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Understanding Clustering Significance in HDLSS Data through Statistical Analysis

This study explores the rigorous analysis of clustering significance within High-Dimensional Low-Sample Size (HDLSS) data. Focusing on the implications of outliers, we delve into the SigClust method to identify when clusters can be claimed to be statistically significant. The application of the 2-means cluster index and Gaussian null distribution helps quantify cluster presence. We address common challenges in estimating covariance matrices and background noise, providing insights into determining true cluster existence amidst complex data distributions.

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Understanding Clustering Significance in HDLSS Data through Statistical Analysis

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  1. Object Orie’d Data Analysis, Last Time • Clustering • Quantify with Cluster Index • Simple 1-d examples • Local mininizers • Impact of outliers • SigClust • When are clusters really there? • Gaussian Null Distribution • Which Gaussian for HDLSS settings?

  2. 2-means Clustering

  3. 2-means Clustering

  4. 2-means Clustering

  5. 2-means Clustering

  6. SigClust • Statistical Significance of Clusters • in HDLSS Data • When is a cluster “really there”? From Liu, et al. (2007)

  7. Interesting Statistical Problem For HDLSS data: • When clusters seem to appear • E.g. found by clustering method • How do we know they are really there? • Question asked by Neil Hayes • Define appropriate statistical significance? • Can we calculate it?

  8. Simple Gaussian Example • Clearly only 1 Cluster in this Example • But Extreme Relabelling looks different • Extreme T-stat strongly significant • Indicates 2 clusters in data

  9. StatisticalSignificance of Clusters Basis of SigClust Approach: • What defines: A Cluster? • A Gaussian distribution (Sarle & Kou 1993) • So define SigClust test based on: • 2-means cluster index (measure) as statistic • Gaussian null distribution • Currently compute by simulation • Possible to do this analytically???

  10. SigClust Statistic – 2-Means Cluster Index Measure of non-Gaussianity: • 2-means Cluster Index: Class Index Sets Class Means “Within Class Var’n” / “Total Var’n”

  11. SigClust Gaussian null distribut’n Estimated Mean (of Gaussian dist’n)? • 1st Key Idea: Can ignore this • By appealing to shift invariance of CI • When data are (rigidly) shifted • CI remains the same • So enough to simulate with mean 0 • Other uses of invariance ideas?

  12. SigClust Gaussian null distribut’n Challenge: how to estimate cov. Matrix? • Number of parameters: • E.g. Perou 500 data: Dimension so But Sample Size • Impossible in HDLSS settings???? Way around this problem?

  13. SigClust Gaussian null distribut’n 2nd Key Idea: Mod Out Rotations • Replace full Cov. by diagonal matrix • As done in PCA eigen-analysis • But then “not like data”??? • OK, since k-means clustering (i.e. CI) is rotation invariant (assuming e.g. Euclidean Distance)

  14. SigClust Gaussian null distribut’n 2nd Key Idea: Mod Out Rotations • Only need to estimate diagonal matrix • But still have HDLSS problems? • E.g. Perou 500 data: Dimension Sample Size • Still need to estimate param’s

  15. SigClust Gaussian null distribut’n 3rd Key Idea: Factor Analysis Model • Model Covariance as: Biology + Noise Where • is “fairly low dimensional” • is estimated from background noise

  16. SigClust Gaussian null distribut’n Estimation of Background Noise : • Reasonable model (for each gene): Expression = Signal + Noise • “noise” is roughly Gaussian • “noise” terms essentially independent (across genes)

  17. SigClust Gaussian null distribut’n Estimation of Background Noise : Model OK, since data come from light intensities at colored spots

  18. SigClust Gaussian null distribut’n Estimation of Background Noise : • For all expression values (as numbers) • Use robust estimate of scale • Median Absolute Deviation (MAD) (from the median) • Rescale to put on same scale as s. d.:

  19. SigClust Estimation of Background Noise

  20. Q-Q plots An aside: Fitting probability distributions to data • Does Gaussian distribution “fit”??? • If not, why not? • Fit in some part of the distribution? (e.g. in the middle only?)

