779 research outputs found
Microarray missing data imputation based on a set theoretic framework and biological knowledge
Gene expressions measured using microarrays usually suffer from the missing value problem. However, in many data analysis methods, a complete data matrix is required. Although existing missing value imputation algorithms have shown good performance to deal with missing values, they also have their limitations. For example, some algorithms have good performance only when strong local correlation exists in data while some provide the best estimate when data is dominated by global structure. In addition, these algorithms do not take into account any biological constraint in their imputation. In this paper, we propose a set theoretic framework based on projection onto convex sets (POCS) for missing data imputation. POCS allows us to incorporate different types of a priori knowledge about missing values into the estimation process. The main idea of POCS is to formulate every piece of prior knowledge into a corresponding convex set and then use a convergence-guaranteed iterative procedure to obtain a solution in the intersection of all these sets. In this work, we design several convex sets, taking into consideration the biological characteristic of the data: the first set mainly exploit the local correlation structure among genes in microarray data, while the second set captures the global correlation structure among arrays. The third set (actually a series of sets) exploits the biological phenomenon of synchronization loss in microarray experiments. In cyclic systems, synchronization loss is a common phenomenon and we construct a series of sets based on this phenomenon for our POCS imputation algorithm. Experiments show that our algorithm can achieve a significant reduction of error compared to the KNNimpute, SVDimpute and LSimpute methods
Challenges of Big Data Analysis
Big Data bring new opportunities to modern society and challenges to data
scientists. On one hand, Big Data hold great promises for discovering subtle
population patterns and heterogeneities that are not possible with small-scale
data. On the other hand, the massive sample size and high dimensionality of Big
Data introduce unique computational and statistical challenges, including
scalability and storage bottleneck, noise accumulation, spurious correlation,
incidental endogeneity, and measurement errors. These challenges are
distinguished and require new computational and statistical paradigm. This
article give overviews on the salient features of Big Data and how these
features impact on paradigm change on statistical and computational methods as
well as computing architectures. We also provide various new perspectives on
the Big Data analysis and computation. In particular, we emphasis on the
viability of the sparsest solution in high-confidence set and point out that
exogeneous assumptions in most statistical methods for Big Data can not be
validated due to incidental endogeneity. They can lead to wrong statistical
inferences and consequently wrong scientific conclusions
Covariance Estimation in High Dimensions via Kronecker Product Expansions
This paper presents a new method for estimating high dimensional covariance
matrices. The method, permuted rank-penalized least-squares (PRLS), is based on
a Kronecker product series expansion of the true covariance matrix. Assuming an
i.i.d. Gaussian random sample, we establish high dimensional rates of
convergence to the true covariance as both the number of samples and the number
of variables go to infinity. For covariance matrices of low separation rank,
our results establish that PRLS has significantly faster convergence than the
standard sample covariance matrix (SCM) estimator. The convergence rate
captures a fundamental tradeoff between estimation error and approximation
error, thus providing a scalable covariance estimation framework in terms of
separation rank, similar to low rank approximation of covariance matrices. The
MSE convergence rates generalize the high dimensional rates recently obtained
for the ML Flip-flop algorithm for Kronecker product covariance estimation. We
show that a class of block Toeplitz covariance matrices is approximatable by
low separation rank and give bounds on the minimal separation rank that
ensures a given level of bias. Simulations are presented to validate the
theoretical bounds. As a real world application, we illustrate the utility of
the proposed Kronecker covariance estimator for spatio-temporal linear least
squares prediction of multivariate wind speed measurements.Comment: 47 pages, accepted to IEEE Transactions on Signal Processin
Measurement Error in Lasso: Impact and Correction
Regression with the lasso penalty is a popular tool for performing dimension
reduction when the number of covariates is large. In many applications of the
lasso, like in genomics, covariates are subject to measurement error. We study
the impact of measurement error on linear regression with the lasso penalty,
both analytically and in simulation experiments. A simple method of correction
for measurement error in the lasso is then considered. In the large sample
limit, the corrected lasso yields sign consistent covariate selection under
conditions very similar to the lasso with perfect measurements, whereas the
uncorrected lasso requires much more stringent conditions on the covariance
structure of the data. Finally, we suggest methods to correct for measurement
error in generalized linear models with the lasso penalty, which we study
empirically in simulation experiments with logistic regression, and also apply
to a classification problem with microarray data. We see that the corrected
lasso selects less false positives than the standard lasso, at a similar level
of true positives. The corrected lasso can therefore be used to obtain more
conservative covariate selection in genomic analysis
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