The maize dataset, consisting of nine Affymetrix microarrays, was generated to investigate the gene transcription activity in three maize root tissues with three replicates for each tissue: the proximal meristem (PM), the quiescent center (QC) and the root cap (RC) 23. 2092 significantly differentially expressed genes have been identified and categorized into 6 classes of expression patterns 23. Here we used these genes to illustrate the property of the proposed transformations with comparison to the traditional PCA.

Firstly, we applied the transformation employed in TransChisq to the data. Figures 1(a)–(c) plot the expression profiles of the genes in this new space. The blue and red genes are from the two dominant classes (RC up- or down-regulated genes account for 94% of all genes) and the other four colors (orange, green, pink, light blue) correspond to the other four small classes (up- or down-regulated genes in QC or PM account for 6% of all genes). The three plots show that the six classes can be recognized explicitly in any of the three subspaces of dimension 2.

We then applied the transformation suggested by a parametric covariance matrix to the same data (see Methods). Figures 1(d)–(f) plot the expression profiles of the genes in this new space. We can see that the second and the third component separate all six classes in Figure 1(e) correctly. The description of the six class separating regions, whose centers are the dotted lines in Figure 1(e), is in Table 1 (e.g., the genes around the line PC2 = 3 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabiodaZaWcbeaaaaa@2DBB@·PC3 < 0 are expected to be PM up-regulated). A convenient common property of this transformation, and the one in TransChisq, is that the information carried by each component is explicit, and hence the region in the new space corresponding to each class can be clearly determined.

For comparison, we performed a traditional PCA analysis to the same data. Figures 1(g)–(i) plot the expression profiles of the genes in the principal component space. We can see that the direct application of the PCA can separate the two dominating expression patterns. But it fails to recognize the other patterns, even when exhausting all principal components. The poor performance of PCA could be attributed to the use of empirical sample covariance matrix in determining the principal components. In the maize dataset, about 94% genes are RC up- or down-regulated genes, which cause most of the variance. The principal components, determined by this sample covariance matrix thus largely capture the two dominating clusters, yet miss the meaningful class information for the other four small groups.

This example demonstrates the advantage of the proposed new data transformations over the traditional PCA in keeping class information intact.

We applied TransChisq to a simulation dataset to evaluate its performance. For comparison purposes, other modified K-means algorithms, i.e. PCAChisq, PoissonC, PearsonC, and Eucli were also applied to the same dataset.

The simulation dataset consists of 46 vectors of dimension 5 and the components are independently generated from different Normal distributions. The mean (μ) and variance (σ²) of the Normal distributions are constrained by σ² = 3μ and described in Table 2. Based on the Normal distributions they are generated from, the 46 vectors are put into six groups, i.e., A, B, C, D, E, and F, whose sizes are 3, 6, 6, 9, 7, and 15 respectively. The motivation and guideline on choosing the various parameters related to this simulation datasets are presented in Additional file 1. Genes with a similar expression shape are considered to be in the same group. Although the expression magnitude in the dataset is not a critical factor for determining the gene clusters, its information is useful and should be taken into account when comparing the profile shapes.

The clustering results from different methods are shown in Figure 2. The horizontal axis represents the index of the 46 genes that belong to six groups (designated A, B, C, D, E and F, and marked at the top of the figure). The vertical axis represents the index of the cluster to which each gene has been assigned by a particular algorithm. Only TransChisq correctly categorized the genes into six groups. PCAChisq, PoissonC, and PearsonC mixed up group A and group B. Eucli clustered genes mainly by the magnitude of the gene expression values rather than the changes of the profile shapes. To reduce the effects from the magnitude, we further applied Eucli to the rescaled data. The rescaling was performed in a way so that the sum of the components within each vector was set the same. The clustering result of Eucli on the rescaled data (Figure 2(f)) is better, but not perfect.

We performed an additional 100 replications of the above simulation. TransChisq, PCAChisq and PoissonC correctly clustered 75, 37 and 43 of the 100 replicate simulation datasets, while PearsonC, Eucli and Eucli on rescaled data cannot generate correct clusters. We also tried PCAChisq on different combinations of principal components to optimize the clustering results. These different combinations, however, are not helpful to identify all the six groups.

