Fourier Harmonic Approach for Visualizing Temporal Patterns of GeneExpression Data

Li Zhang and Aidong ZhangDepartment of Computer Science and Engineering

State University of New York at BuffaloBuffalo, NY 14260

lizhang,azhang@cse.buffalo.edu

Murali RamanathanDepartment of Pharmaceutical SciencesState University of New York at Buffalo

Buffalo, NY 14260murali@acsu.buffalo.edu

Abstract

DNA microarray technology provides a broad snapshotof the state of the cell by measuring the expression levelsof thousands of genes simultaneously. Visualization tech-niques can enable the exploration and detection of patternsand relationships in a complex dataset by presenting thedata in a graphical format in which the key characteris-tics become more apparent. The purpose of this study is topresent an interactive visualization technique conveying thetemporal patterns of gene expression data in a form intuitivefor non-specialized end-users.

The first Fourier harmonic projection (FFHP) was in-troduced to translate the multi-dimensional time series datainto a two dimensional scatter plot. The spatial relationshipof the points reflect the structure of the original dataset andrelationships among clusters become two dimensional. Theproposed method was tested using two published, array-derived gene expression datasets. Our results demonstratethe effectiveness of the approach.

Keywords: visualization, gene expression, time series,Fourier harmonic projection

1 INTRODUCTION

Knowledge of the spectrum of genes expressed at a cer-tain time or under given conditions proves instrumental tounderstand the working of a living cell. DNA microarraytechnology allows measurements of expression levels forthousands of genes simultaneously [9]. Extensive researchhas been conducted on the study of temporal patterns ofgene expressions [10, 16]. Clustering methods which groupgenes or samples with similar patterns have become main-stream analysis tool [14]. Visualization can facilitate thediscovery of structures, features, patterns, and relationshipsin data and may provide more insightful information than

traditional numerical methods. By visualization, we hopeto gain some intuition regarding the data, but more impor-tantly, we would like to understand the relationships be-tween data points and detect the intrinsic structure, or possi-ble cluster tendencies. Visualization is especially importantin the early stages of data analysis in which qualitative anal-ysis is primary to quantitative. Early success will enhancethe users’ performance in the remaining stages of analy-sis. Array-derived gene expression datasets present analysisand visualization challenges because of their dimensional-ity, noisy environment, and pattern varieties.

The parallel coordinates approach [18] is perhaps thesimplest method to display patterns of gene expression pro-files. Here the data in each dimension are plotted along aseparate axis. Holter et al. [8] have used parallel coordi-nates to visualize the temporal progression in the yeast cellcycle data. Self-organizing maps (SOM) [12] is another ap-proach. More recently, Hautaniemi et al. [6] have presenteda heat map-based strategy for visualizing the U-matrix fromSOM. The most prominent visualization-enhanced analy-sis tool for gene expression data is TreeView [5], whichprovides a user-friendly computational and graphical en-vironment for assessing the results from hierarchical clus-tering. The graphical presentations from TreeView includea dendrogram to reflect the distance relationships betweenclusters and a heat plot to visually convey gene expressionchanges between samples. The heat plot can be viewed asvariation of the parallel coordinates plot in which color isused to convey dimension values.

Here, we present an alternative mapping for multi-dimensional data that is based on the first harmonic of thediscrete Fourier transform. The mapping has interestingproperties and preserves certain key characteristics of a va-riety of data sets, especially time series data. Unlike par-allel coordinates and heat plot which display all individualdimensional information, our approach uses a two dimen-sional point to represent each gene over the time at the com-

putational cost ofO(N log N). It focuses on one very im-portant aspect of the visualization: revealing the structure ofthe entire dataset. Tested using two published, array-derivedgene expression time series datasets, our results indicatedthat temporal patterns were well reflected in the visualiza-tion: cluster relationship became two dimensional, clusterswere apparent, and outliers were clear. An interactive visu-alization tool, VizStruct, was implemented to perform thevisualization.

The remainder of this paper is organized as follows. Sec-tion 2 presents the model of visualization. In section 3, weshow our analysis results. The final section discusses otherissues in this approach. Proofs for all mapping propositionsare included as appendix.

