ABC random forests for model choice and parameters estimation

• Python
• Usage
• Model Choice
• Parameter Estimation
• Various
• TODO
• References

Random forests methodologies for :

• ABC model choice (Pudlo et al. 2015)
• ABC Bayesian parameter inference (Raynal et al. 2018)

Libraries we use :

• Ranger (Wright and Ziegler 2015) : we use our own fork and have tuned forests to do “online”¹ computations (Growing trees AND making predictions in the same pass, which removes the need of in-memory storage of the whole forest)².
• Eigen3 (Guennebaud, Jacob, et al. 2010)

As a mention, we use our own implementation of LDA and PLS from (Friedman, Hastie, and Tibshirani 2001, vol. 181, 114), PLS is optimized for univariate, see 5.1. For linear algebra optimization purposes on large reftables, the Linux version of binaries (standalone and python wheel) are statically linked with Intel’s Math Kernel Library, in order to leverage multicore and SIMD extensions on modern cpus.

There is one set of binaries, which contains a Macos/Linux/Windows (x64 only) binary for each platform. There are available within the “Releases” tab, under “Assets” section (unfold it to see the list).

This is pure command line binary, and they are no prerequisites or library dependencies in order to run it. Just download them and launch them from your terminal software of choice. The usual caveats with command line executable apply there : if you’re not proficient with the command line interface of your platform, please learn some basics or ask someone who might help you in those matters.

The standalone is part of a specialized Population Genetics graphical interface DIYABC-RF, presented in MER (Molecular Ecology Resources, Special Issue), (Collin et al. 2021).

Python

Installation

pip install pyabcranger

Notebooks examples

• On a toy example with $MA(q)$, using (Lintusaari et al. 2018) as graph-powered engine.
• Population genetics demo, data from (Collin et al. 2021), available there

Usage

 - ABC Random Forest - Model choice or parameter estimation command line options
Usage:
  ../build/abcranger [OPTION...]

  -h, --header arg        Header file (default: headerRF.txt)
  -r, --reftable arg      Reftable file (default: reftableRF.bin)
  -b, --statobs arg       Statobs file (default: statobsRF.txt)
  -o, --output arg        Prefix output (modelchoice_out or estimparam_out by
                          default)
  -n, --nref arg          Number of samples, 0 means all (default: 0)
  -m, --minnodesize arg   Minimal node size. 0 means 1 for classification or
                          5 for regression (default: 0)
  -t, --ntree arg         Number of trees (default: 500)
  -j, --threads arg       Number of threads, 0 means all (default: 0)
  -s, --seed arg          Seed, generated by default (default: 0)
  -c, --noisecolumns arg  Number of noise columns (default: 5)
      --nolinear          Disable LDA for model choice or PLS for parameter
                          estimation
      --plsmaxvar arg     Percentage of maximum explained Y-variance for
                          retaining pls axis (default: 0.9)
      --chosenscen arg    Chosen scenario (mandatory for parameter
                          estimation)
      --noob arg          number of oob testing samples (mandatory for
                          parameter estimation)
      --parameter arg     name of the parameter of interest (mandatory for
                          parameter estimation)
  -g, --groups arg        Groups of models
      --help              Print help

• If you provide --chosenscen, --parameter and --noob, parameter estimation mode is selected.
• Otherwise by default it’s model choice mode.
• Linear additions are LDA for model choice and PLS for parameter estimation, “–nolinear” options disables them in both case.

Model Choice

Terminal model choice

Example

Example :

abcranger -t 10000 -j 8

Header, reftable and statobs files should be in the current directory.

Groups

With the option -g (or --groups), you may “group” your models in several groups splitted . For example if you have six models, labeled from 1 to 6 `-g “1,2,3;4,5,6”

Generated files

Four files are created :

• modelchoice_out.ooberror : OOB Error rate vs number of trees (line number is the number of trees)
• modelchoice_out.importance : variables importance (sorted)
• modelchoice_out.predictions : votes, prediction and posterior error rate
• modelchoice_out.confusion : OOB Confusion matrix of the classifier

Parameter Estimation

Terminal estim param

Composite parameters

When specifying the parameter (option --parameter), one may specify simple composite parameters as division, addition or multiplication of two existing parameters. like t/N or T1+T2.

