Chapter 4 4.1. Introduction Time series analysis approach for analyzing and understanding real world problems such as climatic and financial data is quite popular in the scientific world (Addison (2002), Feder (1988), Kumar and Foufoula (1993a), Kumar and Foufoula (1997), Lafreniere and Sharp (2003), Mandelbrot and Hudson (2004), Meyer (1998), Rangarajan and Sant (2004)). Till a decade ago, statistical and Fourier analysis methods were quite popular for studying the behavior of time series climatic data. However, recently wavelet and fractal methods are applied for a better understanding of the behavior of such series (Addison (2002), Arneodo et al. (2003), Hu and Nitta (1996), Kulkarni (2000), Kumar and Foufoula (1993b, 1993c), Mallat (1999), Rangarajan and Sant (2004), Turiel et al. (2006)). India is the biggest democratic country whose economy and political system is closely related to agriculture. Agriculture production is mainly dependent on rainfall particularly during rainy season June-July-August-September (abbreviated as JJAS) and winter season October- November-December (abbreviated as OND). In this context months January and February will be abbreviated as JF. Months March-April and May will be abbreviated as MAM. This has been the topic of intense research for more than 100 years applying classical techniques; see for example Singh and Sontakke (1999). The main objective of the work in this chapter is to present the work by Manchanda, Kumar, Khene and Siddiqi on the study of Indian rainfall data during 1813 to 1995 by applying wavelets and wavelet based multifractal formalism. This chapter has been divided into four sections. In the first section above, we have reviewed the work done by various researchers to study the behavior of time series by using wavelet and fractal methods. Section two is divided into five subsections. In the first subsection we discuss the concepts of multifractal, singularity, self-affine function, Hurst exponent and its connection to fractal dimension. In the next three subsections, we have discussed singularity spectrum, Legendre transform, partition function and numerical procedure for calculating the singularity spectrum. Smooth perturbations are discussed in the last subsection. The third section deals with the 94
analysis of rainfall data. We have divided this section into four subsections. In the first subsection, we present the wavelet analysis of Indian rainfall data (annual as well as seasonal) with db2, db3 and Coif 5 wavelets. In the second subsection multifractal analysis of Indian rainfall data is performed to calculate box dimension and regularization dimension for different seasons. In the third subsection regularity analysis of the data under consideration is performed. For the regularity analysis we have used wavelet transform based parametric and non parametric approach. The fourth subsection deals with the wavelet based multifractal analysis of the rainfall time series data using Morlet wavelet. Conclusions to the analysis are made in the last section this chapter. This work presented in this chapter is published in Manchanda, P. et al. (2007a). 4.2. Preliminaries 4.2.1. Wavelet Based Multifractal Formalism Sharp signal transitions create large amplitude wavelet coefficients. Singularities are detected by following across scales the local maxima of the wavelet transform. Fractals describe objects that are too irregular to fit into traditional geometrical settings. Various phenomena display fractal features when plotted as functions of time. Some common examples are atmospheric pressure, water level in a reservoir and prices of the stock market, usually when recorded over fairly long time spans. The zooming capability of the wavelet transform not only locates isolated singular events, but can also characterize more complex multifractal signals having non isolated singularities. Multifractals are fractal objects which cannot be completely described using a single fractal dimension (monofractals).they have in fact an infinite number of dimension measures associated with them. The wavelet transform takes advantage of multifractal self-similarities, in order to compute the distribution of their singularities. This singularity spectrum is used to analyze multifractal properties. Signals that are singular at almost every point are multifractals and they appear in the maintenance of economic records, physiological data including heart records, electromagnetic fluctuations in galactic radiation noise, textures in images of natural terrain and variations of traffic flow, etc. 95
