Bioinformatics Advance Access originally published online on August 18, 2005
Bioinformatics 2005 21(20):3887-3895; doi:10.1093/bioinformatics/bti634
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Non-linear PCA: a missing data approach
1Max Planck Institute of Molecular Plant Physiology Potsdam, Germany
2University of Florida, Plant Molecular and Cellular Biology Program, Department of Environmental Horticulture Gainesville, Florida 32611, USA
3University of Potsdam Bioinformatics, Germany
*To whom correspondence should be addressed.
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Abstract |
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 TOP Abstract 1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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Motivation: Visualizing and analysing the potential non-linear structure of a dataset is becoming an important task in molecular biology. This is even more challenging when the data have missing values.
Results: Here, we propose an inverse model that performs non-linear principal component analysis (NLPCA) from incomplete datasets. Missing values are ignored while optimizing the model, but can be estimated afterwards. Results are shown for both artificial and experimental datasets. In contrast to linear methods, non-linear methods were able to give better missing value estimations for non-linear structured data.
Application: We applied this technique to a time course of metabolite data from a cold stress experiment on the model plant Arabidopsis thaliana, and could approximate the mapping function from any time point to the metabolite responses. Thus, the inverse NLPCA provides greatly improved information for better understanding the complex response to cold stress.
Contact: scholz{at}mpimp-golm.mpg.de
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1 INTRODUCTION |
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TOPAbstract1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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Non-linear principal component analysis (NLPCA) is generally seen as a non-linear generalization of standard linear principal component analysis (PCA) (Jolliffe, 1986; Diamantaras and Kung, 1996). The principal components are generalized from straight lines to curves. Here, we focus on a neural network based NLPCA, the auto-associative neural network (Kramer, 1991; DeMers and Cottrell, 1993; Hecht-Nielsen, 1995; Kirby and Miranda, 1996; Malthouse, 1998). It is successfully applied in the fields of atmospheric and oceanic sciences (Hsieh, 2004; Monahan et al., 2003), in astronomy and even in biomedical research. In Scholz and Vigário (2002) a hierarchically extended version of NLPCA was applied to spectral data from stars and to electromyographic (EMG) recordings for different muscle activities.
There is a wide variety of methods for visualizing data and extracting meaningful components (also termed features, factors or sources) in a non-linear way. Locally linear embedding (LLE) (Roweis and Saul, 2000; Saul and Roweis, 2004) and Isomap (Tenenbaum et al., 2000) were developed to visualize high dimensional data by projecting (embedding) them into a two- or low-dimensional space. A mapping function as a non-linear model is not explicitly given. Principal curves (Hastie and Stuetzle, 1989) and self-organizing maps (SOM) (Kohonen, 2001) are useful for detecting non-linear curves and two-dimensional non-linear planes. Both methods are limited to extraction of two components at most, due to high computational costs. Kernel PCA (Schölkopf et al., 1998), when used as pre-processing, can improve classification results.
Here, we consider the neural network approach. It provides a non-linear model of the mapping function and we will show that it can be applied to incomplete datasets by modelling only the second part of the auto-associative network, the reconstruction or generation part. The difficulty is to estimate both the model weights and the inputs which are now the required components.
For this approach Hassoun and Sudjianto (1997) optimized the weights and the inputs in two alternate steps by minimization of an error function which is equivalent to maximum likelihood. A similar approach was also used by Oh and Seung (1998). As the inputs can be represented by weights, we propose to optimize the inputs and weights simultaneously.
