Ridge estimation of the VAR(1) model and its time series chain graph from multivariate time‐course omics data

Miok, Viktorian; Wilting, Saskia M.; Wieringen, Wessel N. van

doi:10.1002/bimj.201500269

Cited by 6 publications

(24 citation statements)

References 35 publications

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“…This is facilitated by the methodology presented in Miok et al. (), which also describes how inferred model and graph may be exploited to deduce tangible implications for the medical researcher. Here, this is briefly recapitulated.…”

Section: The Var(1) Modelmentioning

confidence: 99%

“…The estimators of the VAR(1) model parameters

normalA

and

Ω_{ε}

are obtained through ridge penalized log‐likelihood maximization (see Miok et al., for details). The estimator of autoregression parameters

normalA

(fixing

Ω_{ε}

) is:

\begin{matrix} true \\ right vec false[\overset{A}{true} (λ_{a}) false] & = & left {true[λ_{a} I_{p^{2} \times p^{2}} + \overset{Γ}{true} (0) \otimes Ω_{ε} true) true]}^{- 1} \times {λ_{a} vec false(A_{0} false) + vec false[Ω_{ε} \overset{Γ}{true} (- 1) false]}, \end{matrix}

where

0true \overset{Γ}{true} (0) = \frac{1}{n false(scriptT - 1 false)} \sum_{i = 1}^{n} \sum_{t = 2}^{T} Y_{*, t - 1, i} Y_{*, t - 1, i}^{⊤}

and

0true \overset{Γ}{true} (- 1) = \frac{1}{n false(scriptT - 1 false)} \sum_{i = 1}^{n} \sum_{t = 2}^{T} Y_{*, t, i} Y_{*, t - 1, i}^{⊤}

.…”

Section: The Var(1) Modelmentioning

confidence: 99%

“…Table of Miok et al. () presents these for the most interesting genes from the P53 signaling pathway model, defined by a large number of out‐ and in‐degree edges and, henceforth, for convenience referred to as “regulators” and “regulatees” genes, respectively. For these genes centrality measures, mutual information and impulse response are given.…”

Section: The Var(1) Modelmentioning

confidence: 99%

“…Finally, path decomposition analysis unravels the association between two nodes at different time points in terms of the paths connecting them. Hereto the covariance between

Y_{j_{1}, t, i}

and

Y_{j_{2}, t + τ, i}

for

τ \geq 0

can be decomposed as a sum over all paths connecting j 1 and j 2 in the time‐series chain graph (Miok et al., ):

\begin{matrix} true \\ right Cov false(Y_{j_{1}, t, i}, Y_{j_{2}, t + τ, i} false| Y_{*, t - 1, i} false) = {({normalΣ}_{ε} {normalA}^{τ})}_{j_{1}, j_{2}} = \sum_{j^{'} = 1}^{p} {({normalΣ}_{ε})}_{j_{1}, j^{'}} {({normalA}^{τ})}_{j^{'}, j_{2}}, \end{matrix}

where

{false(Σ_{ε} false)}_{j_{1}, j^{'}}

can be further decomposed in terms of the contemporaneous paths connecting j 1 and

j^{'}

(see Jones and West, ). As such it guides in the search for the most important paths by which a molecular signal travels between these two nodes.…”

Section: The Var(1) Modelmentioning

confidence: 99%

“…The methodology underlying the first analysis is fully described in Miok et al. () but recapped here as the other analyses extend it. Next, the other network analyses, each incorporating more information through a more complex model, are presented.…”

Section: Introductionmentioning

confidence: 99%

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Ridge estimation of network models from time‐course omics data

2018

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Time-course omics experiments enable the reconstruction of the dynamics of the cellular regulatory network. Here, we describe the means for this reconstruction and the downstream exploitation of the inferred network. It is assumed that one of the various vector-autoregressive models (VAR) models presented here serves as a reasonably accurate description of the time-course omics data. The models are estimated through ridge penalized likelihood maximization, accompanied by functionality for the determination of optimal penalty paramaters. Prior knowledge on the network topology is accommodated by the estimation procedures. Various routes that translate the fitted models into more tangible implications for the medical researcher are described. The network is inferred from the-nonsparse-ridge estimates through empirical Bayes probabilistic thresholding. The influence of a (trait of a) molecular entity at the current time on those at future time points is assessed by mutual information, impulse response analysis, and path decomposition of the covariance. The presented methodology is applied to the omics data from the p53 signaling pathway during HPV-induced cellular transformation. All methodology is implemented in the ragt2ridges package, freely available from the Comprehensive R Archive Network.

