Limitations of Clustering with PCA and Correlated Noise

Abstract

It is now common to have a modest to large number of features on individuals with complex diseases. Unsupervised analyses, such as clustering with and without preprocessing by Principle Component Analysis (PCA), is widely used in practice to uncover subgroups in a sample. However, in many modern studies features are often highly correlated and noisy (e.g. SNP’s, -omics, quantitative imaging markers, and electronic health record data). The practical performance of clustering approaches in these settings remains unclear. Through extensive simulations and empirical examples applying Gaussian Mixture Models and related clustering methods, we show these approaches (including variants of kmeans, VarSelLCM, HDClassifier, and Fisher-EM) can have very poor performance in many settings. We also show the poor performance is often driven by either an explicit or implicit assumption by the clustering algorithm that high variance features are relevant while lower variance features are irrelevant, called the variance as relevance assumption. We develop practical pre-processing approaches that improve analysis performance in some cases. This work offers practical guidance on the strengths and limitations of unsupervised clustering approaches in modern data analysis applications.

1. Introduction

For complex diseases with data such as genomic/metabolomic signatures, imaging data, or clinical data from the electronic health record, it is often of scientific interest to identify groupings of individuals, often called subtyping. Identified subgroups in the sample are a priori unknown and so unsupervised learning approaches, known collectively as clustering methods, are needed [19]. These types of data are now common in biomedical research and well-known to present with analytic challenges such as high correlation among features and high dimension.

For high dimension problems, dimension reduction is often desired prior to applying clustering approaches. Dimension reduction is achieved using principal component analysis (PCA) through the variance-as-relevance assumption that high variance signals are typically relevant and should be retained or more carefully modeled while low variance signals should be discarded or more crudely modeled. Even when PCA is not applied, many clustering approaches assume that high variance features are most likely to discriminate clusters. In supervised contexts, the variance-as-relevance assumption is more reasonable as the features being investigated are typically those already known or reasonably suspected to be related to a known outcome of interest. However, in unsupervised contexts, we lack contextual, theoretical, and often empirical reason to believe the highest variance signals in data are discriminatory for the as-yet unidentified structure [12,50]. For example, variation in whole lung imaging data may primarily reflect healthy variation in size, shape, and structure of the lungs or variability in image acquisition protocols rather than as-yet unidentified disease-related variation. Similarly, top principal components of genetics data have long been known to be sensitive to broad population structure in the senses of ancestry and geography [28,37], which need not be relevant to a disease of interest.

Despite abstract theoretical [12] and empirical [1,50] evidence that caution against assuming variance-as-relevance, these papers individually do not fully address practicalities of applied statistical work, especially with higher dimension data, or approaches to avoiding assuming variance-as-relevance. The communication gap between the theoretical and applied fields means that PCA is still used to process high dimension data prior to unsupervised clustering to consolidate signals and reduce dimensionality [15,33]. However, many of these applications have had little or no investigation of practical performance when variance-as-relevance does not hold. Thus, unsupervised analyses in typical modest to high dimension feature spaces common in biomedical research may be being applied in situations ripe for poor statistical performance, which may contribute to the lack of reproducibility of many analyses. Recent methodological developments continue to incorporate PCA in this manner [21,23,51] or appear unaware of these important drawbacks of PCA in discussion of established approaches [2,20]. Acknowledgment of known drawbacks in applied contexts are more rare [26].

The primary purpose of this paper is to investigate the impacts of correlation, dimension, and methodological approach on clustering performance for a variety of PCA-based pre-processing methods and Gaussian Mixture Model (GMM) analytic approaches and offer solutions to challenges encountered. In particular, we conduct an extensive simulation study designed for many situations that are now common in biomedical research and compare the different clustering approaches across several real biomedical data examples. We find that the variance-as-relevance assumption, as made indirectly or directly by clustering methods, was a clear methodological obstruction to good clustering performance. As such, we also develop new practical pre-processing solutions in an attempt to alleviate poor performance due to deviations from the variance-as-relevance assumption. One approach, the Shapiro-Wilk (SW) filter, an alternative to reducing dimension after PCA, shows particular promise in preemptively countering the variance-as-relevance assumption.

The paper is organized as follows: in Section 2, we introduce four motivating data sets which are later used in designing the simulation scenarios and in real data comparisons. In Section 3 we briefly review the clustering methods considered in this work and demonstrate the variance-as-relevance assumption as made by clustering methods through a diagnostic simulation. Then in Section 4.1 we present different existing and new approaches to pre-processing data for clustering and discuss how they relate to the variance-as-relevance assumption: PCA (existing), PCA with the Shapiro-Wilk filter (new), and a decorrelation filter. In Section 4.2, we consider three simulation contexts that focus on clustering data having large variance noise PCs, demonstrate standalone performance of the SW filter, and investigate combinations of clustering and pre-processing methods. In Section 4.3, we consider sensitivity of real data clustering results to clustering and pre-processing methods. Results are given in Section 5 and discussion and conclusions are offered in Section 6.

2. Motivating Data

Imaging, gene expression, and metabolomics data are three examples of typically high dimension, highly correlated data which might be considered for exploratory analyses such as clustering. Both the dimensionality of the data and the typical levels of correlation suggest such data contain large variance latent signals (e.g. PCs) which are nevertheless unimportant for discriminating disease subtypes, thereby potentially violating the variance-as-relevance assumption.

We consider first a radiomics panel computed from high resolution lung CT scans of sarcoidosis patients enrolled in the GRADS study ($n=321$ observations of $p=566$ features) [35]. Radiomics measures, such as Haralick measures [18], are engineered features commonly used for analysis in place of raw data to reduce dimensionality of high resolution biomedical images while retaining textural information. The sarcoidosis data we consider consist of such a radiomics panel computed using the packages lungct (https://github.com/ryansar/lungct) and RIA [24,25] in R. The maximum pairwise correlation in absolute value of features in these data is exactly 1 as the radiomic features include linear rescalings of features. The radiomics panel contains 9706 pairs of features which are correlated beyond 0.9 in absolute value, 30 of which are effectively identical pairs.

We also consider chronic obstructive pulmonary disease metabolite data (COPDGene) previously studied in [16] ($n=1130$, $p=995$). We use the continuous metabolite data as prepared for analysis in that work after imputation and log transform. The maximum pairwise correlation in absolute value of features in these data is 0.994. The data contain 86 pairs of features correlated beyond 0.9 in absolute value.

The two classic examples in the clustering literature, albeit labeled data sets, are the Golub gene expression data ($n=38$, $p=3051$, 2 classes by type of Leukemia) [17] available through the multtest package [38] in R, and part of The Cancer Genome Atlas (TCGA) gene expression RNA-seq data (available: $n=801$, $p=20531$, 5 classes by tumor type: BRCA, KIRC, COAD, LUAD and PRAD) [47] available through the UCI Machine Learning Repository [dataset][13]. In this work, the TCGA data are additionally filtered to remove all features with constant value across all observations or with 10% or more 0 values (used: $n=801$, $p=15832$, 5 classes). The maximum pairwise correlation in absolute value of features in the TCGA and Golub data is 0.986 and 0.998 respectively. The TCGA and Golub data respectively contain 1850 and 115 pairs of features correlated beyond 0.9 in absolute value.

