Clone: Clone a network.
In GiulioCostantini/TOMproject: Simulation of Topological Overlap
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
Description
This function starts from an original network, represented by a partial correlation matrix (pcm), and from a vector (assi. Assi is the output of function Assign) and specifies, for each node in the pcm, how many clones will correspond to it. The result is a new network in which each node is a clone of a node in the original network. The output is not in the form of a network, but of a data matrix.
Usage
1
Arguments
pcm
Matrix of numerics. Partial correlation matrix.
assi
Vector of integers, same size as a dimension of pcm. The i-th element specifies the number of clones of the i-th node of the pcm.
n
Integer. Desired number of observations.
furthersharedvar
Numeric in [0,1]. Portion of the residual variance of the original node that is kept intact in the clones. See details.
rndvar
Numeric in [0,1]. Portion of the variance of the clones that will be composed by random noise. See details.
p
Numeric in [0,1]. Probability that a neighbor is considered in the process of clonation.
keeporig
Logical. Should the original nodes be preserved in the cloned network.
Details
The function takes a partial correlation matrix in input and returns a data matrix and a matrix of clones.
The backbone of the function (i.e., if parameter p==1 and parameter rndvar==0) proceedes in this way:
1) Generate a data matrix from the pcm (orig), with M=0 and SD=1, using Cholesky decomposition (This matrix is returned in output).
2) Pick the i-th variable (starting from the first) in orig and decompose it in in two parts: The "systematic" one is the variance shared with other nodes in the network. The "residual" one is its unique variance. This split is performed using multiple regression and saving the fitted values ("systematic" variance) and the residuals ("residual" variance).
3) Compose assi[i] clones of the i-th variable, by assembling their variance in this way:
(a) The systematic component, identical to the "systematic" variance of the original node (i.e., R^2 in the multiple regression of the original node).
(b) The remaining variance, equal to err=1-R^2, is made by two parts.
(b1) A shared residual component: this amounts to err*furthershared and is equal to the residual component of the original node, multiplied by sqrt((1-R^2)*furthershared) to get the desired variance. This furhter component is shared by all the clones of the same node.
(b2) The remaining variance, err*(1-furthershared), is selected to be perfectly unique for each clone, i.e., completely linearly independent from each of the other nodes and clones in the network and from the clones of the same node. No linear dipendence is allowed, not even by chance.
4) Repeat the steps 2-3 for the next node, until all the nodes in pcm have been cloned. In decomposing the variance of the node, consider the original ndoes if they have not been cloned yet, their clones otherwise.
As a result a matrix of clones (cloned, see output) is returned, with n rows and sum(assi) columns.
This function is very generic, allowing different kinds of clones to be returned: we describe some important kinds of clones.
Duplicates) If parameter p = 1, parameter rndvar = 0, and parameter furthersharedvar = 1, the clones of a node are exact replicas of the node and therefore the off-diagonal elements of the resulting correlation network are identical to the corresponding ones in the original network. The parameter furthersharedvar = 1 determines that all the original variance of the node is kept in the clones.
Same Matrix) If parameter p = 1, parameter rndvar = 0, and parameter furthersharedvar = 0, the correlations among the clones have the two following properties: the correlation among two clones of two different nodes i and j is exactly identical to that of the original i and j. The correlation among two clones of the same nodes is the minimum possible such as the first property is obtained and is equal to the R^2 of the multiple regression at step 2.
Using parameter furthersharedvar it is possible to create a bridge between the ideal cases "Duplicates" and "Same Matrix". For instance, if furthersharevar=.5, then the 50% of the error variance of the original nodes is preserved in their clones, therefore the off-diagonal elements of the resulting correlation matrix are equal to those of the original network, but the node share some variance that is irrelevant to strictly obtain this aim.
Parameter p allows to modify the multiple regression at Step 2, such as instead of considering all the predictors (i.e., neighbors of the node in the correlation matrix), each predictor is considered with probability p. If a predictor is selected, but it has been already cloned, its clones are considered instead. Notice that however at least one predictor is always guaranteed, even if zero predictors are extracted. Parameter p has been inspired by the partial duplication process described in Chung et al. (2003); however while in Chung's network each new node (corresponding to our clones) inherits only a subset of the neighbors of the original node, in our procedure each clone "is influenced" only by a subset of the neighbors of the original node. If the network is dense enough however, it is very likely that it will correlate also with the other neigbors (by spurious correlations). If parameter p is < 1, then the component (a) of the variance of each clone, as detemrmined at step 3, is less than R^2. The remaining variance is "filled" with random variance, perfecly unique for each clone.