2 METHODS AND SYSTEM

The Mapping

Mapping converts multi-dimensional data to two-dimensions for visualization. An array-derived profilefor M genes withN measurements results inM N -dimensional data point containing real valued numericaldata. Time series data in its simplest form is merely a setof datayt, t = 0, . . . , N − 1 where the subscriptt indi-cates the time at which the datumyt was observed [4]. Onthe other hand, a discrete-time real signal onN evenly dis-tributed time points is represented as an indexed sequenceof N real numbers0, . . . , N − 1 denoted byx[n] [1]. Eachterm ofx[n] is denoted byx[n]. The denotation similaritybetween time series and digital signal suggests that we canview each data point in a time series as a discrete-time realsignal (It is not necessary for the signal’s time index to com-ply with the actual time points). In this scenario, the prob-lem of a two dimensional visualization of the time series istransformed into the problem of finding a two-dimensionalpoint estimation for signals (data points).

The frequency domain representation of discrete-timesignals is through discrete-time Fourier transform, or DFT[13]. The DFT of aN -point signalx[n] is a frequency se-quence withN complex values:F(x[n]) = [Fk(x[n])],where each

Fk(x[n]) =N−1∑n=0

x[n]WnkN , k = 0, . . . , N−1. (1)

WN = e−i2π/N is called twiddle factor.Each harmonicFk in the DFT is a measure of thekth

sinusoidal frequency component in the signal. For exam-ple, the zero harmonic,F0, is the mean value, the first har-monicF1 measures the base frequency component, the sec-ond harmonicF2 measures the component in the signal thatis twice the base frequency, and so forth. Because Fourier

harmonics, in general, are complex numbers, they providethe two-dimensional point estimate for mapping a multi-dimensional signal. For this reason, we refer to the mappingas the Fourier harmonic projections. In particular, we areinterested in the first Fourier hamonic projection (FFHP):

F1(x[n]) =N−1∑n=0

x[n]WnN =

N−1∑n=0

x[n]e−i2πn/N . (2)

The relationship between the DFT and the mapping al-lows the fast Fourier transform algorithm (FFT), originallydiscovered by Cooley and Tukey [13], to be used for com-putation. The FFT is a computationally efficient algorithmand has a complexity ofO(N log N).

The complex number ofF1(x[n]) in Equation (2) can beexpressed in terms of magnituder and phaseθ to provide auseful geometric interpretation of the mapping. The data setwas normalized so that the range of values of each dimen-sion across the dataset was0 to 1. For a data point withNdimensions, the complex exponential divides a unit circlecentered at the origin of the complex plane intoN equallyspaced angles. The value of the first dimension is projectedon the radial line corresponding toθ = 0 and, similarly, thevalue of thekth dimension is projected on to the radial linecorresponding to theθ = 2π(1−k)/N radians. The overalltwo-dimensional FFHP mapping is the complex sum of allN projections from a data point. Figure 1 illustrates the ge-ometric interpretation for a point containing6 dimensions.

unit circle

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Figure 1. A geometric interpretation of thefirst Fourier harmonic projection. A normal-ized 6-dimensional data point is shown on theright by the stem plot. The powers of twid-dle factor W6 divide the unit circle centeredat the origin into 6 equal angles and each di-mension of the data point is projected onto adifferent radial angle (open circle). The pro-jections are taken complex number sum togive a 2-dimensional image (indicated by afilled circle).

amplitude shifting amplitude scaling

+ a x a

origin

time shifting time shifting

origin

(A) (B)

Figure 2. Illustration of the effect of (A) amplitude shifting and multiplying, and (B) time shifting ofthe first Fourier harmonic projection.

Properties of FFHP

The first Fourier harmonic projection has useful prop-erties that preserve the correlation between dimensions inthe multi-dimensional data point. We summarize them aspropositions below. Detailed proofs for propositions areprovided as the appendix.

1. Data points with equal values for all the dimensionsare mapped to the origin. Ifx[n] = [a, . . . , a], thenF1(x[n]) = 0.

2. Two data points whose dimension values differ due to theamplitude shifting of a constant are mapped to the samepoint. If y[n] = x[n] + a, thenF1(y[n]) = F1(x[n]).

3. Two data points whose dimension values differ due to theamplitude multiplying a constant are mapped to the twopoints on a line through the origin. Ify[n] = ax[n], thenF1(y[n]) = aF1(x[n]). See Figure 2A.

4. Two data points whose dimension values are transposingeach other, i.e. symmetric regarding the middle time point,are mapped to the points symmetric to the real axis. Ify[n] = x[N − n− 1], thenF1(y[n]) = F1(x[n]).