For all of these data, a primary goal of analysis is to identify disease subtypes through unsupervised clustering. In labeled data sets, we also investigate the extent to which subtypes identified through clustering correspond with known subtypes. Such a correspondence of generated subtypes with known subtypes may permit construction of reliable biomarkers of disease condition in applied contexts.

3. Relevant Background: Gaussian Mixture Models (GMMs) and Variance as Relevance Assumption

3.1. GMM and Clustering Theory

We briefly review GMM and clustering theory and discuss methods considered in this work. A summary of methods may be found in Table 1.

Table 1.

Clustering Methods Considered.

Method	R Package	Description	K Esti mation Method	Variable Selection	Versions Consid ered
K-means [30]	stats [39]	Standard K-means. No dimension reduction/variable selection.	Gap statistic or silhouette statistic.		Gap vs silhouette
Sparse K-means [48,49]	sparcl	A weighted version of standard K-means with an L1 restriction on weights to induce variable selection.	BCS gap statistic* or silhouette statistic.	L1 restriction	Gap vs silhouette
Variable Selection for Model-Based Clustering [31,32]	VarSelLCM	Diagonal GMM with models indexed by variable relevance. Irrelevant variables are assumed to have common mean and variance across clusters	BIC	Model selection (BIC)
High Dimension Classifier [5,9]	HDclassif	GMM assuming each cluster occurs in a latent subspace consisting of high variance PCs. Two implementations considered were default (fits most general model) and exhaustive (fits 14 candidate models).	BIC		Default model vs all models
Fisher-EM [6,7]	FisherEM	GMM assuming all clusters occur in a common latent subspace identified using Fisher discrimination theory. Two implementations considered were LASSO and elastic net penalized fits.	BIC	LASSO or elastic net penalties in fitting	LASSO vs EN

^*The BCS-based gap statistic [48] was developed for selection of a tuning parameter in SK-means, but can also be used simultaneously for the selection of the number of clusters.

The general GMM may be defined as follows: Let $\mathbf{x}$ be a vector of $p$ features, such as an individual’s gene expression profile. Then a GMM is given by probability density function

$$f(\mathbf{x})=\sum _{k=1}^K{\pi }_k\varphi (\mathbf{x}\mid {\mu }_k,{\Sigma }_k)$$

where ${\pi }_k$ is the probability of a subject belonging to cluster $k$ and $\varphi (\mathbf{x}\mid \mu ,\Sigma )$ is a multivariate Gaussian density having mean $\mu$ and covariance $\Sigma$. A specific parameter is called common if it does not differ by cluster $k$.

Moderate and high-dimension GMM methods typically make simplifying assumptions on the covariance structure [11] or use notions of latent spaces, resulting in intentional model misspecification as a trade-off for performance. A GMM method may then apply information criteria to select an appropriate parsimonious structure and number of clusters $K$.

The GMM-related methods we consider are not model-based, and so required pairing with a separate method for the selection of $K$. Two such methods considered here are the gap statistic [45,48] and the silhouette statistic [41].

The GMM and GMM-related methods investigated in this study were identified primarily via surveys [14] and [8], which review recent developments in GMM and GMM-related methods. We only investigated methods with a readily available implementation in R [39] as they are most likely to be used in practice (Table 1). GMMs considered include FisherEM [6,7], HDclassif [5,9], and VarSelLCM [31,32]. GMM-related methods include K-means [30] and SK-means [48,49]. In particular, K-means is associated with a GMM as a common variance proportional to the identity is taken to 0. SK-means is a weighted and penalized version of K-means which is similarly but more loosely associated with a GMM having a common diagonal covariance matrix.

FisherEM and HDclassif actively model correlated observations using latent spaces and parsimonious covariances. VarSelLCM assumes independence of variables as a simplifying assumption and SK-means is associated with a GMM which assumes independence, but both are intended for application to correlated data as well. Additionally, SK-means, FisherEM, and VarSelLCM are methods which perform variable selection and clustering simultaneously.

All methods considered are intended for high-dimension data except K-means, which is included as a benchmark. We also note that the review papers and the papers developing these methods tend to cover more straightforward high-dimensional applications than considered in this work.

3.2. Variance-as-Relevance

Formally, the variance-as-relevance assumption is the assumption that higher variance signals are more relevant and should be retained or more carefully modeled in analysis, while lower variance signals are less relevant and should be discarded or more crudely modeled. In the context of clustering, this assumption can lead to higher variance features being used to identify clusters rather than lower variance features even when, in truth, lower variance features containing more discriminative information (Figure 1, discussed below).

Figure 1.

Diagnostic simulation for variance-as-relevance assumption: A single sample of 1000 observations from a well-separated 2-component mixture of Gaussians with common standard deviation 0.25, equal proportions, and centers separated by 2 was generated with one noise variable x representing a noisy high-variance PC independent of one clustering variable y representing a lower variance PC with signal. This one data set was repeatedly scaled to have larger noise variance and clustered according to five methods with default settings. Data are plotted to scale and colored according to fit class labels.

HDclassif makes the assumption explicitly by assuming high-variance cluster-specific probabilistic principal components are discriminative. SK-means makes the assumption by implementing a variable weighting scheme with weights proportional to variable specific raw between-cluster-sum-of-squares in absence of an L1 restriction. K-means makes the assumption through its relation to a spherical GMM, effectively assuming all clusters are spherically shaped. While the FisherEM method does not appear to make the assumption in theory, the assumption is made by the initialization algorithm used by default in its implementation (K-means). Informal simulations suggest that random initialization might aid the FisherEM implementation in overcoming the assumption, though many more fittings are required (data not shown). Only VarSelLCM does not make the assumption.

Since the variance-as-relevance assumption is made mostly implicitly and in a multitude of ways, we designed a diagnostic simulation to both demonstrate the detrimental impact of the assumption on clustering performance and aid in future determination of whether a method makes the assumption implicitly.

In particular, a single data set from a well-separated bivariate 2-component spherical GMM is sampled such that the two cluster centers are on the $y$-axis. Multiple related data sets are then derived by re-scaling the $x$-dimension (the noise dimension) to be larger. Here, we might loosely understand the $x$ and $y$ dimensions to represent two large principal components (PCs) of a dataset, with the first and largest PC $x$ being purely noise and the other smaller PC $y$ containing the cluster structure. This simulation structure highlights the impact of increasing the variance of the noise relative to the signal. Each data set is identical except for scale.

