Tests for the elliptical symmetry of hyperspectral imaging
data
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Citation
Sidi Niu, Vinay K. Ingle, Dimitris G. Manolakis and Thomas W.
Cooley, "Tests for the elliptical symmetry of hyperspectral
imaging data", Proc. SPIE 7812, 78120D (2010);
doi:10.1117/12.861055 © 2010 COPYRIGHT SPIE
As Published
http://dx.doi.org/10.1117/12.861055
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SPIE
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Final published version
Accessed
Fri May 27 00:18:40 EDT 2016
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http://hdl.handle.net/1721.1/60938
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Detailed Terms
Tests for the Elliptical Symmetry of Hyperspectral Imaging
Data
Sidi Niua , Vinay K. Inglea , Dimitris G. Manolakisb and Thomas W. Cooleyc
a Northeastern
University, 360 Huntington Avenue, Boston, MA 02115;
Lincoln Laboratory, 244 Wood Street, Lexington, MA 02420;
c Air Force Research Laboratory, 3550 Aberdeen Ave SE, Kirtland AFB, MN 87117.
b MIT
ABSTRACT
Accurate statistical models for hyperspectral imaging (HSI) data distribution are useful for many applications.
A family of elliptically contoured distribution (ECD) has been investigated to model the unimodal ground cover
classes. In this paper we propose to test the elliptical symmetry of real unimodal HSI clutters which will answer
the question whether the family of ECD will provide an appropriate model for HSI data. We emphasize that the
elliptical symmetry is an inherent feature shared by all ECDs. It is a prerequisite that real HSI clutters must
pass these elliptical symmetry tests, so that the family of ECD can be qualified to model these data accurately.
Keywords: hyperspectral imaging, modeling, elliptical symmetry testing
1. INTRODUCTION
Accurate statistical models for HSI data distribution are useful for many applications. These models provide
the foundation for development and evaluation of reliable algorithms for detection, classification, clustering, and
estimation. A typical hyperspectral image usually contains several different ground cover classes and consequently
can be modeled by a finite mixture statistical model. Each component density function of the mixture model is
supposed to characterize a single unimodal ground clutter. A family of ECD has been investigated to model the
unimodal ground cover classes. Regardless of the mean vector, an ECD can be determined by the scale/covariance
matrix and its generating function which are assumed to be independent from each other. Since the covariance
matrix can be estimated from the sample covariance matrix (sometimes regularized sample covariance matrix),
the only flexibility left is finding an appropriate generating function to characterize real HSI data behaviors.
Previous work has focused on fitting a specific ECD in the Mahalanobis distance sense to capture the heavy
tail behavior of real HSI data. In this paper we stress on the other issue that testing the elliptical symmetry
of real unimodal HSI clutters will answer the question whether the family of ECD will inherently provide an
appropriate model for HSI data. Several graphical testing schemes are proposed. Instead of testing elliptical
symmetry directly, HSI data are first prewhitened and then checked for their spherical symmetry. A t-plot and a
β-plot correlates the spherical symmetry to graphical linearity. Pairwise scatter plots and angular goodness-of-fit
test evaluate the symmetry behavior in polar coordinates. Numerical measures are also calculated in each case
as auxiliaries to graphical plots. We emphasize that the elliptical symmetry is an inherent feature shared by all
ECDs. It is a prerequisite that real HSI clutters must pass these elliptical symmetry tests, so that the family of
ECD can qualify to model these data accurately.
The organization of this paper is as follows. In Section 2, we first discuss the mathematical background of
statistical modeling as well as definition and properties of ECDs. In Section 3, the proposed graphical elliptical
symmetry tests derived from the unique properties of ECDs are discussed and illustrated by synthetic data.
These tests are then applied to real clustered HSI data in Section 4. The main results and conclusions are
summarized in Section 5.