5. Data points that differ only because they are “time-shifted”by d dimensions relative to each other are mapped to thecircumference of the circle that is concentric with the unitcircle and the angle between the points in the visualizationis φ = 2πd/N . If y[n] = x[n − d], thenF1(y[n]) =F1(x[n])W d

N . This property is illustrated in Figure 2B.

6. Letw[n] = x[n] − y[n] be the difference between the twoN -dimensional points,x[n] and y[n]. The distance be-tween these two points in the visualization is:

‖F1(w[n])‖2 = g0N

(1 + 2

N−1∑

k=1

rk cos(2πk/N)

)(3)

Theory in Practice

We will demonstrate in the next section that the relativelocations of temporal profiles’ mapping images can be pre-dicted by Propositions 1–5. For example, genes with rela-tively low levels of expression throughout the time line aremapped close to the origin. Genes that steadily increaseover the time and genes that steadily decrease over the timeline are likely mapped symmetric to the real axis.

In Equation (3), theg0 is the variance ofw[n], rk isthe kth-sample autocorrelation coefficient ofw[n]. It canbe shown [4] that−1 ≤ rk ≤ 1 and for mutually inde-pendent random sequences or white noise,rk = 0. Equa-tion (3) provides insight into the cluster delineation capa-bilities of the first Fourier harmonic projection. The vari-ance andkth-sample autocorrelation coefficients of the dif-ference between two points within a given cluster are likelyto be small because they will share similarities across manydimensions. Thus, points within cluster are likely to mapclose to each other in the visualization.

The first Fourier harmonic projection is well suited forvisualizing temporal patterns. FFHP has a better class sepa-ration capability when the first harmonic is more dominantamong all harmonics. Since the first harmonic is a measureof the base sinusoidal frequency component in the signal, alower frequency signal tends to have a larger first harmonic.This is the case for a large number of time series data wheremost trend patterns show low frequency variations (not hav-ing multiple cycles). The first Fourier harmonic projectionis sensitive to dimension order. For time series data, thetime points (dimensions) are naturally ordered.

VizStruct System

The first Fourier harmonic projection approach is im-plemented as a visualization tool called VizStruct, whichis written in Java and is available on request from the firstauthor (lizhang@cse.buffalo.edu). The name VizStruct em-

(A) (B) (C)

Figure 3. Three snapshots from a dimension tour through a synthetic three-dimensional data set con-taining 25, 000 points. The parameter settings for (A)–(C) were 〈0.5, 0.5, 0.5〉, 〈0.3, 0.5, 1〉, and 〈0.05, 1, 1〉,respectively.

phasizes its capability of visualizing the structure of thedataset.

Dimension Tour

The dimension tour is a feature of VizStruct that allowsthe user to interact with the data via dynamic animations.It is analogous to the grand tour [2] and the user interactswith the visualization by changing the dimension parame-ter associated with each dimension. The default value foreach dimension parameter is0.5 and the individual param-eters can be changed over the range−1 to +1 either manu-ally or systematically using the program. Because no two-dimensional mapping can capture all the interesting prop-erties of the original multi-dimensional space, two pointsthat are close in the visualization can theoretically be farapart in the multi-dimensional space. The dimension tour,which creates animations that explore dimension parameterspace, can reveal structures in the multi-dimensional inputthat were hidden due to overlap with other points in the vi-sualization.

Figure 3 illustrates the capabilities of the dimension tourusing a synthetic dataset containing25, 000 points in threedimensions. At the default settings of dimensional param-eters (Figure 3A),5 clusters are apparent. However, dur-ing the course of the animations (Figures 3B and 3C), themulti-layered structures of the original5 clusters becomeincreasingly apparent.

3 RESULTS

Data Sets for Visualization

Our approach was tested using two published array-derived data sets. Therat kidneyarray dataset of Stuart etal. [16] containes measurements of gene expressions duringrat kidney organogenesis. The data were downloaded fromhttp://organogenesis.ucsd.edu/data.html. It consists of873genes which vary significantly during kidney developmentat 7 different time points: gestational day13, 15, 17, 19;newborn (N);1 week (W); and nonpregnant adult (A).

Thefibroblastsdataset of Iyer et al. [10] is the result of astudy of the response of human fibroblasts to serum. It con-sists of gene expressions measuring the temporal changesin mRNA levels of517 human genes at13 time points,ranging from15 minutes to24 hours after serum stim-ulation. The data were downloaded from http://genome-www.stanford.edu/serum.

