Introducing interval time series: accuracy
measures ∗
Javier Arroyo1 and Carlos Maté2
1
2
Departamento de Sistemas Informáticos, Universidad Complutense, Profesor
Garcı́a-Santesmases s/n, 28040 Madrid, Spain. javier.arroyo@fdi.ucm.es
Instituto de Investigación Tecnológica, ETSI (ICAI), Universidad Pontificia
Comillas, Alberto Aguilera 25, 28015 Madrid, Spain. cmate@upcomillas.es
Summary. The confluence between time series analysis and symbolic data analysis
lead to a promising area: symbolic time series. In these kind of series the considered
variable is a symbolic one (e.g. a histogram or an interval variable). This paper
focuses on interval time series, which are useful to describe the evolution through
time of the range of variation of a phenomenon (e.g. the flow of a river). Accuracy
measures for these series based on distances for interval data will be proposed.
Finally, an example illustrates how to forecast interval time series in a simple way.
Key words: interval data, error measures, forecasting, symbolic data analysis, time
series analysis
1 Introduction
For the time being, quantitative forecasting methods have focused on single-valued
time series, i.e. series where every observation at each time point is a single value.
The analysis of this kind of time series is a mature field where enormous progress
have been made from the seventies [DH06]. Single-valued time series are useful to
represent time-varying contexts, however, in some situations, other kinds of time
series are better suited. For example, consider the daily electric demand in a region:
in this case a classical time series with the total demand each day can describe the
phenomenon, but it would not report on the intra-daily variability of the demand;
however, a histogram-valued or a boxplot-valued time series describing the distribution of the hourly demand each day would be more appropriate as they would report
on the distribution shape. Similarly, the lower and upper monthly water levels of
a river at a given location can be suitably represented by an interval-valued time
series, as it reports on the variability of the river flow. Both of them are examples
of symbolic-valued time series.
Symbolic-valued time series arise as the combination of time series analysis and
symbolic data analysis. Symbolic data analysis states that symbolic variables (lists,
∗
This work is funded by Universidad Pontificia Comillas (PRESIM project).
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Javier Arroyo and Carlos Maté
intervals, frequency distributions, etc) are better suited than single-valued variables
for faithfully describing complex real-life situations. According to [BD03], there is a
great need of development of methodologies dealing with symbolic variables. However, thus far, symbolic-valued time series have not been tackled. There is an approach for dealing with symbolic data in a temporal setting [GP00], but it is not
related with forecasting but with three-way data analysis. There is a field also called
symbolic time series analysis [DFT03], where series are sequences of a finite set of
symbols. However, we are referring to series that are sequences of numerical observations taken at regular time intervals, where the observed variable is not single-valued
but symbolic, such as distribution-, interval- or histogram-valued variables.
This paper will only be focused on interval time series (ITS). Notation for this
new entity will be proposed and several sources of ITS will be commented. The core
of the paper is devoted to propose an approach to measure forecasting errors in an
ITS context. Finally, an example to illustrate an straightforward way of forecasting
ITS is shown.
2 Notation for Interval Time Series
A variable Y is termed interval variable and denoted by [Y ], if all elements i of a
set E take values in the domain B = {[α, β], −∞ < α ≤ β < ∞}. The particular
value of [Y ] for the ith element is denoted by [Y ]i = [Yi , Yi ]. An interval is defined
by its interval bounds (minimum and maximum). Equivalently, it can be defined by
its midpoint (center) and its radius, which are
ci = mid[Y ]i =
Yi + Yi
Yi − Yi
and ri = rad[Y ]i =
,
2
2
(1)
respectively. Thus, the value of an interval variable for the ith individual can also
be denoted by the midpoints-radii notation: [Y ]i = hci , ri i. Minimum, maximum,
midpoint and radius can be considered as interval attributes.
An interval time series {[Y ]t } is a time series where the variable observed through
time t = 1, ..., n is an interval variable, the value of the variable in each instant of time
t is expressed as [Y ]t = [Yt , Yt ] = hct , rt i. However, in order to improve the legibility
of subsequent formulae the following notation will be used: [Y ]t = [Yt,L , Yt,U ] =
hYt,C , Yt,R i. In order to denote a forecasted value, a hat will be placed above the
variables: [Ŷ ]t = [Ŷt,L , Ŷt,U ] ≡ hŶt,C , Ŷt,R i.