Each scaled data set is then clustered by each method with default settings (with no L2 penalty for FisherEM and all models considered for HDclassif). A clustering method which does not make the variance-as-relevance assumption is expected to correctly identify the cluster regardless of scaling. A clustering method which does make the variance-as-relevance assumption is expected to correctly identify the cluster structure when the noise scale is small to moderate, but switch to incorrectly splitting this high variance noise dimension when re-scaled to be sufficiently large (Figure 1). Note that while the figure depicts a single run of this simulation, results are representative of repeated runs and reflect large sample behavior (data not shown).

Now that we understand one reason unsupervised clustering methods might fail, we might consider two practical pre-processing approaches to potentially improve performance. First, if clustering methods are prone to focus on high variance signals regardless of whether these signals are discriminative for the purposes of clustering, we should attempt to identify and remove such signals in pre-processing. Many unsupervised variable selection techniques already exist [44]. Identification of a technique which does not make the variance-as-relevance assumption and which is appropriate for contexts of interest is an open question. As proof of concept, we introduce the Shapiro-Wilk filter in this work to demonstrate how a variable selection method might function to counteract the variance-as-relevance assumption as made by clustering methods.

Second, truly redundant features might be removed. Redundance can inflate noise variance which may unduly influence or distort results. To this end, we introduce a decorrelation filter for identifying subsets of features which are not redundant with respect to a user-specified maximal pairwise correlation. This filter might be considered an incremental improvement over some comparable established methods in that it guarantees all discarded features are represented by (AKA redundant for) kept features and permits specification of secondary priorities, such as greedy selection of small representative sets.

4. Methods

Throughout this work, we denote by “raw” those data which have been standardized to have mean 0 and feature variance 1. For comparability, all other pre-processed data sets are obtained by applying PCA and filters to “raw” data. All pre-processing methods are listed in Table 2. We first introduce the pre-processing approaches and then present the method for obtaining our numerical experiments (e.g. simulated datasets).

Table 2.

Pre-processing Methods Considered

Method	Description
Raw	Standardize all variables to have observed mean 0 and variance 1
PCs_all	PCs of Raw data without dimension reduction
PCs_90	PCs with standard dimension reduction such that 90% of variability is retained
PCs_80	PCs with standard dimension reduction such that 80% of variability is retained
SW, SW_15	PCs which produce Shapiro-Wilk normality test p-value less than 0.15
decor	Raw features selected by the decorrelation filter with small set ranking and tolerance $t=0.9$
decorSW	PCs selected by the Shapiro Wilk filter ($p<0.15$) from PCs computed from decorrelated data ($t=0.9$)

4.1. Preprocessing approaches

4.1.1. PCA

PCA is a standard pre-processing technique for moderate to high dimension data with correlation. A rotation is used to produce consolidated signals (PCs) which are uncorrelated. We consider standard PCA-based pre-processing techniques which either don’t include dimension reduction or use the standard dimension reduction technique for PCA whereby lowest variance PCs are discarded until either 80% or 90% of variability as measured by sums of squares is retained. For this work, we chose to use principal components and not unit variance principal eigenvectors.

A plethora of more sophisticated PCA-based and non-PCA-based approaches to unsupervised feature selection might have been considered [44], but standard PCA-based techniques are still sufficiently widely used in practice to warrant investigation, especially in light of concerns over variance-as-relevance. In particular, standard dimension reduction in PCA is based on the variance-as-relevance assumption, while the rotation step of PCA may consolidate a strong latent noise signal into a single high-variance noise feature which then dominates other features.

The scale function of the base [39] package in R was used to standardize data prior to PCA. The prcomp function of the stats [39] package in R was used to perform PCA.

4.1.2. Shapiro-Wilk

The results of the variance-as-relevance diagnostic simulation suggested that implementation of a discriminative filter after PCA might greatly improve clustering performance by consolidating and removing non-discriminative/irrelevant high variance signals. Filtering PCs in particular as opposed to raw data is supported by [50], though in a supervised fashion. For this purpose, we simplify and adapt a filter introduced in the work of [34] on the EMMIX-GENE method for clustering microarray expression data by independently testing features for multiple components.

Under the strong assumption of GMM data, a normality test is equivalent to a test for multiple components. Thus, a normality test might be used to identify discriminative features in GMM data. We use the Shapiro-Wilk test for normality [43] since it is the most powerful among a studied collection of normality tests [40]. In the GMM context, we understand the Shapiro-Wilk test resulting in a low p-value as indicative of multiple components, i.e. cluster structure. Thus, the discriminative filter we call the Shapiro-Wilk (SW) filter is as follows:

1. Compute PCs from raw data.

2. Apply a Shapiro-Wilk test to each PC.

3. Discard all PCs with resulting p-value greater than a pre-specified cutoff while retaining all other PCs.

See [22] for a highly related use of normality testing for filtering raw features prior to PCA and clustering. Note that unlike our approach, filtering prior to PCA does not benefit from the consolidation of latent signals provided by PCA prior to signal detection via normality testing.

The standalone performance of the SW filter as measured by Shapiro-Wilk p-values computed for relevant vs irrelevant PCs is considered in Section 4.2 when applied to simulated data. As these data are simulated from GMMs, the Shapiro-Wilk filter is a well-specified test and is able to identify discriminative signal in PCs marginally when present as expected according to classical hypothesis testing theory and the specified test level (p-value cutoff).

Since we consider primarily clustering methods which simultaneously perform feature selection, it is appropriate to be conservative in discarding data prior to clustering. As such, users might select a more generous cutoff p-value. Larger cutoff values are more generous in the sense that more features are retained by the filter. For demonstration, we select 0.15 in this work.

The performance of clustering analyses when paired with the Shapiro-Wilk filter as a pre-processing step is considered in Sections 4.2 and 5.2.

The Shapiro-Wilk test was applied via the shapiro.test function in the R package stats [39].

4.1.3. Decorrelation Filter

We introduce a simple decorrelation filter for variable selection. This filter guarantees “representation” of discarded variables by kept variables and a lack of “redundance” among kept variables according to a correlation tolerance $t$. This filter allows for user specification of secondary priorities, including greedy production of small sets of kept variables for dimension reduction and production of random sets for sensitivity analyses. The findCorrelation function from the caret package [27] in R might be considered an alternative decorrelation approach. After introducing our decorrelation filter, we briefly compare the performance of these two approaches to assess the validity of our decorrelation filter for meeting the goals of representation and lack of redundance.

Formally, let $A$ be the set of variables kept and $D$ be the set of variables discarded by the filter. Although the word representation has other definitions in other contexts such as mathematics, for this work we define redundance of a kept variable $x\in A$ and the representation of a discarded variable $x^{\prime }\in D$ as

$$\text{Redundance of}x=\underset{\tilde{x}\in A-\{x\}}{\max }\mid \text{Corr}(x,\tilde{x})\mid \text{Representation of}{x}^{\prime }=\underset{\tilde{x}\in A}{\max }\mid \text{Corr}(x^{\prime },\tilde{x})\mid .$$

For a specified tolerance $t$ and ranking function $R$, the decorrelation filter is designed to produce a subset of variables such that redundance is less than $t$ for all kept variables $x$ and representation is no less than $t$ for all discarded variables $x^{\prime }$, with a secondary preference for retaining high rank features.