Rat Kidney Dataset

In the rat kidneydataset, there are5 discrete patterns orgroups of gene expression during nephrogenesis. Figure 4illustrates these temporal profiles characterized by the ide-alized gene expressions.

Figure 5A shows how genes are classified by a hierar-chical clustering algorithm. It copies the Figure3 in [16].Figure 5B shows the parallel coordinates of the dataset. Pat-terns of genes in each group comply to the profiles depictedin Figure 4 (with some noise).

13151719 N W A0

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13151719 N W A

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Figure 4. Idealized temporal gene expression profiles during kidney development. The groups werenamed 1 through 5 based on the timing of their peak expression during development. Seven timepoints were 13, 15, 17, 19 embryonic days; N , newborn; W , 1 week old; A, adult.

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Figure 5. Visualization of the rat kidneydataset in heat plot and parallel coordinates. (A) Dendrogramand a heat plot from hierarchical clustering algorithm. (B) Parallel coordinates for the entire datsetand each of the gene groups.

Figures 6A-B show the visualization of therat kidneydataset in VizStruct under the first Fourier harmonic pro-jection for two dimension parameter settings. There are5sets of colored symbols for each of the5 gene groups. Eachsymbol represents one gene across7 time points. In Figure6A, two big clusters are clearly apparent from the visualiza-tion. The top cluster consists of genes from groups3, 4, and5. The bottom cluster is comprised of genes from groups1and2. Furthermore, genes from each group are aggregated.

The formation of two large clusters can be interpreted bythe temporal profiles. Groups1 and 2 with genes whichhave very high relative levels of expression in early de-velopment are quite different from groups3, 4, and5 forgenes that have a relatively steady increase in expressionthroughout development. The visualization also indicatesthat points in the upper cluster are symmetric to the pointsin the lower cluster. Properties of FFHP may suggest thereason. Temporal profiles of groups1 and4 suggest that

they are somewhat symmetric to the middle time point (ges-tational day19). By Proposition4, they would be mappedto points symmetric to the real axis. On the other hand,groups4 and group5 are mapped closely since they havesimilar profiles except for the significantly up-regulated inthe last time point. The same arguments can be applied inthe case of group1 vs. group2, or group3 vs. group4.

Due to the noise, most boundaries between groups arenot very clear. However, the separation between group4and group5 improves in Figure 6B compared to Figure 6A.

Fibroblast Dataset

Temporal patterns are slightly complicated in thefibrob-last dataset. Data has been classified into10 groups usingthe hierarchical clustering algorithm by the original author[10]. Figure 7A shows the result of the hierarchical cluster-ing. It is a copy of Figure3 from [10]. Figure 7B gives the

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Figure 6. Visualization of the rat kidney dataset in VizStruct. Five gene groups were representedby blue plus symbols, red circles, green triangles, magenta stars, and black cross symbols. Thedimension parameters for (A) and (B) were 〈0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5〉 and 〈1, 0.5,−1,−0.5, 0.5, 1,−1〉,respectively.

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Figure 8. Visualization of the fibroblastdataset in VizStruct. 11 colored numbers were used for eachof the gene groups. (A) The mapping of the entire dataset. (B) Enlarged portion of the visualizationin (A).

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parallel coordinates the entire dataset and each gene group.

Figure 8A shows the visualization of thefibroblastsdataset in VizStruct under the first Fourier harmonic pro-jection. Colored numbers1 through11 were used for eachgene group. The visualization reveals several outliers andno distinct clusters. The majority of data are too dense to beseen. By zooming technology, the enlarged detail is shownin Figure 8B. More numbers spread out, but most blue num-bers (1) were still covered by numbers2 and3.

