nsprcomp: Non-Negative and Sparse PCA
In nsprcomp: Non-Negative and Sparse PCA
Description
Usage
Arguments
Details
Value
Note
References
See Also
Examples
Description
Performs a constrained principal component analysis,
where non-negativity and/or sparsity constraints are enforced on the principal axes.
The result is returned as an object of class nsprcomp, which inherits from
prcomp.
Usage
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nsprcomp(x, ...)
## Default S3 method:
nsprcomp(x, retx = TRUE, ncomp = min(dim(x)),
omega = rep(1, nrow(x)), k = ncol(x), nneg = FALSE, center = TRUE,
scale. = FALSE, tol = NULL, nrestart = 5, em_tol = 0.001,
em_maxiter = 100, partial_model = NULL, verbosity = 0, ...)
## S3 method for class 'formula'
nsprcomp(formula, data = NULL, subset, na.action, ...)
Arguments
x
a numeric matrix or data frame which provides the data
for the principal component analysis.
...
arguments passed to or from other methods.
retx
a logical value indicating whether the principal components, i.e.
x projected into the principal subspace, should be returned.
ncomp
the number of principal components (PCs) to be computed. With
the default setting, PCs are computed until x is fully deflated.
ncomp can be specified implicitly if k is given as a vector.
omega
a vector with as many entries as there are data samples, to
perform weighted PCA (analogous to weighted least-squares regression). The
default is an equal weighting of all samples.
k
either a scalar or a vector of length ncomp, specifying the
upper bounds on the cardinalities of the principal axes (PAs).
nneg
a logical value indicating whether the loadings should be
non-negative, i.e. the PAs should be constrained to the non-negative
orthant.
center
a logical value indicating whether the empirical mean of (the
columns) of x should be subtracted. Alternatively, a vector of
length equal to the number of columns of x can be supplied. The
value is passed to scale.
scale.
a logical value indicating whether the columns of x
should be scaled to have unit variance before the analysis takes place. The
default is FALSE for consistency with prcomp. Alternatively,
a vector of length equal to the number of columns of x can be
supplied. The value is passed to scale.
tol
a threshold indicating the magnitude below which components should
be omitted. Components are omitted if their standard deviations are less
than or equal to tol times the standard deviation of the first
component. With the default NULL setting, no components are omitted.
With tol = 0 or tol = sqrt(.Machine$double.eps), essentially
constant components are omitted.
nrestart
the number of random restarts for computing the principal
component via expectation-maximization (EM) iterations. The solution
achieving maximum standard deviation over all random restarts is kept. A
value greater than one can help to avoid poor local maxima.
em_tol
If the relative change of the objective is less than
em_tol between iterations, the EM procedure is asssumed to have
converged to a local optimum.
em_maxiter
the maximum number of EM iterations to be performed. The EM
procedure is terminated if either the em_tol or the
em_maxiter criterion is satisfied.
partial_model
NULL or an object of class nsprcomp. The
computation can be continued from a partial model by providing a
nsprcomp object (either from a previous run of this function or from
asdev) and setting ncomp to a value greater than the
number of components contained in the partial model. See the examples for
an illustration.
verbosity
an integer specifying the verbosity level. Greater values
result in more output, the default is to be quiet.
formula
a formula with no response variable, referring only to numeric variables.
data
an optional data frame (or similar: see
model.frame) containing the variables in the
formula. By default the variables are taken from
environment(formula).
subset
an optional vector used to select rows (observations) of the
data matrix x.
na.action
a function which indicates what should happen
when the data contain NAs. The default is set by
the na.action setting of options, and is
na.fail if that is unset. The ‘factory-fresh’
default is na.omit.
Details
nsprcomp computes a principal component (PC) using expectation-maximization
iterations, where non-negativity of the loadings is achieved by projecting
the principal axis (PA) into the non-negative orthant, and sparsity of the
loadings is achieved by soft thresholding (Sigg and Buhmann, 2008).
Because constrained principal axes no longer correspond to true eigenvectors
of the covariance matrix and are usually not pairwise orthogonal, special
attention needs to be paid when computing more than a single PC. The
algorithm implements the generalized deflation method proposed by
Mackey (2009) to maximize the additional variance of each
component. Given a basis of the space spanned by the previous PAs, the
variance of the PC is maximized after projecting the current PA to the
ortho-complement space of the basis. This procedure maximizes the
additional variance not explained by previous components, and is
identical to standard PCA if no sparsity or non-negativity constraints
are enforced on the PAs.
