=Paper=
{{Paper
|id=Vol-3105/paper43
|storemode=property
|title=Adaptation of Compositional Data Analysis in Deep Learning to Predict Pasture Biomass Proportions
|pdfUrl=https://ceur-ws.org/Vol-3105/paper43.pdf
|volume=Vol-3105
|authors=Badri Narayanan,Mohamed Saadeldin,Paul Albert,Kevin McGuinness,Noel E. O'Connor,Brian Mac Namee
|dblpUrl=https://dblp.org/rec/conf/aics/NarayananSAMON21
}}
==Adaptation of Compositional Data Analysis in Deep Learning to Predict Pasture Biomass Proportions==
https://ceur-ws.org/Vol-3105/paper43.pdf
Adaptation of Compositional Data Analysis in
Deep Learning to Predict Pasture Biomass
Proportions
Badri Narayanan1,3,4 , Mohamed Saadeldin1,3,4 , Paul Albert2,3,4 , Kevin
McGuinness2,3,4 , Noel E. O’Connor2,3,4 , and Brian Mac Namee1,3,4
1
School of Computer Science, University College Dublin
2
School of Electronic Engineering, Dublin City University
3
Insight SFI Centre for Data Analytics
4
VistaMilk
badri.narayanan@insight-centre.org
Abstract. Dry biomass weight measurements from a quadrat in a pad-
dock for grass, clover and weeds when expressed as percentages of total
dry herbage mass are compositional in nature. Unlike real valued regres-
sion problems, prediction of compositional data is handled differently
in statistics because of its closure property where the components of
the composition are positive data adding up to a constant sum and is
therefore constrained in the simplex space, in our case 100%. Our motiva-
tion in this paper was to study whether the adaptation of compositional
data analysis (CoDa) techniques in deep learning improves the prediction
results over the best performing deep learning model we used in our earlier
paper [Narayanan et al., 2021]. Although the log ratio transformation
of targets is an appropriate adaptation of CoDa and is interesting for
Biomass prediction, our study indicates that the CoDa adaptation does
not improve the prediction errors over our earlier method.
Keywords: Deep Learning · Compositional Data Analysis · Isometric
Log Ratio · Simplex · Softmax
1 Introduction
The dairy industry uses clover and grass as fodder for cows. Grass and clover
are grown together in fields to improve the consistency of high-quality biomass
yield and to reduce the need for external fertilizers. Accurate estimation of the
dry biomass percentages of grass and clover species (as well as weeds) in fields
is very important for determining optimal seeding density, fertilizer application
and elimination of weeds. The dry biomass weights of the individual components,
when expressed as percentages of overall weight of the harvested and dried
biomass, are compositional in nature.
Compositional data are positive data summing to a constant value, and
measure relative changes in the components. They are constrained in the simplex
space. Standard multivariate statistical analysis and regression techniques assume
Copyright 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)
2 B. Narayanan et al.
the sample space to be R. The sample space for compositional data, however, is
restricted in the simplex due to the sum constraint. Compositional data analysis
(CoDa) Aitchison [2005] is a set of mathematical techniques that helps in analysing
the relative proportions of individual components.
In this paper, we examine the applicability of the principles of CoDa to
the problem of predicting biomass composition from farm imagery using deep
learning. We present an adaptation of the approach used in statistical CoDa
where the compositional data is transformed from the simplex space to the real
space using the isometric log ratio (ILR) transformation [Egozcue et al., 2003].
These transformed values are used as targets with a model that predicts the
dry mass fractions of grass, clover and weeds from images of a section of grass
marked with a square frame (known as a quadrat). We compare the prediction
results to the best performing model from our previous paper [Narayanan et al.,
2021] that uses the composition data directly as a target. This comparison shows
that the addition of the approaches from CoDa do not improve the performance
of the model.
In the rest of this paper, Section 2 outlines related work for CoDa techniques
in applied statistics and highlights the few available adaptations by the machine
learning community. This is followed by a description of our experimental design
in Section 3 and a discussion of the results in Section 4. Finally, Section 5
summarises the findings from this paper and suggests directions for future work.
2 Related Work
This section introduces compositional data analysis, establishes the relationship
between the softmax transformation and the simplex, and reviews the limited
applications of CoDa to machine learning and biomass composition prediction.