Further author information: (Send correspondence to Sidi Niu)
Sidi Niu: E-mail: niusidi@gmail.com, Telephone: 1 617 710 0991
Imaging Spectrometry XV, edited by Sylvia S. Shen, Paul E. Lewis, Proc. of SPIE Vol. 7812,
78120D · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.861055
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2. MATHEMATICAL BACKGROUND
2.1 Statistical modeling
A typical HSI image of remote sensing, for example a down-looking airborne image, usually includes large area
and thus contains different ground covers, called clutters. Hence the whole image has unimodal distribution which
is hard to represent. We assume that the whole image can be well classified into finite number of distributional
unimodal clutters. Each clutter can thus be modeled separately. The multimodal probability density function
(pdf) representing the whole image can be obtained by the finite mixture model1
f (x; Ψ) =
K
πi fp (x; Θi )
(1)
i=1
where Ψ is the set including all parameters, Ψ = [π1 , . . . , πK , Θ1 , . . . , ΘK ]. The mixture weights πi , or priors,
K
satisfy the constraints, 0 ≤ πi ≤ 1 and i=1 πi = 1, and fp (x; Θi ) is a single unimodal pdf characterized by
parameter Θi with a dimensionality of p. The finite mixture model describes the whole hyperspectral image
as a mixture of pixels from K different spectrally homogeneous distributions, with the probability of a pixel
belonging to one of the distributions equals to its weight πi . In this paper, the constitutional pdfs fp (x; Θi ) are
assumed to be ECDs, that is fp (x; Θi ) = ECDp (x; μ, Σ, q). The graphical symmetry tests proposed in Section 3
will attempt to verify this ECD assumption. We first discuss the definition as well as some key properties of
ECDs.
2.2 Elliptically Contoured Distribution
A random vector x = [x1 , . . . , xp ]T , with dimensionality p, is said to have distribution belonging to the family
of elliptically contoured distribution, denoted by x ∼ ECDp (x; μ, Σ, q) if its pdf, when exists, can be expressed
as a function of the quadratic form (x − μ)T Σ−1 (x − μ) and is given by2
p
1
fx (x) = (2π)− 2 (det Σ)− 2 q((x − μ)T Σ−1 (x − μ)),
(2)
where μ = [μ1 , . . . , μp ]T is the location parameter, Σ is the scale matrix (symmetric and positive definite), and
q(·) is called generating function which categorizes subset distributions within the family. Since the location
parameter can be set to be zero by changing the origin, an ECD can be fully characterized by its scale matrix
Σ and generating function q(·) without loss of generality.
The ECD family is a generalization of the widely-used normal or Gaussian distribution by introducing extra
degrees of freedom. In fact, a representation theorem3 exists for all ECDs which is summarized below.
Theorem 1. If a random vector is an ECD random vector, then there exists a nonnegative random variable s
such that the pdf of the random vector conditioned on s is a multivariate normal pdf.4
Given the representation theorem, any ECD can be generated by modulation of normal distribution. Let
z = [z1 , z2 , . . . , zp ]T denote a real, zero mean, normal random vector with covariance matrix C. Let s denote a
nonnegative random variable with pdf fs (s), called characteristic pdf. Consider the product defined by x = s z.
For the problem of background clutter modeling and simulation, it is desirable to independently control the nonGaussian clutter envelop pdf and its correlation properties. Therefore, z and s are assumed to be statistically
independent.4 The generating function q(·) is solely determined by the characteristic pdf fs (s), while the scale
matrix Σ is related to the covariance matrix C by a positive scaler α > 0, that is Σ = α C. Hence, the scale
matrix Σ and the generating function q(·) characterizes a particular ECD independently. We emphasize that
the correlation property given by the scale matrix Σ does not categorize which subclass of an ECD belongs to,
it is the generating function q(·) which discriminates different subclasses within the ECD family.
Since the same scale matrix Σ can be shared by any ECD, in order to unveil the symmetry property, without
generality, a whitening process is applied to an ECD. Correspondingly, a spherical distribution is obtained with
the same generating function but with zero mean and identity matrix as its covariance matrix. In this case,
each marginal distribution is an identical independent distribution (i.i.d.). Important insights can be gained if
generalized spherical coordinate representation is applied.5
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Theorem 2. A random vector z = [z1 , z2 , . . . , zp ]T , which has zero-mean and identity covariance matrix, is
ECD if and only if there exist random variables r ∈ (0, ∞), θk ∈ (0, π), 1 ≤ k ≤ p − 2 and θp−1 ∈ (−π, π) such
that when the components of z are expressed in the generalized spherical coordinates:4
r = z12 + z22 + . . . + zp2 ,
(3a)
cos θ1 = z1 /r,
0 < θ1 < π,
zk
,
0 < θk < π,
cos θk =
zk−1 tan θk−1
−π < θp−1 < π.
tan θp−1 = zp /zp−1 ,
(3b)
2 ≤ k ≤ p − 2,
(3c)
(3d)
then the random variables r and θk , 1 ≤ k ≤ p − 1 are mutually statistically independent with pdfs of the form:
rp−1
q(r2 ),
2(p/2)−1 Γ(p/2)
Γ[(p − k + 1)/2]
fk (θk ) = √
sinp−1−k (θk ),
πΓ[(p − k)/2]
f (r) =
fp−1 (θp−1 ) = (2π)−1 ,
(4a)
0 < θk < π,
1 ≤ k ≤ p − 2,
−π < θp−1 < π,
(4b)
(4c)
where q(·) is the same generating function as in Eq. (2) and Γ(v) is the Euler’s Gamma function.