See the references for further details.
Value
nsprcomp returns a list with class (nsprcomp, prcomp)
containing the following elements:
sdev
the additional standard
deviation explained by each component, see asdev.
rotation
the matrix of non-negative and/or sparse loadings,
containing the principal axes as columns.
x
the scores matrix
X*W containing the principal components as columns
(after centering and scaling if requested). For the formula method,
napredict is applied to handle the treatment of values
omitted by the na.action.
center, scale
the centering and
scaling used, or FALSE
xp
the deflated data matrix
corresponding to x
q
an orthonormal basis for the principal
subspace
Note
The PCA terminology is not consistent across the literature. Given a
zero mean data matrix X (with observations as rows) and a
basis W of the principal subspace, we define the scores
matrix as Z=X*W which contains the principal
components as its columns. The columns of the pseudo-rotation matrix
W are called the principal axes, and the elements of
W are called the loadings.
Deflating the data matrix accumulates numerical errors over successive PCs.
References
Sigg, C. D. and Buhmann, J. M. (2008) Expectation-Maximization
for Sparse and Non-Negative PCA. In Proceedings of the 25th
International Conference on Machine Learning (pp. 960–967).
Mackey, L. (2009) Deflation Methods for Sparse PCA. In
Advances in Neural Information Processing Systems (pp. 1017–1024).
See Also
Examples
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if (requireNamespace("MASS", quietly = TRUE)) withAutoprint({
set.seed(1)
# Regular PCA, with the tolerance set to return five PCs
prcomp(MASS::Boston, tol = 0.36, scale. = TRUE)
# Sparse PCA with different cardinalities per component. The number of components
# is derived from the length of vector k.
nsprcomp(MASS::Boston, k = c(13, 7, 5, 5, 5), scale. = TRUE)
# Non-negative sparse PCA with four components. Note that the principal axes
# naturally have a high degree of orthogonality, because each component
# maximizes the additional variance not already explained.
set.seed(1)
nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE)
# The optimization can get stuck in local optima. Increase the number of
# random restarts or the number of power iterations to likely obtain decreasing
# standard deviations.
set.seed(1)
(nspc <- nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE,
nrestart = 10, em_tol = 1e-4, verbosity = 1))
# continue the computation of components from a partial model
nsprcomp(MASS::Boston, k = 3, ncomp = 5, nneg = TRUE, scale. = TRUE, partial_model = nspc)
# The reconstruction error for each sample can be influenced using the
# weighting vector omega. To reconstruct the data, the generalized
# inverse of the pseudo-rotation matrix has to be used, because the constrained
# principal axes are in general not pairwise orthogonal.
set.seed(1)
X <- matrix(runif(5*10), 5)
nspc <- nsprcomp(X, omega = c(5, 1, 1, 1, 5), ncomp = 2, nneg = TRUE)
X_hat <- predict(nspc)%*%MASS::ginv(nspc$rotation) + matrix(1,5,1)%*%nspc$center
rowSums((X - X_hat)^2)
})
Example output
> set.seed(1)
> prcomp(MASS::Boston, tol = 0.36, scale. = TRUE)
Standard deviations (1, .., p=14):
[1] 2.5585132 1.2843410 1.1614241 0.9415625 0.9224421 0.8124105 0.7317177
[8] 0.6348831 0.5265582 0.5022524 0.4612919 0.4277704 0.3660733 0.2456149
Rotation (n x k) = (14 x 5):
PC1 PC2 PC3 PC4 PC5
crim 0.242284451 -0.065873108 0.395077419 -0.100366211 0.004957659
zn -0.245435005 -0.148002653 0.394545713 -0.342958421 0.114495002
indus 0.331859746 0.127075668 -0.066081913 0.009626936 -0.022583692
chas -0.005027133 0.410668763 -0.125305293 -0.700406497 -0.535197817
