Fourier Harmonic Approach for Visualizing Temporal Patterns of Gene
Expression Data
Li Zhang and Aidong Zhang
Department of Computer Science and Engineering
State University of New York at Buffalo
Buffalo, NY 14260
{lizhang,azhang}@cse.buffalo.edu
Abstract
DNA microarray technology provides a broad snapshot
of the state of the cell by measuring the expression levels
of thousands of genes simultaneously. Visualization techniques can enable the exploration and detection of patterns
and relationships in a complex dataset by presenting the
data in a graphical format in which the key characteristics become more apparent. The purpose of this study is to
present an interactive visualization technique conveying the
temporal patterns of gene expression data in a form intuitive
for non-specialized end-users.
The first Fourier harmonic projection (FFHP) was introduced to translate the multi-dimensional time series data
into a two dimensional scatter plot. The spatial relationship
of the points reflect the structure of the original dataset and
relationships among clusters become two dimensional. The
proposed method was tested using two published, arrayderived gene expression datasets. Our results demonstrate
the effectiveness of the approach.
Keywords: visualization, gene expression, time series,
Fourier harmonic projection
1 INTRODUCTION
Knowledge of the spectrum of genes expressed at a certain time or under given conditions proves instrumental to
understand the working of a living cell. DNA microarray
technology allows measurements of expression levels for
thousands of genes simultaneously [9]. Extensive research
has been conducted on the study of temporal patterns of
gene expressions [10, 16]. Clustering methods which group
genes or samples with similar patterns have become mainstream analysis tool [14]. Visualization can facilitate the
discovery of structures, features, patterns, and relationships
in data and may provide more insightful information than
Murali Ramanathan
Department of Pharmaceutical Sciences
State University of New York at Buffalo
Buffalo, NY 14260
murali@acsu.buffalo.edu
traditional numerical methods. By visualization, we hope
to gain some intuition regarding the data, but more importantly, we would like to understand the relationships between data points and detect the intrinsic structure, or possible cluster tendencies. Visualization is especially important
in the early stages of data analysis in which qualitative analysis is primary to quantitative. Early success will enhance
the users’ performance in the remaining stages of analysis. Array-derived gene expression datasets present analysis
and visualization challenges because of their dimensionality, noisy environment, and pattern varieties.
The parallel coordinates approach [18] is perhaps the
simplest method to display patterns of gene expression profiles. Here the data in each dimension are plotted along a
separate axis. Holter et al. [8] have used parallel coordinates to visualize the temporal progression in the yeast cell
cycle data. Self-organizing maps (SOM) [12] is another approach. More recently, Hautaniemi et al. [6] have presented
a heat map-based strategy for visualizing the U-matrix from
SOM. The most prominent visualization-enhanced analysis tool for gene expression data is TreeView [5], which
provides a user-friendly computational and graphical environment for assessing the results from hierarchical clustering. The graphical presentations from TreeView include
a dendrogram to reflect the distance relationships between
clusters and a heat plot to visually convey gene expression
changes between samples. The heat plot can be viewed as
variation of the parallel coordinates plot in which color is
used to convey dimension values.
Here, we present an alternative mapping for multidimensional data that is based on the first harmonic of the
discrete Fourier transform. The mapping has interesting
properties and preserves certain key characteristics of a variety of data sets, especially time series data. Unlike parallel coordinates and heat plot which display all individual
dimensional information, our approach uses a two dimensional point to represent each gene over the time at the com-
putational cost of O(N log N ). It focuses on one very important aspect of the visualization: revealing the structure of
the entire dataset. Tested using two published, array-derived
gene expression time series datasets, our results indicated
that temporal patterns were well reflected in the visualization: cluster relationship became two dimensional, clusters
were apparent, and outliers were clear. An interactive visualization tool, VizStruct, was implemented to perform the
visualization.
