Comparative study of different wavelets for hydrologic forecasting

Computers & Geosciences 46 (2012) 284–295

Contents lists available at SciVerse ScienceDirect

Computers & Geosciences journal homepage: www.elsevier.com/locate/cageo

Comparative study of different wavelets for hydrologic forecasting R. Maheswaran n, Rakesh Khosa Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi, 110016, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 25 August 2011 Received in revised form 25 December 2011 Accepted 26 December 2011 Available online 20 January 2012

Use of wavelets in the areas of hydrologic forecasting is increasing in appeal on account of its multi resolution capabilities in addition to its ability to deal with non-stationarities. For successful implementation of wavelets based forecasting methodology, selection of the appropriate mother wavelet form and number of decomposition levels plays an important role. Wavelets based forecasting methodologies have been discussed extensively in published literature but discussion on some key issues of concern such as selection of mother wavelets is rather meager. Appropriately, therefore, this paper presents a comparative evaluation of different wavelet forms when employed for forecasting future states of various kinds of time series. The results suggest that those wavelet forms that have a compact support, for example the Haar wavelet, have a better time localization property and show improved performance in the case of time series that have a short memory with short duration transient features. In contrast, wavelets with wider support, for example db2 and spline wavelets, yielded better forecasting efficiencies in the case of those time series that have long term features. Results further suggest that db2 wavelets perform marginally better as compared to the spline wavelets. It is hoped that this study would enable a reasoned selection of mother wavelets for future forecasting applications. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Wavelets Forecasting Selection of mother wavelets Stream flow

1. Introduction Hydrological forecasting has always been somewhat of a challenge for researchers who acknowledge its fundamental role in real time operation of controls for efficient water management as well as in disaster management. Recent developments in wavelets have led to a renewed effort to design better forecasting frameworks for non-stationary hydrologic and other environmental systems. Studies such as Kim et al. (2003), Partal and Kisi (2007), Zhou et al. (2008), Kisi (2009, 2010), Adamowski and Sun (2010), Tiwari and Chatterjee (2010) are some recent examples where wavelets have been used in the area of state forecasting and these studies do indeed establish the advantage of using wavelets for improving forecasting efficiencies. However, it is important to note, most of these aforementioned studies use wavelets in a generic way and no attempt is made to develop a systematic reasoning for the choice of specific wavelets that have been used in these studies. In addition to the choice of an appropriate mother wavelet, these studies are also silent on issues such as the choice of the appropriate wavelet algorithm, selection of the number of important scale levels for decomposition and, importantly, treatment of the boundary problems. For example, studies by Kisi (2009, 2010) and Tiwari and Chatterjee (2010) employ

n

Corresponding author. Tel.:þ 91 11 2659 6366. E-mail addresses: [email protected] (R. Maheswaran), [email protected] (R. Khosa). 0098-3004/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2011.12.015

the Discrete wavelet transform (DWT) whereas Kim et al. (2003) and ´ trous wavelet transform for Nourani et al. (2009) have applied the a wavelet decomposition and, in contrast, the Continuous Wavelet Transform (CWT) based approach has been used as a method of choice by Adamowski and Sun (2010) for flood forecasting. Farge (1992) gives a detailed analysis of different complex continuous wavelets and concludes that many different wavelets have to be tested to get an optimized choice of wavelet for the given problem. Kumar and Georgiou (1997) conclude that the Haar wavelet often provides a good and simple choice for applications where the process has sharp and compact variations whereas higher order wavelets are better when frequency localization is required. De Chazal et al. (2000) applied different wavelets for classification of ECG signals and suggests that the Haar, alternatively also referred to as the Daubechies 1, wavelet is best suited to the classification task as it requires least computation. Khosa et al. (2005), in their study of non-stationary time series, found that there was no single wavelet form that could be presented as a general solution for picking up the various kinds of non-stationarities that may be expected in for a geophysical time series. Ahuja et al. (2005) also underscore the importance of a rational basis for wavelet selection and accordingly, in their study, the latter authors have provided quantitative guidelines aimed to reduce dependence on a trial-and-error procedure that is often resorted to. Azeem et al. (2006) carried out a comparative study of different wavelets in wavelet networks and concluded that the ‘Mexican Hat’ and ‘Morlet’ wavelet can be used as suitable candidates for wavelet networks. Nourani et al. (2009) applied different wavelets in their Wavelet-Artificial Neural Network model

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

for rainfall runoff modeling and their results indicate that the Haar wavelet performed better than other wavelets used in the latter study. Zhu et al. (2009) proposed a technique based on weighted sum of linear correlation coefficients as an aid to identify a suitable wavelet form amongst various competing alternative forms. In a later study, Stolojescu et al. (2010) tested different wavelet families for internet traffic forecasting and concluded that to obtain a good prediction, it is necessary to use Haar mother wavelets or mother wavelets having good time-frequency localization capabilities. From the studies reviewed, included those cited above, it is a reasonable assertion that the issue of mother wavelet selection as well as selection of a suitable decomposition algorithm is essentially unresolved and a better insight is required especially when seeking to use wavelets for hydrologic forecasting applications. The present paper describes details of a parametric study that has been carried out in which different wavelet forms are investigated in terms of (i) decomposition algorithm, (ii) decomposition level, (iii) forecast performance and (iv) computational effort. The modeling methodology has been implemented on three separate case studies and the results are also presented herein for analysis.