Two observations can be made: (1) a large number ofgenes are close to the center. (2) most clusters have a radialshape emitting from the center. They can be interpreted bythe FFHP properties. (1) Temporal patterns of groupA andgroupB have very flat shape and relative smaller values. ByPropositions1 and2, they tend to be mapped closed to theorigin. (2) In hierarchical clustering, Pearson’s correlationcoefficient was used to measure the similarity. Genes whosetime values differ due to the amplitude shifting or multiply-

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x

y

(A) (B) (C)

Figure 10. Comparison of Sammon’s mapping and the first Fourier harmonic projection. (A) Theresult from Sammon’s mapping of the rat kidneydataset. Five gene groups were represented by 5colored symbols the same as were used in Figure 6. (B)–(C) Comparison of relative gene locationsbetween (B) VizStruct and (C) Sammon’s mapping. Five colors were used for each of the genegroups. The color schema is the same as in (A). Notice that the visualization layout of (A) and (C) isidentical. The purpose of panel (C) is to compare the gene by gene location to the panel (B).

ing a constant have very high coefficient values. By Propo-sitions2 and3, they tend to be mapped closely along a linethough the origin.

Some relative gene group locations can be predicted. Forexample, by Proposition4, groups4 and5 are mapped sym-metric to the real axis. Close inspection suggests that tem-poral pattern of group8 is somewhat a2 or3 right time-shiftfrom the pattern of group6. By Proposition 5, each timepoint shift correspondents to roughly2/13π ≈ 30 rota-tion. In Figure 8B, genes in group8 are mapped about60

clockwise to genes in group6.Lacking clear clusters in the visualization indicates that

different clustering methods may yield different results.Figure 9 shows the grouping aspect of k-means cluster-ing. Euclidean distance was used in k-means as the distancemeasure. The visualization reveals that genes closed to theorigin are grouped as one.

Comparison of FFHP to Other Visualizations

Heat plot and parallel coordinates display all individualdimensional information. In parallel coordinates, a polylinerepresents a gene over time. This format is a concise andintuitive for displaying temporal profiles. However, the par-allel coordinate has obvious drawbacks: when the data sizebecomes larger it becomes increasingly unreadable as indi-cated in the first panel of Figure 5 and Figure 7. Heat plotuses a color mosaic instead of a polyline to overcome over-

lapping. Combined with the dendrogram from hierarchicalclustering, it gives a global view as well as individual clus-ters of the dataset by grouping genes with similar patternstogether. Cluster relationships in heat plot is one dimen-sional: clusters of genes are listed one by one as shown inFigure 5A and Figure 7A.

The first Fourier harmonic projection takes a differentapproach. It uses a two dimensional point to represent eachgene over time. By doing so, it takes advantage of the spa-tial relationship of the points to reflect the structure of theoriginal dataset. Thus the cluster relationships become twodimensional and FFHP has a better capability of displayingoutliers than heat plot. Unlike heat plot, which requires al-gorithms to group similar genes to make a meaningful visu-alization, FFHP directly mapped the multi-dimensional dataonto a two dimensional space without any prior knowledgeof the dataset.

Multidimensional scaling (MDS) [3] is a competing ap-proach for visualizing multi-dimensional data. We com-pared FFHP to Sammon’s mapping [15], a variant of MDSthat optimizes the following stress functionE :

E =1∑

i

∑j<i

d∗ij

∑i

∑j<i

(d∗ij − dij)2

d∗ij, (4)

whered∗ij is the distance between pointsi andj in theN -dimensional space anddij is the distance betweeni andjin the visualization.

Figure 10 shows Sammon’s mapping of therat kidneydataset. A comparison of Figure 10A to Figure 6A re-veals the extensive similarities between FFHP and Sam-mon’s mapping. The relative locations of individual sam-ples are also remarkably similar. This is indicated in Figure10B and Figure 10C.

Sammon’s mapping has some drawbacks: (1) it providesa single final result and the user cannot intervene interac-tively during visualization, (2) the incremental addition ofeven a single point requires a complete repetition of the op-timization procedure and possible extensive reorganizationof all the previously mapped points to new locations, and(3) it requires time-consuming optimization procedures oftime complexityO(N2) or greater.

Our results illustrate some of the strengths and weak-nesses of FFHP. The visualization reflects the structure ofthe dataset: outliers, clusters and their relationships. Com-prehending the structure of the dataset can facilitate thechoice and understanding the results of different clusteringmethods. Although the FFHP uses an approach to mappingmulti-dimensional data that is distinctively different fromSammon’s mapping, it yields results that are consistentlycomparable. However, when the dataset contains a largenumber of patterns, it becomes difficult to separate differentpatterns and the visualization may be difficult to interpret.