The remainder of this paper is organized as follows. Section 2 presents the model of visualization. In section 3, we
show our analysis results. The final section discusses other
issues in this approach. Proofs for all mapping propositions
are included as appendix.
2 METHODS AND SYSTEM
The Mapping
Mapping converts multi-dimensional data to twodimensions for visualization. An array-derived profile
for M genes with N measurements results in M N dimensional data point containing real valued numerical
data. Time series data in its simplest form is merely a set
of data {yt , t = 0, . . . , N − 1} where the subscript t indicates the time at which the datum yt was observed [4]. On
the other hand, a discrete-time real signal on N evenly distributed time points is represented as an indexed sequence
of N real numbers 0, . . . , N − 1 denoted by x[n] [1]. Each
term of x[n] is denoted by x[n]. The denotation similarity
between time series and digital signal suggests that we can
view each data point in a time series as a discrete-time real
signal (It is not necessary for the signal’s time index to comply with the actual time points). In this scenario, the problem of a two dimensional visualization of the time series is
transformed into the problem of finding a two-dimensional
point estimation for signals (data points).
The frequency domain representation of discrete-time
signals is through discrete-time Fourier transform, or DFT
[13]. The DFT of a N -point signal x[n] is a frequency sequence with N complex values: F(x[n]) = [Fk (x[n])],
where each
Fk (x[n]) =
N
−1
X
nk
x[n]WN
,
k = 0, . . . , N −1.
(1)
n=0
WN = e−i2π/N is called twiddle factor.
Each harmonic Fk in the DFT is a measure of the kth
sinusoidal frequency component in the signal. For example, the zero harmonic, F0 , is the mean value, the first harmonic F1 measures the base frequency component, the second harmonic F2 measures the component in the signal that
is twice the base frequency, and so forth. Because Fourier
harmonics, in general, are complex numbers, they provide
the two-dimensional point estimate for mapping a multidimensional signal. For this reason, we refer to the mapping
as the Fourier harmonic projections. In particular, we are
interested in the first Fourier hamonic projection (FFHP):
F1 (x[n]) =
N
−1
X
n
x[n]WN
=
n=0
N
−1
X
x[n]e−i2πn/N .
(2)
n=0
The relationship between the DFT and the mapping allows the fast Fourier transform algorithm (FFT), originally
discovered by Cooley and Tukey [13], to be used for computation. The FFT is a computationally efficient algorithm
and has a complexity of O(N log N ).
The complex number of F1 (x[n]) in Equation (2) can be
expressed in terms of magnitude r and phase θ to provide a
useful geometric interpretation of the mapping. The data set
was normalized so that the range of values of each dimension across the dataset was 0 to 1. For a data point with N
dimensions, the complex exponential divides a unit circle
centered at the origin of the complex plane into N equally
spaced angles. The value of the first dimension is projected
on the radial line corresponding to θ = 0 and, similarly, the
value of the kth dimension is projected on to the radial line
corresponding to the θ = 2π(1 − k)/N radians. The overall
two-dimensional FFHP mapping is the complex sum of all
N projections from a data point. Figure 1 illustrates the geometric interpretation for a point containing 6 dimensions.
W46
W56
unit circle
F1(x[n])
W36
W06
x[n]
2
W6
1
W6
Figure 1. A geometric interpretation of the
first Fourier harmonic projection. A normalized 6-dimensional data point is shown on the
right by the stem plot. The powers of twiddle factor W6 divide the unit circle centered
at the origin into 6 equal angles and each dimension of the data point is projected onto a
different radial angle (open circle). The projections are taken complex number sum to
give a 2-dimensional image (indicated by a
filled circle).
+a
xa
origin
origin
amplitude scaling
amplitude shifting
time shifting
(A)
time shifting
(B)
Figure 2. Illustration of the effect of (A) amplitude shifting and multiplying, and (B) time shifting of
the first Fourier harmonic projection.
Properties of FFHP
Theory in Practice
The first Fourier harmonic projection has useful properties that preserve the correlation between dimensions in
the multi-dimensional data point. We summarize them as
propositions below. Detailed proofs for propositions are
provided as the appendix.