A note about PLS heuristic

The --plsmaxvar option (defaulting at 0.90) fixes the number of selected pls axes so that we get at least the specified percentage of maximum explained variance of the output. The explained variance of the output of the $m$ first axes is defined by the R-squared of the output:

$$Yvar^m = \frac{\sum_{i=1}^{N}{(\hat{y}^{m}{i}-\bar{y})^2}}{\sum{i=1}^{N}{(y_{i}-\hat{y})^2}}$$

where $\hat{y}^{m}$ is the output $Y$ scored by the pls for the $m$th component. So, only the $n_{comp}$ first axis are kept, and :

$$n_{comp} = \underset{Yvar^m \leq{} 0.90*Yvar^M, }{\mathop{\text{argmax}}}$$

Note that if you specify 0 as --plsmaxvar, an “elbow” heuristic is activiated where the following condition is tested for every computed axis :

$$\frac{Yvar^{k+1}+Yvar^{k}}{2} \geq 0.99(N-k)\left(Yvar^{k+1}-Yvar^ {k}\right)$$

If this condition is true for a windows of previous axes, sized to 10% of the total possible axis, then we stop the PLS axis computation.

In practice, we find this $n_{heur}$ close enough to the previous $n_{comp}$ for 99%, but it isn’t guaranteed.

The signification of the noob parameter

The median global/local statistics and confidence intervals (global) measures for parameter estimation need a number of OOB samples (--noob) to be reliable (typlially 30% of the size of the dataset is sufficient). Be aware than computing the whole set (i.e. assigning --noob the same than for --nref) for weights predictions (Raynal et al. 2018) could be very costly, memory and cpu-wise, if your dataset is large in number of samples, so it could be adviseable to compute them for only choose a subset of size noob.

Example (parameter estimation)

Example (working with the dataset in test/data) :

abcranger -t 1000 -j 8 --parameter ra --chosenscen 1 --noob 50

Header, reftable and statobs files should be in the current directory.

Generated files (parameter estimation)

Five files (or seven if pls activated) are created :

• estimparam_out.ooberror : OOB MSE rate vs number of trees (line number is the number of trees)
• estimparam_out.importance : variables importance (sorted)
• estimparam_out.predictions : expectation, variance and 0.05, 0.5, 0.95 quantile for prediction
• estimparam_out.predweights : csv of the value/weights pairs of the prediction (for density plot)
• estimparam_out.oobstats : various statistics on oob (MSE, NMSE, NMAE etc.)

if pls enabled :

• estimparam_out.plsvar : variance explained by number of components
• estimparam_out.plsweights : variable weight in the first component (sorted by absolute value)

Various

Partial Least Squares algorithm (univariate)

1. $X_{0}=X ; y_{0}=y$
2. For $k=1,2,...,s$ :
   1. $w_{k}=\frac{X_{k-1}^{T} y_{k-1}}{y_{k-1}^{T} y_{k-1}}$
   2. Normalize $w_k$ to $1$
   3. $t_{k}=\frac{X_{k-1} w_{k}}{w_{k}^{T} w_{k}}$
   4. $p_{k}=\frac{X_{k-1}^{T} t_{k}}{t_{k}^{T} t_{k}}$
   5. $X_{k}=X_{k-1}-t_{k} p_{k}^{T}$
   6. $q_{k}=\frac{y_{k-1}^{T} t_{k}}{t_{k}^{T} t_{k}}$
   7. $u_{k}=\frac{y_{k-1}}{q_{k}}$
   8. $y_{k}=y_{k-1}-q_{k} t_{k}$

Comment When there isn’t any missing data, stages $2.1$ and $2.2$ could be replaced by $w_{k}=\frac{X_{k-1}^{T} y_{k-1}}{\left|X_{k-1}^{T} y_{k-1}\right|}$ and $2.3$ by $t_{k}=X_{k-1}w_{k}$

To get $W$ so that $T=XW$ we compute : $$\mathbf{W}=\mathbf{W}^{*}\left(\widetilde{\mathbf{P}} \mathbf{W}^{*}\right)^{-1}$$

where $\widetilde{\mathbf{P}}{K \times p}=\mathbf{t}\left[p{1}, \ldots, p_{K}\right]$ and $\mathbf{W}^{*}_{p \times K} = [w_1, \ldots, w_K]$

TODO

Input/Output

• Integrate hdf5 (or exdir? msgpack?) routines to save/load reftables/observed stats with associated metadata
• Provide R code to save/load the data
• Provide Python code to save/load the data

C++ standalone

• Merge the two methodologies in a single executable with the (almost) the same options
• (Optional) Possibly move to another options parser (CLI?)