2. Wavelet analysis Wavelet analysis has become an important milestone in spectral analysis due to its multi-resolution and localization capability both in time and frequency domain and has been extensively applied in the area of time series analysis and prediction. Wavelet decompositions at various scales (frequencies) often reveal underlying low and high frequency components that may be present in the observed series and, importantly, helps in localizing these features in time. Several algorithms have been designed that enable such decompositions and selection of any particular approach depends, understandably, on the application at hand. For example, Continuous Wavelet Transform (CWT), used mainly in the processing of medical images and seismic signals, calculates the wavelet transform as an integral product of the given signal and the wavelet function. The coefficients of the wavelet transform of a square-integrable continuous-time signal, f (t), are defined by the linear integral operator   Z 1 1 tt Cða, tÞ ¼ ð1Þ f ðtÞca, t ðtÞdt where ca, t ðtÞ ¼ pffiffiffi c a a 1 The function c(t), which can be real or complex, plays the role of a convolution-kernel and is called a wavelet. The parameter a can be interpreted as a dilation (a41) or a contraction (a o1) factor of the wavelet function c(t) corresponding to different scales of observation. The parameter t can be interpreted as a temporal translation or shift of the function c(t) and allows discrimination of the signal f (t) locally around time t. The wavelet function c(t) is designed with the following properties (Burrus et al., 1998). (I) The function integrates to zero: Z 1 cðtÞdt ¼ 0

ð2aÞ

285

among the wavelet coefficients. This redundancy, on account of correlation between coefficients, is intrinsic to the wavelet-kernel and not a characteristic of the analyzed signal. As an alternative, for practical applications (as in the study of noise reduction models for communication systems and image and signal compression), Discrete Wavelet Transform (DWT) is usually preferred. The accompanying discussion below compares some important but distinctive modeling issues that are relevant for the various wavelet types commonly used and explored in this study. 2.1. Discrete wavelet transform (DWT) The DWT wavelet coefficients are calculated at every dyadic step i.e., the operation of WT is carried out at dyadic dilations and integer translations. The wavelet function in its dyadic form can be represented as

cj,k ðtÞ ¼ 2ðj=2Þ cð2j tkÞ

ð3Þ

In Eq. (3), c(t) is the mother wavelet and j and k are the translation and dilation indices. The DWT is generally a nonredundant transform and, accordingly, only a minimally required number of wavelet coefficients are preserved at each level of decomposition which, as a further consequence, enables reconstruction of the original signal from a reduced number of wavelet coefficients. While this property is useful in applications such as data and image compression, this type of non-redundant DWT, however, is prone to shift sensitivity and is therefore an undesirable feature when applied to problems related to singularity detection, forecasting and nonparametric regression. The shift variant nature of DWT is demonstrated in Fig. 1 with respect to two shifted step-inputs which are decomposed up to J¼3 levels using the ‘db2’ wavelet. If the DWT were shift invariant then the details and approximations should preserve the same relative ‘shift’ distance as in the original step functions. However, it can be seen from these accompanying details and approximations that the original ‘offset’ distance is not preserved and actually shows unpredictable variations leading to a severely compromised forecasting capability. Inclusion of newer data also presents difficulties as seen in terms of its effect on the decomposed scales. As shown in Fig. 2, a 2nd order Daubechies (db2) wavelet filter is used to compute the transform and inconsistencies are seen to develop in the boundary regions as new data points are used to compute the wavelet transform. The figure shows the details of the signal corresponding to each incremental addition to the data length and it is seen that the coefficients computed with the addition of new data points do not present a smooth continuity from the previous step. This, seemingly inconsistent, boundary region behavior comprises the forecasting performance of the wavelets based model while, at the same time, preventing the model to understand the true nature of the process signal. The study by Mun (2004) presents a more detailed discussion on these issues. 2.2. The a0 trou wavelet transform

1

(II) The function is square integrable or, equivalently, has finite energy Z 1 2 9cðtÞ9 dt o1 ð2bÞ 1

A disadvantage of these non-orthogonal wavelets is that the CWT of a given signal is characterized by redundancy of information

The above mentioned disadvantages of DWT have also been discussed at length in Aussem et al. (1998), Renaud et al. (2005) and Adamowski and Sun (2010) and these authors have also proposed the use of the a’ trou wavelet transform which, while having an intrinsic disadvantage of large redundancy and increased computational complexity, possesses the much desired shift invariant property and therefore is suitable for time series analysis, regression and forecast applications. The alternative, shift-invariant a’ trou wavelet transform algorithm computes the approximation and detail signals at the same spacing across

286

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

APPROXIMATION AT LEVEL 3

2 0 -2

0

50

100

150

200

250

200

250

200

250

150

200

250

150

200

DETAIL AT LEVEL 3

2 0 -2

0

50

100

150

DETAIL AT LEVEL 2

2 0 -2

0

50

100

150

DETAIL AT LEVEL 1

10 5 0

0

50

100 Two shifted signals

6 5 4 3 2 1 0

0

50

100

250

Fig. 1. Shift variant property of DWT.

Detail at Level 1 using db2 4 3

upto t=103

2 uptp t=102

1 0 upto=104

-1

´ trous algorithm and the author has also shown that the a algorithm is in fact a non-orthonormal multiresolution algorithm for which the discrete wavelet transform is exact. ´ trous wavelet transform is to fill the The basic idea of the a resulting gaps using redundant information obtained from the original series with the accompanying advantage that the additional information provides a basis for enhanced forecast accuracy while also promoting a comprehensive understanding of underlying process properties in terms of its observation data. In this approach, as a downside, the number of wavelet coefficients does not shrink across the transformed levels and, due to the redun´ trous wavelet transform has larger storage dancy in coefficients, a requirements and involves more computations. 2.2.1. Implementation of a0 trous algorithm Corresponding to the original series x(t), smoother versions of x(t) are defined at different scales as given by Eqs. (4) and (5).

-2 -3

c0 ðtÞ ¼ xðtÞ 90

95

100

105

cj ðtÞ ¼ Fig. 2. Effect of inclusion of new data points on DWT.

scales using non-orthonormal wavelet bases. Shensa (1992) ´ trous algorithm provides the mathematical derivation of the a along with an investigation of its relationship with the pyramidal

1 X

ð4Þ hðlÞcj ðt þ 2j1 lÞ

ð5Þ

l ¼ 1

In the preceding Eq. (5), j takes values from 1, 2, 3 y. J where J is the level of decomposition and ‘h’ is a low pass filter with compact support such as the B3-spline defined as (1/16, 1/4, 3/8, 1/4, 1/16) and Haar defined as (1/2, 1/2).

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

0.3

D102 D103 D104

0.2

287

filtering is done with the past values of x(t) and not with future ones.

3. Selection of wavelet upto t=103

0.1

upto t=102

upto=104

0

-0.1

-0.2

-0.3 100

100.5

101

101.5

102

102.5

103

103.5

104

Fig. 3. Effect of inclusion of new data points on a trous algorithm.