The same network architecture is also used by Valpola for a non-linear factor analysis (NFA) and a non-linear independent factor analysis (NIFA) (Lappalainen and Honkela, 2000; Honkela and Valpola, 2005), also applicable to incomplete datasets (Raiko and Valpola, 2001). The weights and inputs are optimized by Bayesian learning. The inputs (components) are explicitly modelled by a plain Gaussian distribution in NFA and a mixture of Gaussian distribution in NIFA. Although Bayesian inference in NFA and maximum likelihood in NLPCA often lead to similar results, their conceptual basis is rather different. Maximum likelihood attempts to find a single set of values for the network weights and inputs. In contrast, in the Bayesian approach the weights and inputs are described by posterior probability distributions which lead to a good regularisation. There are some relations: the Gaussian prior distribution for the weights corresponds to the use of a weight-decay regularizer in the maximum likelihood approach. Minimization of a mean square error function is equivalent to one maximum a posteriori (MAP) with additive Gaussian observation noise. In the proposed inverse NLPCA model a single error function is minimized. The model weights and inputs (components) are optimized simultaneously and the model is extended to be applicable to incomplete datasets. There are many methods for estimating missing values (Little and Rubin, 2002). Here, we focus on detecting non-linear components from incomplete datasets, so our approach involves ignoring missing values not a priori estimating them. However, once the non-linear mapping is effectively modelled, the missing values can then be estimated as well. This is shown for an artificial dataset and for experimental data. Estimation results were compared with results of state-of-the-art estimation techniques. There are two PCA based linear techniques: the recently published Bayesian missing value estimation method for gene expressions (Oba et al., 2003) which is based on Bayesian principal component analysis (BPCA) (Bishop, 1999) and probabilistic PCA (PPCA) (Verbeek et al., 2002) based on Roweis, (1997). Furthermore, there are the k-nearest neighbour based approach, KNNimpute (Troyanskaya et al., 2001), and a non-linear estimation by SOM.
There are many other approaches which are not further considered; for example, there are methods based on non-linear regression among variables (Zhou et al., 2003) or on modelling a dynamical system (Simeka and Kimmel, 2003). The latter takes the time information into account. It belongs, therefore, to supervised methods where it is much more difficult to avoid over-fitting than in the previously mentioned unsupervised methods.
Cold stress to the cell can cause rapid changes in metabolite levels. Here, we have analysed the temporal metabolite response to cold stress in the model plant Arabidopsis thaliana. The proposed inverse NLPCA model was applied to these, partly incomplete, metabolite data (Kaplan et al., 2004). Thus, we model the cold stress adaptation by a mapping function from a given time point to the metabolite responses. For each time point we are able to give the metabolites in the order of importance, i.e. the metabolites are ranked by the relative change in their concentration level. This procedure is analogous to ranking in PCA by the eigenvector values (also termed loadings or weights).
The observed experimental time information is not used in this unsupervised model. Thus, the risk of over-fitting is much lower than in a supervised regression model. Furthermore, the response time and developmental state of plant individuals in any experiment differs from the exact physical time measurement. Hence we cannot absolutely trust the physical experimental time for the description of biological experiments. An unsupervised model will be superior in accommodating the unavoidable individual variability of biological samples such as plants.
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2 AUTO-ASSOCIATIVE NEURAL NETWORKS |
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TOPAbstract1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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The NLPCA, proposed by Kramer (1991), is based on a multi-layer perceptron (MLP) with an auto-associative topology, also known as an autoencoder, replicator network, bottleneck or sand glass type network. A good introduction to multi-layer perceptrons can be found in Bishop (1995), Haykin (1998).
The auto-associative network performs the identity mapping, the output 
has to be equal to the input x, by minimizing the square error 
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This is no trivial task, as there is a ‘bottleneck’ in the middle, a layer of fewer nodes than at input or output, where the data have to be projected or compressed into a lower dimensional space Z, (Fig. 1).
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The network can be divided into two parts: the first part represents the extraction function
, whereas the second part represents the inverse function, the generation or reconstruction function 
. A hidden layer in each part enables the network to perform non-linear mapping functions. |
3 INVERSE NLPCA MODEL |
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TOPAbstract1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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The inverse model of NLPCA extracts the required components by only modelling the generation function
of the auto-associative network. It is the inverse function to the component extraction function . The inverse model presents a set of advantages; we only have to train the second part of the auto-associative network, which is more efficient than training both parts. Also, we model the natural process, which has generated the observed samples, hence we can be sure that such a function exists, which is not necessarily the case for the extraction model. And, most importantly, the inverse NLPCA can be extended to handle incomplete datasets, as we do not need the sample data as input, the data are needed only as required output.
As the desired components are now unknown inputs, the blind inverse problem is to estimate both the inputs and the parameters of the model by only given outputs. This makes sense only with the additional constraint of a lower dimensional input.