  21. Q-Q plots Approaches to: Fitting probability distributions to data • Histograms • Kernel Density Estimates Drawbacks: often not best view (for determining goodness of fit)

  22. Q-Q plots Simple Toy Example, non-Gaussian!

  23. Q-Q plots Simple Toy Example, non-Gaussian(?)

  24. Q-Q plots Simple Toy Example, Gaussian

  25. Q-Q plots Simple Toy Example, Gaussian?

  26. Q-Q plots Notes: • Bimodal  see non-Gaussian with histo • Other cases: hard to see • Conclude: Histogram poor at assessing Gauss’ity Kernel density estimate any better?

  27. Q-Q plots Kernel Density Estimate, non-Gaussian!

  28. Q-Q plots Kernel Density Estimate, Gaussian

  29. Q-Q plots KDE (undersmoothed), Gaussian

  30. Q-Q plots KDE (oversmoothed), Gaussian

  31. Q-Q plots Kernel Density Estimate, Gaussian

  32. Q-Q plots Kernel Density Estimate, Gaussian?

  33. Q-Q plots Histogram poor at assessing Gauss’ity Kernel density estimate any better? • Unfortunately doesn’t seem to be • Really need a better viewpoint Interesting to compare to: • Gaussian Distribution • Fit by Maximum Likelihood (avg. & s.d.)

  34. Q-Q plots KDE vs. Max. Lik. Gaussian Fit, Gaussian?

  35. Q-Q plots KDE vs. Max. Lik. Gaussian Fit, Gaussian? • Looks OK? • Many might think so… • Strange feature: • Peak higher than Gaussian fit • Usually lower, due to smoothing bias • Suggests non-Gaussian? • Dare to conclude non-Gaussian?

  36. Q-Q plots KDE vs. Max. Lik. Gaussian Fit, Gaussian

  37. Q-Q plots KDE vs. Max. Lik. Gaussian Fit, Gaussian • Substantially more noise • Because of smaller sample size • n is only1000 … • Peak is lower than Gaussian fit • Consistent with Gaussianity • Weak view for assessing Gaussianity

  38. Q-Q plots KDE vs. Max. Lik. Gauss., non-Gaussian(?)

  39. Q-Q plots KDE vs. Max. Lik. Gau’n Fit, non-Gaussian(?) • Still have large noise • But peak clearly way too high • Seems can conclude non-Gaussian???

  40. Q-Q plots • Conclusion: KDE poor for assessing Gaussianity How about a SiZer approach?

  41. Q-Q plots SiZer Analysis, non-Gaussian(?)

  42. Q-Q plots SiZer Analysis, non-Gaussian(?) • Can only find one mode • Consistent with Gaussianity • But no guarantee • Multi-modal  non-Gaussianity • But converse is not true • SiZer is good at finding multi-modality • SiZer is poor at checking Gaussianity

  43. Q-Q plots Standard approach to checking Gaussianity • QQ – plots Background: Graphical Goodness of Fit Fisher (1983)

  44. Q-Q plots Background: Graphical Goodness of Fit Basis:    Cumulative Distribution Function  (CDF) Probability quantile notation: for "probability” and "quantile" 

  45. Q-Q plots Probability quantile notation: for "probability” and "quantile“ Thus is called the quantile function

  46. Q-Q plots Two types of CDF: • Theoretical • Empirical, based on data

  47. Q-Q plots Direct Visualizations: • Empirical CDF plot: plot vs. grid of (sorted data) values • Quantile plot (inverse): plot vs.

  48. Q-Q plots Comparison Visualizations: (compare a theoretical with an empirical) • P-P plot: plot vs. for a grid of values • Q-Q plot: plot vs. for a grid of values

  49. Q-Q plots Illustrative graphic (toy data set):

  50. Q-Q plots Empirical Quantiles (sorted data points)

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