In the spherical coordinates representation, the random variable r, say the envelop, contains all the distinctive
information characterizing individual ECD member through the generating function q(·), while distributions
in the orthogonal subspace spanned by θ1 , . . . , θp−1 are totally fixed and shared by the whole ECD family.
Consequently, fitting a specified ECD to a particular HSI clutter degrades to a one dimensional problem in the
Mahalanobis square distance r2 = (x − μ)T Σ−1 (x − μ).5 We use the same notation r2 because this distance
equals the Euclidean square distance in spherical coordinates r2 . To answer the question whether the ECD family
will inherently provide an satisfying model for HSI clutters, we need to compare the empirical distributions with
the theoretical ones in the angular subspace spanned by θ1 , . . . , θp−1 .
3. ELLIPTICAL SYMMETRY TESTS
In this section, we introduce several graphical symmetry testing techniques, all of which are based on whitened
spherical distribution mentioned previously.
3.1 Scatter Plot
The primary way to verify a ECD distribution is to use a scatter plot5 which arises from the spherical coordinates
representation. From Theorem 2, whitened ECDs in spherical coordinates share the same angles distributions and
are only identified by unique envelops. We use p × p matrix of plots in which off-diagonal scatter plots investigate
the independence of pairwise coordinates and diagonal histogram plots illustrate the marginal distributions.
Figure 1a is an illustrative scatter plot for synthetic data generated from z ∼ t5 (0, I), where t5 (0, I) denotes
a multivariate t distribution of dimensionality 5 with zero mean and identity matrix as its covariance matrix,
which is known as a member of ECD family. We are most interested here in the off-diagonal scatter plots. A
scatter plot in which the points are distributed evenly indicates that the two correspondent random variables
are uncorrelated which is a looser condition for independence that is hard to verify.
3.2 Angular Goodness-of-fit Plot
In Theorem 2, the pdfs of the angle random variables are explicitly derived which remain unchanged as long as
spherically contoured condition is met. As a supplemental test of diagonal histograms in scatter plot, quantilequantile (Q-Q) plots of theoretical cumulative distribution function (cdf) and empirical cdf are developed. Such
a procedure is valid in exploiting quadratic statistics6 and provides a measure of the difference between the
theoretical cdf and the empirical cdf. As shown in Figure 1b, the empirical cdf plots of synthetic data drawn
from symmetric distribution z ∼ t5 (0, I) overlap the theoretical ones perfectly. One thing to mention here is
that all synthetic data from a particular member of ECD family will theoretically exhibit the exact same results.
Three goodness-of-fit metrics7 are computed as numerical measurements supplemental to the graphical plots.
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2
W = 0.09335; A
1
2
= 0.5561; U
2
2
= 0.09227
W = 0.1949; A
1
1
= 1.082; U
0.6
0.4
Empirical
Theoretical
0
2
0
0.2
0
1
0
1
θ /π
θ /π
1
2
W = 0.08229; A
1
2
= 0.5382; U
2
2
2
= 0.03345
W = 0.2378; A
1
3
0.8
0.8
2
= 1.419; U
2
= 0.2249
4
F
0.6
F
0.6
4
= 0.06199
F
0.8
0.6
F
0.8
0.2
3
2
2
0.4
1
2
0.4
0.4
0.2
0.2
0
0
1
0
−1
θ3/π
(a)
0
1
θ4/π
(b)
Figure 1: Testing results for synthetic data drawn from z ∼ t5 (0, I). (a) evenly distributed points in scatter
plot indicates that the two correspondent random variables are uncorrelated. (b) angular goodness-of-fit plots
compare the deviation between empirical cdfs and theoretical ones. The corresponding angular cdf plot is linked
to its histogram in scatter plot by circled numbers in red.
3.3 t-plot
The scatter plots and angular goodness-of-fit plots examine symmetry behavior in the whole angular subspace.