nox 0.325193880 0.254276363 -0.046475549 -0.053707583 0.194605570
rm -0.202816554 0.434005810 0.353406095 0.293357309 -0.008320481
age 0.296976574 0.260303205 -0.200823078 0.078426326 0.149750092
dis -0.298169809 -0.359149977 0.157068710 -0.184747787 -0.106219480
rad 0.303412754 0.031149596 0.418510334 0.051374381 -0.230352185
tax 0.324033052 0.008851406 0.343232194 0.026810695 -0.163425820
ptratio 0.207679535 -0.314623061 0.000399092 0.342036328 -0.615707380
black -0.196638358 0.026481032 -0.361375914 0.201741185 -0.367460674
lstat 0.311397955 -0.201245177 -0.161060336 -0.242621217 0.178358870
medv -0.266636396 0.444924411 0.163188735 0.180297553 -0.050659893
> nsprcomp(MASS::Boston, k = c(13, 7, 5, 5, 5), scale. = TRUE)
Standard deviations (1, .., p=5):
[1] 2.5584808 1.2472326 1.0405531 0.9642164 0.8752789
Rotation (n x k) = (14 x 5):
PC1 PC2 PC3 PC4 PC5
crim -0.2429863 0.0000000 0.4854165 0.0000000 0.0000000
zn 0.2446861 0.0000000 0.5967689 0.0000000 0.0000000
indus -0.3318673 0.0000000 0.0000000 0.0000000 0.0000000
chas 0.0000000 -0.4315023 0.0000000 0.7490440 -0.4310263
nox -0.3253668 -0.2702262 0.0000000 0.0000000 0.0000000
rm 0.2017390 -0.4401132 0.0000000 -0.4474849 0.0000000
age -0.2967945 -0.2763328 0.0000000 0.0000000 0.2574766
dis 0.2981333 0.3520975 0.2488706 0.0000000 -0.1555882
rad -0.3042870 0.0000000 0.4437986 0.0000000 0.0000000
tax -0.3247282 0.0000000 0.3864504 0.0000000 0.0000000
ptratio -0.2073060 0.3886047 0.0000000 0.0000000 0.0000000
black 0.1975268 0.0000000 0.0000000 0.4110236 0.8419257
lstat -0.3109319 0.0000000 0.0000000 0.1935561 0.0000000
medv 0.2659092 -0.4424227 0.0000000 -0.1796834 0.1219658
> set.seed(1)
> nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE)
Standard deviations (1, .., p=4):
[1] 2.1308851 1.5497667 0.7822017 1.0082622
Rotation (n x k) = (14 x 4):
PC1 PC2 PC3 PC4
crim 0.3126418 0.0000000 0.0000000 0.0000000
zn 0.0000000 0.4900940 0.0000000 0.0000000
indus 0.3990108 0.0000000 0.0000000 0.0000000
chas 0.0000000 0.0000000 0.0000000 0.9669007
nox 0.4027495 0.0000000 0.0000000 0.0000000
rm 0.0000000 0.4480493 0.7706175 0.0000000
age 0.3579659 0.0000000 0.0000000 0.0000000
dis 0.0000000 0.4586857 0.0000000 0.0000000
rad 0.3932454 0.0000000 0.0000000 0.0000000
tax 0.4151053 0.0000000 0.0000000 0.0000000
ptratio 0.0000000 0.0000000 0.0000000 0.0000000
black 0.0000000 0.3145526 0.0000000 0.2551531
lstat 0.3546046 0.0000000 0.0000000 0.0000000
medv 0.0000000 0.4997236 0.6372979 0.0000000
> set.seed(1)
> (nspc <- nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE,
+ nrestart = 10, em_tol = 1e-04, verbosity = 1))
[1] "component 1: maximum objective is 2293 at random restart 0"
[1] "component 1: maximum objective is 2293 at random restart 1"
[1] "component 1: maximum objective is 2293 at random restart 2"
[1] "component 1: maximum objective is 2293 at random restart 3"
[1] "component 1: maximum objective is 2293 at random restart 4"
[1] "component 1: maximum objective is 2293 at random restart 5"
[1] "component 1: maximum objective is 2293 at random restart 6"
[1] "component 1: maximum objective is 2293 at random restart 7"
[1] "component 1: maximum objective is 2293 at random restart 8"
[1] "component 1: maximum objective is 2293 at random restart 9"
[1] "component 2: maximum objective is 1213 at random restart 0"
[1] "component 2: maximum objective is 1213 at random restart 1"
[1] "component 2: maximum objective is 1213 at random restart 2"
[1] "component 2: maximum objective is 1213 at random restart 3"
[1] "component 2: maximum objective is 1213 at random restart 4"
[1] "component 2: maximum objective is 1213 at random restart 5"