3 Sources of Interval Time Series
ITS can describe situations that cannot be described by classical time series, such as
when no precise data is available and inaccuracy or uncertainty must be taken into
account (e.g. when the measurement instrument is not reliable). Another example
is an ITS describing the blood pressure of a person through time. However, inherent
ITS are scarce.
Sampling and aggregation are the main ways of obtaining ITS. These approaches
are also applied to obtain classical time series, but we believe that is worth to use
ITS as they can offer a complementary view of the phenomena. For example, a
Introducing interval time series: accuracy measures
1141
continuous time series produced by a sensor can be sampled recording the lower
and upper values in each hour. It will lead to an ITS describing the hourly range
of values, which would report on the variability of the original series. Something
similar happens when ITS are applied to aggregate across a set of individual time
series measuring the same variable, e.g. a set of series representing the levels of an
air-pollutant in different locations in a city can be aggregated by an ITS representing
the minimum and maximum levels of pollutant in the whole city. In [ZT00] eighteen
time series representing the annual output growth rate of industrialized countries
are aggregated by the median. Curiously, while the article is focused on the median
time series, its charts also represent an ITS describing the interquartile ranges. We
believe that in these cases, it would be interesting to also analyze the ITS. Obviously,
in sampling and aggregation contexts intervals can arise not only from the minimum
and maximum observed values, but from the interquartile range or from the middle
90% of the scores (in order to avoid outliers).
4 Error Measures for Interval Time Series
In classical time series, error measures are based in the difference between the observed and the actual value. In interval algebra, the difference between a pair of
intervals, [A] and [B], is defined as [A] − [B] = [AL − BU , AU − BL ]. Unfortunately,
this operation is not appropriate to define error measures, because it does not faithfully represent the concept of deviation [PL03]. Therefore, we propose to define ITS
error measures based on distances for intervals. Distances objectively measure the
dissimilarity between an observed interval and its forecast. Moreover, distances can
be easily summarized without squaring them or using absolute values as they always
take non-negative values.
Let {[Y ]t } be the observed ITS, and {[Ŷ ]t } be the forecast of this ITS with
t = 1, ..., n, a set of error measures is shown in the next subsections.
4.1 Mean Distance Error based on Hausdorff Distance
Hausdorff proposed a metric to measure distances between compact sets. Intervals
are compact sets identified by ordered couples of values (i.e. their lower and upper
bounds). Given two intervals [A] = [AL , AU ] = hAC , AR i and [B] = [BL , BU ] =
hBC , BR i, the Hausdorff metric for intervals is:
dH ([A], [B]) = max(|AL − BL |, |AU − BU |) = |AC − BC | + |AR − BR |.
(2)
Note that for a pair of degenerate intervals, i.e. [A] = [x, x] and [B] = [y, y], we
have dH ([A], [B]) = |x − y|, which is the usual topology in the real line.
The Mean Distance Error based on Hausdorff metric is defined by:
M DEH =
1
n
Xd
n
t=1
H ([Y ]t , [Ŷ ]t )
=
1
n
X[|Y
n
t=1
t,C
− Ŷt,C | + |Yt,R − Ŷt,R |].
(3)
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Javier Arroyo and Carlos Maté
4.2 Root Mean Squared Distance Error based on Hausdorff metric
There are two generalizations of the Hausdorff distance for interval data in a ndimensional space. They consider that an item described by n interval variables can
be alternatively represented in a n-dimensional space by parallelotopes or by hyperspheres [PI05]. Here, we only consider the parallelotopes approach, which generalizes
the Hausdorff distance by means of the Minkowski metric. The distance with α ≥ 1
for two items, X and Y , described by n interval variables is defined as:
vuX
u
(X, Y ) = t [d
n
dHgen
α
α
H ([X]j , [Y ]j )] .
j=1
An extension of (4) is obtained by introducing weights, wj > 0 with
1, to model the relative importance of the variables:
(4)
P
vuX
u
(X, Y ) = t [w d
n
j=1
wj =
n
dHgen
α
j H ([X]j , [Y ]j )] .
α
(5)
j=1
An interval time series {[Y ]t } with t = 1, ..., n can be considered as an individual
described by n interval variables, that is, represented in a n-dimensional space as a
parallelotope of n faces. Given (5) with α = 2 and assigning equal weights (wj = 1/n)
to all variables, the Root Mean Squared Distance Error based on the Hausdorff
metric is defined as:
vu
1 uX
= t (Y
n
n
RM SDEH
t,C
− Ŷt,C )2 + (Yt,R − Ŷt,R )2 + 2|Yt,C − Ŷt,C ||Yt,R − Ŷt,R |.
t=1
(6)
It can be seen that the first two measure components account for the square difference between midpoints and radii, while the third component accounts for the
combined effect of the error in the midpoints and in the radii.
