KERNEL AND WAVELET SMOOTHING:
Basic theory and examples of practical data analysis1)2)
V t zslav Vesel
Brno, Czech Republic
1991 Mathematics Subject Classi cation. Primary: 62G07; Secondary: 65D10
Key words and phrases: Kernel smoothing, wavelet smoothing
1. INTRODUCTION
Kernel and wavelet smoothing are two modern
tools for data analysis at this time. We shall explain
the basic theoretical ideas of both methods
and confront the results obtained when processing
each of three environmental data series by either
method.
Notation.
N ::: the set of all natural numbers
Z ::: the set of all integers
R ::: the set of all real numbers
L1 ::: the Banach space of all complexvalued
functions which are absolutely
integrable on R in the
Lebesgue sense
L2(I) ::: the Hilbert space of all complexvalued
measurable functions which
are absolutely square-integrable in
the Lebesgue sense on the interval
I R; we denote further
hg1;g2i =
R
I g1(x)g2(x)dx :::the
inner product and
kgk =
p
hg;gi =
qR
I jg(x)j2 dx
:::the norm
L2 := L2(R)
i;j ::: the Kronecker symbol (= 1 for i =
j and zero otherwise)
E ::: the operator of expectation
2. THE BASIC CONCEPT OF KERNEL
SMOOTHING
Problem statement
Let X = (X1;X2;:::;Xn) be sorted values a
X1 < X2 < ::: < Xn b of an independent
variable x 2 R, which are either xed prescribed
or samples of a random variable X, and let Y =
(Y1;Y2;:::;Yn) be samples of a random variable
Y observed for X within the additive model
Yi = m(Xi)+ Ei; i = 1;2;:::;n; n 2 N
where m(x) = E(YjX = x) is an unknown function
to be estimated from a set of n measurements
of m(Xi) which are loaded with independent and
identically distributed errors Ei.
The frequently considered nonparametric estimator
bm(x) of m(x) is usually the weighted average
of response observations Y1;Y2;:::;Yn and a general
formula for this estimator is
m(x) bm(x) =
nX
i=1
Yi Wn;i(x) (1)
where Wn;i(x) is a suitable weight function
depending in general not only on the
tted point x but also on the covariate observations
X1;X2;:::;Xn, i.e. Wn;i(x) =
Wn;i(x;X1;:::;Xn). Typically Wn;i(x) ! 0 as
jx Xij ! 1. Such an estimator is called a linear
smoother because it is linear in the response.
The linear smoothers di er by the choice of the
weight functions Wn;i;i = 1;2;:::;n: (see e.g. 8]),
and a very useful method for the choice of weights
is the kernel smoothing.
De nition.
A real function K(x) is called a kernel function
(kernel) if K(x) 2 L1 and
R 1
1 K(x)dx = 1.
We put also Kh(x) = 1
hK x
h ;h > 0 the widthmodi
ed kernel function which, clearly, pre-
serves
R 1
1 Kh(x)dx = 1.
There are two popular approaches to constructing
kernel estimates.
De nition. (Weight function by NadarayaWatson
(1964) 12, 14])
W(1)
n;i (x) := Khi (x Xi)Pn
i=1 Khi (x Xi) (2)
where K(x) is a continuous kernel with fast decay
limx! 1 xK(x) = 0.
De nition. (Weight function by GasserMuller
(1979) 7])
Put so = a; si = 1
2
(Xi + Xi+1) for i =
1;2;:::;n 1 and sn = b.
1)
Research supported by the GA of the Czech Republic under grant number 201/96/0665.
2)
In: Proceedings and abstracts of ESES'96, Brno (I. Horov , J. Jure kov and J. Vosmansk , ed.), Masaryk
University of Brno, Czech Rep., Aug. 1996, pp. 163{174.
Then
W(2)
n;i(x) :=
Z si
si 1
Khi (x t)dt (3)
for a kernel K(x).
The former (eq. (2)) is based on the choice of
weights bymeansofdirect kernel evaluation|the
so-called Nadaraya-Watson estimators (see 8]).