[1] "component 2: maximum objective is 1213 at random restart 6"
[1] "component 2: maximum objective is 1213 at random restart 7"
[1] "component 2: maximum objective is 1213 at random restart 8"
[1] "component 2: maximum objective is 1213 at random restart 9"
[1] "component 3: maximum objective is 404 at random restart 0"
[1] "component 3: maximum objective is 523.2 at random restart 1"
[1] "component 3: maximum objective is 525.1 at random restart 2"
[1] "component 3: maximum objective is 525.1 at random restart 3"
[1] "component 3: maximum objective is 525.1 at random restart 4"
[1] "component 3: maximum objective is 525.1 at random restart 5"
[1] "component 3: maximum objective is 525.1 at random restart 6"
[1] "component 3: maximum objective is 525.1 at random restart 7"
[1] "component 3: maximum objective is 525.1 at random restart 8"
[1] "component 3: maximum objective is 525.1 at random restart 9"
[1] "component 4: maximum objective is 520.6 at random restart 0"
[1] "component 4: maximum objective is 520.6 at random restart 1"
[1] "component 4: maximum objective is 520.6 at random restart 2"
[1] "component 4: maximum objective is 520.6 at random restart 3"
[1] "component 4: maximum objective is 520.7 at random restart 4"
[1] "component 4: maximum objective is 520.7 at random restart 5"
[1] "component 4: maximum objective is 520.7 at random restart 6"
[1] "component 4: maximum objective is 520.7 at random restart 7"
[1] "component 4: maximum objective is 520.7 at random restart 8"
[1] "component 4: maximum objective is 520.7 at random restart 9"
Standard deviations (1, .., p=4):
[1] 2.130886 1.549822 1.014159 1.011090
Rotation (n x k) = (14 x 4):
PC1 PC2 PC3 PC4
crim 0.3127134 0.0000000 0.0000000 0.0000000
zn 0.0000000 0.4850296 0.0000000 0.0000000
indus 0.3989959 0.0000000 0.0000000 0.0000000
chas 0.0000000 0.0000000 0.9783593 0.0000000
nox 0.4025259 0.0000000 0.0000000 0.0000000
rm 0.0000000 0.4519551 0.0000000 0.0000000
age 0.3572843 0.0000000 0.0000000 0.0000000
dis 0.0000000 0.4529891 0.0000000 0.2019432
rad 0.3936984 0.0000000 0.0000000 0.0000000
tax 0.4155276 0.0000000 0.0000000 0.0000000
ptratio 0.0000000 0.0000000 0.0000000 0.9793972
black 0.0000000 0.3177198 0.0000000 0.0000000
lstat 0.3545021 0.0000000 0.0000000 0.0000000
medv 0.0000000 0.5043192 0.2069133 0.0000000
> nsprcomp(MASS::Boston, k = 3, ncomp = 5, nneg = TRUE, scale. = TRUE, partial_model = nspc)
Standard deviations (1, .., p=5):
[1] 2.1308856 1.5498222 1.0141589 1.0110895 0.9012304
Rotation (n x k) = (14 x 5):
PC1 PC2 PC3 PC4 PC5
crim 0.3127134 0.0000000 0.0000000 0.0000000 0.0000000
zn 0.0000000 0.4850296 0.0000000 0.0000000 0.0000000
indus 0.3989959 0.0000000 0.0000000 0.0000000 0.0000000
chas 0.0000000 0.0000000 0.9783593 0.0000000 0.0000000
nox 0.4025259 0.0000000 0.0000000 0.0000000 0.0000000
rm 0.0000000 0.4519551 0.0000000 0.0000000 0.0000000
age 0.3572843 0.0000000 0.0000000 0.0000000 0.3385724
dis 0.0000000 0.4529891 0.0000000 0.2019432 0.0000000
rad 0.3936984 0.0000000 0.0000000 0.0000000 0.0000000
tax 0.4155276 0.0000000 0.0000000 0.0000000 0.0000000
ptratio 0.0000000 0.0000000 0.0000000 0.9793972 0.0000000
black 0.0000000 0.3177198 0.0000000 0.0000000 0.9025057
lstat 0.3545021 0.0000000 0.0000000 0.0000000 0.2661807
medv 0.0000000 0.5043192 0.2069133 0.0000000 0.0000000
> set.seed(1)
> X <- matrix(runif(5 * 10), 5)
> nspc <- nsprcomp(X, omega = c(5, 1, 1, 1, 5), ncomp = 2, nneg = TRUE)
> X_hat <- predict(nspc) %*% MASS::ginv(nspc$rotation) + matrix(1, 5, 1) %*%
+ nspc$center
> rowSums((X - X_hat)^2)
[1] 0.22883035 0.57480275 0.32494491 0.65648697 0.07084008
nsprcomp documentation built on May 1, 2019, 7:56 p.m.