There are also other techniques, for example some robust statistics, which test whether the data is symmetric in
certain direction. We first begin with the t-statistic which tests the spherical symmetry in the t direction8 . Let
z = [z1 , . . . , zp ]T be the same whitened random vector. It is well known9 that the statistic
√
p z̄k
yk = t(zk ) =
(5)
sk
p
p
where zk = [zk1 , . . . , zkp ]T , z̄k = (1/p) i=1 zki , and s2k = (1/p) i=1 (zki − z̄k )2 . Obviously, y1 , . . . , yN are
i.i.d. according to tp−1 , where N is the total number of samples. We plot ascending ordered y(k) against the
2k−1
2N -quantile of tp−1 , k = 1, . . . , N . If the data distribution is spherically symmetric, then the plot will cling
to the 45 ◦ line through the origin. Both outliers and presence of systematic differences between y(k) and the
2k−1
◦
2N -quantile of tp−1 can be detected by visually comparing the plots to the 45 line. Deviation from linearity
in the t-plot indicates deviation of the underlying distribution from elliptical distributions. As a numerical
measurement of symmetric behavior auxiliary to the graphical plot, we also calculate the correlation coefficient
between ordered empirical y(k) and theoretical ordered univariate t samples. The value of 1 indicates a perfectly
linear plot. An illustrating example of the t-plot can be found in Figure 2a where the symmetric synthetic data
from t5 (0, I) is tested. From the plot, we can see that most of the points fall on the reference line except for few
outlier ones.
3.4 β-plot
Similar to the t-plot, the β-plot employs the calculation of robust β-statistics, which tests the symmetric behavior
in the β direction. Let z = [z1 , . . . , zp ]T be a random vector with zero mean and identity covariance matrix,
then the β-statistics is computed by
ba,p (z) =
Σai=1 zi2
,
Σpi=1 zi2
a = 1, 2, . . . , p
(6)
It can be shown that ba,p (z) is distributed according to beta distribution with parameters a/2 and (p − a)/2,
denoted by Beta(a/2, (p − a)/2). Generally, a good choice for a is a = p2 , where a denotes the largest integer
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20
1
0.9
15
0.8
Empirical Quantiles
Empirical Quantiles
10
5
0
−5
0.7
0.6
0.5
0.4
0.3
−10
0.2
ρ = 0.9985
ρ
= 0.9913
0.05
ρ
= 0.9794
−15
ρ=1
ρ
= 0.9999
0.05
ρ
= 0.9998
0.1
0.01
−20
−20
−15
−10
−5
0
5
10
15
20
0.01
0
0
0.2
Theoretical Quantiles
0.4
0.6
0.8
1
Theoretical Quantiles
(a)
(b)
Figure 2: Testing results for synthetic data drawn from z ∼ t5 (0, I). (a) t-plot employs the evaluation of robust
t-statistics. Symmetry behavior is tested by the degree of alignment between empirical line and reference 45 ◦
line. (b) β-plot employs the evaluation of robust β-statistics. Symmetry behavior is tested by the degree of
alignment between empirical line and reference 45 ◦ line.
which is no more than a. Similar to the t-plot, we can plot a β-probability Q-Q plot by assigning the ordered
empirical β-statistics ba,p (zk ), k = 1, . . . , N , where N is the total number of samples, to the correspondent
kth quantile of theoretical distribution Beta(a/2, (p − a)/2). We also compute the correlation coefficient as an
auxiliary numerical measurement. An illustrating example of the β plot is shown in Figure 2b. In this figure,
symmetric synthetic data from t5 (0, I) is tested and as seen in the plot, the points fall on the reference line
perfectly.
4. EXPERIMENTAL RESULTS
In this section, we now apply the derived graphical symmetry tests to real HSI data. The data set used for
this project, shown in Figure 3a, is from an Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data
measurement over Fort A. P. Hill, Virginia, on 9 November 1999 and 26 September 2001 on a Twin Otter
aircraft flying at an altitude of approximately 12, 000ft. Fort A. P. Hill, located in eastern Virginia, is heavily
forested with natural areas of deciduous, coniferous, and mixed deciduous-coniferous forests. There are also
numerous large plantations of loblolly pine that are intended for harvest as pulp wood. The trees in the loblolly
pine plantations were set in regular rows at close spacing to maximize yield and planted at the same time. The
result is a forest land cover that is spectrally and spatially uniform. By contrast, the natural forests show much
greater spectral and spatial variability. The forest land covers have been mapped by Fort personnel for resource
management and conservation purposes and are contained in a Geographic Information System (GIS) database.