External interfaces

• R package
• Python package

Documentation

• Code documentation
• Document the build

Continuous integration

• Linux CI build with intel/MKL optimizations
• osX CI build
• Windows CI build

Long/Mid term TODO

• methodologies parameters auto-tuning
  • auto-discovering the optimal number of trees by monitoring OOB error
  • auto-limiting number of threads by available memory
• Streamline the two methodologies (model choice and then parameters estimation)
• Write our own tree/rf implementation with better storage efficiency than ranger
• Make functional tests for the two methodologies
• Possible to use mondrian forests for online batches ? See (Lakshminarayanan, Roy, and Teh 2014)

References

This have been the subject of a proceedings in JOBIM 2020, PDF and video (in french), (Collin et al. 2020).

Collin, François-David, Ghislain Durif, Louis Raynal, Eric Lombaert, Mathieu Gautier, Renaud Vitalis, Jean-Michel Marin, and Arnaud Estoup. 2021. “Extending Approximate Bayesian Computation with Supervised Machine Learning to Infer Demographic History from Genetic Polymorphisms Using DIYABC Random Forest.” Molecular Ecology Resources 21 (8): 2598–2613. https://doi.org/https://doi.org/10.1111/1755-0998.13413.

Collin, François-David, Arnaud Estoup, Jean-Michel Marin, and Louis Raynal. 2020. “Bringing ABC inference to the machine learning realm : AbcRanger, an optimized random forests library for ABC.” In JOBIM 2020, 2020:66. JOBIM. Montpellier, France. https://hal.archives-ouvertes.fr/hal-02910067.

Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. 2001. The Elements of Statistical Learning. Vol. 1. 10. Springer series in statistics New York, NY, USA:

Guennebaud, Gaël, Benoît Jacob, et al. 2010. “Eigen V3.” http://eigen.tuxfamily.org.

Lakshminarayanan, Balaji, Daniel M Roy, and Yee Whye Teh. 2014. “Mondrian Forests: Efficient Online Random Forests.” In Advances in Neural Information Processing Systems, 3140–48.

Lintusaari, Jarno, Henri Vuollekoski, Antti Kangasrääsiö, Kusti Skytén, Marko Järvenpää, Pekka Marttinen, Michael U. Gutmann, Aki Vehtari, Jukka Corander, and Samuel Kaski. 2018. “ELFI: Engine for Likelihood-Free Inference.” Journal of Machine Learning Research 19 (16): 1–7. http://jmlr.org/papers/v19/17-374.html.

Pudlo, Pierre, Jean-Michel Marin, Arnaud Estoup, Jean-Marie Cornuet, Mathieu Gautier, and Christian P Robert. 2015. “Reliable ABC Model Choice via Random Forests.” Bioinformatics 32 (6): 859–66.

Raynal, Louis, Jean-Michel Marin, Pierre Pudlo, Mathieu Ribatet, Christian P Robert, and Arnaud Estoup. 2018. “ABC random forests for Bayesian parameter inference.” Bioinformatics 35 (10): 1720–28. https://doi.org/10.1093/bioinformatics/bty867.

Wright, Marvin N, and Andreas Ziegler. 2015. “Ranger: A Fast Implementation of Random Forests for High Dimensional Data in c++ and r.” arXiv Preprint arXiv:1508.04409.

Footnotes

1. The term “online” there and in the code has not the usual meaning it has, as coined in “online machine learning”. We still need the entire training data set at once. Our implementation is an “online” one not by the sequential order of the input data, but by the sequential order of computation of the trees in random forests, sequentially computed and then discarded. ↩

2. We only use the C++ Core of ranger, which is under MIT License, same as ours. ↩