The latter (eq. (3)) is based on weights which
are a convolution of the kernel with a histogram
representing the data (see 11]). It is typical
for both of these methods that the weights Wn;i
depend on some function K called kernel and
on parameters hi called smoothing parameters
or bandwidths. The kernel K and the bandwidths
hi determine the local properties of the estimator
bm(x) at the point x. If the bandwidths
h := h1 = h2 = ::: = hn are xed, depending
neither on the location of x nor on the covariate
values X1;X2;:::;Xn, the informationprovided by
the density of data points is not fullyincorporated
in the estimator. That is why Fan and Gijbels
introduced a new type of kernel estimator with
variable bandwidth (see 6]). Their estimator
is based on the Cleveland idea (see 1]) to obtain a
linear smoothervia alocallinearapproximationto
the regression function. For an expository paper
on the variable bandwidth method see 10].
We can also guess the xed bandwidth by the trial
and error method or in some optimal way by minimizing
a suitable error measure, the mean-square
error estimate being the typical choice.
In addition also the so-called optimal kernels
may be used. These kernels are compactly supported
on 1;1] where they coincide with a polynomial
of a given degree which smoothly decays
to zero at interval bounds 1 and 1 with order of
smoothness being prescribed.
3. THE BASIC CONCEPT OF WAVELET
SMOOTHING
Problem statement
Consider the additive model
Y(x) = m(x)+ e(x); x 2 R
where Y(x) are observed values, m(x) 2 L2 is an
unknown real function to be estimated and e(x)
is the white noise. In the discrete setting Y(x)
are of course observed only at a nite discrete set
x = X1;X2;:::;Xn.
On the contrary to kernel smoothing which yields
a direct estimate bm(x) of m(x), in the case of
wavelet smoothing(for basic wavelet theory see
for example the monographs 2, 9]) we estimate
m(x) indirectly by nding estimates bcj;k of coe cients
cj;k;j;k 2 Z in the expansion
m(x) =
1X
j= 1
1X
k= 1
cj;k j;k(x) (4)
where j;k(x) = 2j=2 x k=2j
1=2j is a speci c basis
in L2 generated by dilating and shifting a suitable
function (x) 2 L2;k k = 1 which is called
mother wavelet, clearly k j;kk = 1 is preserved
for each j;k. If this basis is orthonormal in L2
(h j;k; l;mi = j;l k;m) then (x) is called orthogonal
wavelet and cj;k are easily computed
by
cj;k = hm; j;ki =
Z 1
1
m(x) j;k(x)dx (5)
The wavelet expansions are a non-periodic analogy
of Fourier expansions of T-periodic functions
in L2( a;b]);b a = T > 0 where w(x) =
1pT expi2 x=T is the basic complex wave function
playing the role of (x) and generating the
orthonormal Fourier basis fwj(x)gj2Z;wj(x) =
w(jx). Then for T-periodic m(x) 2 L2( a;b]) we
get its Fourier series expansion
m(x) =
1X
j= 1
hm;wjiwj(x) =
1X
j= 1
cj expi2 jx=T
where
cj = 1p
T
hm;wji = 1p
T
Z b
a
m(x)wj(x)dx
= 1
T
Z b
a
m(x)exp i2 jx=T dx
are the well-known Fourier coe cients of m.
Hence it is natural to call the wavelet expansion
the wavelet series and the coe cients cj;k the
wavelet coe cients. The index j determines
the scaling parameter 1=2j which is closely related
to the j-th harmonic frequency fj = j=T associated
with the j-th Fourier coe cient cj. Nevertheless,
one principal di erence between w(x)
and (x) is that 2 L2 ) (x) ! 0 for x ! 1.
In addition the theory impliesalso another natural
condition, namely (x) 2 L1 and
R 1
1 (x)dx = 0
which says that (x) must change its sign. Altogether
(x) is a \damped wave" or wavelet.