4 DISCUSSION

Visualization of microarray data is a challenge becauseof its high dimensionality. In this paper, we have exploreduse of the first Fourier harmonic projection for visualizingmulti-dimensional time course array datasets. Unlike par-allel coordinate or heat plot approach which display all di-mensional information, FFHP uses a two dimensional pointto represent each data point (gene in this case). Our re-sults indicated that temporal patterns were well reflected byspatial relationships: genes with similar pattern were aggre-gated and relative locations of gene groups can be predicted.Moreover, the first Fourier harmonic projection was shownto yield results that were similar to those from Sammon’smapping.

Achieving two dimensional mapping requires a trade-off. The mapping is lossy for detailed dimensional infor-mation. Our approach attempts to preserve to the maximumsemantics of the data points via Fourier harmonic aspect.In particular, characterizing the data using two descriptivemeasurements: the real and imaginary portions of the firstdiscrete Fourier harmonic. A similar approach uses prin-cipal component analysis (PCA) [11]. This visualizationdeploys another two descriptive measurement: the first andsecond principal component.

In addition to the first Fourier harmonic projection,higher Fourier harmonics can also be used as mappings.

It can be shown that for any harmonic(> 1), there exitsan equivalent first harmonic of the original discrete signalwhose time indices are systematically rearranged. At cer-tain conditions (such as temporal patterns with high fre-quency variations, i.e. multiple cycles), higher harmonicprojections enhance substructure separation in the visual-ization. Detailed discussion is beyond the scope of this pa-per due to a length constraint.

The FFHP mapping results in two-dimensional visual-izations that are identical to those of radial coordinate visu-alization techniques, e.g., RadViz [7]. However, rather thanvector notation and thespring paradigmof RadViz, we haveused a complex number notation. This substantive reformu-lation of the mapping provides valuable theoretical insightsand allows important properties of mapping, including itsrelationship to the DFT, to be easily derived. It can alsocreate possible extensions such as higher Fourier harmonicprojections.

The first Fourier harmonic projection requires data with-out missing values. This requirement can be easily metbecause filling in missing values is a mature research field[17].

Gene expression data can be studied in either samplespace or gene space. Here, we have reported only the visual-ization on gene space. In a separate report [19], we appliedthe first Fourier harmonic projection on the sample spaceand performed visualization-driven classifications. Our ex-periments demonstrated that FFHP offered an alterative for-mat of visualization. We believe that using FFHP alone orin combination with heat plot or parallel coordinates wouldgive a biologist additional powerful tools for analyzing andvisualizing microarray data sets.

ACKNOWLEDGEMENTS

This work was supported by grants from the NationalScience Foundation. We also thank the anonymous review-ers for their constructive comments on the manuscript.

References

[1] Cadzow, J. A., Landingham, H. F.Signals, Systems, andTransforms. Prentice-Hall, Inc., Englewood Cliffs, NJ,1985.

[2] Cook, C., Buja, A., Cabrera, J., and Hurley, C. GrandTour and Projection Pursuit.Journal of Computational andGraphical Statistics, 2(3):225–250, 1995.

[3] Davision, M. L.Multidimensional Scaling. Krieger Publish-ing, Inc., Malabar, FL, 1992.

[4] Diggle, P. J.Time Series: A Biostatistical Introduction. Ox-ford University Press, Oxford OX2 6DP, 1990.

[5] Eisen, M. B., Spellman, P. T., Brown, P. O., and Botstein, D.Cluster Analysis and Display of Genome-wide ExpressionPatterns.Proc. Natl. Acad. Sci. USA, Vol. 95:14863–14868,December 1998.

[6] Hautaniemi, S., Yli-Harja, O., Astola, J., Kauraniemi, P.,Kallioniemi, A., Wolf, M., Ruiz, J., Mousses, S., andKallioniemi, O. Analysis and Visualization of Gene Ex-pression Microarray Data in Human Cancer Using Self-Organizing Maps, 2003. To appear in Machine Learning:Special Issue on Methods in Functional Genomics.

[7] Hoffman, P. E., Grinstein, G. G., Marx, K., Grosse, I., andStanley, E. DNA Visual and Analytic Data Mining. InIEEEVisualization ’97, pages 437–441, Phoenix, AZ, 1997.

[8] Holter, N. S., Mitra, M., Maritan, A., Cieplak, M., Banavar,J. R., and Fedoroff, N. V. Fundamental Patterns Underly-ing Gene Expression Profiles: Simplicity from Complex-ity. Proc. Natl. Acad. Sci. USA, Vol. 97(15):8409–8414, July2000.

