CSIS Discussion Paper Series
No. 8

A Robust Exploratory Method for
Qualitative Trend Curve Analysis
against Poor Quality Data

Atsuyuki Okabe* and Atsushi Masuyama**

March 9, 1999

* Center for Spatial Information Science, University of Tokyo** Department of Urban Engineering, University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JapanAbstract
This paper proposes a theoretical method for qualitative trend curve analysis,
supposing the situation that data quality is poor and the amount of data
is very large. Here the ‘trend curve’ means, for example, a sequence ofmonthly temperature over twelve months at a location. The paper first
defines ‘qualitative’ characteristics of a trend curve in terms of isomorphic
relations. Second, these concepts are extended to macroscopic qualitative
similarity. Third, using the macroscopic qualitative similarity, the paper
proposes a method for categorical clustering and a method for measuring
the magnitude of qualitative change in trend curves. Fifth, the proposed
theoretical method is implemented in a GIS environment and an exploratory
tool (a computer program called QuaT) is developed. Sixth, this tool isapplied to the analysis of land cover change in the Persian Gulf Area between
1982 and 1993. Last, the paper discusses the limitations of the proposed
method.
Acknowledgements
This paper has been developed from the theoretical part of Masuyama, Okabe, Sadahiro and Shibasaki (1998) presented at the 7th Annual Conferenceof the GIS Association, Japan. We express our thanks to Y. Sadahiro and
R. Shibasaki for their valuable comments on an earlier development, andto K. Sakai for her interpretation of our numerical analysis in the Persian
Bay Area. This study was partly supported by the National Science Funds,
Japan, 09NP1301 (Islamic Area Studies) and 10202201 (Spatial Information
Science for Human and Social Sciences).

iFigure 1: An illustrative example of trend curves, categorical clustering andqualitative changes

1

Introduction

The objective of this paper is to propose a theoretical as well as practicalmethod for analyzing qualitative characteristics of trend curves that often
appear in spatio-temporal analysis, supposing the situation that data quality is poor and the amount of data is very large (such as remotely sensed
data). Here the ‘trend curve’ means, for example, a sequence of monthly
temperature over twelve months at a location (Figure 1). First, we develop
a method for clustering trend curves with respect to ‘qualitative’ characteristics of the trend curves (the colored circles in Figure 1). Second, wedevelop a method for detecting ‘qualitative’ change in trend curves between
two periods in time, say 1980, and 1990. The main concept used in the thesemethods is ‘qualitative similarity’ of trend curves, and this concept is fully
discussed and formalized mathematically in this paper. Third, we implement
these theoretical methods in a GIS environment and develop a user-friendly
exploratory tool (a computer program, called QuaT).
We develop the above method as the first-phase analysis in the context
of the two-phase analysis: the first-phase analysis is exploratory and the second phase analysis is explanatory. The two-phase analysis is motivated by
the recent progress in modern data acquisition technologies, such as remotesensing, global positioning systems and mobile GIS. They bring us a hugeamount of data every month or everyday. In such an ‘excess’ data situation,
we are often buried in the data and have a difficulty in fixing a suitable explanatory model or an appropriate hypothesis. To overcome this difficulty,

1

we first carry out pre-analysis or data mining (Adriaans and Zantinge, 1996),
in which we attempt to find potential explanatory models using crude data.
The second-phase analysis is the ordinary (or classic) analysis, in which wedevelop an explanatory model based upon the result of the first-phase analysis and test it with well adjusted data.
In the first-phase analysis, data quality is usually very poor, because data
are not adjusted enough, or sometimes the data quality is unknown. In thiswild situation, it is useless to apply ‘sophisticated’ analysis that is sensitive toquantitative values. Rather, we need ‘simple’ analysis that is robust against
poor quality data. We also need efficient computational methods, because
a huge amount data requires much processing time and the first-phase analysis often requires fast processing. The method proposed in this paper isdeveloped for the first-phase analysis to satisfy these requirements.
The related literature is numerous. First, the trend curve analysis is discussed in depth in the time-series analysis (Anderson, 1994) that is applied
to meteorology (for example, Handcock and Wallis, 1994), hydrology (forexample, Rouhani and Wackernagel, 1990) and so forth. We note, however,that those methods mostly deal with quantitative nature of time-series phenomena; less attention is paid to qualitative nature. Second, the exploratory
analysis dates back Tukey (1977) in statistics, and it has been developed in
geographical analysis since the late 1980’s. Examples are Openshaw et al.
(1987), Haslett et al. (1987), Walker and Moore (1988) and Anslein et al.
(1993), among others. These methods (or tools), however, do not deal withtrend curves. Third, trend curves often appear in the analysis of remotelysensed data, because they are periodically provided. A typical analysis ofsuch data is the analysis of seasonal land cover change, for example, DeFries
and Townshend (1994) and Millington et al. (1994). In these studies, the2