If (x) is completely damped (zero outside of
a bounded interval) then we say that (x) is a
compactly supported wavelet. This property
shows to be very useful because each wavelet coe
cient cj;k contributes to the wavelet expansion
of m(x) only in a neighbourhood of x = k=2j, i.e.
its e ect is local compared with the globale ect
of a Fourier coe cient cj.
Wavelet smoothing
Due to the linearity of (5) we get
cY
j;k = cj;k +ce
j;k
where cY
j;k;cj;k and ce
j;k are wavelet coe cients
in the expansions of Y (x);m(x) and e(x), respectively.
The wavelet smoothing techniques are
aimed at nding a suitable modi cation rule ( )
such that (cY
j;k) = bcj;k cj;k is a good estimate
of cj;k. This approach followsagain the analogy
with Fourier-based ltration where we modify
the Fourier coe cients cY
j , an important special
case being the classical linear ltration where
(cY
j ) = jcY
j ; j 0 and f jgj2Z is the socalled
transfer function of the lter. Due to
the locale ect of cj;k for a xed k the wavelet representation
allows one to construct locally adaptive
lters in this way which is an excellent new
featurecomparedwith the classical Fourier lters
where the e ect is global.
There are three wavelet coe cient modi cation
techniques commonly applied.
1. Positive scaling
bcpos
j;k = j;kcY
j;k; j;k 0
which is the direct generalization of the
transfer function mentioned above.
2. Hard thresholding
All wavelet coe cients which are below a
certain threshold level are put to zero:
chard
j;k = 0 for jcj;kj <
cj;k for jcj;kj .
3. Soft thresholding
All wavelet coe cients are reduced by a certain
threshold level :
csoft
j;k = sign(cj;k)max(0;jcj;kj )
Donoho and Johnstone 3, 4, 5] suggested a
method for an optimal (in a certain sense) choice
of the threshold which is either universal or speci
c for each scaling level j ( = j). These and
other similar methods became known as wavelet
shrinkage or wavelet de-noising.
In practical computation we use the discrete setting
where cj;k are evaluated via DWT (Discrete
Wavelet Transform) which is the natural
counterpart of the DFT (Discrete Fourier Transform).
The related fast algorithms are known
as FWT (Fast Wavelet Transform) and FFT
(Fast Fourier transform).
4. EXAMPLES OF PRACTICAL DATA
ANALYSIS
In this section both kernel and wavelet smoothing
will be demonstrated on real time series. All computations
were accomplished in MATLAB 4.2c
and supported by specialized m- le libraries (toolboxes).
While the toolbox for kernel smoothing
has been developed by the author himself,
the results of wavelet smoothing have been obtained
using WavBox 4.3b which is an excellent
Wavelet Toolbox (218 m- les, 850 kB) of
Carl Taswell from the Stanford University, CA,
USA 13]. In particular for wavelet shrinkage we
have applied the WavBox function wdenoise with
the threshold estimator DJE (Donoho-JohnstoneEstimator)
which yields a separate threshold for
each scaling level. As the mother wavelet the
compactly supported orthogonal least asymmetric
wavelet of order 8 from the Daubechies family
has been chosen. For more details the list of main
WavBox object properties follows:
DataDimension = 1
MappingStructureType = DWT
MappingStructureClass = DWT
ObjectStructureType = DWT
ObjectStructureClass = DWT
FilterClass = ORTH
FilterFamily = DOLA
NumberVoices = 1
AnalysisFilterParameter = 8
SynthesisFilterParameter = 8
ConvolutionVersion = CPF
FilterName = DOLA16
FilterLength = 16
Description of data sets
Data set 1 (size 90):
Mean autumn atmospheric temperatures measured
in Hurbanov, Czech republic in 1903{1992.
Data set 2 (size 168):
Seasonal deviations of cloudiness from the mean
observed in the Northern Croatia in 1951{1992.
Data set 3 (size 912):
Monthly mean ow on the river Morava observed
in Krom , Czech Republic in 1916{1991.