The parameter rndvar allows to add further random variance to the clones. For instance, if parameter rndvar = .4, the 60% of the variance of the clones will be the one defined in step 4, while the remaining 40% is random. This random variance however is not cleaned from random covariance with other nodes in the network. It is meant to make the cloned matrices less "perfect" and closer to the real networks and also to make the task to recover the original network structure progressively higher.
Finally, if parameter keeporig = TRUE, the nodes in the original network are kept also in the cloned network. This step is done at the very end, by replacing one clone by node with the original variable in orig.
Value
A list of two matrices.
orig is a data matrix generated from pcm using Cholesky decomposition. It has n observations and as many variables as there are in pcm. A partial correlaiton matrix obtained by these data should reproduce pcm, even though with some minor discrepancies.
cloned is a matrix of cloned nodes, n observations by sum(assi) variables. Each i-th node in pcm has assi[i] clones in cloned.
Author(s)
Giulio Costantini
References
Chung, F., Linyuan, L., Dewey, G., Galas, D. J. (2003), Duplication models for biological networks. Journal of Computational Biology, 10(5), p. 677-687
Examples
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## Not run:
require(adegenet)
# generate a network with 10 (15 edges in the original adjacency matrix)
# and then generate several cloned networks with 30 clones
N <- 10
N2 <- 30
net <- Topology(N=N, m=15, topology="BA", exact.m=F, force.connected=T, negpro=0, pw_ba=1, minpcor=.1, maxpcor=.3)
pcm <- net$pcm
assi <- Assign(N=N, N2=N2, method_assign="pwr", shape1= .0001, shape2 = 1000)
assignments <- c()
for(i in 1:length(assi)) assignments <- c(assignments, rep(i, assi[i]))
par(mfrow = c(4,2))
qgraph(pcm, layout = "spring")
title(sub = "original pcm")
# The extreme case "Duplicates"
duplicates <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=1, p=1)
# verify that the correlation of the clones with the original nodes are all = 1
round(cor(duplicates$cloned, duplicates$orig)^2,2)
qgraph(cor(duplicates$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "duplicates")
# "Duplicates", with .50 noise
noisy50dup <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=.5, furthersharedvar=1, p=1)
qgraph(cor(noisy50dup$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "duplicates with .50 noise")
# "Duplicates", with .90 noise
noisy90dup <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=.9, furthersharedvar=1, p=1)
qgraph(cor(noisy90dup$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "duplicates with .90 noise")
# The extreme case "SameMatrix".
samemat <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=0, p=1)
# verify that the resulting collapsed correlation matrix is identical, besides the diagonal elements, which are the R^2.
cm <- cor(samemat$orig)
cm_cloned <- cor(samemat$cloned)
cm_reprod <- matrix(ncol = N, nrow = N)
for(i in 1:N)
{
for(j in 1:N)
{
submat <- cm_cloned[assignments == i, assignments == j]
if(i == j & assi[i] > 1) submat <- submat[lower.tri(submat)]
cm_reprod[i,j] <- mean(submat)
}
}
round(cm_reprod, 2)
round(cm_reprod-cm, 4)
qgraph(cor(samemat$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "same matrix")
# a Mix of cases Duplicates and SameMatrix obtained by varying parameter furthersharedvar
dup50same50 <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=.5, p=1)
# verify that the resulting collapsed correlation matrix is identical, besides the diagonal elements, which are the higher than R^2.
cm <- cor(dup50same50$orig)
cm_cloned <- cor(dup50same50$cloned)
cm_reprod <- matrix(ncol = N, nrow = N)
for(i in 1:N)
{
for(j in 1:N)
{
submat <- cm_cloned[assignments == i, assignments == j]
if(i == j & assi[i] > 1) submat <- submat[lower.tri(submat)]
cm_reprod[i,j] <- mean(submat)
}
}
round(cm_reprod, 2)
round(cm_reprod-cm, 4)
qgraph(cor(dup50same50$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = ".5 duplicate .5 samemat")
# Use parameter p to consider only the 50% of the predictors for each clone
net50 <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=0, p=.5)
qgraph(cor(net50$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "half predictors by clone")
## End(Not run)
GiulioCostantini/TOMproject documentation built on May 6, 2019, 6:29 p.m.