major effort is address to how to adjust data with ground truth data. We
consider that this type of analyses is carried out in the second-phase. As
mentioned in the above, we are concerned with the trend curve analysis in
the first phase and such an analysis is few in the related literature except for
Eastman and Fulk (1993) and Samson (1993).

2

Qualitative similarity between trend curves

Consider a continuous curve, f(t), defined on [0, T ], which is assumed to besecond order differentiable and d2f(t)/dt2 6= 0 (these assumptions are madejust for theoretical consideration; they will be relaxed in Section 5). We
classify the characteristic of f(t) in a sufficiently small neighborhood of t
according to whether f(t) is non-singular or singular at t, i.e. df(t)/dt 6= 0
or df(t)/dt = 0. When the former holds, we say that the local characteristicof f(t) around t is a slope; when the latter holds, the local characteristic off(t) around t is a flat (Figure 2). Note that the local characteristic means thecharacteristic of f(t) in a sufficiently small neighborhood around t, denotedby Nε (t).
We further classify the flats according to d2 f(t)/dt2 < 0 or d2 f(t)/dt2 >0. When the former holds, we say that the local characteristic of f(t) aroundt is a peak; when the latter holds, the local characteristic of f(t) around t is abottom (Figure 2). In special cases at boundaries t = 0 and t = T , we definea peak by limt→0+ df(t)/dt ≤ 0 and a bottom by limt→0+ df(t)/dt > 0 ; similarly, a peak by limt→T − df(t)/dt ≥ 0 and a bottom by limt→T − df(t)/dt < 0.
For notational convenience we use C(f(t)) = S, F, P, B if the local characteristic of f(t) around t is a slope, flat, peak and bottom, respectively (Figure
2).

3Figure 2: Two trend curves that are weakly isomorphic.
It should be noted that these local characteristics are qualitative in thesense that a slope, a flat, a peak and a bottom remain a slope, a flat, apeak and a bottom, respectively under a broad class of monotonically increasing transformations. Stated precisely, if C(f(t)) = X in Nε (t), thenC(f(g(t))) = X in Nε (g(t)) and C(g(f(t))) = X in Nε (t) (X = S, F, P, B)
for any monotonically increasing function g.
Now let us consider two trend curves, f1(t) and f2 (t) (the suffix i may
indicate different locations or the same location but different points in time),
and let (ti1, . . . , tini ), (ti1 < . . . < tini ) be the sequence of points in [0, T ]at which either C(fi (tij )) = P or B holds, j = 1, . . . , ni , (ti1 = 0, tini =T ), i = 1, 2 (Figure 2). Then we can describe a global characteristic of fi (t)over [0, T ] by the sequence C(fi ) = (C(fi (ti1)), . . . , C(fi (tini ))), i = 1, 2. For
example, in Figure 2, C(f1 ) = (B, P, B, P, B, P, B).
For the two trend curves f1 (t) and f2 (t), we consider
Condition 1: C(f1 ) = C(f2 ).
Condition 1 means that if the jth flat in f1(t) is a peak (bottom), thenthe jth flat in f2(t) is also a peak (bottom) for all j = 1, . . . , n1 (= n2 ). For
example, in Figure 2, C(f1(t11 )) = C(f2(t21 )) = B; C(f1 (t12 )) = C(f2(t22 )) =P and so forth. When Condition 1 holds, we say that the two trend curves
f1 (t) and f2 (t) are weakly isomorphic, and denote this relation by f1 (t) ∼f2 (t). Since C(f1 ) = (B, P, B, P, B, P, B) = C(f2 ) holds in Figure 2, these
two trend curves are weakly isomorphic.
The theoretical notion of the weak isomorphism has practical implications. In the real world we often have data whose acquisition time is in4

