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http://www.aimspress.com/journal/Math
AIMS Mathematics, 7(10): 19202–19220.
DOI: 10.3934/math.20221054
Received: 26 March 2022
Revised: 18 August 2022
Accepted: 26 August 2022
Published: 30 August 2022
Research article
On smoothing of data using Sobolev polynomials
Rolly Czar Joseph Castillo and Renier Mendoza*
Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines
*Correspondence: Email: rmendoza@math.upd.edu.ph.
Abstract: Data smoothing is a method that involves finding a sequence of values that exhibits the
trend of a given set of data. This technique has useful applications in dealing with time series data
with underlying fluctuations or seasonality and is commonly carried out by solving a minimization
problem with a discrete solution that takes into account data fidelity and smoothness. In this paper, we
propose a method to obtain the smooth approximation of data by solving a minimization problem in
a function space. The existence of the unique minimizer is shown. Using polynomial basis functions,
the problem is projected to a finite dimension. Unlike the standard discrete approach, the complexity
of our method does not depend on the number of data points. Since the calculated smooth data is
represented by a polynomial, additional information about the behavior of the data, such as rate of
change, extreme values, concavity, etc., can be drawn. Furthermore, interpolation and extrapolation
are straightforward. We demonstrate our proposed method in obtaining smooth mortality rates for
the Philippines, analyzing the underlying trend in COVID-19 datasets, and handling incomplete and
high-frequency data.
Keywords: data smoothing; Whittaker-Henderson method; Sobolev polynomials; high-frequency
data; approximation; generalized cross validation score
Mathematics Subject Classification: 65K10, 90C23, 35A15
1. Introduction
Data smoothing is a method commonly used to obtain a smooth approximation of a crude data set.
If {f1,f2,..., fn}is a sequence of ndata points, the goal of data smoothing is to find a corresponding
sequence of graduated data points {u1,u2,...,un}that provides a better representation of the underlying
unknown true values [7,33]. Data smoothing is also referred to as data graduation in actuarial science
and is commonly used in actuarial studies to compute smooth mortality rates from crude data.
Goodness-of-fit and smoothness are the two most important criteria in data smoothing [7]. The
Whittaker-Henderson method (WHM), a common non-parametric technique often presented as an
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alternative to the moving average method [18, 20] in data smoothing, generates the smoothed data
points while balancing both criteria. In WHM, the sequence of smoothed data points is the minimizer
u={ui}n
i=1of the function
Q(u)=
n
X
i=1
ωi(ui−fi)2+λ
n
X
i=1
(∆ui)2,(1.1)
where
ωi: positive weight vector associated with the data,
∆ui: Newton’s advancing operator defined by ∆ui:=ui+1−ui,
λ: a positive smoothing parameter.
The first term is a measure of fidelity to the original data, while the second term is a measure of
smoothness. The weights ωi’s are pre-determined and data-dependent. Note that for high values of λ,
the smoothing is favored, while lower values result in smoothed data that is closer to actual values. A
WHM where the order of the difference operator ∆is set to 2 is known as the Hodrick-Prescott (HP)
filter [15].
Several techniques for data graduation can be found in [12,16,33, 43, 44]. These methods treat data
graduation as a minimization problem in Rn. In [26], the WHM in (1.1) is generalized as a minimization
problem in the function space L2(Ω). By treating the data {u1,u2,...,un}as a piecewise linear function,
the data graduation problem in (1.1) is solved by minimizing the functional ˜
J(u) : L2(Ω)−→ Rgiven
by
˜
J(u)=1
2ZΩ
ω(u−f)2dx +λ
2ZΩ
|Du|2dx,(1.2)
where fis an interpolation of the data points {f1,f2,..., fn},ωis an interpolation of the weights
{ω1, ω2, . . . , ωn}, and Dis the derivative operator.
