Network-Based Inference Framework for Identifying Cancer Genes from Gene Expression Data

Algorithm 1

Posteriori score-based boosting regression algorithm for inferring networks as Step 2 of netSAM.

INPUT: 𝑛×𝑝 dimensional data matrix 𝐗=(𝑥₁,…,𝑥_𝑖,…,𝑥_𝑝),

(𝑛 denotes the number of samples and 𝑝 represents the number of

genes, 1<𝑖<𝑝).

OUTPUT: An adjacency matrix of graph 𝐆.

(1) Initialize iterating counter 𝑘=0, coefficient of the

  regression 𝛽_𝑖𝑗=0 for all 1≤𝑖, 𝑗≤𝑝, 𝜆=2, and step

  size 𝜀=0.01, 𝑆=10⋅𝑝.

(2) Standardize column vector 𝐱_𝑖 to zero mean and unit

  norm for gene 𝑖(1<𝑖<𝑝). Set residual 𝐑_𝑖^(𝑘)←𝐱_𝑖.

(3) Fit 𝑝×𝑝−1 regressions

  𝛽_𝑖𝑗=𝐱^𝑇_𝑗𝐑_𝑖^(𝑘)𝑖≠𝑗,1≤𝑖,𝑗≤𝑝, where 𝐱^𝑇_𝑗

  means the transpose of 𝐱_𝑗.

(4) Calculate a posteriori score based on likelihood score and

  informative prior as following

  logposterioriscore_𝑖𝑗=−logRSS_𝑖𝑗+log𝐶²

  𝐶=𝑟log𝜋(𝑑),log(𝑑),

  𝜋(𝑑)=𝑑^−𝜆(𝜆>0),𝑑=∑_{1≤𝑚≤𝑝}𝛽_𝑚𝑗,

  where RSS refers to (3) and 𝑟(⋅,⋅) denotes Pearson's

  correlation coefficient.

(5) Find the edge having the best score of posteriori

  probability

  𝑖,̂𝑗̂=argmax_𝑖,𝑗logposterioriscore_𝑖𝑗

(6) Perform the boosting update

  𝛽̂𝑖_,𝑗̂^(𝑘+1)←𝛽̂𝑖_,𝑗̂^(𝑘)𝑖̂+𝛿_,̂𝑗^(𝑘), and

  𝛿̂𝑖_,𝑗̂^(𝑘)𝛽̂𝑖=𝜀⋅_,̂𝑗⋅sgn𝐱^𝑇̂𝑗𝐑̂𝑖^(𝑘),where𝜀>0

  and set 𝐑̂𝑖^(𝑘+1)𝑖̂←𝐑^(𝑘)̂𝑖−𝛿_,̂𝑗^(𝑘)𝐱̂𝑗 and increment

  counter 𝑘←𝑘+1

(7) Repeat Steps 3 to 6 until 𝑘=𝑆.

(8) Calculate 𝜌_𝑖𝑗 from coefficient matrix 𝜷^(𝑆),

  ifsgn𝛽^(𝑆)_𝑖𝑗=sgn𝛽^(𝑆)_𝑗𝑖,

  𝜌_𝑖𝑗=min1,𝛽^(𝑆)_𝑖𝑗𝛽^(𝑆)_𝑗𝑖

  otherwise,𝜌_𝑖𝑗=0,

  where sgn (·) denotes the sign function.

(9) Return an adjacency matrix of network

𝐆=𝑒_𝑖𝑗∣𝑒_𝑖𝑗∈{0,1},𝑒_𝑖𝑗=0,if𝜌_𝑖𝑗=01,otherwise