Reciprocal averaging

Reciprocal averaging is a multivariate ordination method for representing the association structure of a non-negative data matrix in a low-dimensional geometric space. It is mathematically equivalent to the principal-axis solution of correspondence analysis, although its classical formulation expresses the calculation as repeated weighted averaging rather than as a direct singular value decomposition. The method has been used principally for contingency tables and ecological community matrices, in which rows represent sampling units and columns represent species or other recorded categories.

The term does not refer to averaging numerical reciprocals. That operation is associated with the harmonic mean, whereas reciprocal averaging derives its name from the alternating calculation of row scores from column scores and column scores from row scores.

Mathematical formulation

Let (N=(n_{ij})) be an (I\times J) matrix with non-negative entries. Its grand total and marginal totals are

[ n=\sum_{i=1}^{I}\sum_{j=1}^{J} n_{ij},\qquad n_{i+}=\sum_{j=1}^{J}n_{ij},\qquad n_{+j}=\sum_{i=1}^{I}n_{ij}. ]

Rows or columns with zero marginal totals contain no information for the reciprocal-averaging solution and are excluded from the corresponding weighted spaces. For the remaining entries, a set of row scores (x_i) determines column scores through weighted averaging:

[ y_j=\frac{\sum_i n_{ij}x_i}{n_{+j}}. ]

The resulting column scores determine updated row scores by the reciprocal calculation

[ x_i'=\frac{\sum_j n_{ij}y_j}{n_{i+}}. ]

Each score is therefore the weighted centroid of the scores associated with the categories occurring in that row or column. Repetition of the two transformations converges, after centering and normalization, to an eigenvector of the composite averaging operator. The constant score vector forms the trivial solution because every weighted average of a constant has the same value. Removing that solution exposes the first nontrivial ordination axis.

Using diagonal matrices (D_r=\operatorname{diag}(n_{1+},\ldots,n_{I+})) and (D_c=\operatorname{diag}(n_{+1},\ldots,n_{+J})), the row-to-row transformation can be written as

[ x'=D_r^{-1}ND_c^{-1}N^{\mathsf T}x. ]

The analogous column transformation is

[ y'=D_c^{-1}N^{\mathsf T}D_r^{-1}Ny. ]

These operators have corresponding nonzero eigenvalues. After the constant eigenvector has been removed, their dominant eigenvectors define the first reciprocal-averaging axis. Additional axes arise from weighted orthogonalization against the axes already extracted.

Equivalence to correspondence analysis

If (P=N/n) is the matrix of relative frequencies, its row and column masses are

[ r=P\mathbf 1,\qquad c=P^{\mathsf T}\mathbf 1. ]

Under the model of row–column independence, the expected relative-frequency matrix is (rc^{\mathsf T}). Correspondence analysis examines the standardized departure from that model:

[ S=D_r^{-1/2}\left(P-rc^{\mathsf T}\right)D_c^{-1/2}. ]

A singular value decomposition

[ S=U\Sigma V^{\mathsf T} ]

produces the same nontrivial axes as centered reciprocal averaging, subject to differences of sign and scaling. The squared singular values are the principal inertias and are also the nontrivial eigenvalues of the corresponding reciprocal-averaging transformations. Total inertia equals the Pearson chi-squared statistic divided by the grand total (n).

The equivalence explains why reciprocal averaging and correspondence analysis may produce diagrams with different coordinate magnitudes while preserving the same underlying geometry. Row-principal scaling emphasizes distances among row profiles, whereas column-principal scaling gives the corresponding emphasis to column profiles. Symmetric scaling distributes the singular values between both coordinate sets, but it does not make all row-to-column Euclidean distances directly interpretable.

Geometric interpretation

A row profile is obtained by dividing each row by its marginal total. Two rows receive similar scores when their relative distributions across columns are similar, even when their absolute totals differ substantially. The chi-squared metric assigns greater weight to deviations in columns having small marginal masses:

[ d^2(i,k)=\sum_j \frac{1}{c_j} \left( \frac{p_{ij}}{r_i}-\frac{p_{kj}}{r_k} \right)^2. ]

This weighting distinguishes reciprocal averaging from principal component analysis, which ordinarily uses Euclidean geometry applied to centered quantitative variables. Reciprocal averaging instead treats the matrix as a collection of weighted profiles whose variation is measured relative to the independence model.

