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Table 1 List of mathematical symbols

From: A new Bayesian piecewise linear regression model for dynamic network reconstruction

Symbol

Description

Prior distribution

N

Total number of nodes (genes)

n

Number of potential parent nodes, here \(n=N-1\)

h

Data segment h

H

Total number of data segments

k

Number of covariates in covariate set

t

Data point t

\(\sigma ^2\)

Noise variance parameter

\(\sigma ^{-2}\sim GAM(\alpha _{\sigma },\beta _{\sigma })\)

\(\lambda _c\)

Coupling strength parameter, \(h>1\)

\(\lambda _c^{-1}\sim GAM(\alpha _{c},\beta _{c})\)

\(\lambda _u\)

SNR parameter, \(h=1\)

\(\lambda _u^{-1}\sim GAM(\alpha _{u},\beta _{u})\)

\(\lambda _h\)

hth coupling strength parameter (M4 model)

\(\lambda _h^{-1}\sim GAM(\alpha _{c},\beta _{c})\)

\(\delta _h\)

hth coupling indicator variable (M3 model)

\(\delta _h\sim BER(\text{ p})\), \(\text{ p }\sim BETA(a,b)\)

T

Total number of data points

\(T_h\)

Number of data points in segment h

\(D_i\)

ith data point

\(Z_i\)

ith network node

\({\varvec{\pi }_{i}}\)

Parent (covariate) set of ith node, \(Z_i\)

\(p(|\varvec{\pi }|<=3)=c\), \(p(|\varvec{\pi }|>3)=0\)

\(\varvec{\tau }\)

Changepoint set

\(p(\varvec{\tau }) = (1-p)^{(T-1)-(H-1)} \cdot p^{H-1}\)

\(\tau _h\)

Changepoint h

\(X_i\)

ith covariate

\(\mathbf{X }_{h}\)

Design matrix of segment h

\(\mathbf{y }_{h}\)

Response vector of segment h

\(\mathbf{y }_{h}|(\varvec{{w}}_{h},\sigma ^2) \sim \mathcal {N}(\mathbf{X }_{h} \varvec{{w}}_{h} , \sigma ^2 \mathbf{I })\)

\(\varvec{{w}}_{h}\)

Regression coefficient vector of segment h

\(\varvec{{w}}_{h}|(\varvec{\mu }_h,\varvec{\Sigma }_h,\sigma ^2) \sim \mathcal {N}(\varvec{\mu }_h, \sigma ^2 \varvec{\Sigma }_h )\)

\(\tilde{\varvec{{w}}_{h-1}}\)

Posterior expectation of \(\varvec{{w}}_{h-1}\)