2.1 Compositional Data Analysis
Compositional data of D-parts is constrained in the simplex space S D by a con-
stant sum of the components. The sum constraint induces negative correlations
between variables [Chayes, 1960], which violates the independence assumptions
and the central limit theorem [Aitchison, 1982, 2005]. Therefore, the sum con-
straint needs to be broken before standard statistical methods can be applied for
analysis. This is often achieved by transforming the compositional data from the
simplex space into real space using log ratio transformations [Pawlowsky-Glahn
and Egozcue, 2006].
Aitchison [1982] formalises three key principles of CoDa: scale invariance,
permutation invariance, and subcompositional coherence. Any statistical analysis
of compositional data must conform to these principles. Scale invariance is
characterised by the relative information that the compositional data carry, rather
than the individual size of the components. Permutation invariance mandates that
any statistical inference should be independent of the ordering of the components
within the composition. Finally, subcompositional coherence stipulates that results
Adaptation of Compositional Data Analysis in DL 3
from the analysis of components in a full composition should not contradict the
results from a subcomposition, i.e., the distances between two compositions
should decrease when subcompositions of the original ones are considered, and
that scale invariance is preserved within arbitrary subcompositions.
Log Ratio Transformations The centered log ratio (CLR) and isometric log
ratio (ILR) are the prevalent log ratios used in modern CoDa applications. In
a composition x of D ≥ 2 components, the sum constraint of compositional
data implies that there is at least one component that is negatively correlated
with another in the composition, and that there are at most D − 1 independent
components. The composition is therefore constrained in a D −1 dimension vector
space of S D , defined as a D-part simplex on R. The values of these components,
when scaled by their geometric mean and then log transformed, are mapped to a
hyperplane in RD and referred to as the Centred Log Ratio (CLR):
x1 x2 xD
clr(x) = z = ln , ln , . . . , ln ,
g(x) g(x) g(x)
where g(x) is the geometric mean of the k components of x:
√
g(x) = D x1 x2 . . . xD .
Pawlowsky-Glahn et al. [2007] highlight that the CLR introduces a mathematical
complexity in the form of a singular covariance matrix where the determinant is
zero. Additionally, the CLR transformation is not subcompositionally coherent
as the geometric mean of a subcomposition will differ from that of the whole
composition.
These drawbacks led to the introduction of the isometric log ratio (ILR)
by Egozcue et al. [2003] where an isometry from S D to RD−1 is achieved from
an orthonormal basis derived from Gram-Schmidt orthogonalization. In addition
to being an isometry, the ILR is an isomorphism too, and conforms to the three
CoDa principles outlined above. Following Egozcue et al. ILR can be defined as
follows:
ilr(x) = [hx, e1 ia , hx, e2 ia , . . . , hx, eD−1 ia ],
where [e1 , e2 , . . . , eD−1 ] is an orthonormal basis in the simplex, the default
one being the orthonormal basis built by Egozcue et al. using Gram-Schmidt
orthogonalization. hx, ei ia represents the Aitchison inner product between x and
ei . The inverse of ILR transformation is given by
D−1
M
x = ilr−1 (y) = (hy, ~ei ia ei ),
i=1
L J
where ~ei = ilr(ei ) for all i. and denote the compositional operations of
perturbation and power transformation described in the Aitchison geometry of
the simplex [Pawlowsky-Glahn and Egozcue, 2006].
4 B. Narayanan et al.
Although both the CLR and the ILR are isometric and allow for statistical
operations in the Euclidean space, the ILR is the most prevalent in modern
applications of CoDa, simply because of its representation of the composition in
an orthogonal coordinate system. Unlike CLR, the ILR allows for the association
of angles and distances in the simplex with those in the real space, and adheres
to the 3 key principles of CoDa, thereby making it a better choice. For interested
readers, Tolosana-Delgado [2008] provide a short and comprehensive mathematical
representation of these log ratios and other foundational aspects of CoDa.
Handling zero values Zero values in compositional data, if not handled, can be
problematic. Essential zeros refer to the absence of a component in the observation,
whereas rounded zeros indicate approximate recording of a component below
detection limit [Martı́n-Fernández et al., 2003] and need to be addressed. Rounded
zeros are replaced with a threshold value using a multiplicative replacement
method that maintains the constant sum of the composition.
