_{i})\mid i = 1, . . . , n\}\) where \(y_1 > y_2 > . . . > y_n\). (No ties are assumed for simplicity.) The associated values of X do not necessarily follow the ranking of Y. Intuitively, in case of positive influence of X onto Y , high (resp. low) values of X should be associated to high (resp. low) values of Y and the other way around in case of negative influence.
29In the bottom note to Plate XVII Guerry recalls that:
The number of the term of the ordering series \(\Omega\) after which falls the libration center of an ordered series O is marked at its rank by the letter assigned to this series.
Le numéro du terme de la série ordonnatrice \(\Omega\) après lequel tombe le centre de libration d’une série ordonnée O est marqué à son rang par la lettre affectée à cette série.
30My conjecture is that Guerry computes cumulative sums of the \(x_{i}\) conditionally on the ranked values of y: \(H_{X}(y_{n}) = x_{n}\), \(H_{X}(y_{n-1}) = x_{n}+x_{n-1}, ..., H_{X}(y_{n}) = x_{n} + x_{n-1} + ...+ x_{1} = n\mu_{X}\). Introducing the univariate concept of libration, Guerry looks for a value \(y_{i^{*}}\) of Y such that \(H_{X}(y_{i^{*}}) \approx \frac{1}{2}n\mu_{X}\). Again, the conditioning can be considered top-down: \(H_{X}(y_{1}) = x_{1}, H_{X}(y_{2}) = x_{1} + x_{2}, ..., H_{X}(y_{n}) = x_{1} + x_{2} + ...+ x_{n} = n\mu_{X}\). But this may return a (hopefullynot-too) different value for \(i^{\star}_{X}\). Function \(H_{X}\) is better reformulated as either \[\text{bottom-up: } H_{X}(y_{k}) = \sum_{i=1}^{i=n+1-k} \sum_{j=n}^{n} \frac{1}{n}\frac{x_{j}}{\mu_{X}}1_{\left(y_{i},x_{j})\mid y_{i} \geq y_{k}\right)}\] or \[\text{top-down: } H_{X}(y_{k}) = \sum_{i=1}^{k} \sum_{j=n}^{n} \frac{1}{n}\frac{x_{j}}{\mu_{X}}1_{\left((y_{i},x_{j})\mid y_{i} \leq y_{k}\right)}\] where \(y_{i^{*}}\) corresponds to the .5 quantile.
31Rank \(i^{\star}\) can be introduced in the row visualizing Y and the symbol representing X can be inserted in the rectangle corresponding to \(i^{\star}_{X}\). The 4 shades of blue associated with the scale set on Y facilitate a visual interpretation of the negativeness (lightest blue), or neutrality, or positiveness (darkest blue) of the association between Y and X. For each of the 32 types of crime (the « ordering » series Y ), the libration of the 52 « ordered » series were thus computed. It turns out that for most types and for most of the « ordered » series the values falls near the crime median. But there are a few exceptions which are highlighted in Plate XVII by connecting dashed lines; these show important variations in magnitude and sign.
32In a probabilistic setting, X and Y are two non-negative random variables with joint cumulative distribution \(F_{XY}(x, y)\), joint density \(f_{XY}(x, y)\), and strictly positive expected values \(\mu_{Y}\) and \(\mu_{X}\). Since \(f_{XY}(x, y) = f_{Y}(y)f_{X\mid Y}(x\mid y)\), \[H_{X}(y) = \int_{y}^{\infty}f_{Y}(\eta)\int_{0}^{\infty}\frac{\zeta}{\mu_{X}} f_{X\mid Y} (\zeta\mid \eta)d\zeta d\eta\] Obviously \(H_{X}(0) \geq 0\) and \(H_{X}(\infty) = 1\). Let now \(y^{\star}\) be such that \(H_{X}(y^{\star}) = 1/2\) . It is this value \(y^{\star}\) which is used to evaluate the intensity and sign of the link from X onto Y by introducing it in the scale (min, \(\mu_{Y^{+}}\) , \(\mu_{Y}\) , \(\mu_{Y^{-}}\), max). Again, note that this can be considered bottom-up: \[H_{X}(y) = \int_{0}^{y}f_{Y}(\eta)\int_{0}^{\infty}\frac{\eta}{\mu_{X}} f_{X\mid Y} (\eta\mid \zeta)d\eta d\zeta\] which is more in line with modern conventions.
33In any case, finding a general value \(y^{\star}\) such as \(H_{X}(y^{\star}) = .5\) remains not straightforward. This index is not symmetric (X on Y or Y on X), although Guerry speaks of reciprocal dependency (dépendance réciproque) on bottom page of page LXV of the Introduction; it is invariant under multiplication by positive constants but not under affine transformations.