accurately recorded. For instance, a watch equipped in a data acquisition
device may be inaccurate, or researchers forget to record the exact acquisition date. In such a situation, even if we observe the same phenomenon, we
obtain different trend curves f1 (t) and f2 (s) (t and s are used for indicating different time measures). Although they are different, it is quite likelythat the sequence of events is preserved over different time measures t and s.
Mathematically this implies that the time measure t of f1 (t1j ) and the timemeasure s of f2 (s2j ) have the relation t1j = g(s2j ) where g is a monotonically increasing function. The weakly isomorphic trend curves imply thatthe sequence of local characteristics is the same for the two different timemeasures.
We next introduce a little stronger relation than the weakly isomorphic
relation. We order the attribute values of fi (tij ), j = 1, . . . , ni from thelargest to the smallest, and let R(fi (tij )) be the rank of fi (tij ) (i.e. the valueof fi (tij ) is the R(fi (tij ))th largest value among the values of fi (tij ), j =1, . . . , ni ). Then we may describe a global characteristic of fi (t) over [0, T ]by the sequence R(fi ) = (R(fi (ti1 )), . . . , R(fi (tini ))), i = 1, 2. For example,
in Figure 3, R(f1 ) = (6, 2, 3, 1, 5, 4, 7). In terms of this sequence, we state
Condition 2: R(f1 ) = R(f2 ).
Note that Condition 2 includes Condition 1. Condition 2 implies that
if the jth flat in f1 (t) is the kth highest flat in f1 (t), then the jth flat in
f2 (t) is also the kth highest flat in f2(t), and vice versa. For example, in
Figure 3, R(f1 (t11 )) = R(f2 (t21 )) = 6, R(f1 (t12 )) = R(f2 (t22 )) = 2, and soforth. When the two trend curves f1 (t) and f2 (t) satisfy Condition 2, we
say that f1 (t) and f2 (t) are strongly isomorphic, and denote this relation by
f1 (t) ≈ f2(t). The two trend curves in Figure 2 are not strongly isomorphic,
5

Figure 3: Two trend curves that are strongly isomorphic.
but those in Figure 3 are strongly isomorphic (R(f1 ) = (6, 2, 3, 1, 5, 4, 7)= R(f2 )). Obviously, the strongly isomorphic relation implies the weakly
isomorphic relation but not converse.
The theoretical notion of the strongly isomorphic relation does not remaina theoretical notion. In reality, it is likely that sensitivity of a sensor isdifferent from device to device, or it may deteriorate over time. As a result,the observed attribute value u = f1 (t) at t is not always comparable withthe attribute value v = f2 (t) at t; the measures of u and v are likely to
be different or researchers observe attribute values in different manners. In
such a situation, even if we observe the same phenomenon, the observed
trend curves u = f1 (t) and v = f2 (t) are not identical. It is, however,often plausible to assume that the ranks of attribute values at peaks andbottoms are preserved for two different measures u and v. Mathematicallythis implies that the measures u and v have the relation v = g(u) where g isa monotonically increasing function.
This assumption, however, might not be acceptable when data acquisition
devices are very unstable. In such a wild case, it is safe to assume theweakly isomorphic relation. In this sense, the weakly isomorphic relation isa fundamental relation when we analyze poor quality data.
Both weakly and strongly isomorphic relations show qualitative similarity between two trend curves. In the following discussion, when distinction
between the weakly isomorphic relation and the strongly isomorphic relation
is not necessary or analysis applies to both relations, we just refer to thoserelations as isomorphic relations, or qualitative similarity, and denote it by'.
6

Figure 4: Two trend curves that are not microscopically isomorphic but
macroscopically isomorphic.