The functional (1.2) can be viewed as the infinite-dimensional generalization of (1.1). Note that
the second term in (1.2) is a penalty term for data smoothing. This technique of adding a term for
smoothing is called regularization, which has gained popularity in various applications [2, 30, 31, 34,
39]. In this work, we modify the smoothing term of the functional ˜
Jin (1.2) and consider
J(u)=1
2ZΩ
ω(u−f)2dx +λ
2ZΩ
|Dmu|2dx,(1.3)
where mis the order of the differential operator D. By doing this, the solution gains more regularity.
To minimize Jin (1.3), we project the problem to a finite-dimensional polynomial space. Hence, the
solution that we obtain is a polynomial. This approach is motivated by [23], where it is argued that
the WHM is useful in obtaining smooth data points when these graduated data points approximate a
polynomial. One advantage of having a polynomial solution is that it is easier to analyze the solution.
With a polynomial solution, one can easily obtain extreme values, the relevant points at which the
graduated data are increasing or decreasing, and concavity of the smooth data. Furthermore, having a
continuous solution instead of discrete points makes interpolation and extrapolation much easier.
Our proposed method can be used to understand time series trends by analyzing the resulting
polynomial solution. In [40], the use of polynomials is demonstrated in the analysis of COVID-19
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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cases. By partitioning the daily new COVID-19 cases into sections and fitting these sections into either
a logarithmic, exponential, or linear function, the authors characterized the trend of the disease and
identified periods when the incidence of the disease is rapidly or slowly increasing. However, it is not
clear how the data set is partitioned. Identifying all possible partitions of data points and evaluating the
fit of these points to each of the three functions may be computationally expensive. With polynomials,
partitioning can be set at inflection points.
The function Qin (1.1) uses the Newton’s advancing operator ∆ui, which assumes that data points
of independent variables must be evenly spaced. Hence, the WHM cannot be used for data with
missing terms. Our proposed scheme does not have this limitation because one can still compute the
polynomial interpolation of the data, regardless if it is not evenly spaced or some of its terms are
missing. Another disadvantage of WHM is that it solves a linear system whose dimension is equal to
the number of variables in (1.1). Hence, smoothing high frequency data using WHM requires solving a
high-dimensional linear system. In our proposed method, the number of variables depends only on the
degree of the polynomial. Thus, regardless of having incomplete or high frequency data, the resulting
minimization problem has the same dimension.
This paper is organized as follows: Section 2 discusses the derivation of the minimizer of (1.3)
in a finite-dimensional polynomial space. Section 3 presents our proposed algorithm. In Section 4,
we present the applications of our method on mortality rates and COVID-19 data. We also apply our
method to data with missing terms and high frequency data. Finally, we give our conclusions and
recommendations in Section 5.
2. Theoretical framework
Because we expect the minimizer of (1.3) to be at least m-times differentiable, we consider the
Sobolev space Hm(Ω) [9]. We present a first-order optimality condition for (1.3). We show that solving
this optimality condition is equivalent to minimizing Jin (1.3).
Theorem 1. Suppose f and ωare sufficiently smooth functions. Because ωis an interpolation of the
weights, then we can assume that 0< ω(x)≤¯ω,∀x∈Ω, for some ¯ω > 0. Then, u ∈Hm(Ω)is a
minimizer of (1.3), if and only if, u satisfies
ZΩ
ω(uv)dx +λZΩ
D(m)u·D(m)vdx =ZΩ
f vdx,(2.1)
for all v ∈Hm(Ω)with m ∈N.
Proof. Let v∈Hm(Ω) with m∈N. Define r:R→Rby r(t) :=J(u+tv)−J(u). Suppose uis the
minimizer of (1.3). Then J(u)≤J(u+tv) for any t∈R. Hence, r(t)≥0, ∀t∈Rand r(t)=0 at t=0.
By the definition of J, we can simplify r(t) as
r(t)=t"ZΩ
ω(uv −f v)dx +λZΩ
D(m)u·D(m)vdx#+t2
2ZΩ
ωv2+|Dmv|2dx.
Hence,
r0(t)="ZΩ
ω(uv −f v)dx +λZΩ
D(m)u·D(m)vdx#+tZΩ
ωv2+λ|Dmv|2dx.