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Section: The Var(1) Modelmentioning

confidence: 99%

“…The estimators of the VAR(1) model parameters

normalA

and

Ω_{ε}

are obtained through ridge penalized log‐likelihood maximization (see Miok et al., for details). The estimator of autoregression parameters

normalA

(fixing

Ω_{ε}

) is:

\begin{matrix} true \\ right vec false[\overset{A}{true} (λ_{a}) false] & = & left {true[λ_{a} I_{p^{2} \times p^{2}} + \overset{Γ}{true} (0) \otimes Ω_{ε} true) true]}^{- 1} \times {λ_{a} vec false(A_{0} false) + vec false[Ω_{ε} \overset{Γ}{true} (- 1) false]}, \end{matrix}

where

0true \overset{Γ}{true} (0) = \frac{1}{n false(scriptT - 1 false)} \sum_{i = 1}^{n} \sum_{t = 2}^{T} Y_{*, t - 1, i} Y_{*, t - 1, i}^{⊤}

and

0true \overset{Γ}{true} (- 1) = \frac{1}{n false(scriptT - 1 false)} \sum_{i = 1}^{n} \sum_{t = 2}^{T} Y_{*, t, i} Y_{*, t - 1, i}^{⊤}

.…”

Section: The Var(1) Modelmentioning

confidence: 99%

Section: The Var(1) Modelmentioning

confidence: 99%

“…Finally, path decomposition analysis unravels the association between two nodes at different time points in terms of the paths connecting them. Hereto the covariance between

Y_{j_{1}, t, i}

and

Y_{j_{2}, t + τ, i}

for

τ \geq 0

can be decomposed as a sum over all paths connecting j 1 and j 2 in the time‐series chain graph (Miok et al., ):

\begin{matrix} true \\ right Cov false(Y_{j_{1}, t, i}, Y_{j_{2}, t + τ, i} false| Y_{*, t - 1, i} false) = {({normalΣ}_{ε} {normalA}^{τ})}_{j_{1}, j_{2}} = \sum_{j^{'} = 1}^{p} {({normalΣ}_{ε})}_{j_{1}, j^{'}} {({normalA}^{τ})}_{j^{'}, j_{2}}, \end{matrix}

where

{false(Σ_{ε} false)}_{j_{1}, j^{'}}

can be further decomposed in terms of the contemporaneous paths connecting j 1 and

j^{'}

(see Jones and West, ). As such it guides in the search for the most important paths by which a molecular signal travels between these two nodes.…”

Section: The Var(1) Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ridge estimation of network models from time‐course omics data

2018

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Reconstruction of molecular network evolution from cross‐sectional omics data

2018

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Cross-sectional studies may shed light on the evolution of a disease like cancer through the comparison of patient traits among disease stages. This problem is especially challenging when a gene-gene interaction network needs to be reconstructed from omics data, and, in addition, the patients of each stage need not form a homogeneous group. Here, the problem is operationalized as the estimation of stage-wise mixtures of Gaussian graphical models (GGMs) from high-dimensional data. These mixtures are fitted by a (fused) ridge penalized EM algorithm. The fused ridge penalty shrinks GGMs of contiguous stages. The (fused) ridge penalty parameters are chosen through cross-validation. The proposed estimation procedures are shown to be consistent and their performance in other respects is studied in simulation. The down-stream exploitation of the fitted GGMs is outlined. In a data illustration the methodology is employed to identify gene-gene interaction network changes in the transition from normal to cancer prostate tissue.

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Globaltest confidence regions and their application to ridge regression

Solari

Goeman

2021

Biometrical J

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We construct confidence regions in high dimensions by inverting the globaltest statistics, and use them to choose the tuning parameter for penalized regression. The selected model corresponds to the point in the confidence region of the parameters that minimizes the penalty, making it the least complex model that still has acceptable fit according to the test that defines the confidence region. As the globaltest is particularly powerful in the presence of many weak predictors, it connects well to ridge regression, and we thus focus on ridge penalties in this paper. The confidence region method is quick to calculate, intuitive, and gives decent predictive potential. As a tuning parameter selection method it may even outperform classical methods such as cross‐validation in terms of mean squared error of prediction, especially when the signal is weak. We illustrate the method for linear models in simulation study and for Cox models in real gene expression data of breast cancer samples.

show abstract

Ridge estimation of the VAR(1) model and its time series chain graph from multivariate time‐course omics data

Cited by 6 publications

References 35 publications

Ridge estimation of network models from time‐course omics data

Ridge estimation of network models from time‐course omics data

Reconstruction of molecular network evolution from cross‐sectional omics data

Globaltest confidence regions and their application to ridge regression

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