[9] Ideker, T., Galitski, T., and Hood, L. A New Approachto Decoding Life: Systems Biology.Annu. Rev. GenomicsHum. Genet., 2:343–372, July 2001.

[10] Iyer, V. R., Eisen, M. B., Ross, D. T., Schuler, G., Moore,T., Lee, J. C. F., Trent, J. M., Staudt, L. M., Hudson, J. Jr.,Boguski, M. S., Lashkari, D., Shalon, D., Botstein, D., andBrown, P. O. The Transcriptional Program in the Responseof Human Fibroblasts to Serum.Science, Vol. 283(1):83–87,January 1999.

[11] K. Y. Yeung and W. L. Ruzzo. Principal Component Anal-ysis for Clustering Gene Expression Data.Bioinformatics,Vol. 17(9):763–774, 2001.

[12] Kohonen, T. Self-Organizing Maps, Springer Series in In-formation Sciences, volume Vol. 30. Springer, Berlin, Hei-delberg, New York, 1995.

[13] Morrison, N., editor.Introduction to Fourier Analysis. JohnWiley & Sons, Inc., New York, NY, 1994.

[14] Parmigiani, G., Garrett, E. S., Irizarry, R. A., and Zeger, S.L., editor. The Analysis of Gene Expression Data: Methodsand Software. Springer-Verlag New York, Inc, New York,NY, 2003.

[15] Sammon, J. W. A nonlinear mapping for data structure anal-ysis. IEEE Transactions on Computers, C-18(5):401–409,1969.

[16] Stuart, R. O., Bush, K. T., and Nigam, S. K. Changes inGlobal Gene Expression Patterns During Development andMaturation of the Rat Kidney.Proc. Natl. Acad. Sci. USA,Vol. 98(10):5649–5654, May 2001.

[17] Troyanskaya, O., Cantor, M., Sherlock, G., Brown, P.,Hastie, T., Tibshirani, R., Botstein, D., and Altman, R. Miss-ing Value Estimation Methods for DNA Microarrays.Bioin-formatics, Vol.17(6):520–525, 2001.

[18] Ward, M. O. XmdvTool: Integrating Multiple Methods forVisualizing Multivariate Data. InIEEE Visualization 1994,pages 326–336, 1994.

[19] Zhang, L., Zhang, A., Ramanathan, M. et al. VizCluster andIts Application on Clustering Gene Expression Data.In-ternational Journal of Distributed and Parallel Databases,13(1):79–97, January 2003.

Appendix: Proof for Propositions

Lemma 1

(N−1∑n=0

an

)2

=

N−1∑n=0

a2n + 2

N−1∑

k=1

N−k−1∑t=0

at at+k.

Lemma 2 For any two complex numbersz and w, (1) z + w =z + w, (2) z w = z w, (3) z = z.

Lemma 3 Let j ∈ N, thenN−1∑n=0

e−i2πjn/N

=

N−1∑n=0

cos(2πjn/N) =

N−1∑n=0

sin(2πjn/N) = 0.

Lemma 4 FFHP is homomorphic: F1(ax[n] + by[n]) =aF1(x[n]) + bF1(y[n]).

Proposition 1 (Cancellation) If x[n] = [a, . . . , a], thenF1(x[n]) = 0.

Proof: From the formula in Eq. (2), we get

F1(x[n]) =

N−1∑n=0

a e−i2πn/N = a

N−1∑n=0

e−i2πn/N .

By Lemma 3, letj = 1. F1(x[n]) = 0. ¤

Proposition 2 (Amplitude Shifting) If y[n] = x[n] + a, thenF1(y[n]) = F1(x[n])

From the formula in Eq. (2), we get

F1(y[n]) =

N−1∑n=0

(x[n] + a)e−i2πn/N =

N−1∑n=0

x[n]e−i2πn/N

+

N−1∑n=0

a e−i2πn/N = F1(x[n]) + 0

The second summation is0 by Proposition 1. ¤

Proposition 3 (Amplitude Multiplying) If y[n] = ax[n], thenF1(y[n]) = aF1(x[n]).

From the formula in Eq. (2), we get

F1(y[n]) =

N−1∑n=0

a x[n]e−i2πn/N

= a

N−1∑n=0

x[n]e−i2πn/N = a F1(x[n])

¤

Proposition 4 (Transposing) y[n] = x[N − n − 1], thenF1(y[n]) = F1(x[n]).