Description of gures showing the smoothing
results
Figure 1:
Plots of raw data | data set 1 (top), data set 2
(middle), data set 3 (bottom).
Figures 2{4:
Plots of kernel smoothed data sets 1, 2 and 3 , respectively
using the weights (2) with the quartic
kernel
K(x) =
15
16
(1 x2
)2
for x 2 1;1]
0 for x =2 1;1]
and three bandwidths: optimal (top), small (middle)
resulting in undersmoothing and large (bottom)
resulting in oversmoothing.
Figures 5{7:
Plots of wavelet processed data sets 1, 2 and 3, respectively
showing the DJE smoothed data (top)
along with the wavelet coe cients for the raw data
(bottom left) and smoothed data (bottom right).
In the bottom plots cj;k are shown at positions related
horizontallytok andverticallytoj in reverse
order of level numbers (the nest scaling level 1
corresponds to the largest j).
Figure 8:
Multiresolution analysis of the data set 3. The
six plots when ordered row-by-row by the topdown
and left-right method show data smoothed
via clearing all wavelet coe cients at levels 1,1{2,
:::, 1{6, respectively. The plots visualizethe stepby-step
decrease in the resolution when still more
coe cients at successive levels are put to zero.
5. CONCLUSION
There are two basic factors which control the
smoothing operation. First, both methods allow
for a wide choice among various kernel or
wavelet shapes. Second, both methods are yielding
a great exibility in the choice of the smoothing
strategy ( xed or variable bandwidth, thresholding
method or some other speci c manipulation
with the wavelet coe cients). For example
the gures 2{4 show that the right choice of the
bandwidth magnitude is crucial to the nal e ect.
Both the variable bandwidth and a clever modi cation
of wavelet coe cients are surely a powerful
tool how to adapt to the local data behaviour.
However the grand problem is to nd the `best'
procudure just for the data we have.
Both methods o er certain universal procedures
(optimal bandwidth, DJE thresholding) which
should give an optimal result. But these optimality
criteria are hard to compare, they are more related
to the method itself than to the data being
processed for which only certain general assumptions
should be satis ed which cannot be usually
exactly veri ed (except in simulations where everything
works well). Observe that the optimal
results of wavelet smoothing from gures 5{7 are
not in a good agreement with those of optimal
kernel smoothing in gures 2{4, respectively. In
case of the data set 1 we see that kernel smoothing
with bandwidth three times smaller than the
optimal value gives a result which is clearly closer
to the optimal wavelet result. The plots for the
data set 2 exhibit opposite behaviour giving better
agreement with a bandwidth two times larger
than the optimalone. The data sets 2 and 3 seem
to be extreme cases towards the DJE thresholding.
The wavelet smoothingof the data set 2 (Fig. 6) is
nearly total (close to zero mean) saying that only
a negligible portion of useful information was detected.
The data set 3 is the other extreme (Fig.
7) exhibiting a negligible smoothing e ect. Although
kernel smoothing with optimal bandwidth
follows this trend, too (Fig. 3 and 4) we have
not obtained such extreme results. So the question
about credibility of the results is evident and
one is advised to be very careful with universal
techniques without exploiting any additional information
about the data being processed.
References
1] W. S. Cleveland, Robust locally weighted regression
and smoothing scatterplots, J. Amer.
Statist. Assoc. 74 (1979), 829{836.
2] Ingrid Daubechies, Ten lectures on wavelets,
CBMS-NSF Regional Conference Series in
Applied Mathematics, vol. 61, SIAM,
Philadelphia, Pennsylvania, 1992.
3] David L. Donoho and Iain M. Johnstone,
Adapting to unknown smoothness via wavelet
shrinkage, Technical report, Stanford University,
Department of Statistics, Stanford CA
94305, July 1994.
4] , Ideal spatial adaptation via wavelet
shrinkage, Biometrika 81 (1994), 425{455.
5] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian
and D. Picard, Wavelet shrinkage:
Asymptopia ?, J. Royal Statist. Soc. B 57
(1995), no. 2, 301{337.