1. Data points with equal values for all the dimensions
are mapped to the origin. If x[n] = [a, . . . , a], then
F1 (x[n]) = 0.
2. Two data points whose dimension values differ due to the
amplitude shifting of a constant are mapped to the same
point. If y[n] = x[n] + a, then F1 (y[n]) = F1 (x[n]).
3. Two data points whose dimension values differ due to the
amplitude multiplying a constant are mapped to the two
points on a line through the origin. If y[n] = a x[n], then
F1 (y[n]) = a F1 (x[n]). See Figure 2A.
4. Two data points whose dimension values are transposing
each other, i.e. symmetric regarding the middle time point,
are mapped to the points symmetric to the real axis. If
y[n] = x[N − n − 1], then F1 (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 the
circumference of the circle that is concentric with the unit
circle and the angle between the points in the visualization
is φ = 2πd/N . If y[n] = x[n − d], then F1 (y[n]) =
d
F1 (x[n])WN
. This property is illustrated in Figure 2B.
6. Let w[n] = x[n] − y[n] be the difference between the two
N -dimensional points, x[n] and y[n]. The distance between these two points in the visualization is:
Ã
!
N
−1
X
2
kF1 (w[n])k = g0 N 1 + 2
rk cos(2πk/N ) (3)
k=1
We will demonstrate in the next section that the relative
locations of temporal profiles’ mapping images can be predicted by Propositions 1–5. For example, genes with relatively low levels of expression throughout the time line are
mapped close to the origin. Genes that steadily increase
over the time and genes that steadily decrease over the time
line are likely mapped symmetric to the real axis.
In Equation (3), the g0 is the variance of w[n], rk is
the kth-sample autocorrelation coefficient of w[n]. It can
be shown [4] that −1 ≤ rk ≤ 1 and for mutually independent random sequences or white noise, rk = 0. Equation (3) provides insight into the cluster delineation capabilities of the first Fourier harmonic projection. The variance and kth-sample autocorrelation coefficients of the difference between two points within a given cluster are likely
to be small because they will share similarities across many
dimensions. Thus, points within cluster are likely to map
close to each other in the visualization.
The first Fourier harmonic projection is well suited for
visualizing temporal patterns. FFHP has a better class separation capability when the first harmonic is more dominant
among all harmonics. Since the first harmonic is a measure
of the base sinusoidal frequency component in the signal, a
lower frequency signal tends to have a larger first harmonic.
This is the case for a large number of time series data where
most trend patterns show low frequency variations (not having multiple cycles). The first Fourier harmonic projection
is sensitive to dimension order. For time series data, the
time points (dimensions) are naturally ordered.
VizStruct System
The first Fourier harmonic projection approach is implemented as a visualization tool called VizStruct, which
is written in Java and is available on request from the first
author (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 containing 25, 000 points. The parameter settings for (A)–(C) were h0.5, 0.5, 0.5i, h0.3, 0.5, 1i, and h0.05, 1, 1i,
respectively.
phasizes its capability of visualizing the structure of the
dataset.
3 RESULTS
Data Sets for Visualization
Dimension Tour
The dimension tour is a feature of VizStruct that allows
the user to interact with the data via dynamic animations.
It is analogous to the grand tour [2] and the user interacts
with the visualization by changing the dimension parameter associated with each dimension. The default value for
each dimension parameter is 0.5 and the individual parameters can be changed over the range −1 to +1 either manually or systematically using the program. Because no twodimensional mapping can capture all the interesting properties of the original multi-dimensional space, two points
that are close in the visualization can theoretically be far
apart in the multi-dimensional space. The dimension tour,
which creates animations that explore dimension parameter
space, can reveal structures in the multi-dimensional input
that were hidden due to overlap with other points in the visualization.