The detail component of x(t) at level i is defined as dj ðtÞ ¼ cj1 ðtÞcj ðtÞ

ð6Þ

In Eq. (6), the set {d1, d2 y. dp,cp} represents the additive wavelet decompositions of the data up to resolution level p and cp is the residual component or the approximation. Fig. 3 shows the ´ trous algorithm and these effect of adding new data point on the a results suggest that (i) addition of newer data point does not distort the transform, and, importantly, (ii) the transform at the current step is completely consistent with the transform up to ´ previous time step. Thus, it can be concluded that the latter a trous algorithm is better suited for forecasting applications than the DWT.

2.2.2. Treatment of boundary The treatment of boundary merits special attention when implementing wavelets based approaches for forecasting applications. From Eq. (5), it is seen that for a filter length equal to 2p þ1, estimation of wavelet coefficient at any time t uses observations on past states going back up to (t  p) and the future, unrealized states up to (t þp). In general wavelet applications, various kinds of boundary conditions such as (i) periodic boundary, (ii) reflective boundary extension, and (iii) constant extension are usually used for extending the series up to (t þp) (Strang and Nguyen,1996). However, in the case of forecasting problems, these extensions are not useful as discussed in the simple example where the B3-spline wavelet is used for computing c(t) from the series x(t). At each time step, according to Eq. (5), calculation of c1(n) requires future states x(n þ1) and x(n þ2). Similarly, c2(n) requires x(n þ1), x(nþ2) y y, x(n þ6). In general, computation of ci(n) would require x(t) defined at t ¼nþ 1, n þ2, y y nþ (2i þ 1-2) and clearly are not available prior to their actual realization in time as these refer to future observations. Alternatives that do not involve use of future values while calculating the wavelet coefficients include the use of causal filters as these use only the past values to calculate wavelet coefficients at time ’t’. In this regard, Bakshi (1999), and Renaud et al. (2005) have independently proposed the use of a redundant Haar wavelet transform while Yu et al. (2001) developed a predictor–corrector method for boundary treatment which, however, not only involves a large number of iterative computations but is also computationally taxing. As an alternative methodology, Luan (2005) has proposed a shifted B3-spline wavelet so that the

Selection of an appropriate wavelet function poses significant challenges and is governed largely by the problem at hand and some of the distinctive properties of the wavelet function such as (i) its region of support, and (ii) the number of vanishing moments. The region of support implies the length span of the given wavelet which in turn affects its feature localization capabilities as it is understandable that a long and widely distributed wavelet function will calculate the instantaneous process amplitude while, at the same time, spanning a wider window of the underlying process resulting in a high degree of averaging of the process states. Vanishing moment, on the other hand, limits the wavelet’s ability to suitably represent polynomial behavior or information in a signal. For example, Haar wavelet, with one moment, easily encodes polynomials of one coefficient, or constant signal components. Similarly, the db2 wavelet encodes polynomials with two coefficients i.e., a process having one constant and one linear signal component and the db3 wavelet encodes a process having a constant, linear and quadratic signal components. Within each family of wavelets are wavelet subclasses distinguished by their respective number of coefficients and the number of vanishing moments as discussed below. 3.1. Haar wavelet These are symmetric and non continuous wavelets and the number of vanishing moments in this case is equal to one (1). Haar wavelets have been found appropriate for applications in signals that have sharp changes because of its relatively narrow span over which its energy is distributed. 3.2. Daubechies wavelet These are a family of wavelets and include the Haar wavelet, written as db1, as a special case. These are compactly supported wavelets with extreme phase and highest number of vanishing moments for a given support width. The Daubechies wavelets have associated minimum-phase scaling filters, are both orthogonal and biorthogonal,and do not have an explicit analytic expression except for the db1 (or Haar) form. 3.3. B-spline wavelet The most attractive and distinctive property of B-splines are that they are compactly supported and can be analytically formulated in an explicit form. While the zeroth order B-spline corresponds to the well known Haar wavelet, the mth order B-spline function has the following recursive form (Chui, 1992): Nm ðxÞ ¼

x mx Nm1 ðxÞ þ N m1 ðx1Þ,m Z2 m1 m1

where N1 ðxÞ ¼ w½0,1Þ ðxÞ ¼



1

if x A ½0,1Þ

0

otherwise

Amongst the various basis functions, B-splines are unique because they integrate three remarkable properties namely (i) they have compact support, (ii) can be analytically formulated, and (iii) are ideally suited for multi-resolution analysis (Wei and Billings, 2006).

288

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295 ðnÞ

h2s describe the 2nd order nonlinear relation between the nth input un and y, respectively and the second order cross-kernels ðn ,n Þ h2x 1 2 describe the 2nd order nonlinear interactions between each unique pair of inputs (un1 and un2) as they affect y. Eq. (7) can be simplified by combining the last two terms to yield Eq. (8) and it now remains to estimate kernels h1 and h2.

Table 1 Scaling filters for the different wavelets. Wavelets

Corresponding filters

Haar Db2 Db3 Db4

[0.707 0.707] [0.482 0.836 0.224  0.129] [0.332 0.806 0.459  0.135  0.085 [0.230 0.715 0.631  0.028  0.187  0.011] [  0.076  0.030 0.498 0.804 0.298 0.032] [0.0625 0.25 0.375 0.25 0.0625]

Sym4 B3-spline

0.0352] 0.031 0.033  0.099

yðtÞ ¼

Jþ1 X m X

ðnÞ

h1 ðtÞun ðttÞ

n¼1t¼1

 0.013

þ

Jþ1 X Jþ1 X m m X X

ðn ,n2 Þ

h2 1

ðt1 , t2 Þun1 ðtt1 Þun2 ðtt2 Þ þ xt

n1 ¼ 1 n2 ¼ 1 t1 ¼ 1 t2 ¼ 1

ð8Þ

3.4. Sym wavelet Sym wavelet (or, in a compact form, symlet) is only near symmetric and its other properties are similar to those of the general Daubechies wavelets, dbNs. Symlet has the highest number of vanishing moments for a given support width and its associated scaling filters are near linear-phase filters (Nibhanupudi, 2003). Order N can be 2, 3, y. These wavelets can be both orthogonal biorthogonal and provide compact support.

