About the method

================

MOVE is based on the VAE (variational autoencoder) model, a deep learning model

that transforms high-dimensional data into a lower-dimensional space (so-called

latent representation). The autoencoder is made up of two neural networks: an

encoder, which compresses the input variables; and a decoder, which tries to

reconstruct the original input from the compressed representation. In doing so,

the model learns the structure and associations between the input variables.

In `our publication`_, we used this type of model to integrate different data

modalities, including: genomics, transcriptomics, proteomics, metabolomics,

microbiomes, medication data, diet questionnaires, and clinical measurements.

Once we obtained a trained model, we exploited the decoder network to identify

cross-omics associations.

Our approach consists of performing *in silico* perturbations of the original

data and using either univariate statistical methods or Bayesian decision

theory to identify significant differences between the reconstruction with or

without perturbation. Thus, we are able to detect associations between the

input variables.

.. _`our publication`: https://www.nature.com/articles/s41587-022-01520-x

.. image:: method/fig1.svg

VAE design

-----------

The VAE was designed to account for a variable number of fully-connected hidden

layers in both encoder and decoder. Each hidden layer is followed by batch

normalization, dropout, and a leaky rectified linear unit (leaky ReLU).

To integrate different modalities, each dataset is reshaped and concatenated

into an input matrix. Moreover, error calculation is done on a dataset

basis: binary cross-entropy for binary and categorical datasets and mean squared

error for continuous datasets. Each error :math:`E_i` is then multiplied by a

given weight :math:`W_i` and added up to form the loss function:

:math:`L = \sum_i W_i E_i + W_\textnormal{KL} D_\textnormal{KL}`

Note that the :math:`D_\textnormal{KL}` (Kullback–Leibler divergence) penalizes

deviance of the latent representation from the standard normal distribution. It

is also subject to a weight :math:`W_\textnormal{KL}`, which warms up as the

model is trained.

Extracting associations

-----------------------

After determining the right set of hyperparameters, associations are extracted

by perturbing the original input data and passing it through an ensemble of

trained models. The reason behind using an ensemble is that VAE models are

stochastic, so we need to ensure that the results we obtain are not a product

of chance.

We perturbed categorical data by changing its value from one category to

another (e.g., drug status changed from "not received" to "received"). Then, we

compare the change between the reconstruction generated from the original data

and the perturbed data. To achieve this, we proposed two approaches: using

*t*\ -test and Bayes factors. Both are described below:

MOVE *t*\ -test

^^^^^^^^^^^^^^^

#. Perturb a variable in one dataset.

#. Repeat 10 times for 4 different latent space sizes:

    #. Train VAE model with original data.

    #. Obtain reconstruction of original data (baseline reconstruction).

    #. Obtain 10 additional reconstructions of original data and calculate

       difference from the first (baseline difference).

    #. Obtain reconstruction of perturbed data (perturbed reconstruction) and

       subtract from baseline reconstruction (perturbed difference).

    #. Compute p-value between baseline and perturbed differences with t-test.

#. Correct p-values using Bonferroni method.

#. Select features that are significant (p-value lower than 0.05).

#. Select significant features that overlap in at least half of the refits and

   3 out of 4 architectures. These    features are associated with the

   perturbed variable.

MOVE Bayes

^^^^^^^^^^

#. Perturb a variable in one dataset.

#. Repeat 30 times:

    #. Train VAE model with original data.

    #. Obtain reconstruction of original data (baseline reconstruction).

    #. Obtain reconstruction of perturbed data (perturbed reconstruction).

    #. Record difference between baseline and perturbed reconstruction.

#. Compute probability of difference being greater than 0.

#. Compute Bayes factor from probability: :math:`K = \log p - \log (1 - p)`.

#. Sort probabilities by Bayes factor, from highest to lowest.

#. Compute false discovery rate (FDR) as cumulative evidence.

#. Select features whose FDR is above desired threshold (e.g., 0.05). These

   features are associated with the perturbed variable.