Applications of CoDa Applied statistics has seen many applications of Com-
positional Data Analysis (CoDa) in geostatistics [Tolosana-Delgado et al., 2019],
bioinformatics, environmental science and chemistry [Filzmoser et al., 2010]
where many problems are compositional in nature. Liu et al. [2016] trace the
underlying factors that influence rock weathering and mineralisation in stream
sediments of the Nanling tectono-magmatic belt using robust factor analysis and
compositional data analysis.
In the biomass composition problem studied in this paper the targets are
compositional. Aitchison [2005] presents an example similar to this that quantifies
the extent of dependence of sediment composition on water depth in arctic lakes.
Three mutually exclusive and exhaustive constituents (sand, silt and clay) are
recorded in their proportions by weight for 39 samples at different water depths.
The objective is to quantify the extent of dependence of sediment composition on
water depth and hence, identify the nature of sedimentation process. In another
similar example the relative mass of water, fat and protein in a meat sample is
predicted from its NIR spectrum [Verwaeren, 2014].
2.2 Softmax and the Simplex
In the context of the deep learning experiments in this work, it is necessary to
understand the relationship between the softmax activation function used in
neural networks and the simplex space. The softmax is a mathematical function
most commonly used in the output layers of neural networks for multi-class
classification, and provides a generalisation of the sigmoid function in logistic
regression. The softmax function has been extensively used in state-of-the-art
deep neural network models and has been used very successfully in classification
and regression problems.
In a typical multi-class classification setting, the softmax function converts a
vector of k real values into a vector of k probabilities that sum to 1—a probability
Adaptation of Compositional Data Analysis in DL 5
distribution for the predicted classes in the target. Each of these probabilities is
a proportion of the relative scale of the corresponding individual component of
the input vector:
ezi
σ(zi ) = PK for i = 1, . . . , K and z = (z1 , ..., zK ) ∈ RK .
zj
j=1 e
[Amos, 2019, Theorem 4, pg 13] provides a theorem and proof that establishes
the relationship between the softmax activation function and the simplex, where
the softmax acts as the projection of a point x ∈ Rn onto the interior of the
(n − 1)-simplex.
2.3 Applications of CoDa in Machine Learning & Biomass
Prediction
There are limited examples in the literature of the adaptation of the CoDa
principles to machine learning. The use of random forest models trained on
data pre-processed with log ratio transformations [Harris and Grunsky, 2015;
Talebi et al., 2018] illustrate the few attempts in the use of machine learning;
however, this area remains largely unexplored and there are no specific instances
of literature of experiments / benefits of adapting CoDa techniques with deep
learning.
A body of recent research [Skovsen et al., 2018; Larsen et al., 2018; Sindic
and Riday, 2020; Castro et al., 2020; Sun et al., 2021] employs state-of-the-art
deep learning techniques to predict dry matter yield from proximal and UAV
images of grass paddocks. These works typically rely on transfer learning [Pan
and Yang, 2009] to transfer latent representations that were learnt from large
corpus of images by deep networks like VGG16 [Simonyan and Zisserman, 2014]
and Resnet [He et al., 2016]. To our knowledge, there are no references to the
adaptation of CoDa techniques in these deep learning approaches. Therefore it
is interesting to explore the integrated approach of CoDa principles and deep
learning to solve the biomass composition prediction problem. The next section
describes the design of an experiment to assess the effectiveness of adapting the
concept of isometric log ratio transformation (ILR) introduced by Egozcue et al.
[2003] to the deep learning architecture used in Narayanan et al. [2021].
3 Experimental Design
This section describes the design of a set of experiments undertaken to assess the
effectiveness of adopting CoDa techniques in deep learning models for biomass
composition prediction. The section describes the dataset used, the architecture
of the models built and the experimental method used.
3.1 Data Description
The Grass Clover Image Dataset for the Biomass Prediction Challenge [Skovsen
et al., 2019] provides us with 261 images of quadrats of grass with corresponding
6 B. Narayanan et al.
dry biomass composition of grass, white clover, red clover and weeds. These are
expressed in terms of their weights proportional to the total biomass, and sum
to 1. Five example images and their target values are presented in Table 1.