The decorrelation filter iteratively sorts features into the kept set $A$ and the discarded set $D$ from a candidate set $C$. At iteration $k$, a feature $\tilde{x}$ is selected to keep from the candidate set according to a user specified ranking $R(x\mid k)$ encoding secondary priorities, and all remaining candidate features $x$ which are represented by $\tilde{x}$ relative to $t$ are discarded. By only discarding features represented by a kept feature, representation is guaranteed. By discarding represented candidates after each new addition to the kept set, a lack of redundancy is guaranteed.

In principle, any ranking function with unique maxima may be used. Two ranking functions are natural to consider regardless of context. The first is a small set ranking which leads to greedy selection of small kept sets $A$ by ranking candidate variables higher if they will result in a larger number of discarded candidates when kept. Specifically, for candidate variables $C_k$ remaining at iteration $k$, $R(x\mid k)=\#\{x^{\prime }\in C_k:\mid Corr(x,x^{\prime })\mid \ge t\}$ with ties broken at random. The second is a random ranking, e.g. with $R(x\mid k)$ having i.i.d. standard uniform distribution. This allows for analysis of sensitivity of later analysis outcomes to selection of a representative set $A$.

The algorithm for the decorrelation filter is given formally in Figure 2. Note that step 4 is an optional step which might sometimes be used to improve algorithm efficiency. If left out, $A_{k+1}^{\prime }$ can be replaced by $A_{k+1}$ in step 2. As such, we would recommend specification of a rank which results in the same keep set $A$ regardless of whether step 4 is used. Further, note that we might have initialized $A_0$ to contain some variables of a priori known interest, with $D_0$ defined to be the variables represented by $A_0$ and $C_0$ defined to contain remaining variables. The guarantee of representation would still hold. While the guarantee against redundancy might be violated, any two redundant variables in the final set $A$ would necessarily both be contained in the initial set $A_0$.

Figure 2.

A decorrelation filter algorithm which guarantees a resultant set $A$ of retained variables which are representative of discarded variables as defined by a correlation cutoff $t$ and which greedily selects variables according to a user specified ranking $R$.

Note that neither guarantee regarding representation or redundancy was dependent on the choice of ranking, and so a secondary priority may be implemented through the ranking function, making this algorithm greedy with respect to the ranking. The only technical requirement is that the ranking function have a unique maximum at each iteration $k$ such that a candidate $\tilde{x}$ can be selected to keep. However, any ranking function without a unique maximum for each $k$ can be randomly perturbed to break ties.

Decorrelation Filter Performance

Prior to applying our new filter extensively to the numerical experiments and real data applications, we assessed whether the new decorrelation filter met the goals of reducing redundancy and retaining representation and compared it to other similar correlation filters available in R. Specifically we applied the filter repeatedly to the sarcoidosis radiomics data; see Supplementary Figure 16 for a summary of pairwise correlation among considered radiomic features.

We considered the decorrelation filter with both the small set ranking and the random ranking, as well as an established filter also intended for reducing pairwise correlation in data sets: the findCorrelation function. The findCorrelation filter has a default and exact setting, both of which we considered. While other related functions might be considered, such as the leaps, genetic, and anneal functions from the subselect package [36] in R, these functions typically have more complex goals than simply reduction of pairwise correlation. It is unclear that more complex notions of redundancy are appropriate in the clustering context.

We considered tolerance values $t$ between 0.70 and 0.99 with increments of 0.01. For each value of $t$, the decorrelation filter with random ranking was applied 100 times, and the measures mean, median, min, and max redundance (of kept features) and representation (of discarded variables) were recorded, along with the number of features kept. Over the 100 applications, the min, mean, and max of each measure were reported. For the other three filters, decorrelation with small set ranking and the two versions of findCorrelation, each filter was applied once for each value of $t$ and the measures mean, median, min, and max redundance and representation were reported, along with the number of features kept.

Measures of redundance were fairly similar across filters and tolerance levels; see Figure 3. The findCorrelation filters tended to produce variable subsets with lower redundance, though differences across filters were least in typical use cases ($t\ge 0.9$). The top left panel illustrates the guarantee of no redundance relative to tolerance, specifically that all measures are below the line $y=x$ (dotted black line).

Figure 3.

Measures of redundance and representation of variables subsets selected by each of 4 filters (decorrelation filter with random or small set ranking, findCorrelation filter with and without exact computations) when applied to a radiomics panel with 566 highly redundant features, plotted as a function of specified tolerance. A dotted black line indicates the line $y=x$. As the decorrelation filter was applied 100 times per tolerance value, min, max, and mean measures over the 100 iterations were reported.

Measures of representation were less consistent across filters. The bottom right panel illustrates the guarantee of representation relative to tolerance, specifically that measures should be above the line $y=x$. While the decorrelation filter makes this guarantee, the findCorrelation filter does not. Indeed, approximately half of variables discarded by the findCorrelation filters were not represented relative to tolerance in typical use cases ($t\ge 0.9$; right median panel).

Comparing the decorrelation filter with the small set and random rank functions, we see that the small set rank results in measures of redundance and representation consistent with the distribution of measures resultant of the random rank.

The decorrelation filter with small set rank typically resulted in smaller representative sets than the decorrelation filter with random rank; see Figure 4. However, the small set decorrelation filter still returned about 20 more features than the findCorrelation filters in typical use cases ($t\ge 0.9$).

Figure 4.

Number of features selected by each of 4 filters (decorrelation filter with random or small set ranking, findCorrelation filter with and without exact computations) when applied to a radiomics panel with 566 highly redundant features, plotted as a function of specified tolerance. As the decorrelation filter with random was applied 100 times per tolerance value, min, max, and mean number of selected features over the 100 iterations were reported.

As a violation of the guarantee of representation corresponds with discarding information, we consider the decorrelation filter to be superior to the findCorrelation filter. The performance of clustering analyses when paired with the decorrelation filter as a pre-processing step is considered in Sections 4.2.2 and 5.2. Note, this filter was only applied in the numerical experiment with frequent and extreme feature redundancy (pairwise correlation frequently above 0.9).

4.2. Numerical Experiments

We developed three simulated data contexts in which to compare clustering methods and explore the impacts of the variance-as-relevance assumption and pre-processing approaches.

The first is a moderate dimension context with sparse but consolidated signal and varying levels of compound correlation. Data with compound correlation and without cluster structure are comprised of a single high variance latent signal with remaining signals much lower and homogeneous. With cluster signal added, this structure produces a very high variance noise PC, followed by cluster signal PCs, followed by the more homogeneous low-variance noise (Figure 5), allowing for a targeted investigation of the variance-as-relevance assumption in clustering contexts. A range of correlation levels are considered to demonstrate both a rapid degradation in clustering performance as correlation increases and substantial improvement of clustering performance when variance-as-relevance is addressed via the SW filter in pre-processing.