The representation of Eq. (8) can be further simplified by considering each of the lagged variables u1(t  1), u1(t  t), y, u2(t  1), u2(t  t), y as separate variables d1(t), d2(t), d3(t) y y dNl(t) then, Eq. (8) can be written as yðtÞ ¼

Nl X

h1 ðlÞdl ðtÞ þ

l¼1

Nl Nl X X

h2 ðl1 ,l2 Þdl1 ðtÞdl2 ðtÞ

ð9Þ

l1 ¼ 1 l2 ¼ 1

More clearly, 3.5. Scaling filters for different wavelets In general, wavelet decomposition is implemented using the low pass (scaling) and high pass (wavelet) filter and the accompanying Table 1 shows the scaling filters corresponding to different wavelets.

4. Forecasting methodology 4.1. The reference model Since the aim of this study is not to compare different forecasting methods, a common reference model is considered and performance of different wavelets is compared as implemented within this common reference model. The forecasting methodology used in this paper is based on the Wavelet Volterra Coupled model proposed by Maheswaran and Khosa, in press, under review. In this method, the input signal, time series Y¼ (y1,y,yn 1), is decomposed to obtain ´ trous wavelet wavelet coefficients at different scales using the a transform, the wavelet decomposition at various levels are then integrated using the second order MISO Volterra model to provide the forecast at next step. Denote the wavelet coefficients at each scale as u1, u2, y, uJ and the scaling coefficients as uJ þ 1, where J is the coarsest level of decomposition. The wavelet coefficients and scaling coefficients of the original series are nonlinearly convolved using the second order Volterra representation within a multiple inputs-single output frame work. If J denotes the level of decomposition, N is the number of inputs, m denotes the memory length at each level and xt represents the model noise including modeling errors and the unobservable disturbances, the multiscale nonlinear relationship may be written as: 9 Jþ1 X m > X > ðnÞ > > yðtÞ ¼ h1 ðtÞun ðttÞ > > > > n ¼ 1t ¼ 1 > > > > JX þ1 X m m X ðnÞ = h2s ðt1 , t2 Þun ðtt1 Þun ðtt2 Þ þ > n ¼ 1t1 ¼ 1t2 ¼ 1 > > > > > JX þ 1 nX 1 m m 1 > X X ðn ,n Þ > 1 2 > þ h2x ðt1 , t2 Þun1 ðtt1 Þun2 ðtt2 Þ þ xt > > > ; n 1 ¼ 1 n 2 ¼ 1 t1 ¼ 1 t2 ¼ 1 ð7Þ ðnÞ

First order kernels h1 describe the linear relationship between the nth input un and y, the second order self-kernels

dl ðtÞ ¼ fxk ðtÞ 1 rk rJ þ1; dl ðtÞ ¼ fxk ðt-tÞ1 r kr J þ 1;

1r l r J þ 1 J þ 1 ol rN l ;

t ¼ 1,2,3::::m

t ¼ tth lagged value: J ¼ level of decomposition: Nl ¼ total number of lagged variables: Using the OLS-ERR method of Chen et al. (1989), the significant regressor terms were selected and corresponding kernels estimated. The entire model scheme is shown in Fig. 4. The above model is used as the reference model for a comparative evaluation of the different test wavelets and the methodologies were implemented on the MATLAB 7.6.0 programming platform. 4.1.1. Performance evaluation To evaluate the forecast performance as achieved using the various types of wavelet forms, the following statistical measures of error are considered: 1. Mean absolute error (MAE), P 9ObservedðiÞForecastðiÞ9 MAE ¼ N 2. Root mean square error (RMSE). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 i ¼ 1 ðObservedðiÞForecastðiÞÞ RMSE ¼ N1 3. Nash Sutcliffe criteria (NSC): PN 2 t ¼ 1 ðActualðiÞForecastðiÞÞ NSC ¼ E ¼ 1 P N 2 t ¼ 1 ðActualðiÞActualÞ

ð10Þ

ð11Þ

ð12Þ

Karunanithi et al. (1994) have suggested that RMSE is a good measure for indicating goodness of fit and, in general, RMSEZMAE. Additionally, in the study, computational effort is also measured in terms of the number of the coefficients used in the model where the total number of coefficients in a given model is calculated using the rule: Total no:of coefficients ¼ No: of wavelet coefficients þ approximation coefficients:

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

289

D1 = u (t ) D 2 = u (t ) D 3 = u (t )

Wavelet Decomposition

y

Second order Volterra Model

D 4 = u (t )

k = 1, 2...t − 1

C 4 = u (t )

Model Updating

y (t ) Wavelet Decomposition

Observation y(t) Fig. 4. Wavelet Volterra Coupled Model.

12

The no. of coefficients used to capture the underlying dynamics of the time series depends on the properties of the wavelets and it is well understood that smooth and higher order wavelets represents the data adequately with a relatively lesser number of coefficients (parsimonious representation) whereas, on the other hand, simple wavelets like Haar require more number of wavelet coefficients to describe the data to a satisfactory degree.

10

8

X(t)

6

5. Model application

4

In order to investigate the suitability of wavelets for different time series, three case studies are considered. Case study ’A’ deals with three synthetic time series having a stochastic component combined with non-stationary features such as a trend feature and a periodic component. In case study ’B’, monthly stream flow time series is analyzed and, in the final segment of the study, daily stream flow time series is considered as case study ’C’.

2

0

-2

0

5.1. Case study-A

t 200

X ð3Þ t ¼ X t þ sin

1000

20

ð13Þ t2 500

  3pðt1Þ 180

ð14Þ

15

10

ð15Þ

5

Using the above reference time series, three different synthetic time series were developed having the following form and shown as (Figs. 5–7). Time series 1: (Time series with linear trend and periodicity)

0

X ð4Þ t

800

25

X(t)

X ð2Þ t ¼ 0:01fX t gfX t g þ X t þ

400 600 Time Interval

Fig. 5. Time series with linear trend and periodicity.

Consider a reference time series, Xt, generated synthetically as a stationary AR(2) process with auto regressive coefficients f1 ¼ 0:4 and f2 ¼ 0:2 and sample size of 1000. Different nonstationary processes, X ðiÞ t ,i ¼ 1,2. . ., were obtained by combining different components such as a linear trend, nonlinear trend and a periodic component as given in Eqs. (13)–(15). X ðt1Þ ¼ X t þ

200

ð3Þ ¼ X ð1Þ t þX t

ð16Þ

Time series 2: (Time series with nonlinear component, trend and periodicity) ð2Þ ð3Þ X ð5Þ t ¼ X t þX t Time series 3: (Time series with seasonal component)

ð17Þ

X ð6Þ t ¼ ðX t nsm Þ þ mm

ð18Þ

where mm and sm is the synthetic mean and standard deviation values for period of 12 intervals.