Proportions of weight ILR coefficients
Image Grass White Red Weeds ilr 1 ilr 2 ilr 3
clover clover
0.7648 0.2058 0.0000 0.0294 -3.7659 -3.2462 0.5262
0.9085 0.0655 0.0260 0.0000 -0.6518 -2.5238 4.1137
0.9532 0.0073 0.0000 0.0395 -1.4092 -4.7866 -0.6271
0.2065 0.5590 0.0314 0.2031 -2.0354 -0.3619 -0.2418
0.2747 0.5189 0.2064 0.0000 -0.6518 0.1432 4.9636
Table 1: Five example images from the Grass Clover Image Dataset along with
their biomass compositions and ILR transformed targets. The coefficients ilr 1,
ilr 2 and ilr 3 are obtained from the transformation of the proportions of grass,
white clover, red clover and weeds after replacement of zero values.
The dataset was divided into 209 training examples and 52 validation examples.
All the images were standardized to 500 × 500 pixels and, to ensure adequate
training examples for the network to learn efficiently, the training set images were
subject to 10× expansion through runtime augmentations [Krizhevsky et al., 2012].
The transformations in the augmentation included a rotation (up to 15◦ ), zoom
(±15%), height and width shift (20%), shearing (±15%), horizontal reflections,
channel shift (±50), and image wrapping to minimize loss of information.
We use the Compositional Statistics package5 in Python for the ILR and
inverse ILR transformations in this work. Given the non-zero minimum values of
each component in the dataset presented in Table 2, a threshold value of 0.001
5
Compositional Statistics: https://composition-stats.readthedocs.io/
Adaptation of Compositional Data Analysis in DL 7
Grass White Clover Red Clover Weeds
0.051104 0.001333 0.003056 0.001025
Table 2: Non-zero minimum values of the components
was selected for undetectable measurements. Zero values in the data are then
replaced with this minimum threshold value using the multiplicative replacement
method, while ensuring the sum closure of 1. The transformed targets from the
five examples presented in Table 1 are also shown.
3.2 Model Architecture
The model architecture used in this study is the same convolutional neural
network (CNN) architecture from our previous work [Narayanan et al., 2021]
where we used weak supervision [Zhou, 2018], transfer learning from a VGG-16
model pre-trained on the ImageNet dataset, and a multi-target output layer
with softmax regression trained to minimise root mean squared error (RMSE)
loss. The weak supervision is necessary as the dataset has missing values in 104
examples for the red and white clover subcomposition while the overall clover
values were available, and therefore had to be imputed with their corresponding
mean values and readjusted to match the overall clover proportion in the total
biomass. In doing so, the approximated examples were given a lesser weighting
during the loss calculation in a ratio of 1:1.5 with respect to the examples with
recorded values. Latent feature representations through transfer of weights from
the final convolutional layer of the pre-trained VGG-16 network enabled faster
and better optimization of the two trainable dense layers with 4,096 and 256
neurons. The dense layers were equipped with ReLU activations and uniform
random kernel initialization, and each of the dense layers was followed by a layer
of batch normalization to help prevent overfitting. The network was compiled
with the Adam optimizer with an initial learning rate 0.001 and decay factor
10−3 /200. The output layer had 4 neurons with softmax activation, each neuron
corresponding to regression output for grass, white clover, red clover and weeds. As
a direct interpretation of Amos’s theorem [Amos, 2019], the softmax probabilities
from the output layer can be construed as equivalent to the simplical proportions
of the predicted values of the individual components. This provides a framework
for interpreting the results of the model with softmax outputs in the context of
the simplex. In our current work, we transform the 4 target variables of grass,
white clover, red clover and weeds into 3 ILR coefficients. We modify the output
layer of the network in this experiment to 3 neurons with linear activations for
real valued outputs, and use RMSE as the loss function.
8 B. Narayanan et al.
Fig. 1: Training and validation losses (RMSE) for the ILR transformed data.
4 Results and Discussion
An examination of the training and validation losses during model training,
presented in Figure 1, confirms the ability of the model to learn from the ILR
transformed targets in the training data. The top row in Figure 2 shows scatter
plots from the baseline model results from our previous work, of the actual values
versus predicted values for each component of the biomass composition for data
in the validation set. Similarly, the bottom row shows scatter plots of actual vs
predicted values for the ILR transformed targets. It is interesting to note that
the ILR model predicts the proportions of grass and white clover reasonably
well. In the case of red clover, however, the predictions are generally erroneous.
Prediction of weeds is reasonably accurate when the actual weed percentage is
less than 20% of the composition, but erroneous results can be observed above
this range.