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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Since ris optimal at t=0, then
0=r0(0) ="ZΩ
ω(uv −f v)dx +λZΩ
D(m)u·D(m)vdx#.
Rearranging the above equation gives us (2.1).
Now, suppose usatisfies (2.1). Let z∈Hm(Ω) and define v:=z−u. Then,
J(z)−J(u)="ZΩ
ω(uv −f v)dx +λZΩ
D(m)u·D(m)vdx#+1
2ZΩ
ωv2+λ|Dmv|2dx.
From the assumption, the first two terms of the above equation is 0. Thus,
J(z)−J(u)=1
2ZΩ
ωv2+λ|Dmv|2dx ≥0.
Therefore, for any z∈Hm(Ω), J(z)≥J(u), which means that uis a minimizer of J.
Theorem 1 tells us that it is sufficient to solve (2.1) to minimize Jin (1.3). To show that (2.1) has a
unique minimizer, we first need the following results.
Lemma 1. Let Ω⊂Rdbe bounded with Lipschitz boundary. Suppose g satisfies the following
conditions:
(1) g :Hm(Ω)→[0,+∞)is a seminorm.
(2) There exists a positive constant C such that 0≤g(v)≤CkvkHm(Ω), for all v ∈Hm(Ω).
(3) If v ∈ Pk−1:={space of polynomials of degree k −1}and g(v)=0, then v ≡0. Then the norm
kukHm(Ω):=
X
|αk≤m
kDαukL2(Ω)
1/2
(2.2)
is equivalent to the norm
kuk0
Hm(Ω):=
g2(u)+X
|α|=mZΩ
|Dmu|2dx
1/2
.
(2.3)
Proof. This is a special case of the result proven in [38], where the equivalence was shown on a system
of functionals and on a more general Sobolev space Wm,p(Ω).
Lemma 2. For Ω⊂R, we define
|u|0
Hm(Ω):= ZΩ
ωu2+λ|Dmu|2dx!1/2
.
(2.4)
Then the norm in (2.4) is equivalent to the norm in (2.2), or equivalently, ∃ρ1, ρ2>0such that
ρ1kukHm(Ω)≤ |u|0
Hm(Ω)≤ρ2kukHm(Ω).(2.5)
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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Proof. The norm in (2.4) is equivalent to
|u|0
Hm(Ω):= λZΩ
ω
λu2+|Dmu|2dx!1/2
.
We can set
g(u) := ZΩ
ω
λu2dx!1/2
.
Note that gis a weighted L2(Ω)-norm, and hence the first condition of Lemma 1 is satisfied. Because
ω≤¯ω, for all x∈Ωand from the definition of norm in (2.2), we get
g(v)≤r¯ω
λkvkL2(Ω≤r¯ω
λkvkHm(Ω).
Therefore, the second condition of Lemma 1 is satisfied. The last condition easily follows because
ω(x)
λ>0 for all x∈Ω. Therefore, by Lemma 1, our assertion holds.
Theorem 2. The variational formulation in (2.1) has a unique solution in Hm(Ω).
Proof. We define
a(u,v) :=ZΩ
ω(uv)dx +λZΩ
D(m)u·D(m)vdx
and
b(v) :=ZΩ
f vdx.
We use Lax-Milgram’s Lemma to prove the existence of the unique solution. For the discussion of
this lemma, we refer the readers to [9]. To do this, we need to show that ais bilinear, bounded, and
coercive, and bis linear and bounded. The bilinearity and linearity of aand b, respectively, follows
directly from their respective definitions.
We now show that ais bounded, that is, ∃C1>0 such that |a(u,v)| ≤ C1kukHm(Ω)kvkHm(Ω). We use
triangle inequality, Cauchy-Schwarz inequality, and the definition of the norm in (2.2). Thus,
|a(u,v)| ≤ ZΩ
ω(uv)dx+λZΩ
D(m)u·D(m)vdx
≤¯ωkukL2(Ω)kvkL2(Ω)+λkD(m)ukL2(Ω)kD(m)vkL2(Ω)
≤max( ¯ω, λ)
| {z }
C1
kukHm(Ω)kvkHm(Ω).
