Compositional data: the sample space and its structure

Abstract

The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison’s seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain.

Appendices

Appendices

Data

Mean Household Gross Income (MHGI) in Spain The Instituto Nacional de Estadística (INE) (INE 2016) provides on its webpage an estimation of the MHGI per capita for all Autonomous Communities and Autonomous Cities (ACs) in Spain, for the years 2000 to 2014. According to the webpage, the data are median aggregated data, but the algorithm for doing so is not included. Nevertheless, the result is a non-closed composition, and the compositional tools here presented would lead to exactly the same results whether applied to the given data, or a representation in proportions or percentages or, as can be seen in the webpage of the INE, normalized so that the estimated MHGI for the whole of Spain corresponds to 100%. This data set (19 ACs for 15 years, listed in Table 2) is used here to illustrate the procedures and properties presented. Figure 10 shows the values in Euros. The MHGI in the Balearic Islands (circles), Canary Islands (triangles), Madrid ($+$) and Catalonia ($\times$) are highlighted for discussion.

Fig. 10

MHGI per capita for all ACs in Spain for the years 2000–2014. Four ACs are highlighted: Balearic Islands (4, circles), Canary Islands (5, triangles), Madrid (13, $+$) and Catalonia (9, $\times$); the numbers are placed near the end point of each MHGI. Names are shown in the right panel

In Sect. 3, the MHGI for all ACs is considered as a 19-part composition.

Table 2 MHGI per capita in the period 2000–2014 (INE 2016)

Proofs of the properties of the inequality index $A_I^2$

Consider a D-part composition $\mathbf {s}$ divided into two subcompositions $\mathbf {s}_1$ and $\mathbf {s}_2$ of $d_1$ and $d_2$ shares, respectively, with $d_1 + d_2 = D$ and represented in the same units. Proofs of the properties of $A_I^2$ follow.

Decomposability by subcompositions:

$$\begin{aligned} A_I^2( (\mathbf {s}_1,\mathbf {s}_2) ) = \frac{d_1 A_I^2(\mathbf {s}_1)}{d_1+d_2} + \frac{d_2 A_I^2( \mathbf {s}_2)}{d_1+d_2} + \frac{4\ d_1 d_2}{(d_1+d_2)^2}\ A_I^2(\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2)), \end{aligned}$$

where $(\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2))$ is a 2-part composition whose components are the geometric means of the parts in $\mathbf {s}_1$ and $\mathbf {s}_2$, respectively.

Since for any D-part composition $\mathbf {s}$ it holds that $A_I^2(\mathbf {s})=(1/D)\Vert \mathbf {s} \Vert _a^2$, the index of the composed vector of shares $A_I^2((\mathbf {s}_1,\mathbf {s}_2))$ can be expressed as a sum of squares of orthonormal balances obtained in an SBP. Define a first binary partition, by separating the shares in $\mathbf {s}_1$, marked with a $+1$, and the shares in $\mathbf {s}_2$ marked with a $-1$. Any further partitions within $\mathbf {s}_1$ and within $\mathbf {s}_2$ are then possible to obtain the expression

$$\begin{aligned} \Vert (\mathbf {s}_1,\mathbf {s}_2) \Vert _a^2 = \Vert \mathbf {s}_1 \Vert _a^2 + \Vert \mathbf {s}_2 \Vert _a^2 + \left( \sqrt{\frac{d_1d_2}{d_1+d_2}} \log \frac{\mathrm {g}_\mathrm {m}(\mathbf {s}_1)}{\mathrm {g}_\mathrm {m}(\mathbf {s}_2)} \right) ^2. \end{aligned}$$

(10)

The 2-part composition $(\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2))$ has square norm

$$\begin{aligned} \Vert (\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2)) \Vert _a^2 = \left( \frac{1}{\sqrt{2}} \log \frac{\mathrm {g}_\mathrm {m}(\mathbf {s}_1)}{\mathrm {g}_\mathrm {m}(\mathbf {s}_2)} \right) ^2 =2\ A_I^2((\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2))). \end{aligned}$$

(11)

Dividing Eq. (10) by the appropriate constants to obtain inequality indexes, and substituting the value in Eq. (11) yields

$$\begin{aligned} A_I^2( (\mathbf {s}_1,\mathbf {s}_2) ) = \frac{d_1 A_I^2(\mathbf {s}_1)}{d_1+d_2} + \frac{d_2 A_I^2( \mathbf {s}_2)}{d_1+d_2} + \frac{4\ d_1 d_2}{(d_1+d_2)^2}\ A_I^2((\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2))). \end{aligned}$$

(12)

Population replication Let $\mathbf {p}$ be a composition of shares. The property to prove is $A_I^2(\mathbf {p})=A_I^2((\mathbf {p},\mathbf {p}, \dots ,\mathbf {p}))$.