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Description Usage Arguments Details Value Note References See Also Examples
Description
Performs a constrained principal component analysis,
where non-negativity and/or sparsity constraints are enforced on the principal axes.
The result is returned as an object of class nsprcomp, which inherits from
prcomp.
Usage
1 2 3 4 5 6 7 8 9 10 | nsprcomp(x, ...)
## Default S3 method:
nsprcomp(x, retx = TRUE, ncomp = min(dim(x)),
omega = rep(1, nrow(x)), k = ncol(x), nneg = FALSE, center = TRUE,
scale. = FALSE, tol = NULL, nrestart = 5, em_tol = 0.001,
em_maxiter = 100, partial_model = NULL, verbosity = 0, ...)
## S3 method for class 'formula'
nsprcomp(formula, data = NULL, subset, na.action, ...)
|
Arguments
x |
a numeric matrix or data frame which provides the data for the principal component analysis. |
... |
arguments passed to or from other methods. |
retx |
a logical value indicating whether the principal components, i.e.
|
ncomp |
the number of principal components (PCs) to be computed. With
the default setting, PCs are computed until |
omega |
a vector with as many entries as there are data samples, to perform weighted PCA (analogous to weighted least-squares regression). The default is an equal weighting of all samples. |
k |
either a scalar or a vector of length |
nneg |
a logical value indicating whether the loadings should be non-negative, i.e. the PAs should be constrained to the non-negative orthant. |
center |
a logical value indicating whether the empirical mean of (the
columns) of |
scale. |
a logical value indicating whether the columns of |
tol |
a threshold indicating the magnitude below which components should
be omitted. Components are omitted if their standard deviations are less
than or equal to |
nrestart |
the number of random restarts for computing the principal component via expectation-maximization (EM) iterations. The solution achieving maximum standard deviation over all random restarts is kept. A value greater than one can help to avoid poor local maxima. |
em_tol |
If the relative change of the objective is less than
|
em_maxiter |
the maximum number of EM iterations to be performed. The EM
procedure is terminated if either the |
partial_model |
|
verbosity |
an integer specifying the verbosity level. Greater values result in more output, the default is to be quiet. |
formula |
a formula with no response variable, referring only to numeric variables. |
data |
an optional data frame (or similar: see
|
subset |
an optional vector used to select rows (observations) of the
data matrix |
na.action |
a function which indicates what should happen
when the data contain |
Details
nsprcomp computes a principal component (PC) using expectation-maximization
iterations, where non-negativity of the loadings is achieved by projecting
the principal axis (PA) into the non-negative orthant, and sparsity of the
loadings is achieved by soft thresholding (Sigg and Buhmann, 2008).
Because constrained principal axes no longer correspond to true eigenvectors of the covariance matrix and are usually not pairwise orthogonal, special attention needs to be paid when computing more than a single PC. The algorithm implements the generalized deflation method proposed by Mackey (2009) to maximize the additional variance of each component. Given a basis of the space spanned by the previous PAs, the variance of the PC is maximized after projecting the current PA to the ortho-complement space of the basis. This procedure maximizes the additional variance not explained by previous components, and is identical to standard PCA if no sparsity or non-negativity constraints are enforced on the PAs.
See the references for further details.
Value
nsprcomp returns a list with class (nsprcomp, prcomp)
containing the following elements:
sdev |
the additional standard
deviation explained by each component, see |
rotation |
the matrix of non-negative and/or sparse loadings, containing the principal axes as columns. |
x |
the scores matrix
X*W containing the principal components as columns
(after centering and scaling if requested). For the formula method,
|
center, scale |
the centering and
scaling used, or |
xp |
the deflated data matrix
corresponding to |
q |
an orthonormal basis for the principal subspace |
Note
The PCA terminology is not consistent across the literature. Given a zero mean data matrix X (with observations as rows) and a basis W of the principal subspace, we define the scores matrix as Z=X*W which contains the principal components as its columns. The columns of the pseudo-rotation matrix W are called the principal axes, and the elements of W are called the loadings.