Figure 3 illustrates the capabilities of the dimension tour
using a synthetic dataset containing 25, 000 points in three
dimensions. At the default settings of dimensional parameters (Figure 3A), 5 clusters are apparent. However, during the course of the animations (Figures 3B and 3C), the
multi-layered structures of the original 5 clusters become
increasingly apparent.
Our approach was tested using two published arrayderived data sets. The rat kidney array dataset of Stuart et
al. [16] containes measurements of gene expressions during
rat kidney organogenesis. The data were downloaded from
http://organogenesis.ucsd.edu/data.html. It consists of 873
genes which vary significantly during kidney development
at 7 different time points: gestational day 13, 15, 17, 19;
newborn (N); 1 week (W); and nonpregnant adult (A).
The fibroblasts dataset of Iyer et al. [10] is the result of a
study of the response of human fibroblasts to serum. It consists of gene expressions measuring the temporal changes
in mRNA levels of 517 human genes at 13 time points,
ranging from 15 minutes to 24 hours after serum stimulation. The data were downloaded from http://genomewww.stanford.edu/serum.
Rat Kidney Dataset
In the rat kidney dataset, there are 5 discrete patterns or
groups of gene expression during nephrogenesis. Figure 4
illustrates these temporal profiles characterized by the idealized gene expressions.
Figure 5A shows how genes are classified by a hierarchical clustering algorithm. It copies the Figure 3 in [16].
Figure 5B shows the parallel coordinates of the dataset. Patterns of genes in each group comply to the profiles depicted
in Figure 4 (with some noise).
3 1
2.5
2
1.5
1
0.5
0
13151719 N W A
2
13151719 N W A
3
5
4
13151719 N W A
13151719 N W A
13151719 N W A
Figure 4. Idealized temporal gene expression profiles during kidney development. The groups were
named 1 through 5 based on the timing of their peak expression during development. Seven time
points were 13, 15, 17, 19 embryonic days; N , newborn; W , 1 week old; A, adult.
3
3
3
2
2
2
1
1
1
0
All Data
0
Group 1
0
3
3
3
2
2
2
1
1
1
0
Group 3
0
(A)
Group 4
0
Group 2
Group 5
(B)
Figure 5. Visualization of the rat kidney dataset in heat plot and parallel coordinates. (A) Dendrogram
and a heat plot from hierarchical clustering algorithm. (B) Parallel coordinates for the entire datset
and each of the gene groups.
Figures 6A-B show the visualization of the rat kidney
dataset in VizStruct under the first Fourier harmonic projection for two dimension parameter settings. There are 5
sets of colored symbols for each of the 5 gene groups. Each
symbol represents one gene across 7 time points. In Figure
6A, two big clusters are clearly apparent from the visualization. The top cluster consists of genes from groups 3, 4, and
5. The bottom cluster is comprised of genes from groups 1
and 2. Furthermore, genes from each group are aggregated.
The formation of two large clusters can be interpreted by
the temporal profiles. Groups 1 and 2 with genes which
have very high relative levels of expression in early development are quite different from groups 3, 4, and 5 for
genes that have a relatively steady increase in expression
throughout development. The visualization also indicates
that points in the upper cluster are symmetric to the points
in the lower cluster. Properties of FFHP may suggest the
reason. Temporal profiles of groups 1 and 4 suggest that
they are somewhat symmetric to the middle time point (gestational day 19). By Proposition 4, they would be mapped
to points symmetric to the real axis. On the other hand,
groups 4 and group 5 are mapped closely since they have
similar profiles except for the significantly up-regulated in
the last time point. The same arguments can be applied in
the case of group 1 vs. group 2, or group 3 vs. group 4.
Due to the noise, most boundaries between groups are
not very clear. However, the separation between group 4
and group 5 improves in Figure 6B compared to Figure 6A.