3

Macroscopic qualitative similarity between
trend curves

When we defined the isomorphic relations, we assumed that a peak was apeak no matter how small it was, but this assumption is arguable. For example, consider two trend curves in Figure 4. These two curves are notisomorphic, because the curve in panel (a) has 4 peaks, whereas the curve inpanel (b) has 3 peaks. We feel, however, that these two curves are ‘macroscopically’ isomorphic, because the number of ‘distinct’ peaks are the samefor the both curves.
To represent ‘macroscopic’ isomorphism, we first define the height of apeak explicitly. Suppose that C(f(tj )) = P . From the definition of a peak, it
is obvious that C(f(tj−1)) = B and C(f(tj+1 )) = B hold (Figure 5(a), that
is, a peak is always in between two adjacent bottoms. In relation to thesebottoms, we define the height, h(tj ), of the peak at tj byh(tj ) = f(tj ) − max{f (tj−1 ), f(tj+1 )}(1)(Figure 5(a); Okabe, 1982; Okabe and Masuda, 1984). Then we may define a
distinct peak as the peak whose height is greater than a threshold height, h∗,and we refer to such a peak as an h∗ -distinct peak. When we pay attention toa macroscopic nature of the trend curve fi (t), we ignore peaks whose heightis less than h∗ , or we pay attention to only h∗ -distinct peaks. Mathematicallythis implies that: if h(tj ) ≤ h∗, the curve f(t) in tj−1 ≤ t ≤ tj+1 is replacedwith a monotone curve joining f(tj−1 ) and f(tj+1 ) (Figure 5(b)). The resulting curve is denoted by f(t|h∗ ) and we say that f(t|h∗ ) is the h∗ -distinct
7

Figure 5: An h∗ -distinct trend curve of f(t) ((a) f(t), (b) f(t|h∗ )).
trend curve of f(t).
In terms of f(t|h∗ ), the statement that two trend curves f1 (t) and f2(t)are macroscopically (qualitatively) similar is precisely written as f1 (t|h∗) 'f2 (t|h∗); that is, two h∗ -distinct trend curves are isomorphic. Note thatf(t) = f(t|0). Also note that deleting h∗ -distinct peaks from an original
trend curve is similar to filtering in the Fourier analysis. As will be shown in
Section 5, however, our method is much simpler and faster than the Fourier
method.
The macroscopic similarity changes according to the value of h∗. To
illustrate this change, we depict two trend curves, f1 (t|h∗) and f2(t|h∗ ) in
Figure 6. We notice that f1(t|0) ∼ f2 (t|0). As h∗ increases, the isomorphic
relation changes. For 0 ≤ h∗ < 0.2, f1(t|h∗ ) ∼ f2 (t|h∗) holds, but at h∗ = 0.2,
the isomorphic relation changes to f1 (t|h∗) 6∼ f2(t|h∗ ), and this relation holds
for 0.2 ≤ h∗ < 0.7. For 0.7 ≤ h∗ < 1.3, f1 (t|h∗) ∼ f2(t|h∗ ). For 1.3 ≤ h∗ <1.4, f1(t|h∗) 6∼ f2 (t|h∗). At h∗ = 1.4, two trend curves are isomorphic (oneh∗ -distinct peak).
When we can fix a specific level h∗ of macroscopic similarity, we analyze
trend curves in terms of fi (t|h∗). Sometimes, however, we want to see overall
qualitative similarity from a low level (the lowest is h∗ = 0, i.e. microscopic
similarity) to a high level. To measure this overall qualitative similarity, we
may use the length of h∗ in which two trend curves are isomorphic. For example, in Figure 6 (the lines on the right-hand-side), the heavy line segmentsshows the range in which the isomorphic relation f1 (t|h∗) ' f2(t|h∗ ) holds.
If the heavy line segment is long, we may consider that two trend curves
are qualitatively similar. We hence define the magnitude, M(fi , fk ), of the8Figure 6: h∗ -distinct trend curves with respect to h∗ .
overall qualitative similarity between fi (t|h∗) and fk (t|h∗) byM(fi , fk ) =

1

Z h∗

h∗max

0

where

max

(

δ(fi (t|x), fk (t|x)) =δ(fi(t|x), fk (t|x))dx,(2)

1 if fi (t|x) ' fk (t|x),0 if fi (t|x) 6' fk (t|x),(3)

and h∗max is the maximum value of h∗ . M(fi , fk ) takes a value between 0
and 1; M(fi , fk ) = 0 implies that the isomorphic relation does not hold at
all for any value h∗ in [0, h∗max]; M(fi , fk ) = 1 implies that the isomorphic
relation holds for all values in [0, h∗max] (the two trend curves are qualitaitvely
similar from the lowest level (h∗ = 0) to the highest level (h∗ = h∗max )). In
the example of Figure 6, M(f1 , f2) = 0.57.