Concepts of fractals and multifractals and their relevance to the real world systems were introduced by the Mandelbrot and Hudson (2004). In many real world systems, represented by time series, understanding of pattern of the singularities, that is, the graph of points at which time series changes abruptly is quite a challenging task. The time series of rain fall data usually depict fractal or multifractal features. Time series are commonly called self-affine functions as their graphs are self-affine sets that are similar to themselves when transformed by anisotropic dilations, that is., when shrinking along the x-axis by a factor followed by the rescaling of the increments of the function by different factor. Mathematically, if ( ) is a self-affine function then ( ) ( ) ( ( ) ( )) ( ) The exponent here is called roughness or Hurst exponent. Note that if, then is not differentiable and smaller the exponent, the more singular is. Hurst exponent provides an indication of how globally irregular the function is. Indeed it is related to the fractal dimension as. For estimation and properties of the Hurst parameter or exponent we refer to Addison (2002), Feder (1988), Mandelbrot and Hudson (2004), Mallat (1999) and Siddiqi (2005). Fractal functions can posses multi-affine properties in the sense that their roughness (or regularity) may fluctuate from point to point. To describe these multifractal functions, one thus needs to change slightly the definition of the Hurst regularity of so that it becomes a local quantity: ( ) ( ) ( ) ( ) This local Hurst exponent ( ) is generally called H lder exponent of at the point. A more precise mathematical formulation can be given as: To a given signal ( ) a function ( ), the Holder function of, which measures the regularity of at each point is associated. The point wise H lder of at point is defined as: ( ) * ( ) ( ) + ( ) (Here is not an integer and is non-differential). 96
One may also define a local exponent ( ) as: ( ) * ( ) ( ) + ( ) where and are different in general. For example for ( ) ( ) ( ) They have quite different properties. For instance is stable through differentiation ( ( ) ( ) ), whereas is not. The smaller ( ) is, the more irregular the function is at. Fraclab and Benoit software can be used to estimate H lder exponent. H lder exponent is often called Lipschitz exponent. We now recapitulate in the following subsections, some concepts and results which are required in the multifractal analysis of the rainfall time series. 4.2.2. Singularity Spectrum The pattern of singularities (singularity spectrum) is a graph of points at which a time series changes abruptly. The singularity spectrum measures the global repartition of singularities having different Lipschitz regularity. We recall that if is the set of all points where the pointwise Lipschitz regularity of is equal to, the spectrum of singularity, ( ), of is the fractal dimension of. The support of ( ) is the set of such that is not empty. The singularity spectrum gives the proportion of Lipschitz singularities that appear at any scale. 4.2.3. Partition Function Pointwise Lipschitz (H lder) regularity of a multifractal can not be computed because its singularities are not isolated, and the finite numerical resolution is not sufficient to discriminate them. To overcome this difficulty Arneodo, Bacry and Muzy (Muzy et al. (1994)) have introduced the concept of wavelet transform modulus maximum using a global partition function (for updated references related to this method see Arneodo et al. (2003)). 97
Let be a wavelet with vanishing moments. Mallat has shown that if has a pointwise Holder (Lipschitz) regularity at then the wavelet transform ( ) has a sequence of modulus maxima that converges towards at fine scales. The set of maxima at the scale can thus be interpreted as a covering of the singular support of with wavelets of scale. At these maxima locations ( ) ( ) Let { ( )} be the position of all local maxima of ( ) at a fixed scale. The partition function measures the sum at a power of all these wavelet modulus maxima: ( ) ( ) ( ) For each the scaling exponent ( ) measures the asymptotic decay of ( ) at fine scales : ( ) ( ) ( ) This typically means that ( ) ( ) We recall the following theorems which relates ( ) to the Legendre transform of ( ) for self-similar signals. Theorem 4.2.1 was established in Bacry et al. (1993) for a particular class of fractal signals and generalized by Jaffard (1997). Theorem 4.2.1. (Arneodo, Bacry, Jaffard, Muzy) (Mallat (1999)) Let [ ] be the support of ( ) Let be a wavelet with vanishing moments. If is a self similar signal then ( ) ( ( ) ( )) ( ) Theorem 4.2.2. (Mallat (1999)) (a) The scaling exponent ( ) is a convex and increasing function of. (b) The Legendre transform (4.8) is invertible if and only if ( ) is convex, in which case ( ) ( ( ) ( )) ( ) 98