-5 0

200

400

600

800

1000

Time Interval Fig. 6. Time series with non-linear component, trend and periodicity. ð5Þ and X ð6Þ are used as the base The above time series X ð4Þ t ,X t t time series for testing different wavelets. As stated earlier, the aim of the study is to compare the influence of different wavelet families on forecast accuracy and accordingly, therefore, the study has made a comparative evaluation of the following wavelet families: (i) Daubechies db1, (ii) Daubechies db2, (iii) Daubechies db3, (iv) Daubechies db4, (v) Symlet (sym4) and (vi) Spline-B3.

290

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

1400

Table 3 Performance of different wavelets in forecasting – case study A (Time series II).

1200

1000

Wavelet type

Decomposition level

No of coefficients

RMSE

MAE

NSC

Db1(Haar)

2 3 4 2 3 4 2 3 2 3 2 3 2 3

6 8 10 4 5 8 5 3 3 4 5 6 4 4

0.1601 0.1668 0.1628 0.1692 0.1479 0.1762 0.1627 0.226 0.1848 0.1789 0.1807 0.2565 0.1699 0.1578

0.1280 0.1315 0.1305 0.1338 0.1172 0.1412 0.1293 0.1804 0.1500 0.1308 0.1439 0.2359 0.1350 0.1230

0.9917 0.988 0.9911 0.9908 0.9984 0.9891 0.9920 0.9834 0.9891 0.987 0.989 0.9705 0.989 0.9918

800

X(t)

Db2

600 Db3

400

Db4 Sym4

200

Spline-B3

0

0

200

400

600

800

1000

Time Interval Table 4 Performance of different wavelets in forecasting – case study A (Time series III).

Fig. 7. Synthetic seasonal time series.

Table 2 Performance of different wavelets in forecasting – case study A (Time series I). Type of wavelet.

Decomposition level

No of coefficients

RMSE

MAE

NSC

Db1(Haar)

2 3 4 2 2 3 4 2 3 2 3 2 3 2 3

5 8 8 5 4 6 6 4 8 3 6 3 6 3 6

0.3082 0.8798 0.9105 0.3329 0.3423 0.9089 1.11 0.4760 1.012 0.4684 1.01 0.4106 0.981 0.4660 0.90

0.2522 0.7014 0.65 0.2721 0.2987 0.7112 0.832 0.3970 0.8095 0.3869 0.8556 0.3310 0.8123 0.3804 0.64

0.9368 0.759 0.702 0.9184 0.897 0.7091 0.6100 0.8198 0.6234 0.8114 0.5687 0.8748 0.60 0.8204 0.721

Wavelet type

Decomposition level

No of coefficients

RMSE MAE

NSC

Db1(Haar)

2 3 4 2 3 4 2 3 4 2 3 4 3 4 3

5 10 12 4 6 8 4 5 6 4 6 8 4 6 6

3.16 1.81 1.60 3.00 1.64 1.66 2.33 1.72 1.88 2.55 1.84 1.86 1.78 1.83 1.71

0.8423 0.935 0.9431 0.8582 0.968 0.9554 0.9145 0.9501 0.9421 0.8915 0.942 0.938 0.9438 0.9 0.9526

Db2

Db4 Sym4 spline-B3

The results of the forecasting performance of different wavelets are reported in Tables 2–4.

Db4

Sym4 Spline-B3

Time series I

Db3

Db3

20 10 0 200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

5 0

D2

5 0 -5 5 D1

5.1.1. Synthetic time series I ´ trous wavelet transform of time series Figs. 8 and 9 show the a I using the Haar and db2 wavelets, respectively. Forecast models were developed using various number of wavelet and approximation coefficients for each scale level and the results for some of the models tried are presented in Table 2 for this case study. From Table 2, it is seen that the Haar (also db1) seems to perform better than other wavelets due to their good localization ability. The model using Haar wavelet with 2 levels decomposition has a RMSE equal to 0.3082 whereas the models based on db2 and sym4 wavelets yielded results with higher RMSE values equal to 0.3329 and 0.4106, respectively. Also, it is noted that the model performance with 2 levels of decomposition is better than the higher levels of decompositions because the test signal is made of high frequency components and, therefore, most of its energy is captured in the 1st and 2nd levels with only marginal content remaining for the higher levels to pick.

0 10 A2

Db2

1.877 1.049 1.14 2.19 0.98 1.045 1.57 1.12 1.06 1.669 1.2135 1.269 1.11 1.17 1.10

0 -5 ´ trous decomposition of the Synthetic time series I using Haar. Fig. 8. a

Further, it can be seen that the Haar wavelet is able to achieve good prediction accuracy with least number of coefficients – only 5 – due to its compactness as may be concluded from the high NSC value equal to 0.9368 obtained for this wavelet form as against a value equal to 0.9184 achieved by the db2 wavelet.

10 0 0

200

400

600

800

1000

Time series II

20

20 10 0

A3

Time series I

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

30 20 10 0

5 0

200

400

600

800

0

D1

-2

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

1 0

200

400

600

800

1000

0

2

-1

0

1

-2

200

0 -1

D1

D2

2

0

1

1000

D3

0

D2

A2

10

291

0

200

400

600

800

1000

0 -1

´ trous decomposition of the Synthetic time series I using Db2. Fig. 9. a

Time series II

30 20 10 0

A3

´ trous decomposition of the Synthetic time series II using db2. Fig. 11. a

30 20 10 0

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

D3

1 0 -1 D2

1 0 -1

D1

1 0 -1

´ trous decomposition of the Synthetic time series II using haar. Fig. 10. a

From the time series plot, it is observed that the original time series has an increasing linear tendency and, therefore, requiring mother wavelets with good frequency localization. Also, the variability in the time series seems to be high suggesting a need for mother wavelets with good time localization and, presumably, it may be reasonable to expect that best results would be obtained using mother wavelets with good time–frequency localization such as the db2 and/or sym4 wavelets. However, to the contrary, the Haar wavelet is seen to perform better and may be because the value of RMSE depends on prediction accuracy in the face of process variability rather than the overall trend which implies the need for wavelet having good localization property (Haar) than the one having both.