4

Categorical classification and qualitativechangeWe can utilize the isomorphic relation to classify trend curves categorically.
Let fis (t|h∗ ) be an h∗ -distinct trend curve at a location i = 1, . . . , n in a time
period s = 1, . . . , p. We can classify the trend curves over space (i = 1, . . . , n)into k classes, K = {K1 , . . . , Kk } by the isomorphic relation '. That is,fi (t|h∗) ' fi0 (t|h∗) holds if and only if fi (t|h∗ ) and fi0 (t|h∗) belong to thesame class Kj . In particular, when the isomorphic relation is the weakly
isomorphic relation (∼), trend curves are characterized by the number ofpeaks, and so Kj is the class of trend curves which has j peaks.
We make two remarks on K. First, classes in K are categorical in the sensethat trend curves are classified by the qualitative similarity (the isomorphic
9

Figure 7: Qualitative change occurs in a trend curve.
relations). Second, we can control the number k of classes through h∗. As
the value of h∗ increases, the number of classes decreases.
Now we are ready to define ‘qualitative change’ in the trend curve fis (t|h∗)
at a location i(= 1, . . . , n) in a time period s(= 1, . . . , p). When fis (t|h∗) 6'
fis+1 (t|h∗) holds, we say that qualitative change occurs in the trend curve at
a location i in a time period s (Figure 7).
Sometimes we wish to measure the magnitude of this qualitative change.
When fis (t|h∗) 6∼ fis+1 (t|h∗) holds, recalling that the class Kj is the classof trend curves which has j h∗ -distinct peaks, the magnitude of the qualitative change, D1 (fis , fis+1 ), may be measured by the difference between thenumbers of peaks, i.e.
D1 (fis , fis+1 ) = j − j 0 for fis (t|h∗) ∈ Kj , fis+1 (t|h∗) ∈ Kj 0 .

(4)

In the example of Figure 7, D1 = +1. Note that this measure is applicable
only to the weakly isomorphic relation. Recalling the overall qualitative
similarity defined by equation (2), we can define an alternative measure by
D2 (fis , fis+1 ) = 1 − M(fis , fis+1 ).

(5)

In the example of Figure 7, D2 = 0.7. This measure is more general thanD1 because it can be defined not only for the weakly isomorphic relation
but also for the strongly isomorphic relation. Using these measures, we candetect where qualitative changes occur in a region. An actual example isshown in Section 5.

10

5Implementation: program QuaT

Having shown the theory of how to detect qualitative changes in trend curves
over space and time, we now wish to implement this theory in a GIS environment and develop an exploratory tool, called QuaT.
In the above theory, we assume that the function f(t) is continuous,
second order differentiable and df(t)/dt 6= 0. In practice, however, we usuallyobserve the value of f(t) at a finite number of points, t̂1, . . . , t̂m (filled circles
in Figure 8; note that t̂i used here should be distinguished from ti or tij (thelocation of a peak or a bottom) in the above section). Since this function is
discrete, the above theoretical analysis is not directly applicable. It should
be noted, however, a few modifications make the above theoretical analysis
applicable. First, we can construct a continuous curve from f(t̂1), . . . , f(t̂m )by joining f(t̂j ) and f(t̂j+1 ) by a straight line segment (Figure 8). This
modification implies a linear interpolation. Second, we define a slope, a peak
and a bottom by:
C(f(t̂j )) = S if and only if f(t̂j−1 ) < f(t̂j ) < f(t̂j+1 )or f(t̂j ) > f(t̂j+1 ) > f(t̂j+1 );(6)

C(f(t̂j )) = P if and only if f(t̂j−1 ) < f(t̂j ) and f(t̂j ) > f(t̂j+1 ); (7)C(f(t̂j )) = B if and only if f(t̂j−1 ) > f(t̂j ) and f(t̂j ) < f(t̂j+1 ). (8)In special cases at boundaries t = t̂1 and t = t̂m ,C(f(t̂1)) = P if and only if f(t̂1 ) > f(t̂2)
or f(t̂m−1 ) < f(t̂m );(9)

C(f(t̂1)) = B if and only if f(t̂1) < f(t̂2 )or f(t̂m−1 ) > f(t̂m ).
11

(10)