The spectrum ( ) of self-similar signals is convex. Theorem 4.2.3. (Mallat (1999)) Let ( ) where Gaussian. For any ( ), the modulus maxima of ( ) belong to connected curves that are never interrupted when the scale decreases. For detailed properties of singularities spectrum we refer to Meyer (1998). We first calculate ( ) ( ) then derive the decay scaling component ( ), and finally compute ( ) with a Legendre transform. If then the value of ( ) depends mostly on the small amplitude maxima ( ) 4.2.4. Procedure for Numerical Calculation of Singularity Spectrum ( ) 1. Compute maxima ( ) and the maximum of its absolute value at each scale. 2. Chain the wavelet maxima across scales. 3. Compute the partition function ( ) ( ) ( ) 4. Compute the scaling coefficient ( ) with a linear regression of ( ) as a function of ( ) i.e, ( ) ( ) ( ) ( ) 5. Compute the singularity spectrum ( ) ( ( ) ( )) ( ) 4.2.5. Smooth Perturbations Let be a multifractal whose spectrum of singularity ( ) is calculated from ( ). If a regular signal is added to then the singularities are not modified and the singularity spectrum of remains unchanged. We study the effect of this smooth perturbation on the spectrum calculation. The wavelet transform of is ( ) ( ) ( ) ( ) 99
Let ( ) and ( ) be the scaling exponent of the partition functions ( ) and ( ) calculated from the modulus maxima respectively of ( ) and ( ). The following theorem relates ( ) and ( ) Theorem 4.2.4. (Arneodo, Bacry, Muzy) (Mallat (1999)) Let be a wavelet with exactly vanishing moments. Suppose that is a self similar function. If is a polynomial of degree then ( ) ( ) for all If ( ) is almost everywhere non zero then ( ) ( ) {( ) (4.14) where is defined by ( ) ( ) (4.15) 4.3. Analysis In this section we have performed wavelet analysis, multifractal analysis, regularity analysis and wavelet based multifractal analysis of Indian rainfall data. 4.3.1. Wavelet Analysis of Indian Rainfall Data Long period instrumental series are vital in studies on climate variation, its modeling, monitoring and prediction. Longest seasonal and annual Indian rainfall series have been reconstructed from the past records over different spatial rainfall zones and for the country as a whole. Kumar et al. (2006) have studied Indian rainfall series from 1813-1995 by applying MATLAB wavelet toolbox. Wavelet analysis of the association between southern oscilation and the Indian summer monsoon is discussed by Kulkarni (2000). Wavelet analysis of the summer rainfall over Northern China and India has been carried out by Hu and Nitta (1996). In this section, we visualize time series of the Indian rain fall (seasonal and annual) through the microscope of different wavelets. The discrete wavelet analysis of the Indian meteorological time series data is carried out in terms of decomposition of different rainy seasons in terms of approximations and details. The decomposition analysis of different four rainy seasons as well as the 100
annual rainfall data is shown in figures 1-10. In these figures, the x-axis represents the time period of the data under consideration. Each of these figures consists of six parts. The first figure (on left top) representing original signal for the rainfall time series data, the second one (on right top) gives the approximation. This approximation part corresponds to the amplitude of the signal for respective wavelets used at level 4. The other four parts and represent detail of the signal. Figure 1: DWT decomposition of Jan-Feb season rainfall (Daubechies 2 wavelet). The discrete wavelet analysis for the first two seasons Jan-Feb and March-April-May is performed using two different wavelets db2 and Coif 5 (Fig. 1-4), both at level 4. The rainfall signal for the next two seasons JJAS (June-July-August-September) and OND (Oct-Nov-Dec) as well as annual data (1813-1995) is analyzed using two different wavelets db3 and Coif 5 at level 4 (Fig. 5-10). 101
Figure 2: DWT decomposition of Jan-Feb season rainfall (Coifman 5 wavelet). Figure 3: DWT decomposition of March-April-May season rainfall (Daubechies 2 wavelet). 102