Fibroblast Dataset
Temporal patterns are slightly complicated in the fibroblast dataset. Data has been classified into 10 groups using
the hierarchical clustering algorithm by the original author
[10]. Figure 7A shows the result of the hierarchical clustering. It is a copy of Figure 3 from [10]. Figure 7B gives the
6
12
G2
G4
10
G5
4
G3
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Im[F1]
Im[F1]
G4
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2
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−6
−4
−2
0
(A)
2
Re[F1]
4
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10
(B)
Figure 6. Visualization of the rat kidney dataset in VizStruct. Five gene groups were represented
by blue plus symbols, red circles, green triangles, magenta stars, and black cross symbols. The
dimension parameters for (A) and (B) were h0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5i and h1, 0.5, −1, −0.5, 0.5, 1, −1i,
respectively.
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0
All
Group A
Group B
Group C
Group D
Group E
Group F
Group G
Group H
Group I
Group J
Group U
15
10
5
0
(A)
(B)
Figure 7. Visualization of the fibroblast dataset in heat plot and parallel coordinates. (A) Dendrogram
and a heat plot from hierarchical clustering algorithm. (B) Parallel coordinates for the entire datset and
each of the gene groups. The gene group labelled U consists genes without label after hierarchical
clustering.
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(A)
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5 were used for each
Figure 8. Visualization of the fibroblast dataset in VizStruct. 11 colored numbers
of the gene groups. (A) The mapping of the entire dataset. (B) Enlarged portion of the visualization
10
in (A).
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Figure 9. Visualization of the fibroblast dataset in VizStruct. Genes were grouped by the k-means
clustering method. 11 colored numbers were used for each gene groups. (A) The mapping of the
entire dataset. (B) Enlarged portion of the visualization in (A).
parallel coordinates the entire dataset and each gene group.
Figure 8A shows the visualization of the fibroblasts
dataset in VizStruct under the first Fourier harmonic projection. Colored numbers 1 through 11 were used for each
gene group. The visualization reveals several outliers and
no distinct clusters. The majority of data are too dense to be
seen. By zooming technology, the enlarged detail is shown
in Figure 8B. More numbers spread out, but most blue numbers (1) were still covered by numbers 2 and 3.
Two observations can be made: (1) a large number of
genes are close to the center. (2) most clusters have a radial
shape emitting from the center. They can be interpreted by
the FFHP properties. (1) Temporal patterns of group A and
group B have very flat shape and relative smaller values. By
Propositions 1 and 2, they tend to be mapped closed to the
origin. (2) In hierarchical clustering, Pearson’s correlation
coefficient was used to measure the similarity. Genes whose
time values differ due to the amplitude shifting or multiply-
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−4
5
(B)
−3
−2
−1
0
x
1
2
3
(C)
Figure 10. Comparison of Sammon’s mapping and the first Fourier harmonic projection. (A) The
result from Sammon’s mapping of the rat kidney dataset. Five gene groups were represented by 5
colored symbols the same as were used in Figure 6. (B)–(C) Comparison of relative gene locations
between (B) VizStruct and (C) Sammon’s mapping. Five colors were used for each of the gene
groups. The color schema is the same as in (A). Notice that the visualization layout of (A) and (C) is
identical. 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 Propositions 2 and 3, they tend to be mapped closely along a line
though the origin.
Some relative gene group locations can be predicted. For
example, by Proposition 4, groups 4 and 5 are mapped symmetric to the real axis. Close inspection suggests that temporal pattern of group 8 is somewhat a 2 or 3 right time-shift
from the pattern of group 6. By Proposition 5, each time
point shift correspondents to roughly 2/13π ≈ 30◦ rotation. In Figure 8B, genes in group 8 are mapped about 60◦
clockwise to genes in group 6.
Lacking clear clusters in the visualization indicates that
different clustering methods may yield different results.
Figure 9 shows the grouping aspect of k-means clustering. Euclidean distance was used in k-means as the distance
measure. The visualization reveals that genes closed to the
origin are grouped as one.