Because the clustering by human experts are raw and without any statistical assumption, we also employ
automatic clustering method using iterative Expectation-Maximization (EM)10 to cluster the data pixels under
normal mixture model assumption. The unimodal data sets we tested here are comprised only by the pixels
labeled as the same group by automatic method within the human predefined ground cover regions. We will
show the test results for Data set 2 Coniferous forest, Data set 3 Deciduous forest, and Data set 6 South panel
field in this paper.
Figure 4 and 7 show the test results for Data set 2 Coniferous forest. The scatter plot in Figure 4a shows the
points are distributed evenly so that the uncorrelated condition is met. Only slight deviation can be found in
Figure 4b which means the empirical cdfs are quite similar to the prescribed theoretical ones. Both t-plot shown
in Figure 7a and β-plot shown in Figure 7b exhibit good linearity. The test results verify that the Coniferous
forest data exhibit strong symmetry behavior and thus can be modeled by ECD accurately. Figure 5 and 8 show
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Fort A. P. Hill, Eastern Virginia
Data set 3
10
Data set 5
Data set 2
Data set 4
50
9
100
8
150
7
Data set 6
Data set 1
200
6
250
5
300
Background regions
1. All loblolly pines
2. Coniferous forest
3. Deciduous forest
4. Mixed coniferous forest
5. Mixed pine forest
6. South panel field
350
4
400
3
450
2
500
100
(a)
200
300
400
500
1
(b)
Figure 3: (a) Data set used in this project. The area is heavily forested with natural stands of deciduous,
coniferous, and mixed deciduous-coniferous forests. There are also numerous large plantations of loblolly pine
that are intended for harvest as pulp wood. The whole image has been mapped for different forest land covers by
human experts. (b) Automatic generated clutters using expectation-maximization method under normal mixture
assumption.
the test results for Data set 6 South panel field. The test result is similar to Coniferous forest so that South
panel field data is also distributed symmetrically.
Figure 6 and 9 show the test results for Data set 3 Deciduous forest. The scatter plot in Figure 6a shows some
particular shape of the points distribution which means there are some correlation between the corresponding
random variables. In Figure 6b, some explicit deviation can be seen in the angular Q-Q plots but in the overall
the empirical cdfs are similar to the prescribed theoretical ones. Both in Figure 9a t-plot and in Figure 9b, we can
see some points especially outliner ones deviate explicitly from the reference lines. If we look back at Figure 3b,
the area of deciduous forest is more complex and contains different ground covers. In such case, pixels in this
area would probably include different kinds of trees which may explain the degradation in symmetry behavior.
5. CONCLUSIONS
In summary, we developed three graphical tests to examine the symmetry behavior of unimodal data sets.
According to the experimental results, most of the resulting clutters exhibit strong symmetric behavior which
demonstrates that ECD based statistical model can fit HSI data quite accurately. The physical qualification of
ECD modeling HSI data is discussed by Sangston.11 In order to model heavy tail behavior of HSI data, other
members of ECD family can be employed rather than normal distribution.5, 12 The symmetry performance of
the data depends on the quality of classification. In this work, the performance is improved by applying iterative
clustering methods.
REFERENCES
[1] McLachlan, G. and Peel, D., [Finite Mixture Models], Wiley Series in Probability and Statistics, John Wiley
& Sons, Inc., New York (2000).
[2] Giri, N. C., [Multivariate Statistical Analysis], Marcel Dekker, Inc., New York, 2nd ed. (2004).
[3] Yao, K., “A representation theorem and its application to spherically-invariant random processes,” IEEE
Trans. Inform. Theory 19, 600–608 (May 1973).
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W2 = 0.2976; A2 = 2.108; U2 = 0.173
W2 = 1.528; A2 = 14.59; U2 = 0.6924
1
5
0.8
0.8
0
1
0.6
0.6
F
1
F
10
0.4
0.4
0.5
Empirical
Theoretical
0.2
0
1
0
0
2
2
0
1
θ1/π
0.5
0.2
2
1
θ2/π
2
W = 0.7609; A = 3.618; U = 0.3376
0
0
2
2
W = 0.1934; A = 1.195; U = 0.1504
1
1
0.8
0.8
0.6
0.6
0.5
F
0
1
0
−1
0
5
10
0
0.5
1
0
0.5
1
0
0.5
1 −1
0
F
1
0.4
0.4
0.2
0.2
0
1
0
0
−1
1
θ /π
3
0
θ /π
1
4
(a)
(b)
Figure 4: Testing results for coniferous forest. (a) scatter plot shows the points are distributed evenly so that
the uncorrelated condition is met; (b) only slight deviation is visible in the angular Q-Q plots which means the
empirical cdfs are quite similar to the prescribed theoretical ones.