Deflating the data matrix accumulates numerical errors over successive PCs.
References
Sigg, C. D. and Buhmann, J. M. (2008) Expectation-Maximization for Sparse and Non-Negative PCA. In Proceedings of the 25th International Conference on Machine Learning (pp. 960–967).
Mackey, L. (2009) Deflation Methods for Sparse PCA. In Advances in Neural Information Processing Systems (pp. 1017–1024).
See Also
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 | if (requireNamespace("MASS", quietly = TRUE)) withAutoprint({
set.seed(1)
# Regular PCA, with the tolerance set to return five PCs
prcomp(MASS::Boston, tol = 0.36, scale. = TRUE)
# Sparse PCA with different cardinalities per component. The number of components
# is derived from the length of vector k.
nsprcomp(MASS::Boston, k = c(13, 7, 5, 5, 5), scale. = TRUE)
# Non-negative sparse PCA with four components. Note that the principal axes
# naturally have a high degree of orthogonality, because each component
# maximizes the additional variance not already explained.
set.seed(1)
nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE)
# The optimization can get stuck in local optima. Increase the number of
# random restarts or the number of power iterations to likely obtain decreasing
# standard deviations.
set.seed(1)
(nspc <- nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE,
nrestart = 10, em_tol = 1e-4, verbosity = 1))
# continue the computation of components from a partial model
nsprcomp(MASS::Boston, k = 3, ncomp = 5, nneg = TRUE, scale. = TRUE, partial_model = nspc)
# The reconstruction error for each sample can be influenced using the
# weighting vector omega. To reconstruct the data, the generalized
# inverse of the pseudo-rotation matrix has to be used, because the constrained
# principal axes are in general not pairwise orthogonal.
set.seed(1)
X <- matrix(runif(5*10), 5)
nspc <- nsprcomp(X, omega = c(5, 1, 1, 1, 5), ncomp = 2, nneg = TRUE)
X_hat <- predict(nspc)%*%MASS::ginv(nspc$rotation) + matrix(1,5,1)%*%nspc$center
rowSums((X - X_hat)^2)
})
|
Example output
> set.seed(1)
> prcomp(MASS::Boston, tol = 0.36, scale. = TRUE)
Standard deviations (1, .., p=14):
[1] 2.5585132 1.2843410 1.1614241 0.9415625 0.9224421 0.8124105 0.7317177
[8] 0.6348831 0.5265582 0.5022524 0.4612919 0.4277704 0.3660733 0.2456149
Rotation (n x k) = (14 x 5):
PC1 PC2 PC3 PC4 PC5
crim 0.242284451 -0.065873108 0.395077419 -0.100366211 0.004957659
zn -0.245435005 -0.148002653 0.394545713 -0.342958421 0.114495002
indus 0.331859746 0.127075668 -0.066081913 0.009626936 -0.022583692
chas -0.005027133 0.410668763 -0.125305293 -0.700406497 -0.535197817
nox 0.325193880 0.254276363 -0.046475549 -0.053707583 0.194605570
rm -0.202816554 0.434005810 0.353406095 0.293357309 -0.008320481
age 0.296976574 0.260303205 -0.200823078 0.078426326 0.149750092
dis -0.298169809 -0.359149977 0.157068710 -0.184747787 -0.106219480
rad 0.303412754 0.031149596 0.418510334 0.051374381 -0.230352185
tax 0.324033052 0.008851406 0.343232194 0.026810695 -0.163425820
ptratio 0.207679535 -0.314623061 0.000399092 0.342036328 -0.615707380
black -0.196638358 0.026481032 -0.361375914 0.201741185 -0.367460674
lstat 0.311397955 -0.201245177 -0.161060336 -0.242621217 0.178358870
medv -0.266636396 0.444924411 0.163188735 0.180297553 -0.050659893