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Description Usage Arguments Details Value Author(s) References Examples
Description
This function starts from an original network, represented by a partial correlation matrix (pcm), and from a vector (assi. Assi is the output of function Assign) and specifies, for each node in the pcm, how many clones will correspond to it. The result is a new network in which each node is a clone of a node in the original network. The output is not in the form of a network, but of a data matrix.
Usage
1 |
Arguments
pcm |
Matrix of numerics. Partial correlation matrix. |
assi |
Vector of integers, same size as a dimension of |
n |
Integer. Desired number of observations. |
furthersharedvar |
Numeric in [0,1]. Portion of the residual variance of the original node that is kept intact in the clones. See details. |
rndvar |
Numeric in [0,1]. Portion of the variance of the clones that will be composed by random noise. See details. |
p |
Numeric in [0,1]. Probability that a neighbor is considered in the process of clonation. |
keeporig |
Logical. Should the original nodes be preserved in the cloned network. |
Details
The function takes a partial correlation matrix in input and returns a data matrix and a matrix of clones.
The backbone of the function (i.e., if parameter p==1 and parameter rndvar==0) proceedes in this way:
1) Generate a data matrix from the pcm (orig), with M=0 and SD=1, using Cholesky decomposition (This matrix is returned in output).
2) Pick the i-th variable (starting from the first) in orig and decompose it in in two parts: The "systematic" one is the variance shared with other nodes in the network. The "residual" one is its unique variance. This split is performed using multiple regression and saving the fitted values ("systematic" variance) and the residuals ("residual" variance).
3) Compose assi[i] clones of the i-th variable, by assembling their variance in this way:
(a) The systematic component, identical to the "systematic" variance of the original node (i.e., R^2 in the multiple regression of the original node).
(b) The remaining variance, equal to err=1-R^2, is made by two parts.
(b1) A shared residual component: this amounts to err*furthershared and is equal to the residual component of the original node, multiplied by sqrt((1-R^2)*furthershared) to get the desired variance. This furhter component is shared by all the clones of the same node.
(b2) The remaining variance, err*(1-furthershared), is selected to be perfectly unique for each clone, i.e., completely linearly independent from each of the other nodes and clones in the network and from the clones of the same node. No linear dipendence is allowed, not even by chance.
4) Repeat the steps 2-3 for the next node, until all the nodes in pcm have been cloned. In decomposing the variance of the node, consider the original ndoes if they have not been cloned yet, their clones otherwise.
As a result a matrix of clones (cloned, see output) is returned, with n rows and sum(assi) columns.
This function is very generic, allowing different kinds of clones to be returned: we describe some important kinds of clones.
Duplicates) If parameter p = 1, parameter rndvar = 0, and parameter furthersharedvar = 1, the clones of a node are exact replicas of the node and therefore the off-diagonal elements of the resulting correlation network are identical to the corresponding ones in the original network. The parameter furthersharedvar = 1 determines that all the original variance of the node is kept in the clones.
Same Matrix) If parameter p = 1, parameter rndvar = 0, and parameter furthersharedvar = 0, the correlations among the clones have the two following properties: the correlation among two clones of two different nodes i and j is exactly identical to that of the original i and j. The correlation among two clones of the same nodes is the minimum possible such as the first property is obtained and is equal to the R^2 of the multiple regression at step 2.
Using parameter furthersharedvar it is possible to create a bridge between the ideal cases "Duplicates" and "Same Matrix". For instance, if furthersharevar=.5, then the 50% of the error variance of the original nodes is preserved in their clones, therefore the off-diagonal elements of the resulting correlation matrix are equal to those of the original network, but the node share some variance that is irrelevant to strictly obtain this aim.
Parameter p allows to modify the multiple regression at Step 2, such as instead of considering all the predictors (i.e., neighbors of the node in the correlation matrix), each predictor is considered with probability p. If a predictor is selected, but it has been already cloned, its clones are considered instead. Notice that however at least one predictor is always guaranteed, even if zero predictors are extracted. Parameter p has been inspired by the partial duplication process described in Chung et al. (2003); however while in Chung's network each new node (corresponding to our clones) inherits only a subset of the neighbors of the original node, in our procedure each clone "is influenced" only by a subset of the neighbors of the original node. If the network is dense enough however, it is very likely that it will correlate also with the other neigbors (by spurious correlations). If parameter p is < 1, then the component (a) of the variance of each clone, as detemrmined at step 3, is less than R^2. The remaining variance is "filled" with random variance, perfecly unique for each clone.