We use (2.5) from Lemma 2 to show that ais coercive, that is, ∃C2>0 such that |a(u,u)| ≥ C2kuk2
Hm(Ω)
for all u∈Hm(Ω). Indeed,
|a(u,u)|=ZΩ
ωu2+λ|Dmu|2dx
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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=h|u|0
Hm(Ω)i2
≥ρ1
|{z}
C2
kuk2
Hm(Ω).
Finally, we show that bis bounded, that is, ∃C3>0 such that |b(v)| ≤ C3kvkL2(Ω). Using Cauchy-
Schwarz inequality, we get
|b(v)|≤kfkL2(Ω).
| {z }
C3
kvkL2(Ω).
We have shown in Theorem 1 that we can minimize Jin (1.3) by solving an equivalent variational
formulation, which we have shown in Theorem 2 to have a unique solution in Hm(Ω). Because of the
equivalence of k·kHm(Ω)and |·|0
Hm(Ω), we can use the following inner product for Hm(Ω):
hu,viH:=ZΩ
ω(uv)dx +λZΩ
D(m)u·D(m)vdx,(2.6)
where ωis a continuous function in Ω, and λ > 0. With the above inner product, we can rewrite the
optimality condition in (2.1) as
hu,viH=hf,viL2(Ω)∀v∈Hm(Ω).
In this work, we pose the above variational formulation in a finite-dimensional subspace Sof
Hm(Ω), that is, we solve
hu,viH=hf,viL2(Ω)∀v∈S.
We consider the subspace Swhich is spanned by a set of orthonormal polynomials {p1,p2,...,pl,}
for some l∈N. We refer to {p1,p2,...,pl}as Sobolev polynomials because they are constructed using
the inner product in Eq (2.6), which involves derivatives, such that each polynomial in this subspace are
also in the Sobolev space Wm,p(Ω). We were motivated to use Sobolev polynomials because they are the
best polynomial approximation to a function fwith respect to the L2(Ω) norm [28]. A comprehensive
review of the history and recent development in the study of Sobolev polynomials can be found in [27].
Since S=span{p1,p2,...,pl}, it is sufficient to find u∈Ssuch that
hu,pjiH=hf,pjiL2(Ω)∀j∈ {1,2,...,l}.
If u∈S, then u=
l
P
i=1
uipiand so, we solve
h
n
X
i=1
uipi,pjiH=hf,pjiL2(Ω)∀j∈ {1,2,...,l}.
Equivalently,
n
X
i=1
uihpi,pjiH=hf,pjiL2(Ω)∀j∈ {1,2,...,l}.
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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Because hpi,pjiH=δi j, we obtain
ui=hωf,piiL2(Ω)∀i∈ {1,2,...,l}.(2.7)
Therefore, the solution u∈Sof
hu,viH=hf,viL2(Ω∀v∈S
is given by
u=
l
X
i=1
pihωf,piiL2(Ω).(2.8)
We construct the orthonormal polynomials using Gram-Schmidt orthonormalization process [24].
We will also use polynomial interpolations for ωand fso that the solution uis a linear combination of
polynomials. The Gram-Schmidt process is started using Chebyshev polynomials as the initial basis
functions [36].
3. Proposed algorithm
To compute uin (2.8), we first obtain the polynomial interpolations of ωand f. Then we get
a set of orthonormal polynomials from an independent set of polynomials. In our case, we use the
Chebyshev polynomials {c1,c2,...,cl}, for some user-defined degree l−1. Then, using Gram-Schmidt
orthonormalization process, we construct a set of orthonormal polynomials {p1,p2,...,pl}. The order
of differentiation mis also user-defined. The construction of the Sobolev polynomials is summarized
in Algorithm 1.
Algorithm 1 Creating a set of orthonormal Sobolev polynomials from Chebyshev polynomials
1: Input: Set the desired degree of polynomial l−1, and the endpoints a,bof the interval of interest
according to the crude data points.