34Let (Y, X) be a bivariate normal with mean vector \((\mu_{Y} ,\mu_{X})\). (Note that Guerry would have strongly opposed this illustration!) The bivariate normal model for (Y, X) being introduced here in a situation where all observations are non-negative, it is assumed that the marginal means are strictly positive. This allows to extend the marginal cumulative concentration curves for Y and X. The variances of Y and X is denoted as \(\sigma_{YY}\) , \(\sigma_{XX}\) and their correlation coefficient as \(\rho\). The conditional expectation of X given Y = y, \(\textbf{E}[X\mid Y = y]\), is \(\alpha +\beta.y\) where \(\alpha = \mu_{X} -\beta\mu_{Y}\) and \(\beta = \rho\sqrt{\frac{\sigma_{XX}}{\sigma_{XY}}}\) , the usual least squares regression coefficients of X onto Y . Nowadays, if investigating the influence of X onto Y, it would be more appropriate to consider the conditional expectation of Y given X rather than that of X given Y . But this choice of conditioning results from the assumed interpretation presented in subsection 3.2 above.
35The general bottom-up formula \[H_{X}(y) = \int_{-\infty}^{y}f_{Y}(\eta)\int_{-\infty}^{+\infty}\frac{\zeta}{\mu_{X}} f_{X\mid Y} (\zeta\mid Y=\eta)d\zeta d\eta\] can be written as: \[H_{X}(y) = \frac{1}{\mu_{X}}\int_{-\infty}^{y}f_{Y}(\eta)(\alpha+\beta\eta)d\eta\] or \[H_{X}(y) = \alpha\frac{1}{\mu_{X}}F_{Y}(y)+ \beta\frac{\mu_{Y}}{\mu_{X}}G_{Y}(y)\] where \(G_{Y}\) is the cumulative concentration function associated to the marginal cumulative distribution function \(F_{Y}\) of Y as introduced in subsection 3.1. The formula looks straightforward and has a clear connection with the well known least-square regression coefficients.
36Again, finding a value \(y^{\star}\) such as \(H_{X}(y^{\star}) = .5\) and interpreting it is not straightforward. The variations of function \(H_{X}\) have to be investigated in all possible configurations allowed by the the hypothesis \[\mu_{X} > 0, \mu_{Y} > 0, \beta = \rho \sqrt{\frac{\sigma_{XX}}{\sigma_{YY}}} : \: \alpha > 0 \text{ and } \rho > 0,\: \alpha = 0 \text{ and } \rho > 0,\: \alpha < 0 \text{ and } \rho > 0,\: \alpha > 0 \text{ and } \rho = 0, \: \alpha > 0 \text{ and } \rho < 0\]
37It turns out the solution of the equation \(H_{X}(y) = .5\) is greater (resp. lower) than \(\mu_{Y}\) for \(\rho > 0\) (resp. \(\rho\) < 0) and exactly \(\mu_{Y}\) for \(\rho\) = 0. This fully supports (my conjectural understanding of) Guerry’s intuition in a context not recommended for modeling count data.
38In Plate XVII (see Figure 4), some symbols representing the ordered series are displayed as follows: systematically, 3 covariates are introduced along column 51 (resp. 2) of the 23 types of crime. For example for crime IX, Counterfeiting (Fausse monnaie), one can read \(\lambda\) (Assizes), \(\mu\) (Summary conviction), and b (Population born in the same county) on the left side and \(\gamma\) (Female education from marriage registers), \(\kappa\) (Female education from Assizes and summary convictions), and a (anglican), on the right side. Next to them are the words libration of intersection? How do they « intersect » with crime IX? In the example above, all 6 symbols of the corresponding ordered series are also mentioned in the median rectangle of the row describing the distribution of crime type IX! A libration of intersection is introduced in the table describing the terms and symbols used in the Atlas (see subsection ):
Libration of intersection: value of the ordered series \(O_{a,b,...}\) above the point corresponding to the centre of normal libration \(\cup\cap\) of the ordering series \(\Omega_{I,II,...}\). Variable order (see Plate. XVII)
(Libration d’intersection: Valeur de la série ordonnée \(O_{a,b,...}\). au dessus du point correspondant au centre de libration normale \(\cup\cap\) de la série ordonnatrice \(\Omega_{I,II,...}\) Ordre variable (voy. pl. XVII))
39In the notation introduced in subsection 3.2, this value might be the X coordinate in the couple \(\left(y_{i^{\star}}, x_{i ^{\star}}\right)\) \(i^{\star} = min \left(k \mid \sum_{i=1}^{i=k} y_{i} > \frac{1}{2}nm_{Y}\right) = \cup\cap_{Y}\) (Remember that \(y_1 > y_2 > . . . > y_n\)) Clearly, this unique value of \(x_{i^{\star}}\) is not « robust » against random fluctuations of X. But this remains to be investigated.