This study evaluates the performance of TransChisq on the normally distributed data with Poisson-like property: variance increases with mean. The success of this application sheds a light on applying TransChisq to a microarray dataset in addition to the SAGE data.

For further validation we applied TransChisq, PCAChisq, PoissonC, PearsonC, Eucli and SRC (the K-means algorithm using Spearman Rank correlation as the similarity measure) to a set of mouse retinal SAGE libraries. The raw mouse retinal data consists of 10 SAGE libraries (38818 unique tags with tag counts ≥ 2) from developing retina taken at 2-day intervals. The samples range from embryonic, to postnatal, to adult 21. Among the 38818 tags, 1467 tags that have counts greater than or equal to 20 in at least one of the 10 libraries were selected. The purpose of this selection is to exclude the genes with uniform low expression. To be more effective in comparing the clustering algorithms, a subset of 153 SAGE tags with known biological functions were selected. These 153 tags fall into 5 functional groups: 125 of these genes are developmental genes that can be further categorized into four classes by their activities at different developmental stages; the other 28 genes are not relevant to the mouse retina development (see Table 3). The average expression profile for each of the five clusters is shown in Figure 3.

TransChisq, PCAChisq, PoissonC, PearsonC, Eucli and SRC were applied to group these 153 SAGE tags into five clusters. Here we assumed that the number of the clusters, K, is known. A study to evaluate the performance of different measures in determining K when it is unknown can be found in a later section of this paper. The clustering results showed that TransChisq and PCAChisq outperform others (Table 4): 12, 12, 22, 26 and 38 of the 153 tags are incorrectly clustered by TransChisq, PCAChisq, PoissonC, PearsonC and Eucli on rescaled data respectively. For the results from Eucli on original data, as the correspondence between the predicted clusters and the true clusters is unclear, we cannot report the number of incorrectly clustered tags. We also evaluated the quality of the clustering results against an external criterion, the adjusted Rand Index 26. The adjusted Rand Index assesses the degree of agreement between two partitions of the same set of objects. We compared the clustering results from each algorithm with the true categorizations, and calculated the adjusted Rand Index accordingly. The adjusted Rand Index varies between 1 (when the two partitions are identical) and 0 (when the partitions or the resulted clusters are random). A higher adjusted Rand Index represents the higher correspondence between the two partitions. From Table 4, we can see that the adjusted Rand Index results confirm that TransChisq and PCAChisq perform similarly and have clear advantages over other methods.

To further demonstrate how effective TransChisq is in clustering genes with characterized patterns in a microarray analysis, we applied TransChisq to a microarray yeast sporulation dataset 22. Chu et al. measured gene expressions in the budding yeast Saccharomyces cerevisiae at seven time points during sporulation using spotted microarrays, and identified seven distinct temporal patterns of induction 22. 39 representative genes were used to define the model expression profile for each pattern. Based on their properties, the seven patterns are designated as Metabolic, Early I, Early II, Early-Mid, Middle, Mid-Late and Late. The average expression profiles for these seven patterns are presented in Figure 4. The genes in Early I, Early II, Middle, Mid-Late and Late initiates induction of expression at 0.5 h, 2 h, 5 h, 7 h and 9 h, respectively, and sustains expression through the rest of the time course. The expression of metabolic genes is also induced at 0.5 h as in Early I, but decays afterwards. The expression of genes in Early-Mid is induced not only at the 0.5 h and 2 h as in Early genes, but also at 5 h and 7 h, as in the Middle and Mid-Late genes. This data structure made it difficult to separate the Early-Mid genes from others. The direct clustering analyses using PearsonC or Eucli were not successful.