2: Construct a set of basis Chebyshev polynomials of dimension lover the interval (a,b).
3: Apply the Gram-Schmidt orthogonalization procedure that uses the inner product in Eq (2.6) on
the set of Chebyshev polynomials from step 2.
4: Output: The Gram-Schmidt orthogonalization procedure that uses the specified inner product in
step 3 results in a set of orthonormal Sobolev polynomials {p1,p2,...,pl}.
The user can also specify the value for the smoothing parameter λ. However, by default, the
algorithm uses the approach presented in [12, 42]. The smoothing parameter λis computed as the
minimizer of the generalized cross validation (GCV) score, which is expressed as
GCV (λ)=
n
n
P
i=1
(ˆ
fi−fi)2
n−
n
P
i=1
(1 +λγ2
i)−1!2,(3.1)
where
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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ˆ
f=In+λDTD−1f,
nis the number of data points, γ0
isare the eigenvalues of DTD, and Dis a tridiagonal matrix with
entries given by
Di,i−1=2
hi−1(hi−1+hi),Di,i=−2
hi−1hi
,Di−1,i=2
hi(hi−1+hi),
with hirepresenting the step between ˆ
fiand ˆ
fi+1.
For a detailed discussion on how to use GCV to identify the smoothing parameter, we refer the
readers to [11, 12, 14, 41, 42]. To compute the GCV value numerically, we used the Matlab code
provided in the Appendix of [12]. The GCV function in (3.1) may have multiple local minima so we
have to use a global minimizer. In this study, we use Genetic Algorithm (GA) which has growing
applications in science and engineering because of its capability to estimate the global minimum and
its non-reliance on the derivative of the objective function [21, 22, 37]. We utilize the Matlab built-in
function ga, which only requires the upper and lower bounds for the smoothing parameter. Although
GA can converge to the global minimum, it can still sometimes get stuck at a local minimum. To
guarantee global convergence, we run ga 10 times and store the best solution λGA . To further improve
accuracy, we hybridize GA with fmincon, a Matlab built-in code for interior point algorithm [6], which
is a local search technique. We use λGA as the initial guess of fmincon. We denote λ?as the minimizer
of the hybrid GA-interior point method. Upon obtaining λ?, we can use the orthogonal polynomial
calculated in Algorithm 1 and obtain the smooth polynomial approximation uin (2.8). We summarize
the proposed method in Algorithm 2.
Algorithm 2 Data graduation using Sobolev polynomials
1: Input: The crude data {fi}n
i=1.
2: Set the value for l(dimension of the polynomial space) and m(order of differentiation).
3: Determine the polynomial interpolations, ωand f, of the weights and the data, respectively.
4: Set the bounds for the smoothing parameter [λmin, λmax ].
5: Estimate the minimizer the function GCV in (3.1) over the interval [λmin, λmax]using the Matlab
built-in program ga 10 times. Set λG A as the minimizer among the 10 solutions with the least GCV
score.
6: To get a more accurate global minimizer λ?, implement the interior point method to the GCV
in (3.1) over the interval [λmin, λmax]using the Matlab built-in pogram fmincon.
7: Using the smoothing parameter λ?, obtain a set of orthonormal Sobolev polynomial functions
{p1,p2,...,pl}according to Algorithm 1.
8: Output: The smooth approximation of the data as a polynomial function u=
l
P
i=1
pihωf,piiL2(Ω).
Given a data {f1,f2,..., fn}, a unique smooth approximation uis calculated using (2.8). Note that
the orthogonal polynomials pido not rely on the data. This means that if the data has a particular error
structure, uncertainty might occur in the coefficient terms uiin (2.7). In this study, we rely on a general
bootstrap approach presented in [8, 10] to quantify the uncertainty of the coefficients and construct
confidence intervals. The step-by-step procedure is stated in Algorithm 3.
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Algorithm 3 Uncertainty quantification
1: Input: The coefficients uicalculated from (2.7).
2: Generate Msimulated data ˆ
f1,ˆ
f2,..., ˆ
fnby adding noise assuming an error structure to the
calculated Sobolev polynomial uat points x1,x2,...,xn.