Comparison of FFHP to Other Visualizations
Heat plot and parallel coordinates display all individual
dimensional information. In parallel coordinates, a polyline
represents a gene over time. This format is a concise and
intuitive for displaying temporal profiles. However, the parallel coordinate has obvious drawbacks: when the data size
becomes larger it becomes increasingly unreadable as indicated in the first panel of Figure 5 and Figure 7. Heat plot
uses a color mosaic instead of a polyline to overcome over-
lapping. Combined with the dendrogram from hierarchical
clustering, it gives a global view as well as individual clusters of the dataset by grouping genes with similar patterns
together. Cluster relationships in heat plot is one dimensional: clusters of genes are listed one by one as shown in
Figure 5A and Figure 7A.
The first Fourier harmonic projection takes a different
approach. It uses a two dimensional point to represent each
gene over time. By doing so, it takes advantage of the spatial relationship of the points to reflect the structure of the
original dataset. Thus the cluster relationships become two
dimensional and FFHP has a better capability of displaying
outliers than heat plot. Unlike heat plot, which requires algorithms to group similar genes to make a meaningful visualization, FFHP directly mapped the multi-dimensional data
onto a two dimensional space without any prior knowledge
of the dataset.
Multidimensional scaling (MDS) [3] is a competing approach for visualizing multi-dimensional data. We compared FFHP to Sammon’s mapping [15], a variant of MDS
that optimizes the following stress function E:
1
E = XX
i
X X (d∗ij − dij )2
,
d∗ij
d∗ij i j<i
(4)
j<i
where d∗ij is the distance between points i and j in the N dimensional space and dij is the distance between i and j
in the visualization.
Figure 10 shows Sammon’s mapping of the rat kidney
dataset. A comparison of Figure 10A to Figure 6A reveals the extensive similarities between FFHP and Sammon’s mapping. The relative locations of individual samples are also remarkably similar. This is indicated in Figure
10B and Figure 10C.
Sammon’s mapping has some drawbacks: (1) it provides
a single final result and the user cannot intervene interactively during visualization, (2) the incremental addition of
even a single point requires a complete repetition of the optimization procedure and possible extensive reorganization
of all the previously mapped points to new locations, and
(3) it requires time-consuming optimization procedures of
time complexity O(N 2 ) or greater.
Our results illustrate some of the strengths and weaknesses of FFHP. The visualization reflects the structure of
the dataset: outliers, clusters and their relationships. Comprehending the structure of the dataset can facilitate the
choice and understanding the results of different clustering
methods. Although the FFHP uses an approach to mapping
multi-dimensional data that is distinctively different from
Sammon’s mapping, it yields results that are consistently
comparable. However, when the dataset contains a large
number of patterns, it becomes difficult to separate different
patterns and the visualization may be difficult to interpret.
4 DISCUSSION
Visualization of microarray data is a challenge because
of its high dimensionality. In this paper, we have explored
use of the first Fourier harmonic projection for visualizing
multi-dimensional time course array datasets. Unlike parallel coordinate or heat plot approach which display all dimensional information, FFHP uses a two dimensional point
to represent each data point (gene in this case). Our results indicated that temporal patterns were well reflected by
spatial relationships: genes with similar pattern were aggregated and relative locations of gene groups can be predicted.
Moreover, the first Fourier harmonic projection was shown
to yield results that were similar to those from Sammon’s
mapping.
Achieving two dimensional mapping requires a tradeoff. The mapping is lossy for detailed dimensional information. Our approach attempts to preserve to the maximum
semantics of the data points via Fourier harmonic aspect.
In particular, characterizing the data using two descriptive
measurements: the real and imaginary portions of the first
discrete Fourier harmonic. A similar approach uses principal component analysis (PCA) [11]. This visualization
deploys another two descriptive measurement: the first and
second 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 exits
an equivalent first harmonic of the original discrete signal
whose time indices are systematically rearranged. At certain conditions (such as temporal patterns with high frequency variations, i.e. multiple cycles), higher harmonic
projections enhance substructure separation in the visualization. Detailed discussion is beyond the scope of this paper due to a length constraint.