The geometric means of any number of replications of $\mathbf {p}$ are equal, that is $\mathrm {g}_\mathrm {m}(\mathbf {p}) = \mathrm {g}_\mathrm {m}((\mathbf {p},\mathbf {p},\dots ,\mathbf {p}))$. Applying the previous result of decomposability by subcompositions [Eq. (12)], the value of $A_I^2((\mathrm {g}_\mathrm {m}(\mathbf {s}_1),\mathrm {g}_\mathrm {m}(\mathbf {s}_2)))$ is null. Applying repeatedly the decomposability by subcompositions, the population replication property holds.

Principle of transfers A transfer of $\delta >0$ to the i-th unit increases its share to $p_i+\delta$ being $\delta$ detracted from the share of the j-th unit. If $p_j-\delta > p_i+\delta$, then $A_I^2(\mathbf {p})\ge A_I^2(\mathbf {p}')$ where $\mathbf {p}'$ is the composition after the transfer.

Without loss of generality, take $i=1$, $j=2$. There are positive constants, a, b, such that $p_1'=p_1+\delta = p_1 a$, $p_2'=p_2-\delta =b p_2$, with $a > 1$ and $0< b < 1$, since $p_2-p_1> 2\delta >0$. The new shares $\mathbf {p}'$ are the perturbation

$$\begin{aligned} \mathbf {p}'= \mathbf {a} \oplus \mathbf {p}, \quad \mathbf {a}=(a,b,1,1,\dots ,1). \end{aligned}$$

The statement is proven if the following inner product satisfies

$$\begin{aligned} \langle \mathbf {a}, \mathbf {p}\rangle _a = \langle \mathrm {clr}(\mathbf {a}), \mathrm {clr}(\mathbf {p})\rangle _e \le 0. \end{aligned}$$

After some computation, the inner product is

$$\begin{aligned} \langle \mathbf {a}, \mathbf {p}\rangle _a = \log a \cdot \mathrm {clr}_1(\mathbf {p}) + \log b\cdot \mathrm {clr}_2(\mathbf {p}), \end{aligned}$$

(13)

where $\mathrm {clr}_k(\mathbf {p})= \log p_k - \log \mathrm {g}_\mathrm {m}(\mathbf {p})$, $k=1,2$. This inner product is negative if the two inequalities $\log (ab)> 0$ and $(\log b)/(\log a)> -1$ hold. In fact, since $\log a>0$, the negativeness of Eq. (13) is equivalent to

$$\begin{aligned} \mathrm {clr}_1(\mathbf {p}) + \frac{\log b}{\log a}\cdot \mathrm {clr}_2(\mathbf {p}) < 0, \end{aligned}$$

which derives from $(\log b)/(\log a)> -1$ and $\mathrm {clr}_1(\mathbf {p}) < \mathrm {clr}_2(\mathbf {p})$.

The first inequality derives from

$$\begin{aligned} ab = \frac{(p_1+\delta )(p_2-\delta )}{p_1 p_2} = 1 + \frac{\delta ((p_2-p_1)-\delta )}{p_1p_2} \ \end{aligned}$$

and the assumption that $(p_2-p_1)-2\delta \ge 0$. The second inequality also holds from the assumption $p_1+\delta < p_2-\delta$, since this is $a < b$ and then $\log a < \log b$; finally, $\log a>0$ implies that $(\log b) /(\log a)< -1$.

Supplementary material

This section contains some tables and figures that are not included in the body of the article due to its extension (Tables 3, 4, 5; Figs. 11, 12, 13, 14, 15).

Table 3 Center and normalized variation matrix of MHGI (Appendix A)

Table 4 Signs code for the SBP approaching principal balances for MHGI data

Table 5 Estimated balances of coefficients in model (9)

Fig. 11

Scatterplot of two balances with common parts in the denominator. When the slope is equal to 1, the two parts in the numerator are associated within the considered subcomposition. Left panel: testing association between Gal and Vas (p value 0.21, unit slope not rejected). Right panel: testing association between IBal and ICan (p value 0.0000, unit slope rejected)

Fig. 12

Scree plot for the MHGI data. Full line with circles: proportion of total variance explained by each principal component. Dotted line with triangles: cumulative proportion of total variance explained by the sequence of principal components. Note that only 14 square singular values are reported as there are only 15 data points despite there being 19 ACs

Fig. 13

MHGI per capita data. Form biplot of the subcomposition Cataluña (Cat), Madrid (Mad) and Melilla (Mel), representing 100% of total variance. The numeric markers correspond to years after 2000. The behavior of points is almost linear, as suggested by the biplot in Fig. 6

Fig. 14

Evolution of MHGI represented in the two first principal balances. The numbers correspond to years after 2000

Fig. 15

Form biplot of compositional residuals of regression model (9). Left panel: first and second principal components. Right panel: first and third principal components. No trend is detected. Total variance: residuals $9.097\times 10^{-5}$; MHDI data 0.00234; percentage of total variance explained by the model 96.1%