The output depends on the input z and the network weights w 
W3, W4, as illustrated in Figure 2,
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of the 4th sample x4 is missing. Therefore, the partial error 
is set to zero, identical to ignoring or non-back-propagating.The error can be minimized by a gradient optimization algorithm. e.g. conjugate gradient descent (Hestenes and Stiefel, 1952; Press et al., 1992). The gradients are obtained by propagating the partial errors
back to the input layer. For the input gradients it is simply one step further than usual. The gradients of the weights wij W4, wjk W3 and inputs 
are the partial derivatives:
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A disadvantage of such an inverse approach is that there is no mapping function 
, required for new data x. However, we can approximate the mapping by searching for an optimal input z to a given new sample x. For that, the network weights w have to be fixed and the input z has to be optimized to minimize the square error 
. This is only a line search (in case of one component) or low dimensional optimization with given gradients, efficiently done by a gradient optimization algorithm.
The inverse NLPCA is able to extract components of higher non-linear complexity than the standard NLPCA, even self-intersecting components can be modelled. This is shown in Figure 3 for a circular structure in two dimensions, generated from a uniformly distributed factor t (the angle) and a helical structure embedded in three dimensions, generated from a Gaussian distributed factor t. For the uniformly distributed 100 circular data points (plus noise), a [1-3-2] network is trained in 3000 iterations. The noisy helical structure of 1000 Gaussian distributed data points, is modelled with a [1-8-3] network in 10 000 iterations.
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The inverse NLPCA is not restricted to one component. It can be extended to m components by increasing the number of units in the input layer, the component layer z, to m. With an additional hierarchical error function (Scholz and Vigário, 2002), the non-linear components 1, ..., m can be extracted in a hierarchical order, which is a natural non-linear extension to the hierarchical ordered components of the standard linear PCA.
3.1 Regularization
As we usually have a large number of dimensions (metabolites) and a relatively small number of samples, a regularization of the non-linear model is very important.
Standard methods for regularization in neural networks reduce the number of hidden units or add a weight decay term to the error function. Furthermore, auto-associative neural networks have a kind of self-regularization, caused by the fact that for each mapping function the inverse function has to be estimated as well. A complex function has usually a much more complex inverse function or the inverse function does not even exist. Therefore, the auto-associative neural network is constrained to keep the functions as simple as possible. A similar effect is observed when extracting non-linear components in a hierarchical order, where subsequent components are extracted in respect to the previous components. A complex first component would strongly increase the complexity of the second or later components. Thus, the network is constrained to generate very smooth first components.
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4 MISSING VALUE ESTIMATION |
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TOPAbstract1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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The inverse NLPCA model can be easily extended to be applicable to incomplete datasets. If the ith element
of the nth sample vector xn is missing, the partial error is set to zero before back-propagating; hence this error is ignored, and it has no contribution to the gradients. Thus, the non-linear components are extracted from all the available observations. With these components the original data can be reconstructed, including the missing values. The network output gives the estimation of the missing element .
4.1 Missing data: artificial data
The inverse NLPCA approach was first applied to an artificial dataset and the results were compared with other missing value estimation techniques, the linear techniques BPCA1 and PPCA2, the k-nearest neighbour based approach KNNimpute3, and the non-linear SOM4. The data x lie on a one-dimensional manifold (a helical loop) embedded in three dimensions, plus Gaussian noise with standard deviation 0.05, see Figure 4. 1000 samples x were generated from an uniformly distributed factor t over the range [–1, 1], t represents the angle:
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In Figure 4 and Table 1 it is shown that even if the datasets are incomplete for all samples, the inverse NLPCA model is able to detect the non-linear component and gives a very good missing value estimation. The SOM also gives a reasonably good estimation, but the linear approaches BPCA and PPCA, as well as the k-nearest neighbour based approach KNNimpute, fail with this non-linear dataset.