4.3 Mean Distance Error based on Ichino-Yaguchi distance
Ichino and Yaguchi proposed a generalized Minkowski metric for a multidimensional
space of mixed variables (quantitative, qualitative and structural variables) [IY94].
This metric is based on the cartesian join and meet operators, which for interval
variables are defined as: [A] ⊕ [B] = [A] ∪ [B] = [min(AL , BL ), max(AU , BU )] and
[A] ⊗ [B] = [A] ∩ [B], respectively. Let [A] and [B] be a pair of intervals, the IchinoYaguchi distance is:
dIY ([A], [B]) = w([A]∪[B])−w([A]∩[B])+γ(2w([A]∩[B])−w([A])−w([B])), (7)
where w([X]) denotes the width of the interval [X], i.e. w([X]) = XU − XL , and
γ ∈ [0, 0.5] controls the effects of the inner-side nearness and the outer-side nearness
between [A] and [B]. This measure holds the properties to be considered a distance
[IY94]. The use of γ = 0.5 is suggested in [IY94]:
Introducing interval time series: accuracy measures
1143
dγ=0.5
([A], [B]) = w([A] ∪ [B]) − 0.5[w([A]) + w([B])],
IY
(8)
We believe that, in error measurement, γ = 0.5 is a suitable choice as, in our
experience, is not worse than other γ values, and produces a clearer equation (8)
which, moreover, is equivalent to this meaningful formulae:
dγ=0.5
([A], [B]) = 0.5[|AL − BL | + |AU − BU |].
IY
(9)
The Mean Distance Error based on the Ichino-Yaguchi distance is:
M DEIY =
1
n
X 0.5[|Y
n
t,L
− Ŷt,L | + |Yt,U − Ŷt,U |].
(10)
t=1
4.4 Mean Distance Error based on De Carvalho distance
In [Dec96], De Carvalho proposes a normalization of Ichino-Yaguchi’s distance:
dDC ([A], [B]) =
dγIY ([A], [B])
,
w([A] ∪ [B])
(11)
where dγIY ([A], [B]) is given in (7). The range of (11) is [0, 1], and it satisfies the
properties to be considered a distance [Dec96]. If γ = 0, the measure takes its
maximum value, when the intersection of the intervals is null, not taking into account
the nearness of the intervals. As this is not a desirable feature in error measurement,
we discard this value and propose γ = 0.5 which offers more suitable features:
dDC ([A], [B]) = 1 if and only if the considered intervals are degenerate and not
equal, e.g [A] = [3, 3] and [B] = [4, 4]; dDC ([A], [B]) = 0.5 if the considered intervals
are adjacent, e.g. [A] = [1, 3] and [B] = [3, 7], or if one interval is degenerated and
is contained in the other one, e.g. [A] = [1, 4] and [B] = [2, 2]; dDC ([A], [B]) < 0.5 if
w([A]∩[B])
and only if w([A] ∩ [B]) > 0; dDC ([A], [B]) ≤ 0.25 if and only if w([A]∪[B])
≥ 0.5.
The Mean Distance Error based on De Carvalho distance is defined by:
M DEDC =
1
n
Xd
n
t=1
γ=0.5
([Y ]t , [Ŷ ]t )
IY
w([Yt ∪ Ŷt ])
.
(12)
The M DEDC is a scale-independent error measure with range [0, 1] and, according
to its definition, a M DEDC ≥ 0.5 means a poor forecast record.
4.5 Some issues on the accuracy measures proposed
A classical time series can be seen as a particular case of an ITS whose intervals
are degenerate, [Y ]t = [at , at ], at ∈ R. If the accuracy of a classical time series
is evaluated with ITS error measures, the values of M DEH and M DEIY will be
equivalent to the Mean Absolute Error of the classical time series; while the behavior
of RM SDEH will be similar to the behavior of the Root Mean Square Error. In this
case, M DEDC is not applicable, because De Carvalho’s distance take the value 1, if
the pair of intervals are degenerate and not equal.