Figure 8: A piece-linear trend curve.
By these modifications, we can apply the analysis of a continuous trend curvef(t), 0 ≤ t ≤ T to the analysis of a discrete trend curve (f(t̂1 ), . . . , f(t̂m )).
Given these modifications, we can implement the method by the following
procedure.
Program QuaT
Step 0 (initial setting). Input data are: discrete trend curves fis (t̂j ) (j =1, . . . , m) at locations i = 1, . . . , n in time periods s = 1, . . . , p.
Step 1. Find peaks and bottoms in fis (t̂j |0), j = 1, . . . , m by equations (6)- (10) (i = 1, . . . , n; s = 1, . . . , p).
Step 2. Compute the height of each peak in fis (t̂j |0), j = 1, . . . , m byequation (1) (i = 1, . . . , n; s = 1, . . . , p).
Step 3. Construct fis (t̂j |h∗), j = 1, . . . , m (i = 1, . . . , n; s = 1, . . . , p).
Step 4. Classify fis (t̂j |h∗), j = 1, . . . , m (i = 1, . . . , n; s = 1, . . . , p) according
to an appropriate isomorphic relation (∼ or ≈), and display the resultingclasses over space.
Step 5. Compute the magnitude of qualitative change in fis (i = 1, . . . , n)over time periods s = 1, . . . , p by equation (4) or (5), and display the magnitude over space.
This program runs fast. The order of computational time in Step 1
is O(mnp), and that in Step 2 is also O(mnp). In Step 3, we order theheights of the peak from the lowest to the highest, and this ordering requires O(m log m) time for one trend curve. Thus the order in Step 3 is12

O(npm log m). Step 4 requires O(kn) time where k is the number of classes.
Step 5 requires O(kn). In theory, the order of the total computational time
is max{O(npm log m), O(kn)}. In practice, we suppose that the attributevalues in a region are given by raster type data (such as remotely sensed
data), the order of the number n of pixels representing a region is higher
than that of m, k and p. Thus, the total computational time is dominatedby n, implying that QuaT runs with the linear time order of the number oflocations.

6

Application of QuaT to the analysis of seasonal land cover (NDVI) change in the Persian Gulf Area between 1982 and 1993

We applied QuaT to the analysis of seasonal land cover change in the Persian Gulf Area (E40◦ − 50◦ , N27◦ − 37◦ ) between 1982 and 1993. The data
source was Pathfinder AVHRR Land Data Set (NOAA-7, -9 and -11) in
1982 and 1993 (Smith et al., 1997). The area consists of 100 by 100 pixels; consequently, one pixel represents a 0.1◦ by 0.1◦ region (approximately
8 km by 8 km). We used the NDVI (Normalized Difference Vegetation Index; roughly speaking, NDVI indicates the amount of vegetation) over twelve
months, which gave the trend curves fis (t̂j ), j = 1, . . . , 12, i = 1, . . . , 10000,
s = 1982, 1993.
The quality of the data was not satisfactory. First, we tried to remove theeffect of the clouds by taking the maximum value of NDVI among the NDVI
values obtained at three time-points in a month. This treatment resulted
in unequal time intervals because the maximum value was achieved at adifferent time-point in each month (note that the order of observation times
13

Figure 9: The frequency distribution of peak heights.
Figure 10: Categorical classification of NDVI trend curves in the Persian
Gulf Area.
over twelve months are preserved). Second, it is quite likely that the data
quality changed between 1982 and 1993. Third, usually remotely sensed data
should be adjusted by the ground truth data, but it was difficult to do so
because such data were difficult to obtain in the Persian Gulf Area. Fourth,
the amount of data was huge. Although we finally used only 1982 and 1993,
but on the way of the analysis, we used three sets of data in every month overtwelve months over 12 years; consequently the total data amounted to 10000(pixel) × 3 (times in a month) × 12 (months) × 12 (years) = 4.32 × 106 .
Considering these factors, we considered that QuaT was appropriate for the
first-phase analysis.
The input data were the NDVI values fis (t̂j ), j = 1, . . . , 12 (twelve
months), i = 1, . . . , 10000 (locations), s = 1983, 1992 (years) (Step 0). QuaT
computed peaks and bottoms (Step 1), and gave the heights of the peaks(Step 2). To determine an appropriate value of h∗ , QuaT computed the distribution of peak heights (Step 3). The result is shown in Figure 9. From
this distribution, QuaT chose the value h∗ = 0.08 at which the frequency
drastically changed (we also examined several values and realized that thisvalue was an appropriate value) (Step 3).
For h∗ = 0.08, QuaT classified the NDVI trend curves in the Persian Gulf
Area. The result is shown in Figure 10 (Step 4). Finally, QuaT computed
qualitative change in the NDVI trend curves between 1982 and 1993 using
two measures (Step 5). Figure 11 shows the magnitude of the qualitative

14

Figure 11: Qualitative change in NDVI trend curves between 1982 and 1993
in the Persian Gulf Area measured by D1 given by equation (4).
Figure 12: Qualitative change in land cover (NDVI trend curves) between
1982 and 1993 measured by D2 given by equation (5).
change in terms of D1 given by equation (4), and Figure 12 shows that interms of D2 given by equation (5).