6] J. Fan and I. Gijbels, Variable bandwidth and
local linear regression smoother, The Annals
of Statistics 20 (1992), no. 4, 2008{2036.
7] T. Gasser and H. G. Muller, Kernel estimation
of regression functions, In: Smoothing
Techniques for Curve Estimation (Berlin)
(T. Gasser nad M. Rosenblatt, ed.), Lecture
Notes in Mathematics, vol. 757, SpringerVerlag,
1979, pp. 23{68.
8] W. Hardle, Applied nonparametric regression,
Cambridge University Press, Cambridge,
1990.
9] Charles K. Chui, An introduction to wavelets,
Wavelet Analysis and Its Applications, vol. 1,
Academic Press, Inc., San Diego, CA, 1992.
10] Jaroslav Mich lek, Kernel smoothing with
variable bandwidth, In: Proceedings of the
summer school MATLAB'94, Velk Karlovice
(I. Horov , ed.), Masaryk University, Brno,
Czech Rep., August 1994, in print.
11] Hans-Georg Muller, Nonparametric regression
analysis of longitudinal data, Lecture
Notes in Statistics, vol. 46, Springer-Verlag,
Berlin, 1988.
12] E. A. Nadaraya, On estimating regression,
Theory Probab. Appl. 9 (1964), 141{142.
13] Carl Taswell, WavBox 4.3b (Wavelet Toolbox
for MATLAB 4.2c), January 1996,
(Internet URL: http://www.wavbox.com).
14] G. S. Watson, Smooth regression analysis,
Sankhya A26 (1964), 359{372.
Author's address: V. Vesel , Masaryk University,
Jan kovo n m. 2a, 66295 Brno, Czech Republic
(email: vesely@math.muni.cz)
1910 1920 1930 1940 1950 1960 1970 1980 1990
6
7
8
9
10
11
12
13
14
AUTUMN: Mean seasonal atmospheric temperatures measured at Hurbanov
1955 1960 1965 1970 1975 1980 1985 1990
−2
−1.5
−1
−0.5
0
0.5
1
1.5
year
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
350
year
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Figure 1: Plots of raw data sets 1,2 and 3
12−Jun−96 at 18:42:03
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
1−OPTIM.BANDWIDTH: 21.36 for n=90
12−Jun−96 at 18:42:03
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
1−OPTIM.BANDWIDTH: 21.36 for n=90
1910 1920 1930 1940 1950 1960 1970 1980 1990
7
8
9
10
11
12
13
14
autumn temp., Hurbanov in 1903−1992
12−Jun−96 at 18:49:47
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 7 for n=90
12−Jun−96 at 18:49:47
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 7 for n=90
1910 1920 1930 1940 1950 1960 1970 1980 1990
7
8
9
10
11
12
13
14
autumn temp., Hurbanov in 1903−1992
12−Jun−96 at 18:51:25
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 42 for n=90
12−Jun−96 at 18:51:25
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 42 for n=90
1910 1920 1930 1940 1950 1960 1970 1980 1990
7
8
9
10
11
12
13
14
autumn temp., Hurbanov in 1903−1992
Figure 2: Data set 1 smoothed with optimal, small and large bandwidth (top-down)
13−Jun−96 at 14:55:15
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
1−OPTIM.BANDWIDTH: 3.131 with h(n)=0.5*n^(−1/5) for n=168
13−Jun−96 at 14:55:15
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
1−OPTIM.BANDWIDTH: 3.131 with h(n)=0.5*n^(−1/5) for n=168
1955 1960 1965 1970 1975 1980 1985 1990
−1.5
−1
−0.5
0
0.5
1
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
13−Jun−96 at 14:56:01
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 1 for n=168
13−Jun−96 at 14:56:01
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 1 for n=168
1955 1960 1965 1970 1975 1980 1985 1990
−1.5
−1
−0.5
0
0.5
1
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
13−Jun−96 at 14:56:15
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 6 for n=168
13−Jun−96 at 14:56:15