Prior to analyzing the data we substituted the expression ratios that were below zero with zeros as in Figure 5(a). This truncation of negative values simplifies the expression patterns of the 39 representative genes with the key properties in each pattern being intact. The clustering results are summarized in Table 5. We can see that TransChisq outperforms other methods: 3, 7, 8, 13, 14 and 17 of the 39 genes are incorrectly clustered by TransChisq, PoissonC, Eucli, PearsonC, PCAChisq and Eucli on rescaled data respectively. TransChisq also shows the best adjusted Rand Index. It is interesting to see that the performance of Eucli on rescaled data is worse than it is on original data. This suggests that the magnitude information should be critical and cannot be ignored in determining the seven classes. As we have discussed, all methods fail to discern the genes in Early-Mid from the genes in Early I, Early II, Middle, Mid-Late and Late (Figure 5(b)–(f)). Furthermore, PCAChisq and PoissonC mixed up two different patterns from Metabolic and Early I because of their similar induction time at 0.5 h (Figure 5(c) and 5(d)). PearsonC even splits the Metabolic group further into two separate clusters (Figure 5(e)).

For PCAChisq, we tried different combinations of principal components (PCs) to optimize the clustering results. The best result can be reached when the first 5 PCs were used: 3 out of the 39 genes were incorrectly grouped. This optimal result is the same as the one from TransChisq. However, in practice, it is not feasible to exhaust all possible combinations of PCs to search for the optimal clustering result.

An unsolved issue in K-means clustering analysis is how to estimate K, the number of clusters. In the recent literature the Gap statistic was found useful 2526. The technique of the Gap statistic uses the output of any clustering algorithm to compare the 'between-to-total variance (R²)' with that expected under an appropriate reference null distribution. A high R² value represents high variability between clusters and high coherence within clusters. Below we sketch how to calculate the Gap statistic: Let D_k be the R² measure for the clustering output when the number of clusters is k. To derive the reference expected value of D_k, the elements within each row of original data are permuted to produce the new matrices with random profile patterns. Assume B such matrices are obtained. Then for each matrix, a new R² is calculated based on the original clustering output and the pre-selected similarity measure. The average of these R²'s, denoted by D¯k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabdseaebaadaWgaaWcbaGaem4AaSgabeaaaaa@2F59@, serves as the expectation of D_k. With D_kand D¯k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabdseaebaadaWgaaWcbaGaem4AaSgabeaaaaa@2F59@, the Gap function is defined by

Gap(k)= D_k - D¯k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabdseaebaadaWgaaWcbaGaem4AaSgabeaaaaa@2F59@.

The value of k with the largest Gap value will be selected as the optimal number of clusters in that at this k, the observed between-to-total variance R² is the most ahead of expected.

For comparison, we used different measures including TransChisq, PCAChisq, PoissonC, Pearson, Eucli, and SRC to calculate the Gap statistics for each of the two experimental datasets: microarray yeast sporulation data and mouse retinal SAGE data. For the microarray yeast sporulation data, the Gap values from different measures over different number of clusters are shown in Figure 6. We can see that TransChisq shows the maximum Gap value at k = 7. In other words, TransChisq finds an optimal number of 7 clusters, which agrees with the known functional categorization of the genes. Other measures all produce incorrect estimates of the number of clusters on the same dataset. In a similar analysis of the SAGE data, TransChisq, PCAChisq and PoissonC provide a correct estimate on the number of clusters, 5. PearsonC, Eucli and SRC give an incorrect estimate of 3, 14 and 2 respectively (the gap function curves are not shown here). This study shows that when the number of clusters, K, is unknown, the Gap Statistics can be used to estimate K, and TransChisq is favorable over others on estimating the true number of clusters in both experimental datasets.

SAGE is one of the effective techniques for comprehensive gene expression profiling. The result of a SAGE experiment, called a SAGE library, is a list of counts of sequenced tags isolated from mRNAs that are randomly sampled from a cell or tissue. As discussed in Man et al. 27, the sampling process for tag extraction is approximately equivalent to randomly taking a bag of colored balls from a big box. This randomness leads to an approximate multinomial distribution for the number of transcripts of different types. Moreover, due to the vast amount of varied types of transcripts in a cell or tissue, the selection probability of a particular type of transcript at each draw should be very small. This suggests that the tag counts of sampled transcripts of each type are approximately Poisson distributed. PoissonC and PoissonL were developed under this context 20. The method is summarized below.