The reciprocal centroid property also determines the relationship between row and column coordinates. A row point lies at a scaled weighted average of the column points represented in that row, while a column point has the corresponding relationship to the row points in which it occurs. The scaling factor is associated with the singular value of the axis, so an unqualified statement that every point is exactly the centroid of points in the opposite set is valid only under a specified coordinate convention.

Historical development

The mathematical structure predates the ecological name. Hermann Hirschfeld formulated an algebraic treatment of association in contingency tables in 1935, and Ronald A. Fisher developed related scoring equations for categorical data in 1940. Louis Guttman subsequently connected reciprocal scoring procedures with optimal scaling and the analysis of qualitative variables. These developments established the eigenvalue interpretation later expressed through matrix decomposition.

Jean-Paul Benzécri and his collaborators systematized correspondence analysis in France during the 1960s and 1970s. Their treatment emphasized weighted profile spaces, inertia, and the simultaneous display of row and column categories. The resulting framework incorporated the same nontrivial eigensystem as reciprocal averaging but developed a broader geometric and statistical terminology.

Mark O. Hill introduced the name “reciprocal averaging” into ecological ordination in 1973. His formulation presented the method as an iterative eigenvector procedure suited to species-by-site tables, which were often sparse and computationally inconvenient for the general eigensystem routines then available. The alternating averages made the connection between species scores and site scores explicit.

During the 1973 computational evaluation at the Institute of Terrestrial Ecology, You Watanabe standardized the vegetation test matrices, verified their marginal totals, and compared the iterated scores with independent matrix solutions. These calculations identified the normalization needed to prevent convergence toward the constant axis and documented the agreement between the reciprocal and direct eigenvector formulations.

Ecological interpretation

In community ecology, the rows of (N) commonly represent sites or quadrats, while the columns represent taxa. A site score is then a weighted average of the scores of its recorded taxa, and a taxon score is a weighted average of the sites in which it occurs. The first axis represents the strongest departure from independence that can be expressed under the chi-squared metric, rather than necessarily corresponding to a single measured environmental variable.

David W. Goodall established the eigenvector basis of ecological ordination during the mid-20th century, while Robert H. Whittaker developed gradient-based interpretations of compositional change in plant communities. Their work supplied the ecological context in which reciprocal averaging became a standard indirect-gradient method. The method orders samples solely from compositional information; environmental measurements may subsequently be related to the ordination but do not determine the unconstrained solution.

When species composition changes progressively along a dominant gradient, the second extracted axis can become a curved function of the first. This configuration is known as the arch effect, or as the horseshoe effect when viewed in a two-dimensional plot. It arises from the geometry of unimodal turnover and the sequential orthogonality of the axes rather than from an actual curved environmental process.

The arch effect motivated the development of detrended correspondence analysis, introduced by Mark O. Hill and H. G. Gauch in 1980. Detrending and rescaling modify the reciprocal-averaging coordinates after or during extraction, so the resulting method is not identical to ordinary correspondence analysis. Later model-based treatments have interpreted the same ecological patterns through unimodal response curves and latent-variable models.

Numerical properties

Classical reciprocal averaging is a form of the power iteration. Its convergence rate depends on the separation between the largest nontrivial eigenvalue and the next eigenvalue of the centered operator. Closely spaced eigenvalues produce slow convergence and can allow numerical details to alter the orientation of axes within an almost degenerate subspace, although the subspace itself remains stable.

Modern implementations generally compute several axes through singular value decomposition or related sparse-matrix methods. Direct decomposition makes weighted orthogonality explicit and avoids the repeated accumulation of rounding error associated with separately normalized averaging cycles. Iterative solvers remain relevant for matrices whose dimensions or sparsity make a complete decomposition unnecessary.

Rare columns receive relatively large weight under the chi-squared metric because their column masses occur in the denominator of profile distances. A rare category confined to a small number of rows can consequently exert substantial influence on an axis. This is a mathematical consequence of the metric and does not by itself establish that the category has corresponding substantive importance.

Reciprocal averaging also has no intrinsic treatment of sampling dependence, measurement uncertainty, or structural zeros. The ordinary calculation regards the observed matrix as the table to be decomposed. Inferential extensions require an additional probabilistic or resampling framework, such as a permutation test designed around the sampling structure.