5.1.2. Synthetic time series II ´ trous wavelet transFigs. 10 and 11, respectively show the a form of time series II using Haar and db2 wavelets. Examination of these figures shows that the db2 wavelet is able to extract the seasonality component from the time series more efficiently at D3

level (as it is evident from the displayed D3 that there is no residual seasonal component). However, in the case of Haar there is some seasonal component that is still present in the D3 component indicating the necessity for one more level of decomposition. Different forecast models were developed by using wavelet coefficients at different levels and the model results are reported in Table 3. A study of results reported in Table 3 shows that the model employing db2 wavelet as the mother wavelet turns in a better performance in terms of RMSE and NSC values. This can be explained by noting that the db2 wavelet has a reasonable support and also has good time-frequency localization property and these together enable the model to capture both the underlying trend as well as the short term variablities in the time series better than the Haar wavelet based forecast model. It is also noted that the capture of the underlying nonlinear trend required a minimum of three decomposition levels and its role is more substantive as compared to the other features of periodicity and general variability. It is understandable, therefore, that the Daubechies wavelet, with its better frequency localization coupled with a reseasonable support (length span of the wavelet), is better suited for processes having similar underlying long period components. With regards to computional effort, as the other measure of wavelet desirability, the db2 wavelet yields comparable results (RMSE¼0.1479) with five wavelet coeffients while the spline wavelet, which also has good compressive properties, produced similar results (RMSE ¼0.1578) using only four wavelet coeffients.

5.1.3. Synthetic time series III ´ trous wavelet transform of Figs. 12–14, respectively show the a time series III using the Haar, db2 and spline wavelets. An examination of these figures reveals that the spline and db2 wavelet are able to extract the underlying long term features (these wavelet are able to show these features at A3) more efficiently while, in comparison, Haar wavelet is able to capture these features by resorting to more levels of decomposition with a consequent greater number of wavelet coefficients. The results reported in Table 4 also establish clearly that the db2 and spline wavelets produce better results as expected as these possess good frequency localization capabilities which makes them more effiecient at picking up the underlying long term features. It is also interesting to note that, in comparison to

Time series III

292

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

2000 1000 0

0

200

400

600

800

1000

0

200

400

600

800

1000

A3

500

Haar wavelet with 3 levels of decomposition, the model based on Haar wavelet with 4 levels of decompostions performs better than the former implementation (i.e., Haar at 3 levels) but with the down side that the implementation is non-parsimonious as it requires more number of wavelet coefficients.

0

5.2. Case study-B

D3

500 0 -500

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

D2

500 0 -500

D1

500 0 -500

Time series III

´ trous decomposition of the Synthetic time series III using haar. Fig. 12. a

2000 1000 0

0

200

400

600

800

1000

0

200

400

600

800

1000

0

200

400

600

800

1000

A3

500 0

D3

500 0 -500

D2

500

5.3. Case study-C

0 -500

0

200

400

600

800

1000

0

200

400

600

800

1000

500

D1

In this case study, the sensitivity of the different wavelets are tested using the monthly stream flow data for the period 1916– 1917 to 1989–1990 for M.H. Halli station in River Cauvery in South Peninsular India. A split sample approach was employed to calibrate and validate the models and out of the total data length of 74 years, 70% of the data was used for calibration and the remaining, independent data set, was used for validation purposes Fig. 15. As in the preceding study, different wavelet functions were used to decompose the time series at different levels and, corresponding to each of these decomposition sets, WVC forecast model was developed and used for forecasting the one-month ahead stream flow at the indicated site in the river and the model results for various wavelet functions with different levels are reported in the accompanying Table 5. These results show that the model using db2 wavelet with seven coefficients performs better than models that employ other wavelet functions. For example, the db2 based model reports a RMSE of 53.69 whereas the spline-B3 wavelet with four levels (eight coefficients) yields results with a RMSE of 55.65. Clearly, the sharper peak and accompanying wider supporting span of the db2 wavelets results in the dispersion of its energy content over a relatively wider region leading, in turn, to a consequent reduced energy density. As a result, therefore, this particular wavelet is better able to pick up the long term averaging characteristics of the monthly stream flow values.

0 -500

In this case study, the sensitivity of the different wavelets is tested using the daily flow data observed at Yadgir station in the Bhima river, a tributary of River Krishna in mid Peninsular India. Out of the 5 years of daily data used, 3 years was used for model calibration and 2 years of data was set aside for validation (Fig. 16).

´ trous decomposition of the Synthetic time series III using db2. Fig. 13. a

2000

9

0

8 0

100

200

300

400

500

600

700

800

900

A3

300 200 100

0

100

200

300

400

500

600

700

800

900

500

D3

x 10

1000

0 -500

0

100

200

300

400

500

600

700

800

900

Monthly Streamflow (m3)

Time series III

10

7 6 5 4 3

D2

1000

2

0 -1000

0

100

200

300

400

500

600

700

800

900

D1

2000

0 0

0 -2000

1 100

200

300

400

500

600

700

800

900

Time (months) 0

100

200

300

400

500

600

700

800

900

´ trous decomposition of the Synthetic time series III using spline-B3. Fig. 14. a

Fig. 15. Time series of the monthly discharge observed at M.H. Halli, Cauvery, India.

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

Table 5 Performance of different wavelets in forecasting – case study B. Wavelet type

Decomposition level

No.of coefficients

RMSE MAE

NSC

Db1(Haar)

3 4 3 4 3 4 3 4 3 4 3 4

7 12 6 7 5 8 5 8 4 6 6 8

69.75 59.37 56.23 53.69 57.99 62.78 60.61 65.57 63.93 62.11 62.65 55.65

0.807 0.8232 0.8430 0.8669 0.8331 0.8043 0.8176 0.7865 0.7970 0.8042 0.831 0.850

Db2 Db3 Db4 Sym4 Spline-B3

39.85 37.16 35.44 34.14 36.89 38.43 38.88 40.32 41.05 39.56 35.35 36.75

293

results in a relatively superior performance with a NSC value of 0.954 while, in comparison, db2 wavelet based model yielded results having a comparatively low NSC value of 0.829. Clearly, and in sharp contrast to a monthly stream flow series, the daily runoff series is characterized by sudden and short duration peaks, and for any process with these features, wavelets with a smaller support width as, for example, the Haar wavelet, is better suited than wavelets with longer supports because wavelets that have larger supports would average the peaks leading, thereby, to reduced peak values in the forecasts. These results are consistent with the general understanding that wavelets with good time localization and compact support (like that of Haar) are indeed better suited to model a time series that has short memory transient features as is normally expected in a time series of daily stream flows.