Let Y_i(t) be the count of tag i in library t, and Y_i = (Y_i(1),..., Y_i(T)) be the vector of counts of tag i over a total of T libraries. Y_i(t) is assumed to be Poisson distributed with mean γ_it. To model the magnitude and shape of the expression profile separately, Cai et al. 20 further parameterized the Poisson rate as γ_it = λ_i(t)θ_i, where θ_i is the expected sum of counts of tag i over all libraries, and λ_i (t) is the contribution of tag i in library t to the sum θ_i expressed in percentage. The sum of λ_i(t) over all libraries equals to 1. So λ_i(t)θ_i redistributes the tag counts according to the expression shape parameter (λ_i(t)'s) but keeps the sum of counts over libraries constant. The genes with similar λ_i(t)'s over t are considered to be in the same cluster.

For a cluster consisting of tags 1,2,..., m with the common shape parameter λ = (λ(1),..., λ(T)), the joint likelihood function for Y₁, Y₂,...,Y_m is

L ( λ , θ | Y ) ∝ f ( Y 1 , ... , Y m | λ , θ 1 , ... , θ m ) = ∏ i = 1 m ∏ t = 1 T exp ⁡ ( − λ ( t ) θ i ) ( λ ( t ) θ i ) Y i ( t ) Y i ( t ) !       ( 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8B55@

The maximum likelihood estimates of λ and θ₁,..., θ_m are

θ _ i = ∑ i Y i ( t ) ,  and  λ ^ ( t ) = ∑ i = 1 m Y i ( t ) / ∑ i = 1 m θ _ i = ∑ i = 1 m Y i ( t ) / ∑ i = 1 m ∑ t Y i ( t ) .       ( 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=H7aXbGaayPadaWaaSbaaSqaaiabdMgaPbqabaGccqGH9aqpdaaeqbqaaiabdMfaznaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemiDaqNaeiykaKcaleaacqWGPbqAaeqaniabggHiLdGccqGGSaalcqqGGaaicqqGHbqycqqGUbGBcqqGKbazcqqGGaaidaWcgaqaaiqb=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@82AF@

Formula (2) forms the basis of the following two measures for evaluating how well a particular tag fits in a cluster. One natural measure is to use the log-likelihood function: log f(Y_i|λ, θ_i). The larger the log-likelihood is, the more likely the observed counts are generated from the expected Poisson distributions. So for a cluster consisting of tags 1,2,..., m, a likelihood based measure is defined as

L = − log ⁡ f ( Y 1 , ... , Y m | λ ^ , θ ^ ) = ∑ i = 1 m ∑ t = 1 T ( λ ^ ( t ) θ ^ i − Y i ( t ) log ⁡ ( λ ^ ( t ) θ ^ i ) + log ⁡ ( Y i ( t ) ! ) ) .       ( 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGmbatcqGH9aqpcqGHsislcyGGSbaBcqGGVbWBcqGGNbWzcqWGMbGzcqGGOaakieWacqWFzbqwdaWgaaWcbaacbeGae4xmaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab=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@89D1@

The other measure is based on the Chi-square statistic, a well known statistic for evaluating the deviation of the observations from the expected values. It is defined as

D = ∑ i = 1 m ∑ t = 1 T ( Y i ( t ) − λ ^ ( t ) θ ^ i ) 2 / ( λ ^ ( t ) θ ^ i ) .       ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebarcqGH9aqpdaaeWaqaamaaqadabaWaaSGbaeaacqGGOaakcqWGzbqwdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdsha0jabcMcaPiabgkHiTmaaHaaabaacciGae83UdWgacaGLcmaacqGGOaakcqWG0baDcqGGPaqkdaqiaaqaaiab=H7aXbGaayPadaWaaSbaaSqaaiabdMgaPbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaOqaaiabcIcaOmaaHaaabaGae83UdWgacaGLcmaacqGGOaakcqWG0baDcqGGPaqkdaqiaaqaaiab=H7aXbGaayPadaWaaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaaaaleaacqWG0baDcqGH9aqpcqaIXaqmaeaacqWGubava0GaeyyeIuoaaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabd2gaTbqdcqGHris5aOGaeiOla4IaaCzcaiaaxMaadaqadaqaaiabisda0aGaayjkaiaawMcaaaaa@5F7F@