Figure 5.

Simulation 1: 500 sets of PCs_all data with 50 noise features for each correlation level ($\rho =0$, 0.1, 0.25, 0.5) from the compound correlation simulation were simulated. For each PC, the quartiles from the 500 values of the PC variance, the R-square value obtained from regressing the PC onto the true cluster labels, and the negative log10 of the Shapiro-Wilk p-value obtained from the PC are plotted, stratified by correlation level. A line plot is used for visual ease.

Remaining contexts are designed to mimic real-world data. The second context is another moderate dimension context with sparse consolidated signal, but the correlation structure is now sampled from the empirical correlation structure of the radiomics panel described in Section 2. The third is a high dimension context with very diffuse signal and a complex correlation structure sampled in part from the COPDGene data described in Section 2.

Clustering and pre-processing methods used in simulations are summarized in Tables 1 and 2. A supervised assessment of signal as contained by PCs generally and as identified by the Shapiro-Wilk filter is also provided in each simulation.

Throughout these simulations and the real data analyses in Section 5.2, we refer occasionally to ‘failed’ fittings. In this context, failure to fit refers to a clustering algorithm failing to return cluster labels, typically due to lack of algorithm convergence. Failed fittings do not include failure to detect cluster structure, i.e. estimating a single cluster, or analyses which were terminated early due to excessive run time.

4.2.1. Simulation 1: Moderate Dimension with Consolidated Signal

Data were simulated from 8 case scenarios consisting of combinations of two noise levels and four correlation levels. All simulated data consisted of $n=300$ observations of either $p=65$ or 215 variables. 15 variables were ‘relevant’ and contained cluster structure while the remaining 50 or 200 were ‘irrelevant,’ purely noise variables. Relevant variables were sampled independently of irrelevant variables. The relevant variables and the irrelevant variables were each sampled with a compound correlation structure with common correlation $\rho$ and variance 1. We considered $\rho =0$ (independent), 0.1, 0.25, or 0.5.

All simulated data had $K=6$ components with 50 observations each. Mean structure was specified with 5 identically distributed sets of 3 variables. For a given set of 3 variables, the 6 cluster centers were specified as (−2,2,0,0,0,0) for variable 1, (0,0,−2,2,0,0) for variable 2, and (0,0,0,0,−2,2) for variable 3, corresponding with the 6 vertices of a regular octahedron.

First, we investigated PCA and the Shapiro-Wilk filter alone. For each of the 8 case scenarios, 500 raw data sets were simulated and PCs were obtained without dimension reduction (PC_all). For each PC, the PC variance, the r-squared value associated with regressing the PC onto the (categorical) true cluster label, and the Shapiro-Wilk test p-value associated with the PC were obtained. Across the 500 data sets, quartiles 1, 2, and 3 were computed.

Then, we investigated clustering performance. For each combination of the 8 case scenarios, 4 pre-processing steps (Raw, PC_all, PC_90, and PC_80; see Table 2), and clustering methods, 100 data sets each were simulated and clustered when the true number of clusters $K$ was assumed known and again when assumed unknown but no greater than 10. SW_15 was also considered as a pre-processing approach in the more difficult context of unknown $K$.

We report the average adjusted Rand index (ARI) to quantify the level of agreement between the fit cluster labels and the true cluster labels after adjusting for agreement due to chance. The empirical distribution of estimated numbers of clusters is also reported when assumed unknown.

We used the rand.index function in the fossil package [46] in R and the adjustedRandIndex function in the mclust package [42] in R.

4.2.2. Simulation 2: Moderate Dimension with Correlation from Radiomics

This simulation is intended to be most directly comparable to the radiomics panel described in Section 2, and so correlation structure is sampled directly from the data and the number of noise features is expanded to reach the same dimensionality as the real data. We otherwise adopt the moderate dimension, consolidated signal simulation structure from the first simulation with a stronger cluster signal.

Based on earlier simulations, we consider the Shapiro-Wilk filter after PCA and do not consider PCA with standard dimension reduction. We also consider application of the decorrelation filter, as correlation in this simulation is extremely high.

Data sets were simulated from a 6 component GMM with 50 observations per cluster. As in the moderate dimension, consolidated signal simulation, 15 variables were used to encode cluster structure, now with cluster centers a distance of 5 from the origin rather than 2. Similarly, an additional 50, 200, or 550 pure noise variables were also simulated. Unlike in the previous simulation, correlation structure was not compound but rather sampled randomly from the observed correlation structure of the radiomics panel, and all variables were correlated. For each of the three data settings (50, 200, or 550 noise variables), 100 data sets were simulated.

As in the previous simulation, we first investigated PCA and the Shapiro-Wilk filter alone for each of the 3 case scenarios and report the same metrics.

Then we investigated clustering performance. We considered 4 pre-processing types: raw, decor, SW, and decorSW; see Table 2. For each simulated data set, the 4 pre-processing steps were applied. To each pre-processed data set, 5 clustering methods (VarSelLCM, SKmeans with BCS gap, Kmeans with silhouette, HD with all models, and FisherEM with elastic net) were applied with the number of clusters estimated to be no more than 10.

We report number of failed fittings, estimated number of clusters, and ARI of fit cluster labels as compared to true simulated cluster labels.

4.2.3. Simulation 3: High Dimension with Diffuse Signal

This simulation is intended to be comparable to metabolomics data more broadly. As such, simulated data are comprised of feature blocks with compound correlation as well as a large block with empirical correlation sampled from the COPDGene data described in Section 2. Unlike previous simulations, we simulate a low-strength diffuse clustering signal in the data.

As the larger dimension of the problem and diffuse nature of the signal had the potential to impact pre-processing decisions, we continued to consider both PCA with standard dimension reduction and the Shapiro-Wilk filter. The correlation level was insufficient to warrant the decorrelation filter.

Data sets were simulated from a 2 component GMM with 150 observations per cluster ($n=300$). Features were comprised of 5 blocks of 50 features each with compound correlation 0.5 and one block of 500 features with correlation randomly sampled from the empirical correlation of the COPDGene data described in Section 2 ($p=750$). The six blocks were simulated to be independent with unit variance. 75 randomly selected variables out of the 750 were used to encode cluster structure, with the two cluster centers separated by a distance of $\mu$ for each feature.

We considered the three settings, $\mu =1$, 1.25, 1.5. The lowest was chosen from candidate values $\mu =0.1,0.2,0.3,\dots ,2.0$ to be minimal such that Kmeans applied to the 75 relevant features with number of clusters known to be 2 resulted in cluster labels with ARI> 0.9 when compared to true labels in 50 out of 50 preliminary simulations (data not shown). The highest considered signal strength ($\mu =1.5$) was chosen to be minimal such that Kmeans applied to all 750 features with number of clusters known to be 2 resulted in cluster labels with ARI> 0.9 when compared to true labels in at least 25 out of 50 preliminary simulations (data not shown).

