Description Usage Arguments Details Value Note Author(s) References See Also Examples

This function use MM algorithm to fit a piece-wise constant curve to each of the signal sequences.

1 2 |

`Y` |
A matrix of original signal, where each column corresponds to a sequence and each row correspond to a marker. |

`Delta` |
A matrix of the same dimension as |

`sigma` |
A vector of standard deviations for each column of |

`rho1,rho2,rho3` |
Factors to be set in the tuning parameters of lambda1, lambda2, and lambda3. See details. |

`obj_c` |
Stopping criterion based on the size of improvement of objective function. |

`max_iter` |
Maximum iteration of MM algorithm to be used to solve the GFL model. |

`verbose` |
Logical. It indicates whether display the intermediate diagnosis imformation. Defautl is |

In order to fit a piece-wise constant curve to each of the signal sequences, we try to minimize the following objective function

*loss function
+ lambda1 * lasso penalty + lambda2 * fused lasso penalty + lambda3 * group fused lasso penalty
*

The optimal solution is approached via an iteration based algorithm called Majorization-Minimization (MM) algorithm developed by Kenneth Lange (2004). The choices of tuning parameters of the model are suggested as follows:

*λ_{1,i} = c_1 σ_i*

*λ_{2,i} = ρ(p) c_2 σ_i √{\log N}*

*λ_{3,i} = [1-ρ(p)] c_3 σ_i √{pM} √{\log N}*

where *σ_i* is signal noise level of each sequence, *M* is the number of sequences, *N* is the number of markers and *c_1*, *c_2*, *c_3*, *ρ*, and *p* are properly chosen contants, which are absorbed in *ρ_1*, *ρ_2*, and *ρ_3* respectively. More details are referred to Zhang et al. (2012).

All outputs are collected in a list:

`obj` |
A vector of values of objective function at each MM iteration. |

`Beta` |
A matrix of the same dimension as |

`Y`

and `Delta`

must be of the class matrix. If only one signal sequence is to be analyzed, they should be also coerced to matrix with only one column.

Zhongyang (Thomas) Zhang, zhangzy@ucla.edu

Kenneth Lange. (2004)

*Optimization*. Springer, New York.Zhongyang Zhang, Kenneth Lange, Roel Ophoff, and Chiara Sabatti. (2010) Reconstructing DNA copy number by penalized estimation and imputation.

*The Annals of Applied Statistics*, 4(4): 1749-1773.Zhongyang Zhang, Kenneth Lange, and Chiara Sabatti. (2012) Reconstructing DNA copy number by segmentation of multiple sequences.

*Submitted*.

See `FL`

for segmentation of only one sequence of signals.

1 2 3 4 5 6 7 8 9 | ```
## Jointly segment 2 sequences of signals with 100 markers
## Duplications are superimposed on both sequences
Y <- matrix(rnorm(200,0,0.15),100,2)
Y[41:60,] <- rnorm(40,0.3,0.2)
Delta <- matrix(1,100,2)
sigma <- apply(Y,2,FUN="mad")
res <- GFL(Y=Y, Delta, sigma, rho1 = 0.01, rho2 = 0.5*2, rho3 = 0.5*2,
obj_c = 1e-4, max_iter = 1000, verbose = FALSE)
plot(res$Beta[,1],type="s")
``` |

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