The parameter rndvar allows to add further random variance to the clones. For instance, if parameter rndvar = .4, the 60% of the variance of the clones will be the one defined in step 4, while the remaining 40% is random. This random variance however is not cleaned from random covariance with other nodes in the network. It is meant to make the cloned matrices less "perfect" and closer to the real networks and also to make the task to recover the original network structure progressively higher.
Finally, if parameter keeporig = TRUE, the nodes in the original network are kept also in the cloned network. This step is done at the very end, by replacing one clone by node with the original variable in orig.
Value
A list of two matrices.
orig is a data matrix generated from pcm using Cholesky decomposition. It has n observations and as many variables as there are in pcm. A partial correlaiton matrix obtained by these data should reproduce pcm, even though with some minor discrepancies.
cloned is a matrix of cloned nodes, n observations by sum(assi) variables. Each i-th node in pcm has assi[i] clones in cloned.
Author(s)
Giulio Costantini
References
Chung, F., Linyuan, L., Dewey, G., Galas, D. J. (2003), Duplication models for biological networks. Journal of Computational Biology, 10(5), p. 677-687
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | ## Not run:
require(adegenet)
# generate a network with 10 (15 edges in the original adjacency matrix)
# and then generate several cloned networks with 30 clones
N <- 10
N2 <- 30
net <- Topology(N=N, m=15, topology="BA", exact.m=F, force.connected=T, negpro=0, pw_ba=1, minpcor=.1, maxpcor=.3)
pcm <- net$pcm
assi <- Assign(N=N, N2=N2, method_assign="pwr", shape1= .0001, shape2 = 1000)
assignments <- c()
for(i in 1:length(assi)) assignments <- c(assignments, rep(i, assi[i]))
par(mfrow = c(4,2))
qgraph(pcm, layout = "spring")
title(sub = "original pcm")
# The extreme case "Duplicates"
duplicates <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=1, p=1)
# verify that the correlation of the clones with the original nodes are all = 1
round(cor(duplicates$cloned, duplicates$orig)^2,2)
qgraph(cor(duplicates$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "duplicates")
# "Duplicates", with .50 noise
noisy50dup <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=.5, furthersharedvar=1, p=1)
qgraph(cor(noisy50dup$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "duplicates with .50 noise")
# "Duplicates", with .90 noise
noisy90dup <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=.9, furthersharedvar=1, p=1)
qgraph(cor(noisy90dup$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "duplicates with .90 noise")
# The extreme case "SameMatrix".
samemat <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=0, p=1)
# verify that the resulting collapsed correlation matrix is identical, besides the diagonal elements, which are the R^2.
cm <- cor(samemat$orig)
cm_cloned <- cor(samemat$cloned)
cm_reprod <- matrix(ncol = N, nrow = N)
for(i in 1:N)
{
for(j in 1:N)
{
submat <- cm_cloned[assignments == i, assignments == j]
if(i == j & assi[i] > 1) submat <- submat[lower.tri(submat)]
cm_reprod[i,j] <- mean(submat)
}
}
round(cm_reprod, 2)
round(cm_reprod-cm, 4)
qgraph(cor(samemat$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "same matrix")
# a Mix of cases Duplicates and SameMatrix obtained by varying parameter furthersharedvar
dup50same50 <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=.5, p=1)
# verify that the resulting collapsed correlation matrix is identical, besides the diagonal elements, which are the higher than R^2.
cm <- cor(dup50same50$orig)
cm_cloned <- cor(dup50same50$cloned)
cm_reprod <- matrix(ncol = N, nrow = N)
for(i in 1:N)
{
for(j in 1:N)
{
submat <- cm_cloned[assignments == i, assignments == j]
if(i == j & assi[i] > 1) submat <- submat[lower.tri(submat)]
cm_reprod[i,j] <- mean(submat)
}
}
round(cm_reprod, 2)
round(cm_reprod-cm, 4)
qgraph(cor(dup50same50$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = ".5 duplicate .5 samemat")
# Use parameter p to consider only the 50% of the predictors for each clone
net50 <- Clone(pcm=pcm, assi=assi, n=1000, rndvar=0, furthersharedvar=0, p=.5)
qgraph(cor(net50$cloned), layout = "spring", colors = adegenet::fac2col(assignments), labels = assignments)
title(sub = "half predictors by clone")
## End(Not run)
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