> nsprcomp(MASS::Boston, k = c(13, 7, 5, 5, 5), scale. = TRUE)
Standard deviations (1, .., p=5):
[1] 2.5584808 1.2472326 1.0405531 0.9642164 0.8752789
Rotation (n x k) = (14 x 5):
PC1 PC2 PC3 PC4 PC5
crim -0.2429863 0.0000000 0.4854165 0.0000000 0.0000000
zn 0.2446861 0.0000000 0.5967689 0.0000000 0.0000000
indus -0.3318673 0.0000000 0.0000000 0.0000000 0.0000000
chas 0.0000000 -0.4315023 0.0000000 0.7490440 -0.4310263
nox -0.3253668 -0.2702262 0.0000000 0.0000000 0.0000000
rm 0.2017390 -0.4401132 0.0000000 -0.4474849 0.0000000
age -0.2967945 -0.2763328 0.0000000 0.0000000 0.2574766
dis 0.2981333 0.3520975 0.2488706 0.0000000 -0.1555882
rad -0.3042870 0.0000000 0.4437986 0.0000000 0.0000000
tax -0.3247282 0.0000000 0.3864504 0.0000000 0.0000000
ptratio -0.2073060 0.3886047 0.0000000 0.0000000 0.0000000
black 0.1975268 0.0000000 0.0000000 0.4110236 0.8419257
lstat -0.3109319 0.0000000 0.0000000 0.1935561 0.0000000
medv 0.2659092 -0.4424227 0.0000000 -0.1796834 0.1219658
> set.seed(1)
> nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE)
Standard deviations (1, .., p=4):
[1] 2.1308851 1.5497667 0.7822017 1.0082622
Rotation (n x k) = (14 x 4):
PC1 PC2 PC3 PC4
crim 0.3126418 0.0000000 0.0000000 0.0000000
zn 0.0000000 0.4900940 0.0000000 0.0000000
indus 0.3990108 0.0000000 0.0000000 0.0000000
chas 0.0000000 0.0000000 0.0000000 0.9669007
nox 0.4027495 0.0000000 0.0000000 0.0000000
rm 0.0000000 0.4480493 0.7706175 0.0000000
age 0.3579659 0.0000000 0.0000000 0.0000000
dis 0.0000000 0.4586857 0.0000000 0.0000000
rad 0.3932454 0.0000000 0.0000000 0.0000000
tax 0.4151053 0.0000000 0.0000000 0.0000000
ptratio 0.0000000 0.0000000 0.0000000 0.0000000
black 0.0000000 0.3145526 0.0000000 0.2551531
lstat 0.3546046 0.0000000 0.0000000 0.0000000
medv 0.0000000 0.4997236 0.6372979 0.0000000
> set.seed(1)
> (nspc <- nsprcomp(MASS::Boston, k = c(7, 5, 2, 2), nneg = TRUE, scale. = TRUE,
+ nrestart = 10, em_tol = 1e-04, verbosity = 1))
[1] "component 1: maximum objective is 2293 at random restart 0"
[1] "component 1: maximum objective is 2293 at random restart 1"
[1] "component 1: maximum objective is 2293 at random restart 2"
[1] "component 1: maximum objective is 2293 at random restart 3"
[1] "component 1: maximum objective is 2293 at random restart 4"
[1] "component 1: maximum objective is 2293 at random restart 5"
[1] "component 1: maximum objective is 2293 at random restart 6"
[1] "component 1: maximum objective is 2293 at random restart 7"
[1] "component 1: maximum objective is 2293 at random restart 8"
[1] "component 1: maximum objective is 2293 at random restart 9"
[1] "component 2: maximum objective is 1213 at random restart 0"
[1] "component 2: maximum objective is 1213 at random restart 1"
[1] "component 2: maximum objective is 1213 at random restart 2"
[1] "component 2: maximum objective is 1213 at random restart 3"
[1] "component 2: maximum objective is 1213 at random restart 4"
[1] "component 2: maximum objective is 1213 at random restart 5"
[1] "component 2: maximum objective is 1213 at random restart 6"
[1] "component 2: maximum objective is 1213 at random restart 7"
[1] "component 2: maximum objective is 1213 at random restart 8"
[1] "component 2: maximum objective is 1213 at random restart 9"
[1] "component 3: maximum objective is 404 at random restart 0"