The separation between the two clusters induced by these signal strengths may be seen in Supplementary Figure 21. A heatmap of the correlation in a single simulated data set with lowest signal strength may be seen in Supplementary Figure 22.

As in the previous simulation, we first investigated PCA and the Shapiro-Wilk filter alone for each of the 3 case scenarios and report the same metrics.

Then we investigated clustering performance. We considered 4 pre-processing types: raw, PCs_all, PCs_80, and SW (see Table 2); and 5 clustering methods: VarSelLCM, SKmeans with BCS gap, Kmeans with silhouette, HD with all models, and FisherEM with elastic net. For each combination of data setting, pre-processing type, and method, 100 data sets were simulated and clustered with the number of clusters estimated to be no more than 4.

We report number of failed fittings, estimated number of clusters, and ARI of fit cluster labels as compared to true simulated cluster labels.

4.3. Real Data

We analyzed each of the radiomics panel, the Golub data, and the TCGA data (Section 2) for clusters in 40 ways. The 40 analysis approaches were comprised of combinations of 5 clustering methods (VarSelLCM, SK-means with BCS-based gap, K-means with average silhouette, HD classifier with all models, and FisherEM with elastic net), 4 pre-processing methods (raw, decor, decorSW, and SW), and 2 treatments of the number of clusters $K$ (assumed known or unknown). For the labeled Golub and TCGA data, the number of clusters was either fixed at the true number of classes or estimated to be no greater than twice the true number. For the unlabeled radiomics panel, the number of clusters was either arbitrarily fixed at 5 or estimated to be no greater than 10.

For the labeled Golub and TCGA data, the ARI of successfully fit cluster labels compared with true class labels is reported for each analysis. For each data set, the ARI of pairs of successfully fit cluster labels from each analysis is also reported. Additionally, we report failed fittings and estimated number of clusters for the 20 analyses which made such estimates (Table 3).

Table 3.

Number of clusters estimated by each method with each pre-processing step when applied to the sarcoidosis radiomics panel, the Golub data, and the TCGA data in turn. An entry of - indicates failure to fit, i.e. that analysis code completed without error but did not produce results. An entry of * indicates analysis was prematurely terminated due to run time exceeding 10 days. An entry of NA indicates analysis was not run due to comparable or strictly faster analyses being prematurely terminated.

	Radiomics				Golub				TCGA
	Pre-Processing Step
Method	raw	decor	SW	decorSW	raw	decor	SW	decorSW	raw	decor	SW	decorSW
K-means	2	2	2	2	2	2	4	2	7	7	7	7
VarSelLCM	10	10	1	2	4	4	1	1	10	10	1	1
SK-means	2	2	8	8	2	2	3	4	*	*	*	NA
HDclassif	5	5	7	8	1	1	4	4	2	2	10	10
FisherEM	7	5	2	9	-	-	2	2	NA	NA	NA	NA

5. Results

5.1. Results: Numerical Experiments

5.1.1. Simulation 1

Across simulation contexts, PCA consolidated signal well into three linearly independent signals (PCs 2, 3, and 4) according to r-square values consistently well above 0; see Figure 5. The first PC was noise once correlation was introduced, and variance of this PC began to dominate as correlation increased. The Shapiro-Wilk filter correspondingly performed well at distinguishing PCs with cluster signal from those without. For the second and third signaling PCs, PC3 and PC4, signal strength as measured by r-square values and significance as measured by Shapiro-Wilk p-values appeared to increase with correlation. Significance decayed for the top PC with signal (PC2) as correlation increased, corresponding with a decrease in signal as measured by r-square values. Results for data with 50 and 200 noise variables were similar.

For the remainder of this section, we discuss clustering results and refer the reader to Figure 6 summarizing successfully fit analyses for $K$ assumed unknown. Successfully fit analyses for $K$ assumed known are summarized in Supplementary Figure 17. All methods succeeded in at least 92 simulations out of 100 in each context. Estimates of the number of clusters as well as the number of clusters and ARI jointly are considered in Supplementary Figures 18 and 19.

Figure 6.

Simulation 1: Boxplots with overlaid jitterplots of ARI from completed simulations where $K$ is unknown stratified by clustering method (see Table 1), number of noise variables (50 or 200), correlation level ($\rho =0$, 0.1, 0.25, 0.5), and pre-processing step (see Table 2).

We discuss first all those ARI results with K assumed known, then results with K assumed unknown not including the Shapiro-Wilk filter, and then the Shapiro-Wilk filter results.

K Known:

The increase in the dimension of noise from 50 to 200 degraded performance of all methods in all cases for all pre-processing steps. For most methods for most levels of correlation, performance as measured by ARI was at or below 0.25 when the noise dimension was 200. Most methods were able to adequately recover cluster structure when features were independent ($\rho =0$) with typically rapid degradation in performance as correlation increased. VarSelLCM and HD_all exhibited performance improvements as correlation increased when applied to PCs rather than raw data, especially when PCA was paired with dimension reduction. PCA did not impact the performance of Kmeans, and greatly degraded performance of SKmeans in correlated contexts. PCA appeared to improve performance of FisherEM with no apparent impact of dimension reduction. Overall, VarSelLCM clearly outperformed other methods in high correlation contexts and was best able to take advantage of PCA when paired with significant dimension reduction.

For FisherEM, performance differed only marginally between penalty methods, with elastic net performing slightly better. Only elastic net was considered in later simulations and analyses. HD classifier performed substantially better when considering all candidate models than when considering only the default most general candidate. Only HD_all was considered in later simulations and analyses.

K Unknown:

Results when $K$ was estimated followed similar patterns to when $K$ was assumed known but were generally worse overall. Besides VarSelLCM, most methods failed to recover any notable cluster structure (ARI<0.15) for moderate and high correlation cases with high dimension noise regardless of processing step. VarSelLCM clearly outperformed other methods in high correlation and in high dimension contexts, but optimal processing method shifted with correlation.

For SK-means, the BCS-based gap statistic typically outperformed the average silhouette statistic according to ARI, though advantage lessened as correlation increased. For K-means, the average silhouette statistic typically outperformed the traditional gap statistic according to ARI. In later simulations and analyses, only the BCS-based gap for SKmeans and the average silhouette statistic for Kmeans were considered.

SW Filter:

Use of the Shapiro-Wilk filter greatly improved performance of all methods except VarSelLCM in high-dimension or high correlation contexts. All methods achieved an average ARI between about 0.25 and 0.8 in the highest correlation high-dimension context when paired with the SW filter, as compared to no method achieving an average ARI of even 0.1 in the same context when applied to raw data. Most methods achieved at least modest performance in the majority of correlated contexts (ARI>0.5).

For VarSelLCM, the clustering performance was not uniformly improved by use of the SW filter for any given case scenario relative to the optimal pre-processing step of those considered. However, the SW filter was the only pre-processing step which allowed VarSelLCM to perform adequately in all contexts.