3: Recalculate the coefficients ˆuifrom (2.7) given the simulated data ˆ
f1,ˆ
f2,..., ˆ
fn.
4: From the recalculated Mset of values of ˆui, characterize the distribution of the coefficients and
calculate the confidence interval.
5: Output: Histograms to display the empirical distributions of the coefficients and the corresponding
confidence intervals.
4. Results and discussion
In this section, we test the proposed algorithm to determine the graduated values of different
datasets. All numerical simulations were performed in Matlab 2021a on a computer with Intel(R)
Core(TM) i7-8550U CPU clocked at 1.80GHz and 1.99 GHz with 8 GB of RAM that runs Windows
10 OS. By default, we set the order of derivative to 10 and the degree of the polynomial to 8. In all
simulations, the bounds for the smoothing parameter is set to [λmin, λmax ]:=[0.1,5].
For the first application, we test our method to calculate the graduated value of male mortality
rates. We obtained the crude mortality rates from the 2017 Philippine Intercompany Mortality Study
of the Actuarial Society of the Philippines [1]. The illustration of our proposed scheme is shown
in the left panel of Figure 1. The red curve represents the linearly interpolated data and the black
dashed curve shows our proposed method. The smooth approximation using Whittaker-Henderson
graduation is shown in green. It can be seen how our proposed method obtained similar results as the
Whittaker-Henderson method. Both methods produced mortality rates that corrected the fluctuations
observed in the crude data. Moreover, plots produced from both methods also continuously increase
across all ages. While graduated data produced by the discrete Whittaker-Henderson method may
appear smooth because of interpolation, the graduated data produced by our method is guaranteed
to be smooth because it is represented by a polynomial. Note that the crude data for the mortality
rates for ages 80 to 85 are not available. The Whittaker-Henderson method cannot be used to obtain
the graduated values outside the range of the data set. In [1], extrapolation was used to calculate the
graduated values for ages 80–85. The authors used the Gompertz-Makeham model (blue curve) to do
this. An advantage of our proposed method is that extrapolation is straightforward. One simply needs
to evaluate the polynomial at the points outside the range of the data set. The computed graduated
value using the Gompertz-Makeham model and our method are both shown in Figure 1. The right
panel shows the plot of the GCV function in (3.1) for λ∈[0.1,5]. Observe that the plot (blue curve)
is nonlinear and multi-modal which justifies the use of the hybrid GA and interior point method as a
global minimization algorithm. The red dot shows the global minimize (λ?=0.5285), which we set as
the smoothing parameter for this problem. Figure 2 presents our uncertainty analysis for this problem
using Algorithm 3. For this simulation, we assume a Gamma error structure [5]. The uncertainty in our
calculated coefficient values translates into the confidence bounds (red dashed lines) around the smooth
approximation of the data. Since the degree of the polynomial is 8, the dimension of the subspace S
is 9, which means that we calculate 9 coefficients ui’s. The histograms and confidence intervals are
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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presented in the bottom panels of Figure 2. Observe that all the computed coefficients fall within their
corresponding 95% confidence intervals.
In the proposed algorithm, the user can choose the degree of polynomial. The left panel of Figure 3
demonstrates our next example to determine the graduation of male mortality rates for ages 0–30
using various degrees of polynomial while fixing the order of derivative to 10. For this example,
we calculated λ?=1.3798. As presented, the graduation of mortality rates can produce a range of
polynomial functions that practitioners can choose from depending on their purpose. However, it is
also possible to set the degree of polynomial based on the fit of the smoothing curve. One approach
is to iterate over a set of positive integers and select the degree that produces the polynomial that has
least distance to the piecewise linear interpolation of data. The right panel of Figure 3 demonstrates
this and shows that a polynomial with degree 8 (l=9) has the best fit. This polynomial is shown as the
red curve in the left panel.