W2 = 2.911; A2 = 15.19; U2 = 2.185
W2 = 1.439; A2 = 9.422; U2 = 0.1311
1
5
0.8
0.8
0
1
0.6
0.6
F
1
F
10
0.4
0.4
0.5
Empirical
Theoretical
0.2
0
1
0
0
0.5
θ1/π
0.2
0
1
W2 = 0.3738; A2 = 2.122; U2 = 0.08566
1
0
0
θ2/π
1
W2 = 0.7243; A2 = 3.523; U2 = 0.4964
1
1
0.8
0.6
0.6
F
0
1
0.8
0
−1
0
5
10
0
0.5
1
0
0.5
1
0
0.5
1 −1
0
1
F
0.5
0.4
0.4
0.2
0.2
0
0
θ /π
0
−1
1
3
(a)
0
θ /π
1
4
(b)
Figure 5: Testing results for south panel field. (a) scatter plot shows the points are distributed evenly so that
the uncorrelated condition is met; (b) only slight deviation is visible in the angular Q-Q plots which means the
empirical cdfs are quite similar to the prescribed theoretical ones.
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W2 = 0.4764; A2 = 2.872; U2 = 0.3433
5
0.8
0
1
0.6
W2 = 0.6288; A2 = 3.594; U2 = 0.4419
1
Empirical
Theoretical
0.8
0.6
F
1
F
10
0.4
0.4
0.2
0.2
0.5
0
1
0
0
2
0
1
θ1/π
0.5
2
2
1
θ2/π
2
W = 1.304; A = 6.368; U = 0.4979
0
0
2
2
W = 0.3003; A = 2.445; U = 0.2238
1
1
0.8
0.8
0.6
0.6
0.5
F
0
1
0
−1
0
5
10
0
0.5
1
0
0.5
1
0
0.5
1 −1
0
1
F
1
0.4
0.4
0.2
0.2
0
0
0
−1
1
θ /π
0
1
θ /π
3
4
(a)
(b)
Figure 6: Testing results for deciduous forest. (a) scatter plot shows some particular shape of the points
distribution which means there are some correlation between the corresponding random variables; (b) some
explicit deviation can be seen in the angular Q-Q plots but in the overall the empirical cdfs are similar to the
prescribed theoretical ones.
6
0.9
0.8
4
2
Empirical Quantiles
Empirical Quantiles
0.7
0
−2
0.6
0.5
0.4
0.3
0.2
−4
ρ = 0.9967
ρ0.05 = 0.9982
= 0.9965
ρ
ρ = 0.9998
ρ0.05 = 0.9991
= 0.9988
ρ
0.1
0.01
−6
−6
−4
−2
0
2
4
0.01
6
0
0
Theoretical Quantiles
0.2
0.4
0.6
0.8
1
Theoretical Quantiles
(a)
(b)
Figure 7: Testing results for coniferous forest. (a) t-plot exhibits only slight deviation from perfect linearity; (b)
β-plot is also linear except for few outliner points.
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6
0.9
4
0.8
0.7
Empirical Quantiles
Empirical Quantiles
2
0
−2
−4
−6
0.5
0.4
0.3
0.2
ρ = 0.9958
ρ0.05 = 0.998
ρ0.01 = 0.9966
−8
−10
−10
0.6
−8
−6
−4
−2
0
2
4
ρ = 0.9976
ρ0.05 = 0.9992
ρ0.01 = 0.9988
0.1
0
6
0
0.2
0.4
Theoretical Quantiles
0.6
0.8
1
Theoretical Quantiles
(a)
(b)
Figure 8: Testing results for south panel field. (a) t-plot exhibits only slight deviation from perfect linearity; (b)
β-plot is also linear except for few outliner points.
5
0.7
4
0.6
3
0.5
Empirical Quantiles
Empirical Quantiles
2
1
0
−1
−2
0.4
0.3
0.2
−3
ρ = 0.9796
ρ0.05 = 0.9907
= 0.9807
ρ
−4
ρ = 0.9917
ρ0.05 = 0.9932
= 0.9894
ρ
0.1
0.01
−5
−5
0
0.01
5
0
0
Theoretical Quantiles
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Theoretical Quantiles
(a)
(b)
Figure 9: Testing results for deciduous forest. (a) t-plot shows some explicit deviation from linearity; (b) β-plot
shows many points especially outliner ones deviate from the reference line.
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