By Lemma 2, we have

F1(x[n]) =

N−1∑n=0

x[n]e−i2πn/N =

N−1∑n=0

x[n]ei2πn/N

=

N−1∑n=0

x[n]ei2πn/N = F1(x[−n])

However, whenx[n] is a real signal,x[n] = x[n]. Then we have

F1(x[n]) = F1(x[−n]). i.e., F1(x[n]) = F1(x[−n]).

Sincex[N − n− 1] = x[−n] thenF1(y[n]) = F1(x[n]) ¤

Proposition 5 (Time Shifting) If y[n] = x[n − d], thenF1(y[n]) = Wd

N F1(x[n]).

Proof: Assume0 ≤ n < N , let l = n− d, thenn = l + d. Whenn = 0, l = −d and whenn = N − 1, l = N − 1− d. From theformula in Eq. (2), we get

F1(y[n]) = F1(x[n− d]) =

N−1−d∑

l=−d

x[l]e−i2π(l+d)/N

=

N−1−d∑

l=−d

x[l]e−i2πl/Ne−i2πd/N = WdN

N−1−d∑

l=−d

x[l]e−i2πl/N

However,ei2πn/N = ei2π(n+N)/N andx[n] = x[n + N ],

N−1−d∑

l=−d

x[l]e−i2πl/N =

−1∑

l=−d

x[l + N ] e−i2π(l+N)/N

+

N−1−d∑

l=0

x[l] e−i2πl/N

Let t = l + N for the first summation andt = l for the secondsummation, we get

N−1−d∑

l=−d

x[l]e−i2πl/N =

N−1∑

t=N−d

x[t]e−i2πt/N

+

N−1−d∑t=0

x[t]e−i2πt/N

=

N−1∑t=0

x[t]e−i2πt/N = F1(x[n])

Therefore,F1(y[n]) = F1(x[n])WdN . ¤

Definition 1 The mean of a signalx[n] is defined asx =∑N−1n=0 x[n]/N . Thek-th sample autocovariance coefficient of a

signalx[n] is defined asgk =∑N−1−k

n=0 (x[n]− x)(x[n + k] −x)/N . g0 is called the variance ofx[n]. Thek-th sample autocor-relation coefficient is defined asrk = gk/g0.

Proposition 6 (General Distance)Letw[n] = x[n]−y[n] be thedifference betweenx[n] andy[n]. The distance betweenF1(x[n])andF1(y[n]) is

‖F1(w[n])‖2 = g0N

(1 + 2

N−1∑

k=1

rk cos(2πk/N)

).

Proof: By Lemma 4, the distance betweenF1(y[n]) andF1(x[n]) is ‖F1(w[n])‖. From Eq. (2), we get

‖F1(w[n])‖ = ‖N−1∑n=0

w[n]e−i2πn/N‖

= ‖N−1∑n=0

w[n] cos(2πn/N)− iw[n] sin(2πn/N)‖

Let ω = 2π/N , by Lemma 3, we haveN−1∑n=0

cos(nω) =

N−1∑n=0

sin(nω) = 0. Now add a termw, the mean ofw[n],

‖F1(w[n])‖2 =

(N−1∑n=0

w[n] cos(nω)

)2

+

(N−1∑n=0

w[n] sin(nω)

)2

=

(N−1∑n=0

(w[n]− w) cos(nω)

)2

+

(N−1∑n=0

(w[n]− w) sin(nω)

)2

Expending each squaring term by Lemma 1, we get

N−1∑n=0

(w[n]− w)2(cos2(nω) + sin2(nω))

+ 2

N−1∑

k=1

N−1−k∑t=0

[(w[t]− w)(w[t + k]− w)Ω]

whereΩ = cos(tω) cos((t + k)ω) + sin(tω) sin((t + k)ω).By trigonometry identitycos θ cos φ + sin θ sin φ = cos(φ −

θ), we haveΩ = cos(kω). Now

‖F1(w[n])‖2 =

N−1∑n=0

(w[n]− w)2

+ 2

N−1∑

k=1

N−1−k∑t=0

[(w[t]− w)(w[t + k]− w) cos(kω)]

= N(g0 + 2

N−1∑

k=1

gk cos(kω))

= g0N

(1 + 2

N−1∑

k=1

rkcos(2πk/N)

). ¤