The results from our previous work is the baseline for comparison with the
results of the CoDa adaptation experiment. Table 3 compares the performance of
this baseline model against the model trained to predict ILR transformed targets.
It is clear that the baseline model outperforms the CoDa-inspired model.
This experiment shows convincing evidence of the ability to learn from ILR
transformed compositional data. Nevertheless, the CoDa adaptation results do
not improve upon the performance of this baseline model. We surmise that
there are two reasons for this. First, it is interesting to note that the softmax
function projects the real valued output vector of the network onto the simplex
(as explained in Section 2.2). In our case the training targets too are in the same
simplical dimensions, and therefore, it is effective within the simplex. Second,
Adaptation of Compositional Data Analysis in DL 9
Fig. 2: Scatter plots of actual vs predicted values for the components.
the premise of CoDa is to ensure data transformations that will satisfy the
requirements of standard statistical analyses, like the central limit theorem and
conformance to the rules of linear independence. On the contrary, deep neural
networks do not require such assumptions and have the ability to effectively
approximate a non-linear estimation function to fit an unknown distribution of
the target data. The problem that CoDa is designed to solve using standard
statistical methods does not exist in deep neural networks. Therefore, the CoDa
adaptation for deep neural networks using the ILR transformation of the targets is
an additional step over a network with an intrinsic ability to learn these non-linear
functions. We believe that these two reasons provide a plausible explanation for
Validation metrics RMSE, MAE
Grass White clover Red clover Weeds Overall
Model RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE
Baseline 8.00 6.21 7.44 5.99 7.33 5.63 5.68 4.20 7.11 5.51
ILR 8.87 7.04 12.64 8.94 11.98 7.39 5.63 2.95 9.78 6.58
Table 3: Comparison of validation metrics RMSE and MAE with the baseline
from our earlier work.
10 B. Narayanan et al.
the better performance of the network with softmax activation over the ILR
transformed approach.
5 Conclusion
In this paper we explored the usefulness of techniques from statistical compo-
sitional data analysis in a deep learning context. In particular, we tested this
with a pasture biomass prediction problem, which is compositional in nature. We
presented an approach that transformed the biomass composition data using the
isometric log ratio (ILR) from the simplex space onto the real space and used
these transformed targets for training a deep network. This paper demonstrates
that it is possible to train a reasonably accurate prediction model using this
approach. Nevertheless, based on the evidence of the results, we conclude that
the softmax works better in the deep learning context than a model trained to
predict targets transformed using ILR. This suggests that it is not useful to adapt
techniques from statistics CoDa to deep learning models. Our further work will
focus on improving the prediction for red clover and weeds.
Acknowledgements
This publication has emanated from research conducted with the financial support
of Science Foundation Ireland under Grant number [16/RC/3835]. For the purpose
of Open Access, the author has applied a CC BY public copyright licence to any
Author Accepted Manuscript version arising from this submission.
Our sincere thanks to Prof. Claire Gormley, School of Mathematics and
Statistics, University College Dublin, for her suggestion to explore Compositional
Data Analysis.
Bibliography
Aitchison, J.: The statistical analysis of compositional data. Journal of the Royal
Statistical Society: Series B (Methodological) 44(2), 139–160 (1982)
Aitchison, J.: A Concise Guide to Compositional Data Analysis p. 134 (2005)
Amos, B.: Differentiable optimization-based modeling for machine learning. Ph.D.
thesis, PhD thesis. Carnegie Mellon University (2019)
Castro, W., Marcato Junior, J., Polidoro, C., Osco, L.P., Gonçalves, W., Ro-
drigues, L., Santos, M., Jank, L., Barrios, S., Valle, C., et al.: Deep learning
applied to phenotyping of biomass in forages with uav-based rgb imagery.
Sensors 20(17), 4802 (2020)
Chayes, F.: On correlation between variables of constant sum. Journal of Geo-
physical research 65(12), 4185–4193 (1960)
Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G., Barcelo-Vidal, C.: Iso-
metric logratio transformations for compositional data analysis. Mathematical
geology 35(3), 279–300 (2003)
Adaptation of Compositional Data Analysis in DL 11
Filzmoser, P., Hron, K., Reimann, C.: The bivariate statistical analysis of
environmental (compositional) data. Science of The Total Environment
408(19), 4230–4238 (Sep 2010), https://linkinghub.elsevier.com/retrieve/pii/
S0048969710004845
Harris, J., Grunsky, E.: Predictive lithological mapping of Canada’s North using
Random Forest classification applied to geophysical and geochemical data.