Using Chi-square statistic as a similarity measure, the penalty for the deviation from large expected count is smaller than that for small expected count. It is consistent with the above likelihood-based measure in that the variance of a Poisson variable equals to its mean. In general, the smaller the value of L or D, the more likely the tags belong to the same cluster. We should also note that the statistics in measure (3) and measure (4) consider both the shape and magnitude information when measuring the cluster dispersion, i.e., the cluster is specified by the shape parameter λ, but the relationship of a tag to a certain cluster is determined by the deviation of observed counts (θ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=H7aXbGaayPadaaaaa@2F2B@_i λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGadiab=T7aSbGaayPadaaaaa@2F2A@_i) from the expected values (θ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=H7aXbGaayPadaaaaa@2F2B@_i λ). Here λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGadiab=T7aSbGaayPadaaaaa@2F2A@_i is the estimated profile shape of tag i (λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGadiab=T7aSbGaayPadaaaaa@2F2A@_i = (λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=T7aSbGaayPadaaaaa@2F29@_i (1),...,λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=T7aSbGaayPadaaaaa@2F29@_i (T)) and λ^i(t)=Yi(t)/∑tYi(t)=Yi(t)/θ^i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcgaqaamaaHaaabaacciGae83UdWgacaGLcmaadaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iabdMfaznaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemiDaqNaeiykaKcabaWaaabeaeaacqWGzbqwdaWgaaWcbaGaemyAaKgabeaaaeaacqWG0baDaeqaniabggHiLdaaaOGaeiikaGIaemiDaqNaeiykaKIaeyypa0ZaaSGbaeaacqWGzbqwdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdsha0jabcMcaPaqaamaaHaaabaGae8hUdehacaGLcmaadaWgaaWcbaGaemyAaKgabeaaaaaaaa@4F25@). A measure that ignores magnitude would take the difference between λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGadiab=T7aSbGaayPadaaaaa@2F2A@_i and λ directly.

Cai et al. 20 have employed the above measures into a K-means clustering algorithm to perform clustering analysis. K-means clustering procedure 5 generates clusters by assigning each object to one of K clusters so as to minimize a measure of dispersion within the clusters. The algorithm is outlined below:

1. All SAGE tags are assigned randomly to K sets. Estimate initial parameters θi(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF4oqCdaqhaaWcbaGaemyAaKgabaGaeiikaGIaeGimaaJaeiykaKcaaaaa@3291@ and λk(0)=(λk(0)(1),...,λk0(T)) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiWacqWF7oaBdaqhaaWcbaGaem4AaSgabaGaeiikaGIaeGimaaJaeiykaKcaaOGaeyypa0JaeiikaGccciGae43UdW2aa0baaSqaaiabdUgaRbqaaiabcIcaOiabicdaWiabcMcaPaaakiabcIcaOiabigdaXiabcMcaPiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiab+T7aSnaaDaaaleaacqWGRbWAaeaacqaIWaamaaGccqGGOaakcqWGubavcqGGPaqkcqGGPaqkaaa@4969@ for each tag and each cluster by formula (2).

2. In the (b+1)th iteration, assign each tag i to the cluster with minimum deviation from the expected model. The deviation is measured by either Li,k(b)=−log⁡f(Yi|λk(b),θi(b)) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGmbatdaqhaaWcbaGaemyAaKMaeiilaWIaem4AaSgabaGaeiikaGIaemOyaiMaeiykaKcaaOGaeyypa0JaeyOeI0IagiiBaWMaei4Ba8Maei4zaCMaemOzayMaeiikaGccbmGae8xwaK1aaSbaaSqaaiabdMgaPbqabaGccqGG8baFiiWacqGF7oaBdaqhaaWcbaGaem4AaSgabaGaeiikaGIaemOyaiMaeiykaKcaaOGaeiilaWccciGae0hUde3aa0baaSqaaiabdMgaPbqaaiabcIcaOiabdkgaIjabcMcaPaaakiabcMcaPaaa@4F85@ or Di,k(b)=∑t(Yi(t)−λk(b)(t)θi(b))2/(λk(b)(t)θi(b)) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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H7aXnaaDaaaleaacqWGPbqAaeaacqGGOaakcqWGIbGycqGGPaqkaaGccqGGPaqkaaaaleaacqWG0baDaeqaniabggHiLdaaaa@639B@.