The FFHP mapping results in two-dimensional visualizations that are identical to those of radial coordinate visualization techniques, e.g., RadViz [7]. However, rather than
vector notation and the spring paradigm of RadViz, we have
used a complex number notation. This substantive reformulation of the mapping provides valuable theoretical insights
and allows important properties of mapping, including its
relationship to the DFT, to be easily derived. It can also
create possible extensions such as higher Fourier harmonic
projections.
The first Fourier harmonic projection requires data without missing values. This requirement can be easily met
because filling in missing values is a mature research field
[17].
Gene expression data can be studied in either sample
space or gene space. Here, we have reported only the visualization on gene space. In a separate report [19], we applied
the first Fourier harmonic projection on the sample space
and performed visualization-driven classifications. Our experiments demonstrated that FFHP offered an alterative format of visualization. We believe that using FFHP alone or
in combination with heat plot or parallel coordinates would
give a biologist additional powerful tools for analyzing and
visualizing microarray data sets.
ACKNOWLEDGEMENTS
This work was supported by grants from the National
Science Foundation. We also thank the anonymous reviewers for their constructive comments on the manuscript.
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Appendix: Proof for Propositions
Lemma 1
ÃN −1
X
!2
an
=
N
−1
X
n=0
a2n + 2
n=0
−k−1
N
−1 N X
X
at at+k .
t=0
k=1
Lemma 2 For any two complex numbers z and w, (1) z + w =
z + w, (2) z w = z w, (3) z = z.
Lemma 3 Let j ∈ N, then
=
N
−1
X
cos(2πjn/N ) =
n=0
N
−1
X
e−i2πjn/N
n=0
N
−1
X
sin(2πjn/N ) = 0.
n=0
Lemma 4 FFHP is homomorphic: F1 (a x[n] + b y[n]) =
a F1 (x[n]) + b F1 (y[n]).
Proposition 1 (Cancellation) If x[n]
F1 (x[n]) = 0.
=
[a, . . . , a], then
Proof : From the formula in Eq. (2), we get
F1 (x[n]) =
N
−1
X
a e−i2πn/N = a
n=0
N
−1
X
e−i2πn/N .
n=0
By Lemma 3, let j = 1. F1 (x[n]) = 0.
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Proposition 2 (Amplitude Shifting) If y[n] = x[n] + a, then
F1 (y[n]) = F1 (x[n])
From the formula in Eq. (2), we get
F1 (y[n])
=
N
−1
X
(x[n] + a)e−i2πn/N =
n=0
+
N
−1
X
N
−1
X
x[n]e−i2πn/N
n=0
a e−i2πn/N = F1 (x[n]) + 0
n=0
The second summation is 0 by Proposition 1.
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Proposition 3 (Amplitude Multiplying) If y[n] = a x[n], then
F1 (y[n]) = a F1 (x[n]).
From the formula in Eq. (2), we get
F1 (y[n])
=
N
−1
X
a x[n]e−i2πn/N
n=0
=
a
N
−1
X
x[n]e−i2πn/N = a F1 (x[n])
n=0
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Proposition 4 (Transposing) y[n] = x[N − n − 1], then
F1 (y[n]) = F1 (x[n]).
Proof : By Lemma 4, the distance between F1 (y[n]) and
F1 (x[n]) is kF1 (w[n])k. From Eq. (2), we get
By Lemma 2, we have
F1 (x[n])
=
N
−1
X
x[n]e−i2πn/N =
n=0
=
N
−1
X
N
−1
X
x[n]ei2πn/N
n=0
kF1 (w[n])k
x[n]ei2πn/N
=
k
= F1 (x[−n])
N
−1
X
w[n]e−i2πn/N k
n=0
n=0
=
However, when x[n] is a real signal, x[n] = x[n]. Then we have
k
N
−1
X
w[n] cos(2πn/N ) − iw[n] sin(2πn/N )k
n=0
F1 (x[n]) = F1 (x[−n]). i.e., F1 (x[n]) = F1 (x[−n]).
