# Towards Confidence Estimates in Cascade Networks using the SelectBoost Package

#### Frédéric Bertrand and Myriam Maumy-Bertrand

#### 2022-11-29

#Introduction

## Aims of the vignette

Extending results from the `Cascade` package: reverse engineering with selectboost to compute confidence indices for a fitted model. We first fit a model to a cascade network using the `Cascade` package `inference` function then we compute confidence indices for the inferred links using the `Selecboost` algorithm.

If you are a Linux/Unix or a Macos user, you can install a version of SelectBoost with support for `doMC` from github with:

    devtools::install_github("fbertran/SelectBoost", ref = "doMC")

## References

- Reference for the Cascade modelling: Vallat, L., Kemper, C. a., Jung, N., Maumy-Bertrand, M., Bertrand, F., Meyer, N., Pocheville, A., Fisher, J. W., Gribben, J. G. et Bahram, S. (2013). Reverse-engineering the genetic circuitry of a cancer cell with predicted intervention in chronic lymphocytic leukemia. Proceedings of the National Academy of Sciences of the United States of America, 110(2), 459-64.

- Reference for the Cascade package: Jung, N., Bertrand, F., Bahram, S., Vallat, L. et Maumy-Bertrand, M. (2014). Cascade : A R package to study, predict and simulate the diffusion of a signal through a temporal gene network. Bioinformatics.

# Code

Code to reproduce the datasets saved with the package and some the figures of the article Bertrand et al. (2020), Bioinformatics. https://doi.org/10.1093/bioinformatics/btaa855

## Data simulation

    library(Cascade)

We define the F array for the simulations.

    T<-4
    F<-array(0,c(T-1,T-1,T*(T-1)/2))

    for(i in 1:(T*(T-1)/2)){diag(F[,,i])<-1}
    F[,,2]<-F[,,2]*0.2
    F[2,1,2]<-1
    F[3,2,2]<-1
    F[,,4]<-F[,,2]*0.3
    F[3,1,4]<-1
    F[,,5]<-F[,,2]

We set the seed to make the results reproducible

    set.seed(1)
    Net<-Cascade::network_random(
      nb=100,
      time_label=rep(1:4,each=25),
      exp=1,
      init=1,
      regul=round(rexp(100,1))+1,
      min_expr=0.1,
      max_expr=2,
      casc.level=0.4
    )
    Net@F<-F

We simulate gene expression according to the network Net

    M <- Cascade::gene_expr_simulation(
      network=Net,
      time_label=rep(1:4,each=25),
      subject=5,
      level_peak=200)

## Network inference

We infer the new network.

    Net_inf_C <- Cascade::inference(M,cv.subjects=TRUE)
    #> We are at step :  1
    #> The convergence of the network is (L1 norm) : 0.0068
    #> We are at step :  2
    #> The convergence of the network is (L1 norm) : 0.00121
    #> We are at step :  3
    #> The convergence of the network is (L1 norm) : 0.00096

Heatmap of the coefficients of the Omega matrix of the network. Run the code to get the graph.

    stats::heatmap(Net_inf_C@network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

    Fab_inf_C <- Net_inf_C@F

## Compute the confidence indices for the inference

    library(SelectBoost)
    set.seed(1)

By default the crossvalidation is made subjectwise according to a leave one out scheme and the resampling analysis is made at the .95 `c0` level. To pass CRAN tests, `use.parallel = FALSE` is required. Set `use.parallel = TRUE` and select the number of cores using `ncores = 4`.

    net_pct_selected <- selectboost(M, Fab_inf_C, use.parallel = FALSE)
    #>
    #> We are at peak :  2
    #> .........................
    #> We are at peak :  3
    #> .........................
    #> We are at peak :  4
    #> .........................

    net_pct_selected_.5 <- selectboost(M, Fab_inf_C, c0value = .5, use.parallel = FALSE)
    #>
    #> We are at peak :  2
    #> .........................
    #> We are at peak :  3
    #> .........................
    #> We are at peak :  4
    #> .........................

    net_pct_selected_thr <- selectboost(M, Fab_inf_C, group = group_func_1, use.parallel = FALSE)
    #>
    #> We are at peak :  2
    #> .........................
    #> We are at peak :  3
    #> .........................
    #> We are at peak :  4
    #> .........................

Use `cv.subject=FALSE` to use default crossvalidation

    net_pct_selected_cv <- selectboost(M, Fab_inf_C, cv.subject=FALSE, use.parallel = FALSE)
    #>
    #> We are at peak :  2
    #> .........................
    #> We are at peak :  3
    #> .........................
    #> We are at peak :  4
    #> .........................

## Analysis of the confidence indices

Use plot to display the result of the confidence analysis.

    plot(net_pct_selected)

Run the code to plot the other results.

    plot(net_pct_selected_.5)

    plot(net_pct_selected_thr)

    plot(net_pct_selected_cv)

Run the code to plot the remaning graphs.

Distribution of non-zero (absolute value > 1e-5) coefficients

    hist(Net_inf_C@network[abs(Net_inf_C@network)>1e-5])

Plot of confidence at .95 resampling level versus coefficient value for non-zero (absolute value > 1e-5) coefficients

    plot(Net_inf_C@network[abs(Net_inf_C@network)>1e-5],net_pct_selected@network.confidence[abs(Net_inf_C@network)>1e-5])

    hist(net_pct_selected@network.confidence[abs(Net_inf_C@network)>1e-5])

Plot of confidence at .5 resampling level versus coefficient value for non-zero (absolute value > 1e-5) coefficients

    plot(Net_inf_C@network[abs(Net_inf_C@network)>1e-5],net_pct_selected_.5@network.confidence[abs(Net_inf_C@network)>1e-5])

    hist(net_pct_selected_.5@network.confidence[abs(Net_inf_C@network)>1e-5])

Plot of confidence at .95 resamling level with groups created by thresholding the correlation matrix versus coefficient value for non-zero (absolute value > 1e-5) coefficients.

    plot(Net_inf_C@network[abs(Net_inf_C@network)>1e-5],net_pct_selected_thr@network.confidence[abs(Net_inf_C@network)>1e-5])

    hist(net_pct_selected_thr@network.confidence[abs(Net_inf_C@network)>1e-5])

Plot of confidence at .95 resampling level versus coefficient value for non-zero (absolute value > 1e-5) coefficients using standard cross-validation.

    plot(Net_inf_C@network[abs(Net_inf_C@network)>1e-5],net_pct_selected_cv@network.confidence[abs(Net_inf_C@network)>1e-5])

    hist(net_pct_selected_cv@network.confidence[abs(Net_inf_C@network)>1e-5])

## Further improvements

For further recommandations on the choice of the `c0` parameter, for instance as a quantile, please consult the SI of Bertrand et al. 2020.