Gradient -\(\nabla Q\) of the negative E-step function -Q of the expectation-maximization algorithm

internal

A random-effects stochastic model that allows quick detection of
clonal dominance events from clonal tracking data collected in gene therapy studies. Starting from the Ito-type equation describing the dynamics of cells duplication, death and differentiation at clonal level, we first considered its local linear approximation as the base model.
The parameters of the base model, which are inferred using a maximum likelihood approach,
are assumed to be shared across the clones. Although this assumption makes inference easier,
in some cases it can be too restrictive and does not take into account possible scenarios of clonal dominance.
Therefore we extended the base model by introducing random effects for the clones.
In this extended formulation the dynamic parameters are estimated using a tailor-made
expectation maximization algorithm. Further details on the methods can be found in L. Del Core et al., (2022) <doi:10.1101/2022.05.31.494100>.

Luca Del Core

RestoreNet

Random-Effects Stochastic Reaction Networks

Marco Grzegorczyk

Ernst Wit

ngQ function

<dl><dt>theta</dt>
<dd>p-dimensional vector parameter.</dd>
<dt>euy_curr</dt>
<dd>current value of the conditional expectation \(E[u \vert y]\) of u given y,
where u and y are the latent and observed states respectively.</dd>
<dt>vuy_curr</dt>
<dd>current value of the conditional variance \(V[u \vert y]\) of u given y,
where u and y are the latent and observed states respectively.</dd>
<dt>M</dt>
<dd>A \(n \times K\) dimensional (design) matrix.</dd>
<dt>M_bdiag</dt>
<dd>A\(n \times Jp\) dimensional block-diagonal design matrix.
Each j-th block (\(j = 1,\dots,J\)) is a \(n_j \times p\) dimensional design matrix for the j-th clone.</dd>
<dt>y</dt>
<dd>n-dimensional vector of the time-adjacent cellular increments</dd>
<dt>V</dt>
<dd>A \(p \times K\) dimensional net-effect matrix.</dd>
<dt>VCNs</dt>
<dd>A n-dimensional vector including values of the vector copy number corresponding to the cell counts of y.</dd>
<dt>nObs</dt>
<dd>A K-dimensional vector including the frequencies of each clone k (\(k = 1,\dots,K\)).</dd>
<dt>dW</dt>
<dd>p-dimensional list of the partial derivatives of W w.r.t. theta.</dd></dl>

Arguments

Gradient of the E-step function Q — ngQ

<dl>

<dt>theta</dt>
<dd>p-dimensional vector parameter.</dd>


<dt>euy_curr</dt>
<dd>current value of the conditional expectation \(E[u \vert y]\) of u given y,
where u and y are the latent and observed states respectively.</dd>


<dt>vuy_curr</dt>
<dd>current value of the conditional variance \(V[u \vert y]\) of u given y,
where u and y are the latent and observed states respectively.</dd>


<dt>M</dt>
<dd>A \(n \times K\) dimensional (design) matrix.</dd>


<dt>M_bdiag</dt>
<dd>A\(n \times Jp\) dimensional block-diagonal design matrix.
Each j-th block (\(j = 1,\dots,J\)) is a \(n_j \times p\) dimensional design matrix for the j-th clone.</dd>


<dt>y</dt>
<dd>n-dimensional vector of the time-adjacent cellular increments</dd>


<dt>V</dt>
<dd>A \(p \times K\) dimensional net-effect matrix.</dd>


<dt>VCNs</dt>
<dd>A n-dimensional vector including values of the vector copy number corresponding to the cell counts of y.</dd>


<dt>nObs</dt>
<dd>A K-dimensional vector including the frequencies of each clone k (\(k = 1,\dots,K\)).</dd>


<dt>dW</dt>
<dd>p-dimensional list of the partial derivatives of W w.r.t. theta.</dd>

</dl>

ngQ: Gradient of the E-step function Q

Description

Usage

Value

Arguments