Short Bio

I am an ecological engineer and scientist motivated by the development of knowledge in bioressource production and Anthropocene landscapes. I am interested in using the many possibilities offered by computer science to provide meaningful perspectives that actually work and can be used by practitioners. My researches are currently focused on the application of mathematical ecology to agriculture through plant nutrition and soil conservation.

We should use balances and machine learning to diagnose ionomes

Plant concentrations of elements found in living tissues (also called the ionome by \citet{Salt_2008}) has been linked to crop yield for nearly two centuries. Such concept emerged from the widely known *law of minimum* developed by Carl Sprengel around 1830 (wrongly attributed to Justus von Liebig, who rather popularized it). In the 70s, Beaufils found that using nutrients alone was somewhat unstable \cite{Walworth_1987}, so he developed the DRIS (diagnosis and recommendation integrated system) with an arguably naive mathematical framework, which was difficult to compute at that time. Following Aitchisons recommendations, Léon Etienne Parent (my father) used centered log-ratios and developed the CND, for compositional nutrition diagnosis \cite{dafir1992}.

Since about 2011, I have published many papers on data analysis of plant ionomics (listed on ResearchGate). It began with the application of nutrient balances in leaves of several fruit crop species \cite{Parent_2013}. After submitting the manuscript to three major journals, who rejected it one after the other (sometimes with depreciation), it was finally published in Frontiers in Plant Science and now reaches close to 9000 views. The approach was to free nutrient concentrations from their 0 to 100% closed space with a well-known transformation technique named the isometric log-ratio \cite{Egozcue_2003}. Indeed, concentrations are compositional data, and not transforming this kind of data will likely return fallacious statistics \cite{Aitchison_1986}. Recent works in biology confirmed these problems \cite{Silverman_2017,Mandal_2015,Friedman_2012}.

Even when compositions are transformed to adequate variables, how a *healthy state* is defined has a fundamental importance on health diagnostics and on the correction measures to be prescribed. After trying different approaches, I figured out that since I was facing complex patterns, machine learning techniques may handle this situation better than other techniques we explored in \citet{Parent_2013a}. What's more: machine learning algorithms are nowadays fairly easy to use.

Finally, the correction needed can be summarized by a translation from a bad nutritional status to a healthy one. A translation from a compositional state to another is called a *perturbation*. A perturbation indicates the change needed, not how to achieve this change: correcting the nutritional status of plants is a fairly complex issue implying fertilization trials contextualized both by data science and experience.

This article describes the logic behind the approach I do recommend to correctly diagnose plant ionomes. I will present why we should not use ranges and linear statistics, then show how machine learning and perturbation can be helpful to assess nutrient imbalance.

Advection-Diffusiuon: weak form

\[\begin{aligned} R(u) = 1 + \frac{\rho}{n} \times \frac{\partial S}{\partial u}\end{aligned}\]

\[\begin{aligned} S = k_p u^b \\ \frac{\partial S}{\partial u} = b k_p u^{b-1}\end{aligned}\]

\[\begin{aligned} \vec{v} = \frac{\vec{v_e}}{R(u)}\end{aligned}\]

\[\begin{aligned} D = \frac{D_l}{R(u)}\end{aligned}\]

Constants: \(b\), \(k_p\), \(D_l\)

Velocity field: \(\vec{v_e}\)

Mass conservation law, assuming that \(\vec{v}\) and \(D\) are functions of \(u\).

\[\begin{aligned} \frac{\partial u}{\partial t} = - \nabla \cdot \left( \vec{v} u \right) + \nabla \cdot \left( D \nabla u \right) \\\end{aligned}\]

Where \(u\) is concentration and \(t\) is time.

Integrate on volume \(\Omega\) and multiply by the test function \(s\).

\[\begin{aligned} \int_{\Omega}\frac{\partial u}{\partial t} s d\Omega + \int_{\Omega}\nabla \cdot \left(\vec{v} u \right) s d\Omega - \int_{\Omega} \nabla \cdot \left( D \nabla u \right) s d\Omega = 0\end{aligned}\]

Integrate by parts the last term of the left-had side.

\[\begin{aligned} \int_{\Omega} \nabla \cdot \left( D \nabla u \right) s d\Omega = \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega - \int_{\Omega} D \nabla u \nabla s d\Omega \end{aligned}\]

Apply Gauss divergence theorem on the first part of the left-hand side.

\[\begin{aligned} \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N} \cup \Gamma_{D}} s D \nabla u \cdot n d\Gamma\end{aligned}\]

Because \(s = 0\) on \(\Gamma_{D}\).