40In the graphic display of Plate XVII, Guerry also draws a « curve of positive coincidence » and a « curve of negative coincidence ». As specified in the margin, Guerry recalls that:
The two curves refer to the intersection libration. They are superimposed with this table of the normal libration, without being connected to it by their graphic construction.
(Les deux courbes se rapportent à la libration d’intersection. Elles sont superposées avec ce tableau de la libration normale, sans s’y rattacher par leur construction graphique.)
Both curves clearly envelop his representation of the distribution of the « ordering » series but how were they practically constructed?
41As seen in subsection 3.2, Plate XVII records the results of an enormous number of elementary computations for positioning in the distribution of each type of crime the symbols associated with the covariates. Remember that there are 32 types of crime (the 23 + 9 so-called ordering series) and 52 covariates (the 52 so-called ordered series) and that, for each ordering series, Guerry did compute 52 value of the libration index of link between the ordered series and the ordering series. The computation of each libration is trivial (cumulative sums) but their number (32 × 52) required a good management of the data and of the conditional sorting. To this end Guerry even designed and used an ad hoc sorting machine which he named ordonnateur analytique. His data management and computation process are not known in details but a tentative an partial reconstruction can be attempted.
42A description of the data frame can be attempted from the crude description given by Guerry (see (Guerry 1864) p. LV). It consisted in a two-way table crossing 54 columns (the 52 counties + 2) and 159 rows (tentatively, 1 row for the name of county, 32 for the types of crime, 32 for their associated ranks, 52 columns for the covariates, and 42 unexplained). The two extra columns were possibly used to store the row total or its half value and possibly some other index (see subsection 3.1). The data frame was then sliced row wise, each row containing all the data describing the associated county in the form of a long strip three meters long (see (Guerry 1864) p. LV). This format simplified the sorting of the rows. In passing, it is hard to resist mentioning Jacques Bertin (1918-2010) who introduced the use of permutations of rows and columns for the visualization of a « physical » data matrix (see (Bertin 1970), Fig. 16).
- 10 The rows were sorted by batches of 10 to facilitate hand computations
43The ordonnateur is presumably a rudimentary device which helped to compute by hand10 the cumulative sum of an ordered series sorted according to the ordering series (see (Friendly and de Saint Agathe 2012) and (Friendly 2022)). By joining the term analytique to the word ordonnateur Guerry makes an implicit reference to Babbage’s Analytical Engine which he knew. For both devices, no consistent understanding can be obtained from the descriptions at hand. However an idea of the mechanisms involved can be obtained from the figures reproduced in a paper by (Shilov and Silantiev 2016)) where they review the life and scientific activity of the Russian Semen Korsakov (1788-1853). Korsakov had invented several machines intellectuelles as soon as 1832. A recent reconstruction of Bertin’s « domino machine » is described on a web site (See (Perrin, Dragicevic, and Fekete 2000)). Note that all these tasks are more or less straightforward in modern computing environments.
44Curiously, a hundred years later, in 1955, IBM France started to sell its Electronic computer IBM 650 under the name Ordinateur IBM 650 (see (Depecker 2015)). But the name ordinateur had been suggested by Jacques Perret (1906-1992), philologist, without any reference to Guerry’s ordonnateur.
45This article does not fully explain all the indices that Guerry used in his 1864 work, as some librations resisted my interpretation. But the quantification of associations between non-negative variables remains an actual question and an approach by rank considerations is fully relevant nowadays. It must be noted nevertheless that neither the name of Guerry’s libration indices, nor the name of his calculating machine have survived him. Worse, if an evolutionary tree of statistical atlases was available, Guerry’s 1864 Atlas would be on a dead branch. My harsh judgment is based on the overwhelming accumulation of information in each figure of the Atlas and, in particular, on its Plate XVII. A rigid application of the principle of non-interpretation of data on the grounds of objectivity has certainly led Guerry to stay too close to his exceptional data base. Still, the introduction of the term libration tells something interesting. Even more, it is the sign of a clever intuition. Guerry was convinced that there was something in the analyzes of the distributions of crime counts which had to go beyond the computation of mere averages, the fashion of his time (see (Desrosières 1988)). The analogy with the astronomical concept of libration is clear. From the Earth, the variability of the trajectory of the moon allows to observe and see more than expected.
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