6. Discussion 7000

6000

Streamflow (m3/s)

5000

4000

3000

2000

1000

0 0

500

1000

1500

2000

2500

Time (days) Fig. 16. Time series of the daily discharge observed at Yadgir, Bhima, India.

Table 6 Performance of different wavelets in forecasting – case study C. Wavelet type

Decomposition level

No. of coefficients

RMSE

MAE

NSC

Db1(Haar)

1 2 3 1 2 1 2 3 1 2 2

3 4 8 2 4 2 4 8 3 4 3

198.77 189.70 365.23 335.23 334.27 355.04 336.79 535.23 393.40 634.45 350.23

73.67 70.32 117.23 110.22 112.935 116.37 129.23 200.12 136.67 222.96 107.35

0.932 0.954 0.7988 0.828 0.8293 0.8078 0.8268 0.5688 0.763 0.402 0.841

Db2 Db3

Sym4 Spline-B3

This paper examines the utility of non-redundant Discrete ´ trou wavelet algorithm for Wavelet Transforms (DWT) and a forecasting applications. A distinctive feature of a DWT is the latter’s lack of shift variant property and, as a result, has a limited appeal when required to be applied to forecasting problems. In ´ trou wavelet comparison, however, as revealed by the study, the a transform is more advantageous than DWT in forecasting applications. Further, three different commonly used wavelet families (Daubechies, Symmlets and Spline) were tested for hydrological forecasting applications having different characteristics. In the first case study, performance of the various candidate wavelet functions was evaluated in terms of various prescribed measures of forecast efficiency when implemented on three synthetically generated time series and each time series provided with different types of transient features such as (i) linear trend, (ii) nonlinear trend, and (iii) periodic component. The results of this part of the overall study show that a time series with a dominant high frequency component is best modeled using the db1 (Haar) wavelet for improved forecast results. On the other hand, the db2 wavelet performed better than other wavelets for the time series with long term non-linear trend and periodic component and, finally, for the synthetic time series with seasonal component, the forecast model based on db2 wavelet with 4 levels decomposition produced better results as compared to other wavelet functions. In the second case study, forecast models were designed with various wavelet functions for monthly stream flows for the M.H. Halli site on Hemavathy, a tributary of Cauvery. Results show that the db2 wavelet based forecast model turned in the best performance with a NSC value of 0.8669 followed closely by B3-spline wavelet based model that yielded forecast efficiency with a NSC value of 0.850. A third case study was implemented in order to obtain daily stream flow forecasts and results suggest that the model implementation that utilized the Haar (db1) wavelet decomposition performed better with a NSC value of 0.954 as compared to other test wavelets. 6.1. Depth of decomposition

As before, various candidate wavelet functions were used to decompose the time series at different levels, and using each of these decomposition sets, a WVC forecast model was developed to obtain one-day ahead stream flow forecasts at the same site as indicated above and the results, derived for various wavelet functions with different levels, are reported in Table 6. These results indicate that the Haar wavelet with four coefficients

The study shows that the level of decomposition is governed mainly by three factors namely (i) order of the time series involved, (ii) wavelet used and its order of the wavelet, and (iii) scale at which the dominant dynamics are taking place in the given time series. For example, for a monthly stream flow time series, the dominant feature was the annual cycle which

294

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

corresponds to the 4th level decomposition (8–16 months dynamics) and, therefore, the model with four levels decomposition performed better. However, the same underlying dynamics may be captured by any other higher order wavelet at lower levels. Similarly, in the case of the daily flow series, the model with only two levels of decomposition performed better showing the short memory (2 – 4 day) nature of the time series. However, care should be taken as for the correspondence between the scales and levels of decomposition will be different for different wavelet depending on their support. 6.2. Computational ease With regards to computational burden, it is seen that both the db wavelet as well as the spline wavelet offer better wavelet bases requiring relatively lesser number of coefficients to capture the essential features of the given time series. This is due to the good compression properties possessed by these wavelets which enable them to capture maximum possible information in the data with a minimum number of wavelet coefficients. The above results and investigations revealed that certain properties of wavelets may be crucial in a particular application depending upon the nature of the time series under consideration. In general, wavelets having compact support have good time localization property (as, for example, Haar wavelet) and accordingly performed better for those time series that have a short duration memory with short duration transient features. On the other hand, wavelets with wider support (db2, spline) are better for time series having long term features. The study also suggests that the spline wavelets, while only next to the db family wavelets in terms of performance, lends itself to a parsimonious implementation on account of its symmetry and, additionally, being a highly regular wavelet (smooth) (Wei and Billings, 2006 ; Chou, 2007). Accordingly, these results suggest that the Haar wavelet be preferred over other wavelet forms when implemented for daily stream flow time series and, for seasonal time series with a pronounced seasonal component, the db or spline wavelet is suggested.

7. Conclusion This paper has investigated various issues that are relevant when developing a wavelet based forecast model for a hydrologic time series and some of the important conclusions derived are given as below: (1) The (non-redundant) DWT implementation of the wavelet decomposition lacks shift invariance property and therefore ´ cannot be used in forecasting applications. Alternatively, the a trous algorithm is seen to be a better choice as it achieves a stretching of each component series (level) to the same lengths by introducing redundant values which increases autocorrelation in the higher level and this, in turn, enhances predictability leading to increased NSC values. ´ trous wavelet transform is the (2) The redundant WT – a most appropriate decomposition algorithm for forecasting purposes. (3) The issue of boundary treatment for wavelet decomposition was clarified and a suitable solution is suggested. (4) The selection of mother wavelet plays an important role in forecasting and the following is concluded from the study: (i) For time series having short memory and transient features, wavelets with compact support and reduced vanishing moments such as the Haar wavelet (db1) is recommended.