Computers & Geosciences 80, 9–25 (Jul 2015), https://linkinghub.elsevier.com/
retrieve/pii/S0098300415000709
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recogni-
tion. In: Proceedings of the IEEE conference on computer vision and pattern
recognition. pp. 770–778 (2016)
Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep
convolutional neural networks. In: Advances in neural information processing
systems. pp. 1097–1105 (2012)
Larsen, D., Steen, K.A., Grooters, K., Green, O., Nyholm, R., et al.: Autonomous
mapping of grass-clover ratio based on unmanned aerial vehicles and convolu-
tional neural networks. In: International Conference on Precision Agriculture.
International Society of Precision Agriculture (2018)
Liu, Y., Cheng, Q., Zhou, K., Xia, Q., Wang, X.: Multivariate analysis for
geochemical process identification using stream sediment geochemical data:
A perspective from compositional data. Geochemical Journal 50(4), 293–314
(2016)
Martı́n-Fernández, J.A., Barceló-Vidal, C., Pawlowsky-Glahn, V.: Dealing with
zeros and missing values in compositional data sets using nonparametric
imputation. Mathematical Geology 35(3), 253–278 (2003)
Narayanan, B., Saadeldin, M., Albert, P., McGuinness, K., Mac Namee, B.:
Extracting pasture phenotype and biomass percentages using weakly supervised
multi-target deep learning on a small dataset. arXiv preprint arXiv:2101.03198
(2021)
Pan, S.J., Yang, Q.: A survey on transfer learning. IEEE Transactions on
knowledge and data engineering 22(10), 1345–1359 (2009)
Pawlowsky-Glahn, V., Egozcue, J.J.: Compositional data and their analysis: an
introduction. Geological Society, London, Special Publications 264(1), 1–10
(2006)
Pawlowsky-Glahn, V., Egozcue, J.J., Tolosana Delgado, R.: Lecture notes on
compositional data analysis (2007)
Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale
image recognition. arXiv preprint arXiv:1409.1556 (2014)
Sindic, C.M., Riday, H.: Using image object recognition to increase biomass in red
clover (trifolium pratense l.) breeding. Crop Science 60(4), 1770–1781 (2020)
Skovsen, S., Dyrmann, M., Eriksen, J., Gislum, R., Karstoft, H., Jørgensen, R.N.:
Predicting dry matter composition of grass clover leys using data simulation and
camera-based segmentation of field canopies into white clover, red clover, grass
and weeds. In: Proceedings of the 14th International Conference on Precision
Agriculture. Montreal, CA: International Society of Precision Agriculture
(2018)
12 B. Narayanan et al.
Skovsen, S., Dyrmann, M., Mortensen, A.K., Laursen, M.S., Gislum, R., Eriksen,
J., Farkhani, S., Karstoft, H., Jorgensen, R.N.: The grassclover image dataset for
semantic and hierarchical species understanding in agriculture. In: Proceedings
of the IEEE/CVF Conference on Computer Vision and Pattern Recognition
Workshops. pp. 0–0 (2019)
Sun, S., Liang, N., Zuo, Z., Parsons, D., Morel, J., Shi, J., Wang, Z., Luo, L., Zhao,
L., Fang, H., et al.: Estimation of botanical composition in mixed clover–grass
fields using machine learning-based image analysis. Frontiers in Plant Science
12, 87 (2021)
Talebi, H., Mueller, U., Tolosana-Delgado, R., Grunsky, E., McKinley, J., Car-
itat, P.d.: Surficial and Deep Earth Material Prediction from Geochemical
Compositions. Natural Resources Research 28 (Oct 2018)
Tolosana-Delgado, R.: Compositional data analysis in a nutshell. University of
Gottingen on-line reference (2008)
Tolosana-Delgado, R., Mueller, U., van den Boogaart, K.G.: Geostatistics for
compositional data: an overview. Mathematical geosciences 51(4), 485–526
(2019)
Verwaeren, J.: Mathematical optimization methods for the analysis of composi-
tional data: subset selection, unmixing and prediction. Ph.D. thesis, Ghent
University (2014)
Zhou, Z.H.: A brief introduction to weakly supervised learning. National Science
Review 5(1), 44–53 (2018)