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 6 for n=168
1955 1960 1965 1970 1975 1980 1985 1990
−1.5
−1
−0.5
0
0.5
1
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
Figure 3: Data set 2 smoothed with optimal, small and large bandwidth (top-down)
5−Jun−96 at 16:09:03
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
1−OPTIM.BANDWIDTH: 0.4838 for n=912
5−Jun−96 at 16:09:03
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
1−OPTIM.BANDWIDTH: 0.4838 for n=912
1920 1930 1940 1950 1960 1970 1980 1990
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
5−Jun−96 at 16:27:35
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 0.16 for n=912
5−Jun−96 at 16:27:35
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 0.16 for n=912
1920 1930 1940 1950 1960 1970 1980 1990
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
5−Jun−96 at 16:22:07
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 1.5 for n=912
5−Jun−96 at 16:22:07
KERNEL: K(x)=(15/16)*(1−x^2)^2 on [−1,1] ... quartic
BANDWIDTH: 1.5 for n=912
1920 1930 1940 1950 1960 1970 1980 1990
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Figure 4: Data set 3 smoothed with optimal, small and large bandwidth (top-down)
1910 1920 1930 1940 1950 1960 1970 1980 1990
7
8
9
10
11
12
13
14
AUTUMN: Mean seasonal atmospheric temperatures measured at Hurbanov
Data smoothed using threshold DJE (WAVELET: ORTH−DOLA−8)
0 0.2 0.4 0.6 0.8 1
4
3
2
1
AUTUMN: Mean seasonal atmospheric temperatures measured at Hurbanov
Level
DWT coefficients of noisy data (WAVELET: ORTH−DOLA−8)
0 0.2 0.4 0.6 0.8 1
4
3
2
1
AUTUMN: Mean seasonal atmospheric temperatures measured at Hurbanov
Level
DWT coefficients of smoothed data (WAVELET: ORTH−DOLA−8)
Figure 5: Data set 1 smoothed via wavelet shrinkage of type DJE
1955 1960 1965 1970 1975 1980 1985 1990
−1.5
−1
−0.5
0
0.5
1
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
Data smoothed using threshold DJE (WAVELET: ORTH−DOLA−8) (year)
0 0.2 0.4 0.6 0.8 1
5
4
3
2
1
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
Level
DWT coefficients of noisy data (WAVELET: ORTH−DOLA−8)
0 0.2 0.4 0.6 0.8 1
5
4
3
2
1
DEVIATIONS OF CLOUDINESS FROM THE MEAN 1951−1992, NORTHERN CROATIA
Level
DWT coefficients of smoothed data (WAVELET: ORTH−DOLA−8)
Figure 6: Data set 2 smoothed via wavelet shrinkage of type DJE
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold DJE (WAVELET: ORTH−DOLA−8) (year)
0 0.2 0.4 0.6 0.8 1
8
7
6
5
4
3
2
1
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Level
DWT coefficients of noisy data (WAVELET: ORTH−DOLA−8)
0 0.2 0.4 0.6 0.8 1
8
7
6
5
4
3
2
1
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Level
DWT coefficients of smoothed data (WAVELET: ORTH−DOLA−8)
Figure 7: Data set 3 smoothed via wavelet shrinkage of type DJE
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold 1 (WAVELET: ORTH−DOLA−8) (year)
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold 1−2 (WAVELET: ORTH−DOLA−8) (year)
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold 1−2−3 (WAVELET: ORTH−DOLA−8) (year)
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold 1−2−3−4 (WAVELET: ORTH−DOLA−8) (year)
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold 1−2−3−4−5 (WAVELET: ORTH−DOLA−8) (year)
1920 1930 1940 1950 1960 1970 1980 1990
0
50
100
150
200
250
300
MONTH MEAN FLOW ON THE RIVER MORAVA 1916−1991, KROMERIZ, CZECH REPUBLIC
Data smoothed using threshold 1−2−3−4−5−6 (WAVELET: ORTH−DOLA−8) (year)
Figure 8: Multiresolution analysis of the data set 3