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4.2 Missing data: metabolite data
The performance of the missing value estimation techniques was also assessed using a real experimental dataset. For that we used a completely available set of 140 metabolites from our cold stress experiment, see section 5 for more details. Different percentages of values were randomly removed and regarded as missing for the estimation techniques. A good overall missing value estimation is obtained for up to 50 percent missing values. This unexpectedly high rate might be caused by the high redundancy in the data, possibly due to high connectivity or dependency among the metabolites. By comparing the different techniques, we first found that BPCA gives the best average over all 140 metabolites, (Fig. 5). But instead of a good average we are interested in a good estimation of the most important metabolites. As our data values are ratios, see section 5.1, a high variance indicates an important metabolite. Therefore, we compared the performance on the first n metabolites of highest variance which mostly also show a strong non-linear behaviour. Now the results are different, (Fig. 6). The inverse NLPCA and SOM, which perform almost equally well, give the best result at the first five most important metabolites, and perform almost as equally well as the result of PPCA with the remaining metabolites.
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4.3 Missing data: gene expression data
To obtain a fair and comprehensive comparison, we also tested the performance of the missing data estimation using a larger set of gene expression data obtained from the same cold stress experiment. The data were again transformed to log2 ratios, relative to the median of control samples at time zero. In total, 16 996 genes were reduced to 1000 of highest log ratio variance. These genes are expected to be most important as they show the largest relative expression change. Twenty-one samples were measured at seven different time points.
Again, instead of a good averaged missing value estimation over all genes, we are interested in a good estimation of the most important genes, those of highest relative variance. Therefore, the cumulative mean square error (MSE) for the first 30 genes of highest ratio variance is shown (Fig. 7). The results differ from those on the metabolite dataset in Figure 6. All methods give quite similar, but significantly better, results than naive substitution by the mean of the residual values of each gene. However, BPCA which was developed for this kind of high-dimensional datasets, gave the best result for both the averaged estimation (not shown) and the estimation for the first n genes as shown in Figure 7. BPCA is successful because it uses principal components in the lower dimensional data space given by the small number of samples and not by the genes. Similar results can therefore also be obtained by the similar technique of PPCA when applied to the transposed dataset. However, the advantage of BPCA is that no parameter k, the number of used components, has to be chosen as is necessary with PPCA. The results of NLPCA were also improved when applied to the transposed matrix, and with the use of more than one non-linear component (k = 4). However, there might be no advantage of a non-linear technique applied to the transposed dataset as a non-linear data structure in gene data space does not necessarily lead to a non-linear structure in sample space (where genes are data points).
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Consequently, for estimating missing values in large gene expression datasets BPCA is a good choice. In datasets with a smaller number of variables, as is typical for metabolite or protein datasets, other methods are more suitable. These include non-linear techniques, such as NLPCA or SOM, when the data are non-linearly distributed. Both the gene expression and metabolite datasets, are available at http://nlpca.mpimp-golm.mpg.de. However, our major objective is to detect non-linear components in incomplete datasets. As these components should explain the experimental factors in the data space given by genes (where samples are data points) a transposed matrix is of no use.
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5 APPLICATION |
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TOPAbstract1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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The proposed inverse NLPCA model was used to analyse the metabolite response of A.thaliana to cold stress at 4°C. This gives us an approximation of the mapping function from a given time point ti to the metabolite responses x, and hence we obtain a ‘noise-free’ model of the biological cold stress response.
5.1 Data acquisition
We have used gas chromatography/mass spectrometry (GC/MS) to measure 497 metabolites at seven different time points, at 0,1,4,12,24,48 and 96 h, time point zero represents the control samples; Only 140 metabolites had available measurements for all samples, these metabolites were used in the previous section 4.2 to test the different methods for missing value estimation. In this experimental section the inverse NLPCA is applied to all metabolites which have <1/3 missing values. After removing 109 metabolites, the final dataset contains 388 metabolites (140 complete, 248 incomplete) and 52 samples at seven different time points (7–8 samples per time point).
The data are transformed to log fold changes (log ratios). All measurements of each metabolite 
are divided by the median of the control samples at time point zero. Consequently, we are analysing ratios of metabolite concentrations with respect to a control time point. The logarithm log2 is used to get symmetric changes: 
.
5.2 Model parameters
As inverse NLPCA model, we have used a network with a [3-20-388] architecture. This means we have extracted three non-linear components; 20 non-linear hidden units were used to perform the non-linear transformation, and 388 metabolites were approximated. The training was done in 300 iterations. To limit the complexity of the model we also added a weight decay term to the error function 
with 
= 0.001 and we have extracted the second and third component in a hierarchical order (Scholz and Vigário, 2002), which stabilizes the first component.