7Concluding discussion

In addition to the above application, we tested the performance of QuaT
in several empirical examples. From these tests, we found the performancequite satisfactory.
First, QuaT runs very fast as was theoretically examined in Section 4.
Second, QuaT is robust against poor quality data. This robustness resultsfrom the property that QuaT deals with qualitative characteristics.
One might question, however, that the isomorphic relations, in particular
the weakly isomorphic relation assumed in QuaT is too crude. At first glance,
it looks so, but it is not so crude as one might feel. One should recall theoverall similarity D2 (fis , fis+1 ) shown in Figure 6. This similarity considers
the height of peaks from the lowest peak to the highest peak, implying thatthe height of the peaks are implicitly taken into account even if we use theweakly isomorphic relation.
Obviously no methods are almighty and QuaT has some limitations. To
see them, we depict Figure 13 which shows a good contrast between QuaT
and an ordinary quantitative method. Figure 13(a) shows three clusters obtained by applying the K-means method (Hartigan, 1975) to the multivariate
15

Figure 13: (a) Clusters obtained by the ordinary clustering method (theK-means method) and (b) those obtained by QuaT.
data (fi (t̂i1 ), . . . , f(t̂i12 )), i = 1, . . . , 10000. Figure 13(b) shows three clusters
obtained by QuaT with h∗ = 0.08. In both figures, the flat trend curves seemto indicate sandy soil areas. Figure 13(a) seems to show two kinds of crops
areas (probably wheat and rice). On the other hand, Figure 13(b) seems toshow single-cropping areas and double-cropping areas. This contrast showsan advantage and a disadvantage of QuaT and the ordinary method. QauT
cannot distinguish one-peak trend curves in Figure 13(a) but the K-means
method can; the K-means method cannot detect double-cropping areas but
QuaT can.
This comparison suggests that QuaT should be applied in the first-phaseanalysis. In the second-phase analysis, the K-means method should be applied to one-peak trend curves obtained by QuaT. Then we can detect quantitative as well as qualitative characteristics of trend curves.
In conclusion, QuaT shows good performance in the first-phase analysis.
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18

f1(t)f2(t)qualitatively
changed ?
Jan Feb ...

... Dect

Jan Feb ...

1980... Dect

1990

location (i,j)

location (i,j)

Figure 1

a peaka slopea bottomt11 t12 t13 t14
C(t1j)=( BPPt15 t16 t17
B P

B )

t

t21 t22 t23 t24 t25 t26 t27
C(t2j)=( B

Figure 2

P

B

P

B P

B)

t

t11 t12 t13
R(t1j)=( 6

2 4

t14

t15 t16 t17 t

1

5 3

7)

t21 t22 t23 t24
R(t1j)=( 6Figure 32

4

1

t25 t26 t27 t
5

3 7)

P

P

P

P

t

Figure 4

P

P P

t

f(t)

t2

t3

t4

B

P

B

t

(a)

f(t￨h*)

t(b)

Figure 5

f1(t￨h*)f2(t￨h*)

0
0.2

0.7

1.3
1.4

h*

Figure 6

f1(t)

D1

D2

+1

0.7

f2(t)

t

t

Figure 7f(t)a peaka bottomt^1t^2... t^m...

Figure 8tfrequency

0.4

0.16

5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0

Figure 9the number of peaks
0
1
2

Figure 10

D1
+2
+1
0
‑1
‑2

Figure 11

D2
1

0

Figure 12

NOV
DEC

NOV
DEC

JUN

MAY

APR

MAR

FEB

JAN

OCT

-0.1
OCT

0
SEP

0.1

SEP

0.2
AUG

0.3

AUG

0.4
JUL

QuaT

JUL

JUN

MAY

APR

MAR

FEB

JAN

0.4

0.3

0.2

0.1

0

-0.1

K-means

Figure 13