5.1.2. Simulation 2

Across simulation contexts, PCA consolidated signal well into PCs with relatively high variance for lower dimension contexts; see Figure 7. However, signal strength appeared to become more variable and decay with increased noise dimension as r-square values decreased overall and inner quartile range increased for signal PCs. Curiously, the lowest variance PC also appeared to develop a consistent, very low strength cluster signal.

Figure 7.

Simulation 2: 500 sets of PCs_all data for each level of noise (50, 200, 550 noise variables) from the radiomics correlation simulation were simulated. For each PC, the quartiles from the 500 values of the PC variance, the R-square value obtained from regressing the PC onto the true cluster labels, and the negative log10 of the Shapiro-Wilk p-value obtained from the PC are plotted, stratified by noise level. A line plot is used for visual ease.

For clustering analyses, failure to fit was common in this simulation for SK-means and FisherEM, especially on raw data; see Supplementary Figure 20, counts of NA. FisherEM in particular struggled to fit raw data with 200 noise variables, failing for all but 2 simulated data sets. VarSelLCM and K-means succeeded in all contexts, while HD classifier failed to fit only one data set in one context (decorSW, 50 noise).

On raw and decor data, only VarSelLCM consistently recovered notable cluster structure and only for 50 noise variables (ARI≈ 0.25); see Figure 8. HD_all is the only other method which occasionally recovered notable cluster structure on raw or decor data, though performance was inconsistent, poor, and limited to lower dimensions. No method performed adequately with 550 noise variables regardless of pre-processing (average ARI < 0.15).

Figure 8.

Simulation 2: Boxplots with overlaid jitterplots of adjusted Rand index from completed simulations stratified by clustering method (see Table 1), pre-processed data type (see Table 2), and number of noise variables (50, 200, or 550).

With 50 or 200 noise variables, methods performed best on either SW or decorSW data. Pre-application of the decorrelation filter aided FisherEM and VarSelLCM for 200 and 550 noise variables, but application of that filter for 50 noise variables or other methods typically reduced or did not notably impact performance.

5.1.3. Simulation 3

Across simulation contexts, PCA consolidated signal poorly according to r-square values; see Figure 9. SW p-value distributions were generally similar for PCs regardless of r-square values. The only PC consistently identified as containing signal by the SW filter was the lowest variance PC, which displayed a very low strength cluster signal according to r-square values. We note that as overall signal strength $\mu$ increased, the individual PCs with the strongest signal according to r-square values appeared to shift bimodally from a range of moderately high variance PCs to the highest variance PCs.

Figure 9.

Simulation 3: 500 sets of PCs_all data for each signal strength ($\mu =1$, 1,.25, 1.5) from the metabolomics simulation were simulated. For each PC, the quartiles from the 500 values of the PC variance, the R-square value obtained from regressing the PC onto the true cluster labels, and the negative log10 of the Shapiro-Wilk p-value obtained from the PC are plotted, stratified by signal strength. A line plot is used for visual ease.

For clustering analyses, failure to fit occurred for SKmeans and FisherEM, with FisherEM succeeding in at least 89 out of 100 simulations in each context and SKmeans succeeding in at least 99 out of 100 simulations in each context; see Supplementary Figure 23. Other methods succeeded in all contexts without fail.

All methods consistently performed worst on SW_15 data; see Figure 10. VarSelLCM, SKmeans, and to a lesser extent FisherEM (for $\mu =1.5$ only), achieved optimal performance as measured by ARI on Raw data. PCA with standard dimension reduction appeared to improve FisherEM performance over application to Raw data in the moderate signal context ($\mu =1.25$). Kmeans achieved similar performance regardless of pre-processing method besides SW_15. HDclassif performed best on PCs_80 data. All clustering methods except VarSelLCM performed well (ARI typically > 0.8) on data with maximal signal strength ($\mu =1.5$) for some pre-processing method. Only VarSelLCM and HDclassif produced fits with ARI above 0.5 in at least 25% of simulations when signal strength was moderate ($\mu =1.25$). No clustering method consistently produced fits with ARI substantially above 0 when signal strength was minimal ($\mu =1$).

Figure 10.

Simulation 3: Boxplots with overlaid jitterplots of adjusted Rand index from simulations stratified by clustering method (see Table 1), pre-processed data type (see Table 2), and signal strength ($\mu =1$, 1.25, 1.5).

5.2. Results: Real Data

5.2.1. Radiomics Panel

Only one fitting failed: VarSelLCM failed to fit to SW data with the number of clusters fixed at 5. The corresponding analysis when the number of clusters was estimated found only 1 cluster. All remaining analyses succeeded in fitting 5 clusters or estimating and fitting between 2 and 10 clusters; see Table 3.

Using ARI to compare analyses, we found fit cluster labels to be typically highly sensitive to all analysis decisions; see Figure 11. In particular, 87% of analysis pairs resulted in a pairwise ARI less than 0.5 and 46% of analysis pairs resulted in a pairwise ARI less than 0.25. Among the clustering methods, Kmeans was the most robust to pre-processing methods. SKmeans and FisherEM were consistent with Kmeans fittings for some choices of pre-processing and treatment of $K$.

Figure 11.

Heatmap of pairwise ARI of fit clusters from each of 39 successfully completed analyses of the radiomics panel, stratified by clustering method (see Table 1), pre-processing method (see Table 2), and treatment of number of clusters $K$ (fixed, estimated). Failed fittings are left blank.

Among analyses with a fixed number of clusters, consistency was lesser but wider spread than among analyses which estimated the number of clusters. Analyses were generally sensitive to treatment of $K$.

5.2.2. Golub

Six Golub fittings failed: FisherEM failed to fit raw and decor data whether the number of clusters was fixed or estimated, and VarSelLCM failed to fit SW and decorSW data when the number of clusters was fixed. Another four fittings estimating the number of clusters found no clustering: HDclassif when applied to raw and decor data and VarSelLCM when applied to SW and decorSW data.

Comparing analyses to known labels, the highest ARI was achieved by VarSelLCM fit to decor data with the number of clusters fixed (ARI=0.70), followed by the same analysis on raw data (ARI=0.53); see Figure 12. VarSelLCM applied to raw data with the number of clusters estimated produced an ARI of 0.33, and all other analyses produced ARI lower than 0.15. Where analyses could be compared with and without application of the Shapiro-Wilk filter, ARI values were generally sufficiently low to preclude practically meaningful differences, though the filter did typically result in lower ARI values.

Figure 12.

Heatmap of ARI of fit clusters compared to true class label from each of 34 successfully completed analyses of the Golub data, stratified by clustering method (see Table 1), pre-processing method (see Table 2), and treatment of number of clusters $K$ (fixed, estimated). Failed fittings are left blank.