RM SDEH is a measure sensitive to outliers (i.e. to extreme midpoint or radius
values) and it remains to be seen if it has good statistical properties as the Root
Mean Square Error in classical time series. In addition, it is not as interpretable as
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Javier Arroyo and Carlos Maté
the other measures proposed. M DEDC is scale independent and is very interpretable
as it ranges from 0 to 1, being 0.5 and 0.25 values with a clear significance. Therefore, it is useful to compare errors in time series that have different scales. M DEH
and M DEIY are scale dependent measures, but their interpretation is clear as they
account for the deviation in midpoints and radii, and in minimums and maximums,
respectively. Their behavior is similar to the one of the Mean Absolute Error in classical time series; thus, they are measures less sensitive to outliers. People familiar
with interval data can feel comfortable using the error measures based on the Hausdorff distance as the Hausdorff distance is the distance applied in Interval Algebra
and taken as a basis in several symbolic methods dealing with intervals [PL03].
The choice of an ITS error measure also should be guided by wether the concept
of accuracy lies on the midpoints and raddi of intervals, or on their lower and upper
bounds. The aspects commented in this section should guide practitioners when
choosing an ITS error measure, but decision should also be oriented by other factors
such as the domain of the problem.
5 A primer approach to forecast interval time series
At the moment, there are no specific forecasting methods proposed for ITS, but
forecasting methods for classical time series can be applied in the following way.
First, ITS should be expressed in terms of their minimum and maximum series, or
of their midpoint and radius series. Each of these series should be independently
analyzed using classical time series analysis methods in order to find the components of its pattern (trend, cycle and seasonality) including possible non-linearities.
Then, it should be determined which pair of series is going to be forecasted and the
forecasting method for each one; moreover, if appropriate, a multivariate method
can be applied. Then, the value of the parameters of the chosen method should be
estimated minimizing an ITS error measure in the training set. Finally, the accuracy
of the calibrated method has to be corroborated with the test set.
Consider an ITS representing the minimum and maximum daily price of the
share of a Spanish bank company from 2005-05-01 until 2005-09-30 (see Fig.1). We
will propose different ways of forecasting this ITS based on exponential smoothing
methods. They are simple methods, but many times have been shown that their
forecasting ability is, on average, as good as that of the more sophisticated ones.
We have proposed three models3 . The midpoint-radius approach, which consists of
a Holt’s Exponential Smoothing method for the midpoints series (α = .99; β =
.02) because there is a trend in the series; and a Single Exponential Smoothing
method for the radius series (α = .16) as it has no trend present. The minimummaximum approach, which consists of the Holt’s Exponential Smoothing method
for the minimum series (α = .99; β = .03) and for the maximum series (α = .99; β =
.14); because both series have trend. The naive method ([Ŷ ]t+1 = [Y ]t ]) will be
applied to determine wether the use of the other two methods is justified or not.
Table 1 shows the one step-ahead forecast performance of the three models in the
test set. The midp-rad approach outperforms the two other approaches, min-max
3
The parameters of the models have been estimated by a genetic algorithm. The
GA have searched good values through the parameter space in order to minimize
the value of the M DEIY in the training set (76 periods).
Introducing interval time series: accuracy measures
1145
Fig. 1. ITS representing the minimum and maximum daily prices of a share
and naive, which obtain quite similar results. Though the conclusion cannot be extrapolated, it agrees with the intuitive notion that it seems more appropriate to deal
with ITS in terms of midpoint and radius than in terms of minimum and maximum,
because it models separately the behavior of the interval location (midpoint) and of
the interval inner variability (radius).
Table 1. Errors of the different forecasting methods in the test set (32 observations)
Approach
M DEH M DSEH
−2
−2
M DEIY
M DEDC
−2
midp-rad 10.22 · 10 2.21 · 10 8.38 · 10 30.97 · 10−2
min-max 11.11 · 10−2 2.35 · 10−2 8.81 · 10−2 33.6 · 10−2
naive
11 · 10−2 2.37 · 10−2 8.5 · 10−2 31.43 · 10−2
6 Conclusions
ITS provides an interesting approach to sample and aggregate massive temporal
data reporting on the range variation through time of the observed variables. This
article has offered a first approach to the field, stressing in accuracy measurement.
However, other matters await further research, for example: the development of new
forecasting methods for ITS along with case studies endorsing their usefulness; the
definition of concepts that allow the description of an ITS; empirical comparisons
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Javier Arroyo and Carlos Maté
of different forecasting methods in order to draw conclusions to guide practitioners;
and so on. Besides ITS, it must be borne in mind that other kinds of symbolic-valued
time series, such as histogram and distribution time series await investigation.
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