3. Set new cluster centers λk(b+1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiWacqWF7oaBdaqhaaWcbaGaem4AaSgabaGaeiikaGIaemOyaiMaey4kaSIaeGymaeJaeiykaKcaaaaa@34C5@ by formula (2).

4. Repeat step 2 till convergence.

Let c(i) denote the index of the cluster that tag i is assigned to. The above algorithm aims to minimize the within-cluster dispersion ∑_iL_i,c(i) or ∑_iD_i,c(i). The algorithm using measure L is called PoissonL, and the algorithm using measure D is called PoissonC. PoissonL and PoissonC perform similarly in applications. But PoissonC is more practical in terms of running time. So we use PoissonC for comparison in this paper.

PoissonC is designed to group the objects by their departure from the expected Poisson distributions. The success of PoissonC has been shown in applications 2021. However, if the clustering purpose is slightly different, some modification on PoissonC may be necessary. For instance, if the shape difference should be more emphasized in determining the relationship, the direction of departure of observed from expected may/should also be considered. As an example, we consider an expression vector Y = (15, 30, 15) and its relationship with two clusters with shape specified by λ₁ = (1/12,5/6,1/12) and λ₂ = (5/12, 1/6, 5/12) respectively. The expectation of Y in cluster 1 is YE1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGymaedaaaaa@301F@ = (5, 50, 5), and in cluster 2, it is YE2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGOmaidaaaaa@3021@ = (25, 10, 25). If more emphasis should be put on the shape change in determining the relationship, Y would be expected to be closer to the first cluster because of the large value observed on the middle component in both Y and YE1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGymaedaaaaa@301F@. PoissonC, however, determines that Y has the same distance to YE1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGymaedaaaaa@301F@ and YE2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGOmaidaaaaa@3021@ (by the measure (4), the distance between Y and YE1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGymaedaaaaa@301F@ is 48, so is the distance between Y and YE2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGOmaidaaaaa@3021@). PoissonC ignores the direction of departure. To address this omission we propose to emphasize the profile shape through suitable data transformations, and to define a distance measure in the transformed space. The construction of a proper feature space under a certain clustering purpose is essential to define an effective distance or similarity measure.

A simple yet natural data transformation to emphasize the expression shape is to consider the mutual differences of the original vector components. Given a gene with expression profile Y_i = (Y_i(1),..., Y_i(T)) the transformed vector Z_i is of dimension T(T-1)/2 with components in the form of Y_i(t₁)-Y_i(t₂) for t₁ = 1,..., T-1 and t₂ = (t₁ + 1),..., T.

According to the Poisson model in the previous section, E(Y_i(t₁)-Y_i(t₂)) = (λ_i(t₁)-λ_i(t₂))θ_i and Var(Y_i(t₁)-Y_i(t₂)) = (λ_i(t₁)+λ_i(t₂))θ_i. For a cluster consisting of tags l, 2,..., m, we can define the following statistic to measure the cluster dispersion:

S t r a n s = ∑ i ∑ t 1 , t 2 ( ( Y i ( t 1 ) − Y i ( t 2 ) ) − E ( Y i ( t 1 ) − Y i ( t 2 ) ) ) 2 / V a r ( Y i ( t 1 ) − Y i ( t 2 ) ) = ∑ i ∑ t 1 , t 2 ( ( Y i ( t 1 ) − Y i ( t 2 ) ) − ( λ ^ ( t 1 ) θ ^ i − λ ^ ( t 2 ) θ ^ i ) ) 2 / ( λ ^ ( t 1 ) θ ^ i + λ ^ ( t 2 ) θ ^ i ) ,       ( 5 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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H7aXbGaayPadaWaaSbaaSqaaiabdMgaPbqabaGccqGHsisldaqiaaqaaiab=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H7aXbGaayPadaWaaSbaaSqaaiabdMgaPbqabaaakiaawIcacaGLPaaaaaGaeiilaWcaaiaaxMaacaWLjaWaaeWaaeaacqaI1aqnaiaawIcacaGLPaaaaaa@D011@

where λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=T7aSbGaayPadaaaaa@2F29@(t) and θ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=H7aXbGaayPadaaaaa@2F2B@_i can be estimated by formula (2). We call the modified K-means algorithm with this measure TransChisq. Applying it to the toy example in the previous section, TransChisq determines that Y is closer to YE1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFzbqwdaqhaaWcbaGaemyraueabaGaeGymaedaaaaa@301F@ as we expected.

To better understand the effects of the proposed data transformation, we performed a simple simulation study and presented the results in Additional file 3.

Now we consider a data transformation determined by a parametric covariance matrix:

R = cov(X) = (γ_ij)_{i,j = 1,..., T}, with γ_ij = α > 0 if i = j and γ_ij = β if i ≠ j,

where X is the data matrix with n observations on the rows and T variables on the columns, and R is the covariance matrix of the T variables. The matrix R in this form implies that the variables have identical variances and covariances with each other. These properties are biologically reasonable in that normalized arrays have identical distributions, hence equal variances. Also all pairs of variables would exhibit equal covariance (or un-correlated when β = 0) if each component had been equally important (or independent) to determine a class.

A data transformation can be defined through the eigenspace of R. One set of column orthonormal eigenvectors, denoted by e₁,e₂,...,e_T, is presented in Additional file 4. Given a gene expression profile Y_i = (Y_i(1),..., Y_i(T)), a transformation based on R is

Z_i = (Z_i1,..., Z_iT) = Y_i (e₁ e₂...e_T).

A convenient property of this transformation is that each component has a clear meaning: with e₁ = [1/T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabdsfaubWcbeaaaaa@2DF8@,...,1/T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabdsfaubWcbeaaaaa@2DF8@]^T, e₂ = [1/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabikdaYaWcbeaaaaa@2DB9@, -1/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabikdaYaWcbeaaaaa@2DB9@,0,...,0]^T and e₃ = [1/6 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabiAda2aWcbeaaaaa@2DC1@,1/6 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabiAda2aWcbeaaaaa@2DC1@,-2/6 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabiAda2aWcbeaaaaa@2DC1@,0,...,0]^T, for a profile Y = (Y₁,..., Y_T), the component associated with e₁ is Ye₁ = (Y₁ + Y₂+...+Y_T)/T MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabdsfaubWcbeaaaaa@2DF8@, which reflects the general expression level; the component associated with e₂ is Ye₂ = (Y₁-Y₂)/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabikdaYaWcbeaaaaa@2DB9@, which reflects the difference between Y₁ and Y₂; the component associated with e₃ is Ye₃ = (Y₁+Y₂-2Y₃)/6 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabiAda2aWcbeaaaaa@2DC1@, which reflects the relationship among Y₁, Y₂ and Y₃.

According to the Poisson model, E(Z_it) = E(Y_i)e_t = (λ_i(1)θ_i,..., λ_i(T)θ_i)e_t, Var(Z_it) = (λ_i(1)θ_i,..., λ_i(T)θ_i)et2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFLbqzdaqhaaWcbaGaemiDaqhabaGaeGOmaidaaaaa@3095@ and Cov(Z_it, Z_ik) = 0 when t ≠ k. Then for a cluster consisting of tags 1, 2,..., m, we can measure the cluster dispersion by:

S t r a n s _ N = ∑ i ∑ t = 1 , .. , T ( Z i t − E ( Z i t ) ) 2 / V a r ( Z i t ) = ∑ i ∑ t = 2 , .. , T ( Z i t − ( λ ^ ( 1 ) θ ^ i , ... , λ ^ ( T ) θ ^ i ) e t ) 2 / ( λ ^ ( 1 ) θ ^ i , ... , λ ^ ( T ) θ ^ i ) e t 2 .       ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B13C@

