one idea is to apply a log-ratio transformation to V and then make the product, another probably better way  is to make the scalar product like on page 28 at this document

http://dugi-doc.udg.edu/bitstream/handle/10256/297/CoDa-book.pdf?sequence=1

--> power-multiplication of a real vector by a compositional

matrix

one idea is to apply a log-ratio transformation to V and then make the product, another probably better way  is to make the scalar product like on page 28 at this document

http://dugi-doc.udg.edu/bitstream/handle/10256/297/CoDa-book.pdf?sequence=1

--> power-multiplication of a real vector by a compositional

matrix

Matthias is right. In general, the advise is: work in coordinates. This is what we call the "Principle of Working in Coordinates". If an object has a constrained sample space, look if it is possible to define an appropriate structure over the real numbers in this sample space, and within this structure if it is possible to build orthonormal coordinates, which are then real. You can apply to them the standard theory. This is the "safe" approach.

Dear Italo,

I agree with the previous answers by Mattias Templ and Vera Pawlowsky-Glahn. I would like to add some details that may be useful.

Compositions, and their space the simplex, as your V, constitute a Euclidean space. The vector space structure includes perturbation and powering by real scalars. Perturbation of a composition by a real vector with positive components can be carried out without closure as perturbation includes it. Powering applies for any real constant. From these operations other derived operations can be applied to compositions. There are mainly two of them: the perturbation-linear combination of compositions and the perturbation-linear transformations of a compositions.

The first one combines a vector of D real constants, say U, with a set of compositions, say V_1, V_2, ... , V_D. The combination is (pseudo-LaTeX)

U \odot (V_1, V_2, ... , V_D) = (u_1\odot V_1)  \oplus (u_2 \odot V_2) \oplus ... \oplus (u_D \odot V_D)

where \odot is powering and \oplus is perturbation in the simplex. I guess that this is not the operation you are looking for.

The perturbation-linear transformation of V is expressed as a matrix product

W = M \boxdot V,

where M is a (D,D) matrix of real entries and V again a composition. It works like an ordinary matrix product but changing products by powering and sums by multiplications. However, The (D,D)-matrix M must satisfy the so called zero-condition to represent a linear operation in the simplex: the sum of each column and each row must add to zero. Otherwise, the operation W = M \boxdot V is a non linear transformation in the simplex.

From your statement I think you are asking about an operation like

diag(U)\boxdot V,

i.e. M=diag(U). This matrix operation is necessarily non linear in the simplex except that all components in U are equal, thus corresponding to a simple powering.

The two commented operations, when carried out using coordinates of the simplex, are translated into linear combinations of coordinates or matrix linear transformations. However, in the case of a matrix M not fulfilling the zero condition, the expression in coordinates is no longer linear.

This issues are developed in

Egozcue, J. J., Barceló-Vidal, C., Martín-Fernández, J. A., Jarauta-Bragulat, E. and Díaz-Barrero, J. L. and Mateu-Figueras, G. (2011):

Elements of simplicial linear algebra and geometry, In: Pawlowsky-Glahn, V. and Buccianti A. (eds.), Compositional Data Analysis: Theory and Applications, Ch. 11, Wiley, Chichester UK

V. Pawlowsky-Glahn, J. J. Egozcue and R. Tolosana-Delgado (2015):

Modeling and analysis of compositional data, John Wiley & Sons, Chichester UK,

Statistics in practice, Ch. 4

Hope these hints are useful!

I would like to thank all of you for your answers, they were indeed helpful.

I work with diffuse reflectance spectroscopy (see attached file for details) and, the equation of interest could be expressed as F = A \cdot (B \oplus C), where F is the property which is measured in laboratory, A and B are the optical constants of components in a mixture (supposedly known), C represents the components' concentrations and \cdot denotes the Euclidean dot product.

Before I came in contact with your work on compositional data, I was solving a system of equations for C using standard linear algebra (there is one such equation per wavelength in which we take measurements). This system is homogeneous unless we add the restriction that all C_i must sum to 1, and even so it is still possible to obtain negative concentrations as a result.

I wonder if there is some way of representing the Euclidean dot product in transformed coordinates, but it seems the matter is a little more complex then I expected.

Article Quantification of Hematite and Goethite Concentrations in Ka...

Hi Ítalo,

There is a problem in using these constrained sum variables (of any kind, including compositional data) in OLS regression because the sum of variance and covariances is zero for the covariance matrix. The Sx matrix is singular and there is no regular inverse. Moreover, for k variables, you force k-1 covariances (and correlations) to be negative, regardless the real relationship of the data.

You probably want to go compositional. See the R package compositions: http://cran.r-project.org/web/packages/compositions/index.html

There are some books in Amazon on compositional data analysis you could benefit from (including the one authored by Drs. Vera and Juan above). See http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=compositional

Best,

Iuri.

 Iuri is right concerning the singularity of the covariance matrix using raw data, but not that I know with the negativ covariances and correlations. The constant sum imposes that some have to be negative but, up to my knowledge, does not specify their number.

In any case, if you take ilr coordinates, you can use standard methods safely, although using clr or alr taking into account that they are not orthonormal in the Aitchison geometry of the simplex is also fine.