We also applied our proposed method to obtain a smooth time series data for daily new COVID-19
cases in Germany [17]. This dataset spans new COVID-19 cases from 1 September 2020 to 31 May
2021. Here, λ?=0.7227. The left panel in Figure 4 presents the crude daily new COVID-19 cases
in Germany as blue line, and the graduated data as red line. As shown, the plot of the graduated data
points exhibits peaks and troughs that follow the crude data. The underlying trend in the data can also
be observed after smoothing.
The right panel in Figure 4 compares the results of smoothing using different methods, implemented
using the Matlab built-in function smoothdata. Simple moving average, gaussian or kernel
smoothing, and smoothing using locally weighted scatter plot smoothing (LOWESS) all use a window
size of 7 because this allows coverage of both the incubation period and the time from the first
appearance of symptoms to diagnosis [19].
Daily new COVID-19 cases can be viewed as the change in the total number of cases for every small
change in time. Hence, by integrating the interpolating polynomial, the cumulative number of COVID-
19 cases can be approximated. As an illustration, we integrate the polynomial obtained in Figure 4 to
compute the cumulative number of cases. In Figure 5 (left panel), we illustrate how the integral of
the polynomial closely approximates the actual cumulative data. The resulting approximation of the
cumulative data is also a polynomial, which makes finding the inflection points and concavity of the
curve easier. These changes in concavity are shown in Figure 5 (right panel). The segments of the plot
in red indicate a deceleration in the number of cases while the blue segments indicate acceleration.
Plots like this are useful in evaluating the effectiveness of policies in containing the pandemic, or in
identifying events that may have contributed to a more rapid spread of the disease.
We also applied our method to obtain the trend in the total COVID-19 deaths in China from 23
January 2020 to 22 February 2020 [29]. For this example, we computed λ?=0.8018. It was shown
in [3] that the growth in the number of deaths related to COVID-19 in the early stages of the pandemic
follows a quadratic trend. By setting the degree of polynomial to 2, we obtain the graduation in
Figure 6, which captures the trend of the data of COVID-19 deaths.
One of the important applications of our proposed method is in handling time series data with
missing terms. Figure 7 demonstrates this in the case of missing time series data points for
Schistosomiasis cases in years 2009–2011 and 2013 [35]. Here, λ?=1.5577. As shown in the
left panel, our method produced graduated data points that are also consistent with the general trend
in the dataset. Note that both WHM and moving average technique require that the time series data
AIMS Mathematics Volume 7, Issue 10, 19202–19220.
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be evenly spaced. We also applied other methods in extrapolating the missing data points. The right
panel of Figure 7 compares the results of our smoothing algorithm when paired with either linear,
spline, piecewise cubic hermite interpolating polynomial (PCHIP), and the modified Akima piecewise
cubic hermite interpolation (MAkima) to fill the missing data. We used the Matlab built-in function
fillmissing using different methods in our numerical experiments. Our method was applied after
extrapolating the data. For this example, smoothing using the points generated by the piecewise linear
interpolation produced the least L2-norm error. The user can choose which approach they prefer when
filling in missing data. This is a pre-processing step that should be implemented before applying our
method to data with missing values.
Our method offers a useful alternative in smoothing high-frequency datasets. WHM entails high
computational cost in smoothing large datasets because it requires solving a linear system whose
dimension is equal to the number of data points. On the contrary, our method is more suitable because
the number of unknowns depends only on the degree of the polynomial used in the graduation. We
demonstrate this in Figure 8 using temperature data from [32] collected every 5 minutes from 24 April
2019 to 1 October 2020. The data set considered includes 140 thousand data points. For this example,
we calculated λ?=0.9586. Applying WHM to this dataset is not possible because our computer can
only handle a linear system of a dimension of at most 30 thousand. Moreover, observe that there are
missing data points shown in Figure 8 as a break in the blue line. Based on the results of Figure 7, we
used linear interpolation to fill the missing data. Our method was able to provide predicted graduated
values for the missing sequence in this high-frequency dataset. Moreover, the smooth approximation
also exhibits the trend in the data.
Figure 1. Graduation of male mortality rates. The black dotted line in the left panel
represents the graduated rates using our method. The minimization of the GCV function
is presented on the right panel.