We should note the connection between this measure and the S_trans in formula (5). As we discussed above, the component associated with e₂ is (Y₁-Y₂)/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiabikdaYaWcbeaaaaa@2DB9@. Thus the new space associated with S_trans is equivalent to the space determined by e₂ and all its row-switching transformations. We can also define a measure similarly through e₃ or other eigenvectors. S_trans seems to have the potential of losing the information carried by e₃ and other eigenvectors. However, applications of TransChisq to a variety of datasets suggested that this potential information loss is minor and can be ignored in most cases in practice. In fact, the row-switching transformations of e₂ make up most of the information included in e₃ and other eigenvectors.

A potential shortcoming of S_{trans_N} comes from the fact that it is defined based on only one set of eigenvectors. The orthonormal eigenspace of a covariance matrix is not unique (e.g., the row switching operation can result in a different set of eigenvectors) and different eigenspaces may result in different values of S_{trans_N} . Although one can consider all possible eigenspaces to overcome the limitation of S_{trans_N}, it is not computationally feasible.

Applying S_{trans_N} to several different datasets, we observed that i) using the eigenvectors e₁,e₂,...,e_T in Additional file 4, S_{trans_N} performs very similarly to S_trans and ii) when a different set of eigenvectors used, the clustering results can be different, though the difference is not obvious. These results are not presented in this paper.

For comparison purposes, we applied PCA to transform the data 19. PCA is useful to simplify the analysis of a high dimensional dataset. Recently, PCA has been explored as a method for clustering gene expression data 282930313233. But a blind application of PCA in clustering analysis is dangerous in that PCA chooses principal component axes based on the empirical covariance matrix rather than the class information, and thus it does not necessarily give good clustering results 293435.

In some theoretical 35 and empirical 29 studies, there have been observations that the first few principal components (PCs) in PCA are not always helpful to extract meaningful signals from data. Thus, we considered all PCs in this study. By substituting the e₁ e₂...e_T in measure (6) by the eigenvectors from the sample covariance matrix, we defined a new measure and implemented it in the PCAChisq. The Results section gives examples showing the positive and negative effects of the PCA transformation. In general, PCAChisq is difficult to use. Firstly, it is unclear what types of variances the principal components are capturing (if it is the within-cluster variance, the principal components would lead to wrong clustering results). Next, it is unclear how many principal components should be used. The optimal number of PCs is unavailable before we compare the results to the ground truth. To be brief, PCAChisq is only efficient when the principal components happen to match the key features that determine a cluster.

We explored the potential application of the proposed measures to a clustering analysis of microarray data. We proposed the following restricted normal model for this purpose. The parameter notations in the Poisson model were adopted. Given a microarray dataset of expressions of n genes in T experiments, the expression of gene i in experiment t, X_i(t), is assumed to be normally distributed with mean μ_i(t) = λ_i(t)θ_i and variance σi2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@30F0@(t) = kλ_i(t)θ_i, where k is an unknown constant. The derivation of the maximum likelihood estimates (MLEs) of λ_i(t) and θ_i under the normal model is rather involved. So we borrowed the estimators in formula (2). It can be shown that θ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=H7aXbGaayPadaaaaa@2F2B@_i in formula (2) is unbiased and λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=T7aSbGaayPadaaaaa@2F29@_t in formula (2) is consistent under the restricted normal model [see Additional file 5]. With θ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=H7aXbGaayPadaaaaa@2F2B@_i and λ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaGGaciab=T7aSbGaayPadaaaaa@2F29@_t available under the normal model, TransChisq, PCAChisq and PoissonC can be applied.

For both oligonucleotide and cDNA microarray data, it is widely observed that there is strong dependence of the variance on the mean: variance increases with mean 3637. So it is reasonable to expect that our restricted normal model is applicable to many microarray datasets. One example of this application on the yeast sporulation dataset has been presented to demonstrate the power of TransChisq in analyzing microarray data (see the Results section). We should also note that TransChisq would deliver less promising results if the assumption on the relationship between the variance and the mean is seriously violated.