\[\begin{aligned} \int_{\Omega} \nabla \cdot \left( s D \nabla u \right) d\Omega = \int_{\Gamma_{N}} s D \nabla u \cdot n d\Gamma\end{aligned}\]

Put the pieces together. \[\begin{aligned} \int_{\Omega}\frac{\partial u}{\partial t} s d\Omega + \int_{\Omega}\nabla \cdot \left(\vec{v} u \right) s d\Omega - \int_{\Gamma_{N}} s D \nabla u \cdot n d\Gamma + \int_{\Omega} D \nabla u \nabla s d\Omega = 0\end{aligned}\]

Water flow through unsaturated soils: variational formulation of the Richards - Buckingham equation

\[\begin{aligned} \theta(\psi) = \theta_{r} + (\theta_{s} - \theta_{r}) (1+(a_{VG} \psi)^{n_{VG}})^{-m_{VG}} \\\end{aligned}\]

\[\begin{aligned} k(\psi) = k_{sat} \frac {(1-((a_{VG} \psi)^{n_{VG}m_{VG}}) (1+(a_{VG} \psi)^{n_{VG}})^{-m_{VG}}))^2} { (1+(a_{VG} \psi)^{n_{VG}})^{m_{VG}l_{VG}}}\end{aligned}\]

Mass conservation law.

\[\begin{aligned} \frac{\partial \theta}{\partial t} = - \nabla \cdot \vec{q} \\\end{aligned}\]

Darcy law.

\[\begin{aligned} \vec{q} = k \left( h, P \right) \nabla H \\\end{aligned}\]

Where h is the pore pressure, P is the position in 3D space and H is the total head.

\[\begin{aligned} P = (x, y, z)\\ h= - \psi \\ \nabla H = \nabla h + \nabla z \\\end{aligned}\]

The partial differential equation can be written as:

\[\begin{aligned} \frac{\partial \theta}{\partial t} = -\nabla \cdot k \left( h, P \right) \nabla H + A, \\\end{aligned}\]

where \(A\) is a source/sink term.

Integrate on both sides and multiply by the test function \(v\).

\[\begin{aligned} \int_{\Omega}\frac{\partial \theta}{\partial t} v\,d\Omega = -\int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega + \int_{\Omega}A v\,d\Omega \\\end{aligned}\]

Integrate by parts.

\[\begin{aligned} \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \\ \int_{\Omega}\nabla \cdot \left( k \left( h, P \right) \nabla H v\right) \,d\Omega \\\end{aligned}\]

Apply Gauss divergence theorem.

\[\begin{aligned} \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \int_{\Gamma_{N} \cup \Gamma_{D}} k \left( h, P \right) \nabla H v n\,d\Gamma \\ \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \int_{\Gamma_{N}} k \left( h, P \right) \nabla H v n\,d\Gamma + \int_{\Gamma_{D}} k \left( h, P \right) \nabla H v n\,d\Gamma\end{aligned}\]

Because \(v = 0\) on \(\Gamma_{D}\).

\[\begin{aligned} \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \int_{\Gamma_{N}} k \left( h, P \right) \nabla H v n\,d\Gamma \\\end{aligned}\]

Put in the initial integral.

\[\begin{aligned} \int_{\Omega} \frac{\partial \theta}{\partial t} v\,d\Omega = \int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega - \int_{\Gamma_{N}} \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + \int_{\Omega}A v\,d\Omega \\\end{aligned}\]

\(\theta \left( h \right)\) is a constitutive relationship. The weak formulation can thus be written on a \(h\) basis.

\[\begin{aligned} \int_{\Omega} \frac{\partial \theta}{\partial t} \times \frac{\partial h}{\partial h} v\,d\Omega = \int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega - \int_{\Gamma_{N}} \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + \int_{\Omega}A v\,d\Omega \\ \\ \frac{\partial \theta}{\partial h} \int_{\Omega} \frac{\partial h}{\partial t} v\,d\Omega = \int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega - \int_{\Gamma_{N}} \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + \int_{\Omega}A v\,d\Omega \\\end{aligned}\]

Integration by parts.

\[\begin{aligned} \int_{\Omega} \left(\nabla \cdot u \right) v\,d\Omega = -\int_{\Omega} u \nabla v\,d\Omega + \int_{\Omega} \nabla \cdot \left( u v \right) n\,d\Omega \\\end{aligned}\]

Gauss divergence theorem.

\[\begin{aligned} \int_{\Omega} \nabla \cdot F\, d\Omega = \int_{\partial \Omega = \Gamma_{N} \cup \Gamma_{D}} F \cdot n\, d\Gamma\end{aligned}\]

The problem in steady-state with no Neumann boundary conditions can be defined in Sfepy.

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