3D reconstruction from Hi-C distances

Description

A function for reconstructing the 3D configuration from pairwise calibrated Hi-C distances through local stress minimization (see details).

Usage

lsmacof(diss, infD, itermax = 10000, eps = 1e-06, init = NULL, k = 3,
verbose = FALSE, infW = NULL)

Arguments

Details

A calibrated distance Hi-C matrix H can be used as input for a 3D reconstruction algorithm. It is important to note that if H would give the true 3D Euclidean distances than the multi-dimensional scaling (MDS) solution (Torgerson, 1952) would recover the underlying 3D configuration, up to a rotation. However, due to Hi-C limitation we rely mostly on local calibrated distances. Denote delta(i,j) the calibrated distance between loci i and j, and d(i,j), their Euclidean distance, in the true underlying 3D configuration. Here, a zero delta[i,j] , for different loci i and j, represents a missing information that was discarded in the calibration step. Our goal is then to minimize the following function, usually termed stress in an MDS setting (Kruskal, 1964): SUM(i<j)[w(i,j)(delta(i,j)-d(i,j)(Y))^2] , where w(i,j) are the weights we assign according to the reliability of delta(i,j). Since we mostly rely on local information, we can use here a local stress function (Chen and Buja, 2009), where missing delta(i,j) are replaced with a constant dinf (dinf >> known delta(i,j)) and w(i,j) take the value of 1/dinf for missing distances and 1 otherwise (for dinf equal or smaller than 1, weights of missing distances should be set to a small constant << 1). Since w(i,j) define an irreducible matrix, the stress minimization could be performed through Scaling by Majorizing a Complicated Function (SMACOF) (De Leeuw, 1977; De Leeuw, 1988; De Leeuw and Heiser 1977), a well-established strategy for this task, which guarantees convergence.

Value

A M*k configuration matrix (k=3 by default) reconstructed from the given M*M Hi-C pairwise distances.

Author(s)

Yoli Shavit

References

Main reference:
Y. Shavit, F.K. Hamey, P. Lio', FisHiCal: an R package for iterative FISH-based calibration of Hi-C data, 2014 (submitted).

References for Details section:
Chen, L. and Buja, A. (2009) Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing, and Proximity Analysis. Journal of the American Statistical Association 104, 209-219.
De Leeuw, J. (1977). Applications of convex analysis to multidimensional scaling. In Barra, J.R et al (Eds.), Recent developments in statistics, 133-145. Amsterdam, The Netherlands: North-Holland.
De Leeuw, J. (1988). Convergence of the majorization method for multidimensional scaling,. Journal of Classification, 5, 163-180.
De Leeuw, J. and Heiser, W.J. (1977). Convergence of correction-matrix algorithms for multidimensional scaling. In Lingoes, J.C., Roskam, E.E., and Borg, I. (Eds.),Geometric representations of relational data, 735-752. Ann Arbor, MI:Mathesis.
Kruskal, J. B. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115-129.
Torgerson, W.S. (1952). Multidimensional scaling: I. Theory and method. Psychometrika, 17, 401-19.

See Also

prepareCalib
calibHiC
calibrate

Examples

 data(calibHiC)
 data(match)
 data(conf)
 predConf = lsmacof(calibHiC, max(match$distances))
 
 # superimpose
 partialPS<-function(m1, m2)
 {
    # translate to origin
    tm1<-scale(m1, scale = FALSE)
    tm2<-scale(m2, scale = FALSE)
    A<-svd(t(tm2)%*%tm1)
    v<-A$u
    w<-A$v
    # update v a det(R) is positive
    k = ncol(m1)
    d = sign(det(t(w)%*%t(v)))
    v[,k] = v[,k]*-1*d
    R<- w%*%t(v)
    return(list(m1=tm1%*%R,m2=tm2))
  }

  res = partialPS(predConf, conf) 
  if (require(rgl))
  {
    plot3d(res$m2, type = "l", axes = FALSE, box = FALSE, xlab = "", ylab = "", zlab = "")
    lines3d(res$m1, col = "red")
  }

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(FisHiCal)
Loading required package: igraph

Attaching package: 'igraph'

The following objects are masked from 'package:stats':

    decompose, spectrum

The following object is masked from 'package:base':

    union

Loading required package: RcppArmadillo
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/FisHiCal/lsmacof.Rd_%03d_medium.png", width=480, height=480)
> ### Name: lsmacof
> ### Title: 3D reconstruction from Hi-C distances
> ### Aliases: lsmacof
> 
> ### ** Examples
> 
>  data(calibHiC)
>  data(match)
>  data(conf)
>  predConf = lsmacof(calibHiC, max(match$distances))
>  
>  # superimpose
>  partialPS<-function(m1, m2)
+  {
+     # translate to origin
+     tm1<-scale(m1, scale = FALSE)
+     tm2<-scale(m2, scale = FALSE)
+     A<-svd(t(tm2)%*%tm1)
+     v<-A$u
+     w<-A$v
+     # update v a det(R) is positive
+     k = ncol(m1)
+     d = sign(det(t(w)%*%t(v)))
+     v[,k] = v[,k]*-1*d
+     R<- w%*%t(v)
+     return(list(m1=tm1%*%R,m2=tm2))
+   }
> 
>   res = partialPS(predConf, conf) 
>   if (require(rgl))
+   {
+     plot3d(res$m2, type = "l", axes = FALSE, box = FALSE, xlab = "", ylab = "", zlab = "")
+     lines3d(res$m1, col = "red")
+   }
Loading required package: rgl
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>