`LOCUS`

Low-rank decomposition of brain connectivity matrices with universal sparsity

**Author:** Yikai Wang, Jialu Ran,
Ying Guo

`LOCUS`

is a blind source separation (BSS) method for
decomposing symmetric matrices such as brain connectivity matrices to
extract sparse latent component matrices and also estimate mixing
coefficients. For brain connectivity matrices, the outputs correspond to
sparse latent connectivity traits and individual-level trait loadings.
The LOCUS method was published in Wang and Guo (2023).

Below is an illustration of the the main function on simulated data.

```
## Simulated the data to use
= 50
V = S2 = S3 = matrix(0,ncol = V,nrow = V)
S1 5:20,5:20] = 4;S1[23:37,23:37] = 3;S1[40:48,40:48] = 3
S1[15:20,] = -3;S2[,15:20] = -3
S2[15:25,36:45] = 3; S3[36:45,15:25] = 3
S3[= rbind(Ltrans(S1,FALSE) , Ltrans(S2,FALSE), Ltrans(S3,FALSE))
Struth set.seed(100)
= matrix(rnorm(100*3),nrow=100,ncol=3)
Atruth = matrix(rnorm(100*dim(Struth)[2]),nrow=100)
Residual = Atruth%*%Struth + Residual
Yraw
## Run Locus on the data
= LOCUS(Yraw,3,V)
Locus_result
## Visualize the result
par(mfrow=c(2,3))
for(i in 1:dim(Struth)[1]){image(Ltrinv(Struth[i,],V,FALSE))}
for(i in 1:dim(Locus_result$S)[1]){image(Ltrinv(Locus_result$S[i,],V,FALSE))}
```

Wang, Y. and Guo, Y. (2023). LOCUS: A novel signal decomposition method for brain network connectivity matrices using low-rank structure with uniform sparsity. Annals of Applied Statistics.