Very similar and even identical fittings were common. Analyses involving K-means or SK-means applied to raw or decor data with the number of clusters estimated or fixed produced identical results (ARI=1); see Figure 13. These methods and FisherEM in application to SW and decorSW data displayed some similar lack of sensitivity to estimating or fixing the number of clusters, but were more sensitive to cluster method and the decorrelation filter. When HDclassif was applied with the number of clusters fixed, application to decorSW data produced an identical fit to FisherEM applied to SW data with or without the number of clusters fixed, and application to SW data produce a similar or identical fit to Kmeans applied to SW data with or without the number of clusters fixed.

Figure 13.

Heatmap of pairwise ARI of fit clusters from each of 34 successfully completed analyses of the Golub data, stratified by clustering method (see Table 1), pre-processing method (see Table 2), and treatment of number of clusters $K$ (fixed, estimated). Failed fittings are left blank.

Fit cluster labels were otherwise highly sensitive to analysis decisions; see Figure 13. In particular, 88% of analysis pairs resulted in a pairwise ARI less than 0.5 and 74% of analysis pairs resulted in a pairwise ARI less than 0.25. Analyses were typically less sensitive to application of the decorrelation filter than application of the Shapiro-Wilk filter.

5.2.3. TCGA

Only 26 of 40 analyses were fit successfully. For the 40 analyses, more than half of analysis pairs include at least one of the 14 analyses that either failed to fit, was prematurely terminated due to excessive runtime, or was not run due to comparable or strictly faster analyses being prematurely terminated.

Of the successfully fit analyses, all methods produced ARI above 0.5 when compared with the gold standard class labels with the number of clusters fixed; see Figure 14. Results were mixed when the number of clusters was estimated. Overall, the decorrelation filter did not appear to substantially impact gold standard ARI, while the impact of the Shapiro-Wilk filter was method specific. Kmeans produced similar ARI (0.73-0.77) regardless of data pre-processing and estimation or fixing of number of clusters. HDclassif produced similar ARI (0.76-0.83) regardless of data pre-processing when the number of clusters was fixed. When the number of clusters was estimated, HDclassif produced much lower ARI (0.18, 0.14) without the SW filter than with the SW filter (0.61, 0.68). SKmeans produced higher ARI without the SW filter (0.80, 0.82) than with the SW filter (0.58, 0.54) when the number of clusters was fixed. On raw and decor data, VarSelLCM produced similar ARI values when the number of clusters was fixed (0.62, 0.60) and similar but slightly lower values when the number of clusters was estimated (0.50, 0.49).

Figure 14.

Heatmap of ARI of fit clusters compared to true class label from each of 26 successfully completed analyses of the TCGA data, stratified by clustering method (see Table 1), pre-processing method (see Table 2), and treatment of number of clusters $K$ (fixed, estimated). Failed fittings as well as prematurely terminated or not run analyses are left blank.

As half of comparisons could not be made, remaining results for sensitivity are reported in Supplementary Section 7.1, along with details of analysis terminations and failures.

6. Discussion

The primary purpose of this paper was to investigate the impacts of correlation, dimension, and methodological approach on clustering performance for a variety of PCA-based pre-processing methods and for Gaussian Mixture Models (GMMs) and related clustering methods. We identified the variance-as-relevance assumption made by GMM clustering methods as related to poor clustering performance. We proposed two pre-processing approaches which may be used to counteract the assumption. In Section 4.2, we tested these ideas on simulated data.

In Simulations 1 and 2, selecting PCs for clustering using the Shapiro-Wilk filter was associated with significant improvement in clustering performance on correlated data. In the high dimension contexts of Simulation 3, the Shapiro-Wilk filter consistently produced the worst clustering results for all clustering methods, with clustering of raw data being typically best. In combination with the degradation of performance of the Shapiro-Wilk filter in Simulation 2 as dimension increased, the Shapiro-Wilk filter and related approaches may perform well in countering the variance-as-relevance assumption in low and moderate dimension problems but not in high dimension contexts.

Poor performance of the filter in high dimension contexts appears related to PCA smoothing signal away in higher dimensions, rendering moot the application of any filter to PCs. Supervised investigation of cluster signal in PCs in data from Simulation 2 suggested that for a given level of signal in a data set, increasing the dimension of the data by appending noisy features results in a decrease in the cluster signal found in PCs marginally; see Figure 7, middle row.

These analyses build empirically upon previous investigations into theoretical limits on structure recovery from GMM data and from more general data using PCA and spectral methods; classically [37] and [3], and more recently [10], [29], and [4]. Such investigations typically center on ‘spiked’ models for which PCs with large variances are precisely those containing signal of interest, and hence meet the variance-as-relevance assumption. Simulations in our work focus on clustered data having large variance noise PCs, which is expected in unsupervised, high dimension contexts and which violate the variance-as-relevance assumption.

In future research, it might be of interest to explore the underlying connections between this smoothing behavior of PCA in higher dimension, lower signal, correlated contexts and the body of theoretical literature exploring spiked eigenvalue models, which focuses on characterizing signal strength thresholds for identifying structure in clustering contexts and via PCA. In particular, high variance noise signals might be considered in the framework of spiked eigenvalue models as an obstacle to relevant signal detection, as high variance noise may be found theoretically in correlated GMMs and practically in high dimension real data but is typically not considered in the current theory. This amounts to investigating violations of the variance-as-relevance assumption.

Performance of the SW filter on real data was variable. A non-parametric test for multiple components might be used in place of the Shapiro-Wilk test for use on real data to improve performance. We leave identification of an optimal discriminative filter for real data applications to future investigations.

Performance was mixed for the decorrelation filter in both simulated and real data contexts. The decorrelation filter may remove not only redundant noise signals but redundant cluster signals, and so the value of the decorrelation filter depends partly on the balance of irrelevant and relevant redundancy in the data, and the sensitivity of the paired clustering method to redundancy and dimension. Simulation 2 in particular is suggestive of its potential value in moderate and high dimension contexts. More study is needed.

In Section 5.2, we assessed the sensitivity of clustering analyses to analysis approach on three real data sets. While patterns of consistency varied across data sets, we observed substantial variability in clustering outcome as measured by ARI across all data sets. While lack of identifiable signal may have hampered gold standard performance, we note that that does not preclude consistency across analyses. Lack of identifiable signal might consistently be indicated through estimation of 1 or 2 clusters by all methods, but this was not the case.

While the sensitivity analyses presented here may be viewed as quite specific to the methods or contexts considered, these methods were a priori reasonable approaches for applied analyses. Results suggest more broadly that clustering analysis outcomes may be highly sensitive to and reflective of analysis decisions regarding pre-processing and clustering methodology. We strongly caution against over-interpretation of such analyses in application without robust sensitivity analyses or strong contextual support for identified structures. More generally, analysis decisions such as selection of clustering and data pre-processing methodologies should be carefully considered and justified in context given their apparent influence on conclusions.

While it may be unsatisfying to provide no firm recommendation of what ought to be done in practice, this paper points to issues that should be discussed as potential limitations when clustering in moderate to high dimensions with clustering. We also highlight important directions for methodologic development towards counteracting and avoiding the variance-as-relevance assumption, and identifying limitations in application of PCA in absence of this assumption.