(ii) Wavelets with wider support and higher vanishing moments such as the db2 and spline-B3 wavelets are recommended for time series having long term memory and nonlinear features as seen from the obtained prediction accuracy. (5) The depth of time series decomposition must be chosen after an investigation for the underlying dominant features in order to enable an unambiguous detection of these features. (6) Overall, the results show that there is no universal wavelet function which suits all type of time series. Designing of new wavelets having more generic features is also recommended as a future focus.

References Adamowski, J., Sun, K., 2010. Development of a coupled wavelet transform and neural network method for flow forecasting of non-perennial rivers in semiarid watersheds. Journal of Hydrology 390 (1–2), 85–91. Ahuja, N., Lertrattanapanich, S., Bose, N.K., 2005. Properties determining choice of mother wavelet. Vision, Image and Signal Processing, IEE Proceedings 152 (5), 659–664. Aussem, A., Cambell, J., Murtagh, F., 1998. Wavelet based feature extraction and decomposition strategies for financial forecasting. Journal of Computational Intelligence in Finance 6, 5–12. Azeem, M.F., Banakar, A., Kumar, V., 2006. Comparative study of different types of wavelet functions in neural network. In Proceedings of the International Joint Conference on Neural Networks, IJCNN ’06. Bakshi, B.R., 1999. Multiscale analysis and modeling using wavelets. Journal of Chemometrics 13 (3–4), 415–434. Burrus, C., Gopinath, R., Guo, H., 1998. Introduction to wavelets and wavelet transforms, Prentice Hall, Upper Saddle River, NJ. Chen, S., Billings, S.A., Luo, W., 1989. Orthogonal least squares methods and their application to non-linear system identification. International Journal of Control 50 (5), 1873–1896. Chou, C.M., 2007. Efficient nonlinear modeling of rainfall-runoff process using wavelet compression. Journal of Hydrology 332 (3–4), 442–455. Chui, C.K., 1992. An Introduction to Wavelets. Academic Press. De Chazal, P., Celler, B., Reilly, R., 2000. Using wavelet coefficients for the classification of the electrocardiogram. In: Proceedings of the 22nd Annual International Conference on Engineering in Medicine and Biology Society, Vol. 61, pp. 64–67. Farge, M., 1992. Wavelet transforms and their applications to turbulence. Annual Review of Fluid Mechanics 24, 395–458. Karunanithi, N., Grenney, W.J., Whitley, D., Bovee, K., 1994. Neural networks for river flow prediction. Journal of Computing in Civil Engineering 8 (2), 201–220. Khosa, R., Joshi, S.D., Kiran, Gopi, Al Mohseen, Kamal, 2005. Identification of nonstationarities in hydrological time series using wavelets. Hydrology Journal 28 (1-2), 1–18. Kim, H.S., Kang, D.S., Kim, J.H., 2003. The BDS statistic and residual test. Stochastic Environmental Research and Risk Assessment 17, 104–115. Kisi, O., 2009. Neural networks and wavelet conjunction model for intermittent streamflow forecasting. Journal of Hydrologic Engineering 14, 773–782. Kisi, O., 2010. Wavelet regression model for short-term stream flow forecasting. Journal of Hydrology 389 (3–4), 344–353. Kumar, P., Foufoula-Georgiou., 1997. Wavelet analysis for geophysical applications. Reviews of Geophysics, 35 (4), 385–412. Luan, Y., 2005. Multiresolution traffic prediction: combine RLS algorithm with wavelet transform, Information Networking. Convergence in Broadband and Mobile Networking. Lecture Notes in Computer Science. Springer Berlin/ Heidelberg, pp. 321–331. Maheswaran, R., Khosa, R. Multiscale nonlinear model for monthly stream flow forecasting: a wavelet based approach. Journal of Hydroinformatics, doi:10. 2166/hydro.2011.130, in press. Maheswaran, R., Khosa, R. Wavelet Volterra coupled model for stream flow forecasting. Journal of Hydrology. (under 2nd review). Mun, F.K., 2004. Time series forecasting using wavelet and support vector machine, MS Thesis, National University of Singapore. Nibhanupudi, S., 2003. Signal Denoising Using Wavelets. PhD Thesis, University of Cincinnati. Nourani, V., Komasi, M., Mano, A., 2009. A multivariate ANN-wavelet approach for rainfall–runoff modeling. Water Resources Management 23 (14), 2877–2894. Partal, T., Kisi, O., 2007. Wavelet and neuro-fuzzy conjunction model for precipitation forecasting. Journal of Hydrology 342 (1-2), 199–212. Renaud, O., Starck, J., Murtagh, F., 2005. Wavelet-based combined signal filtering and prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 35 (6), 1241–1251. Shensa, M.J., 1992. The discrete wavelet transform: wedding the a trous and mallat algorithms. IEEE Transactions on Signal Processing 40 (10), 2464–2482. Stolojescu, C., Railean, I., Moga, S., Isar, A., 2010. Comparison of wavelet families with application to WiMAX traffic forecasting. In: Proceedings of 12th

R. Maheswaran, R. Khosa / Computers & Geosciences 46 (2012) 284–295

International Conference on Optimization of Electrical and Electronic Equipment (OPTIM), pp. 932–937. Strang, G., Nguyen, T., 1996. Wavelets and Filter Banks. Wellesley, Cambridge Press. Tiwari, M.K., Chatterjee, C., 2010. Development of an accurate and reliable hourly flood forecasting model using Wavelet-Bootstrap-ANN hybrid approach. Journal of Hydrology 394, 458–470. Wei, H., Billings, S., 2006. Long term prediction of non-linear time series using multiresolution wavelet models. International Journal of Control 79, 569–580.

295

Yu, P., Goldberg, A., et al. 2001. Time series forecasting using wavelets with predictor-corrector boundary treatment, In: The Proceedings of 7th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. Zhu, Y., Qian, J., Fan, Q., Wan, D., Li, S., 2009. Study on Hydrology time series prediction based on wavelet-neural networks. In: Proceedings of 8th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2009. Zhou, H.C., Peng, Y., Liang, G.-H., 2008. The research of monthly discharge predictor–corrector model based on wavelet decomposition. Water Resources Management 22 (2), 217–227.