The inverse NLPCA model gives us a non-linear transformation from three estimated non-linear components to a 388 dimensional metabolite dataset. This is shown in Figure 8 for the top three metabolites of highest variance.
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5.3 Results
The extracted first non-linear component is directly related to the experimental time factor, see Figure 9. This means that the global or main information, represented by variance, is the metabolite change over time. This time trajectory clearly has a non-linear behaviour, see Figure 10. The time component gives a strong curve in the original metabolite data space. It can be seen as a noise-reduced representation of the cold stress response. The inverse model gives us a mapping function
1 
388 from a time point t to the response x of all considered 388 metabolites x=(x1, ..., x388)T. Thus, we can analyse the approximated response curves for each metabolite, shown in Figure 11. The cold stress is reflected in almost all metabolites; however, the response behaviour is quite different. Some metabolites have a very early positive or negative response, e.g. maltose and raffinose, whereas other metabolites show only a moderate increase.
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In classical PCA we can select the metabolites that are most important to a specific component by a rank order of the absolute values from the corresponding eigenvector, also termed loadings or weights. As the components are curves in non-linear PCA, no global ranking is possible. The rank order is different for different positions on the curved component, hence different at different time points in our case. However, we can give a rank order for each individual time point by computing the gradient
on the non-linear time curve at this time point. The rank order of the top 20 metabolites is shown in Table 2 for an early time point t1 and a late time point t2. The influence values 
are the l2-normalized gradients qi, 
. The gradient curves over time are shown in Figure 11. We found that even at the last time point of the experiment, 96 hours, there are still some metabolites with significant changes in their concentrations.
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6 CONCLUSIONS |
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TOPAbstract1 INTRODUCTION2 AUTO-ASSOCIATIVE NEURAL...3 INVERSE NLPCA MODEL4 MISSING VALUE ESTIMATION5 APPLICATION6 CONCLUSIONSREFERENCES |
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NLPCA was achieved by an inverse neural network model that was applicable to incomplete datasets. With this inverse NLPCA we were able to extract non-linear (curved) components from datasets with a large number of missing values. These extracted components can be used, together with the model, to reconstruct the original data, including the missing values. We have shown that in the case of non-linearly structured datasets, both non-linear techniques, the inverse NLPCA and SOM, can improve the missing value estimation performance on the most important metabolites. We have shown that in the case of non-linearly structured datasets, both non-linear techniques, the inverse NLPCA and SOM, can improve the missing value estimation performance for the most important metabolites of the lower dimensional metabolite dataset. In the larger gene expression dataset the best missing data estimations were obtained by BPCA and PPCA.
Applied to our cold stress experiment, the first non-linear component was directly related to the experimental time factor. Thus, the inverse NLPCA model gives us the continuous metabolite response over the time frame of the experiment. This trajectory over time is helpful to get a better understanding of the cold stress response. For each time point, including interpolated time points, we are able to give a ranked list of the most important metabolites, analogous to a global ranking in PCA.
The cold stress response clearly showed a non-linear behaviour over time, at the metabolite level (Kaplan et al., 2004). A similar non-linear behaviour was also found in gene expression data from the same cold stress experiment (data not shown). This non-linear analysis can therefore be done in the same way for such data.
Non-linearities are not restricted to temporal experiments, they can also be caused by other continuously changing factors, e.g. different temperatures at a fixed time point. Even natural phenotypes often take the form of a continuous range (Fridman et al., 2004), where non-linearities could exist.
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Acknowledgments |
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We would like to thank John Lunn for helpful comments on the manuscript. Funding to pay the Open Access publication charges for this article was provided by the Max Planck Society.
Conflict of interest: none declared.
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Footnotes |
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1http://hawaii.aist-nara.ac.jp/~shige-o/tools
2http://staff.science.uva.nl/~jverbeek/software/
3http://bioinfo.cnio.es/~jherrero/downloads/KNNimpute/
4http://www.cis.hut.fi/projects/somtoolbox/
Received on January 5, 2005; revised on August 2, 2005; accepted on August 15, 2005
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