Since x[N − n − 1] = x[−n] then F1 (y[n]) = F1 (x[n])
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Let ω = 2π/N , by Lemma 3, we have
N
−1
X
cos(nω) =
n=0
Proposition 5 (Time Shifting) If y[n]
d
F1 (y[n]) = WN
F1 (x[n]).
=
x[n − d], then
N
−1
X
sin(nω) = 0. Now add a term w,
b the mean of w[n],
n=0
Proof : Assume 0 ≤ n < N , let l = n − d, then n = l + d. When
n = 0, l = −d and when n = N − 1, l = N − 1 − d. From the
formula in Eq. (2), we get
NX
−1−d
F1 (y[n]) = F1 (x[n − d]) =
kF1 (w[n])k
2
=
NX
−1−d
d
x[l]e−i2πl/N e−i2πd/N = WN
l=−d
x[l]e−i2π(l+d)/N
=
NX
−1−d
x[l]e−i2πl/N
+
=
l=−d
−1
X
N
−1
X
l=−d
+
x[l] e−i2πl/N
=
l=−d
N
−1
X
x[t]e−i2πt/N
2
N
−1 N X
−1−k
X
+
=
(w[n] − w)
b sin(nω)
kF1 (w[n])k2
x[t]e−i2πt/N
[(w[t] − w)(w[t
b
+ k] − w)Ω]
b
where Ω = cos(tω) cos((t + k)ω) + sin(tω) sin((t + k)ω).
By trigonometry identity cos θ cos φ + sin θ sin φ = cos(φ −
θ), we have Ω = cos(kω). Now
=
N
−1
X
(w[n] − w)
b 2
n=0
t=0
N
−1
X
!2
t=0
k=1
t=N −d
NX
−1−d
ÃN −1
X
(w[n] − w)
b 2 (cos2 (nω) + sin2 (nω))
+
Let t = l + N for the first summation and t = l for the second
summation, we get
x[l]e−i2πl/N
(w[n] − w)
b cos(nω)
n=0
l=0
NX
−1−d
ÃN −1
X
w[n] sin(nω)
n=0
!2
Expending each squaring term by Lemma 1, we get
x[l + N ] e−i2π(l+N )/N
NX
−1−d
+
!2
n=0
However, ei2πn/N = ei2π(n+N )/N and x[n] = x[n + N ],
x[l]e−i2πl/N
w[n] cos(nω)
ÃN −1
X
n=0
l=−d
NX
−1−d
!2
n=0
l=−d
=
ÃN −1
X
+
x[t]e−i2πt/N = F1 (x[n])
2
N
−1 N X
−1−k
X
t=0
d
Therefore, F1 (y[n]) = F1 (x[n])WN
.
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Definition
1 The mean of a signal x[n] is defined as x
b =
PN −1
autocovariance coefficient of a
n=0 x[n]/N . The k-th sample
P −1−k
signal x[n] is defined as gk = N
(x[n] − x
b)(x[n + k] −
n=0
x
b)/N . g0 is called the variance of x[n]. The k-th sample autocorrelation coefficient is defined as rk = gk /g0 .
Proposition 6 (General Distance) Let w[n] = x[n]−y[n] be the
difference between x[n] and y[n]. The distance between F1 (x[n])
and F1 (y[n]) is
Ã
!
N
−1
X
kF1 (w[n])k2 = g0 N 1 + 2
rk cos(2πk/N ) .
k=1
=
N (g0 + 2
Ã
=
g0 N
[(w[t] − w)(w[t
b
+ k] − w)
b cos(kω)]
t=0
k=1
N
−1
X
gk cos(kω))
k=1
1+2
N
−1
X
k=1
!
rk cos(2πk/N ) .
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