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

Randomized accelerated implementation of SPCA, using variable projection as an optimization strategy.

1 2 3 |

`X` |
array_like; |

`k` |
integer; |

`alpha` |
float; |

`beta` |
float; |

`center` |
bool; |

`scale` |
bool; |

`max_iter` |
integer; |

`tol` |
float; |

`o` |
integer; |

`q` |
integer; |

`verbose` |
bool; |

Sparse principal component analysis is a modern variant of PCA. Specifically, SPCA attempts to find sparse
weight vectors (loadings), i.e., a weight vector with only a few 'active' (nonzero) values. This approach
leads to an improved interpretability of the model, because the principal components are formed as a
linear combination of only a few of the original variables. Further, SPCA avoids overfitting in a
high-dimensional data setting where the number of variables *p* is greater than the number of
observations *n*.

Such a parsimonious model is obtained by introducing prior information like sparsity promoting regularizers.
More concreatly, given an *(n,p)* data matrix *X*, SPCA attemps to minimize the following
objective function:

* f(A,B) = \frac{1}{2} \| X - X B A^\top \|^2_F + ψ(B) *

where *B* is the sparse weight (loadings) matrix and *A* is an orthonormal matrix.
*ψ* denotes a sparsity inducing regularizer such as the LASSO (*l1* norm) or the elastic net
(a combination of the *l1* and *l2* norm). The principal components *Z* are formed as

*Z = X B*

and the data can be approximately rotated back as

*Xtilde = Z t(A)*

The print and summary method can be used to present the results in a nice format.

`spca`

returns a list containing the following three components:

`loadings` |
array_like; |

`transform` |
array_like; |

`scores` |
array_like; |

`eigenvalues` |
array_like; |

`center, scale` |
array_like; |

This implementation uses randomized methods for linear algebra to speedup the computations.
*o* is an oversampling parameter to improve the approximation.
A value of at least 10 is recommended, and *o=20* is set by default.

The parameter *q* specifies the number of power (subspace) iterations
to reduce the approximation error. The power scheme is recommended,
if the singular values decay slowly. In practice, 2 or 3 iterations
achieve good results, however, computing power iterations increases the
computational costs. The power scheme is set to *q=2* by default.

If *k > (min(n,p)/4)*, a the deterministic `spca`

algorithm might be faster.

N. Benjamin Erichson, Peng Zheng, and Sasha Aravkin

[1] N. B. Erichson, P. Zheng, K. Manohar, S. Brunton, J. N. Kutz, A. Y. Aravkin. "Sparse Principal Component Analysis via Variable Projection." Submitted to IEEE Journal of Selected Topics on Signal Processing (2018). (available at 'arXiv https://arxiv.org/abs/1804.00341).

[1] N. B. Erichson, S. Voronin, S. Brunton, J. N. Kutz. "Randomized matrix decompositions using R." Submitted to Journal of Statistical Software (2016). (available at 'arXiv http://arxiv.org/abs/1608.02148).

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
# Create artifical data
m <- 10000
V1 <- rnorm(m, 0, 290)
V2 <- rnorm(m, 0, 300)
V3 <- -0.1*V1 + 0.1*V2 + rnorm(m,0,100)
X <- cbind(V1,V1,V1,V1, V2,V2,V2,V2, V3,V3)
X <- X + matrix(rnorm(length(X),0,1), ncol = ncol(X), nrow = nrow(X))
# Compute SPCA
out <- rspca(X, k=3, alpha=1e-3, beta=1e-3, center = TRUE, scale = FALSE, verbose=0)
print(out)
summary(out)
``` |

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