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Figure 2. Graduation of male mortality rates (red solid line) with the quantified uncertainty
(uppermost panel). The blue circles are the mortality data. The cyan lines are the 10000
realizations of the mortality data assuming Gamma error structure. The red dashed lines
illustrate the 95% confidence bands around the smooth approximation. The bottom panels
show the histogram that shows the empirical distribution of the coefficients ui.
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Figure 3. Graduation of male mortality rates for age group 0–30. The left panel presents the
smoothing of data using different dimensions of basis (or l−1 degree of polynomial), while
the right panel presents the L2-norm error corresponding to each dimension of basis. The
polynomial with degree equal to 8, which is shown as red curve in the left panel produced
the best fit according to its L2-norm error, shown as red bar in the right panel.
Figure 4. Data smoothing of daily new COVID-19 cases in Germany, September 2020–May
2021. The red line in the left panel represents the graduated new cases. The right panel
compares our proposed method with other smoothing techniques.
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Figure 5. Cumulative COVID-19 cases in Germany, September 2020–May 2021. The left
panel compares the cumulative COVID-18 data with the proposed method. The right panel
visualizes the time intervals when cases are increasing and decreasing based on the concavity
of the calculated Sobolev polynomial.
Figure 6. Graduation of total COVID-19 deaths in China, 23 January–22 February 2020.
The red line represents the graduated total number of deaths due to COVID-19, assuming
that the data follow a quadratic trend [3].
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Figure 7. Schistosomiasis cases, 2000–2019. The left panel presents the smooth data using
different techniques to fill the missing data while the right panel shows the corresponding
L2-norm error.
Figure 8. Graduation of water temperature, 24 April 2019–1 October 2020. Blue line
represents the crude data. Red line represents the smoothed data.
5. Conclusions
In this paper, we presented an alternative method in data graduation that uses Sobolev polynomials.
We have proven that the resulting minimization problem has a unique solution in a suitable function
space. Furthermore, we formulated an approach using the Gram-Schmidt orthogonalization process
to find the solution in an approximate polynomial space. We applied this method in obtaining the
graduated values of male mortality rates in the Philippines, and COVID-19 data. We also demonstrated
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the usefulness of our method in addressing missing data points and smoothing high frequency data.
Because the obtained results are polynomials, interpolation and extrapolation become straightforward.
Furthermore, inflection points and concavity of the the graduated values are easier to identify. The
regularity of polynomials makes the analysis simpler.
Our study has some limitations that can be explored in future research. First, although one can
calculate the smoothing parameter by minimizing the GCV score, the order of the derivative and the
degree of polynomial are entirely user-defined. If the user does not want to use the default values, the
user can fix the order of the derivative and choose the degree of the polynomial that will yield the least
L2-error. This can be time-consuming especially if the user sets the order of the derivative to a high
value. As a future study, we can explore how both of these relevant parameters can be chosen based
solely on data. Second, our study is posed in one-dimensional time series data. For future work, one
can extend our method to solve multi-dimensional data smoothing problems. For example, one can
consider mortality rates that depend on age and policy duration. Third, one can also consider using
other regularization terms (e.g., total variation) and other non-polynomial basis functions. Fourth, in
the case of high-frequency data, one can cut the computational cost if the interval is subdivided into
segments to reduce the data size [15]. Once the data is segmented, the smooth approximation can
be implemented using parallel computing. This means that the solution in each segment represents a
spline of the entire smooth approximation. However, this will require continuity conditions between
segments, which needs a rigorous theoretical analysis. Fifth, another area that can be explored is the
relation of our method with space-state models. Since we are treating the smoothing as a minimization
in an infinite dimensional Lebesgue space, an extensive study on how to formulate the corresponding
difference equations needs to be carried out. These are exciting research directions but demand a
thorough investigation and would require other numerical optimization algorithms.
Acknowledgments
This research was supported by a grant from the Computational Research Laboratory of the Institute
of Mathematics, University of the Philippines Diliman.
Conflict of interest
The authors declare that they have no conflict of interest.
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