As I will show in the case study (following section), the perturbation vector is difficult to interpret either expressed in the ilr scale or in compositional scale.  I'm suggesting the use of another compositional data transformation: the centered log-ratio, computed row-wise as the natural logarithm of the ratio between each component and the geometric mean of the composition:
\(clr_i = log \left( \frac{x_i}{\left(x_1 \times x_2 \times \cdots \times x_n\right)^{\frac{1}{n}}} \right)\)
Clrs are Euclidean coordinates just like ilrs : a Euclidean distance matrix is identical whether computed on clrs or ilrs. You might ask why not using clrs instead of ilrs from the ground up. That's because their are D clrs for D components. Just as components are summing to 1, clrs sum to 0, which means that its covariance is singular. So what? I suggested to compute the Euclidean distance matrix. Now try to compare Mahalanobis distances of ilrs and clrs. Spoiler warning: you can't. The Mahalanobis distance takes into account the covariance, which needs to be inversed. And you can't inverse it with clrs. Such problems with singularity is common when using clrs. But their is another problem: if you wanted to predict the effect of for instance a treatment or a disease on clrs, the result of the prediction is unlikely to sum to 0: you would have to remove one component then compute its value by subtracting from 0 the sum of the other component, a likely biaised approch. This is why I'm suggesting to perform the modeling process with ilrs.

Case study with cloudberry data

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Conclusion

Even though it took me years to figure it out, the approach is fairly simple:
  1. Explode concentrations to balances of concentrations
  2. Identify a healthy zone and map a model to it
  3. Run the model to predict the yield class of new value
  4. If unhealthy, identify the target zone
  5. Draw the path to the target zone using a perturbation vector
The tricky part is the one after step 4. Correcting a plant ionome will need precise fertilization with the right chemicals and, above all, the right professional agronomist.
All computations done for this article can be found in a R Jupyter notebook here.