[1] "component 3: maximum objective is 523.2 at random restart 1"
[1] "component 3: maximum objective is 525.1 at random restart 2"
[1] "component 3: maximum objective is 525.1 at random restart 3"
[1] "component 3: maximum objective is 525.1 at random restart 4"
[1] "component 3: maximum objective is 525.1 at random restart 5"
[1] "component 3: maximum objective is 525.1 at random restart 6"
[1] "component 3: maximum objective is 525.1 at random restart 7"
[1] "component 3: maximum objective is 525.1 at random restart 8"
[1] "component 3: maximum objective is 525.1 at random restart 9"
[1] "component 4: maximum objective is 520.6 at random restart 0"
[1] "component 4: maximum objective is 520.6 at random restart 1"
[1] "component 4: maximum objective is 520.6 at random restart 2"
[1] "component 4: maximum objective is 520.6 at random restart 3"
[1] "component 4: maximum objective is 520.7 at random restart 4"
[1] "component 4: maximum objective is 520.7 at random restart 5"
[1] "component 4: maximum objective is 520.7 at random restart 6"
[1] "component 4: maximum objective is 520.7 at random restart 7"
[1] "component 4: maximum objective is 520.7 at random restart 8"
[1] "component 4: maximum objective is 520.7 at random restart 9"
Standard deviations (1, .., p=4):
[1] 2.130886 1.549822 1.014159 1.011090
Rotation (n x k) = (14 x 4):
PC1 PC2 PC3 PC4
crim 0.3127134 0.0000000 0.0000000 0.0000000
zn 0.0000000 0.4850296 0.0000000 0.0000000
indus 0.3989959 0.0000000 0.0000000 0.0000000
chas 0.0000000 0.0000000 0.9783593 0.0000000
nox 0.4025259 0.0000000 0.0000000 0.0000000
rm 0.0000000 0.4519551 0.0000000 0.0000000
age 0.3572843 0.0000000 0.0000000 0.0000000
dis 0.0000000 0.4529891 0.0000000 0.2019432
rad 0.3936984 0.0000000 0.0000000 0.0000000
tax 0.4155276 0.0000000 0.0000000 0.0000000
ptratio 0.0000000 0.0000000 0.0000000 0.9793972
black 0.0000000 0.3177198 0.0000000 0.0000000
lstat 0.3545021 0.0000000 0.0000000 0.0000000
medv 0.0000000 0.5043192 0.2069133 0.0000000
> nsprcomp(MASS::Boston, k = 3, ncomp = 5, nneg = TRUE, scale. = TRUE, partial_model = nspc)
Standard deviations (1, .., p=5):
[1] 2.1308856 1.5498222 1.0141589 1.0110895 0.9012304
Rotation (n x k) = (14 x 5):
PC1 PC2 PC3 PC4 PC5
crim 0.3127134 0.0000000 0.0000000 0.0000000 0.0000000
zn 0.0000000 0.4850296 0.0000000 0.0000000 0.0000000
indus 0.3989959 0.0000000 0.0000000 0.0000000 0.0000000
chas 0.0000000 0.0000000 0.9783593 0.0000000 0.0000000
nox 0.4025259 0.0000000 0.0000000 0.0000000 0.0000000
rm 0.0000000 0.4519551 0.0000000 0.0000000 0.0000000
age 0.3572843 0.0000000 0.0000000 0.0000000 0.3385724
dis 0.0000000 0.4529891 0.0000000 0.2019432 0.0000000
rad 0.3936984 0.0000000 0.0000000 0.0000000 0.0000000
tax 0.4155276 0.0000000 0.0000000 0.0000000 0.0000000
ptratio 0.0000000 0.0000000 0.0000000 0.9793972 0.0000000
black 0.0000000 0.3177198 0.0000000 0.0000000 0.9025057
lstat 0.3545021 0.0000000 0.0000000 0.0000000 0.2661807
medv 0.0000000 0.5043192 0.2069133 0.0000000 0.0000000
> set.seed(1)
> X <- matrix(runif(5 * 10), 5)
> nspc <- nsprcomp(X, omega = c(5, 1, 1, 1, 5), ncomp = 2, nneg = TRUE)
> X_hat <- predict(nspc) %*% MASS::ginv(nspc$rotation) + matrix(1, 5, 1) %*%
+ nspc$center
> rowSums((X - X_hat)^2)
[1] 0.22883035 0.57480275 0.32494491 0.65648697 0.07084008
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