Figure 4: DWT decomposition of Mar-Apr-May season rainfall (Coifman 5 wavelet). Figure 5: DWT decomposition of Jun-July-Aug-Sept season rainfall (Daubechies 3 wavelet). 103
Figure 6: DWT decomposition of Jun-July-Aug-Sept season rainfall (Coifman 5 wavelet). Figure 7: DWT decomposition of Oct-Nov-Dec season rainfall (Daubechies 3 wavelet). 104
Figure 8: DWT decomposition of Oct-Nov-Dec season rainfall (Coifman 5 wavelet). Figure 9: DWT decomposition of annual rainfall (1813-1995) (Daubechies 3 wavelet). 105
Figure 10: DWT decomposition of annual rainfall (1813-1995) (Coifman 5 wavelet). 4.3.2. Multifractal Analysis of Indian Rainfall Data Fractal dimensions are the best known part of the fractal analysis. Various kinds of dimensions have been defined in different fields. In the wavelet based multifractal study of the Indian rainfall, we only focus on two types of dimension, namely the box dimension and the regularization dimension. These dimensions aim at approximating the more formal Hausdorff dimension. The regularization dimension is estimated by first computing smoother and smoother versions of the original signal, obtained through convolution with a kernel. If the original signal is a fractal, its graph has infinite length, while all regularized versions have finite length. When the smoothing parameter tends to zero, the smoothed version tends to the original signal, and its length will tend to infinity. The regularization dimension measures the speed at which this convergence to infinity takes place. In many cases, this coincides with the usual box dimension. In general, it can be shown that the regularization dimension is more precise than the box dimension, in the sense that it is always smaller, but still larger 106
than Hausdorff dimension. In addition, the regularization dimension lends to more robust estimation procedures for various reasons. One of them is that we may choose the regularization kernel. Also, the smoothed versions are adaptive by construction. Finally, the smoothing parameter can be varied in very small steps, as box sizes have to undergo sudden changes. Another advantage is that, due to the fully analytical definition of the regularization dimension, it is easy to derive an estimator in the presence of noise. Table 1 Box vs. regularization dimensions for seasonal and annual rainfall data Jan-Feb MAM JJAS OND Annual Box method (least squares regression) Regularization method Kernel : Gaussian Regression : Least Squares 1.45 1.3716 1.4527 1.477 1.6299 2.29 2.5359 2.333 2.394 2.4379 4.3.3. Regularity Analysis In this section we estimate the exponents that arise constantly in the multifractal analysis of signals to assess the regularity of a time series. The most used exponent is the Hölder exponent, which characterizes the regularity of the measure/function under consideration at either any given point (pointwise regularity) or around any given point (local regularity). Many techniques have been developed to estimate pointwise (and local) Hölder exponents, none of which give satisfactory results in all cases. Estimating a local regularity indeed on discrete data without any a priori assumption is indeed a difficult task. This can be done using: 1. Parametric approach. Such estimators have been developed mainly in case of Brownian motion (bm) and its extensions such as fractional Brownian motion (fbm). 2. Non Parametric Methods, which are more numerous and generally give the correct estimation only when some technical conditions are met. We focus on the following methods: (a) Method based on the continuous wavelet transform (CWT). (b) Method based on the discrete wavelet transform (DWT). 107
Figures 11 to 13 show the estimation of point wise Hölder regularity of the Indian rainfall data for the period 1813-1995 for annual as well as seasonal data. We used the parametric approach codes using Fraclab. The results of the non parametric approach using the DWT for four different wavelets (db2, db20, Coif 6, Coif 24) are shown in figures 14-16. Figure 11: Parametric pointwise Hölder regularity estimation for the annual rainfall (1813-1995). 108
Figure 12: Parametric pointwise Hölder regularity estimation for the first two seasons. Figure 13: Parametric pointwise Hölder regularity estimation for the last two seasons. 109
Figure 14: Annual non-parametric point wise Hölder regularity estimation using DWT with two different wavelets. Figure 15: Non-parametric point wise Hölder regularity estimation using DWT with two different wavelets (db2 and db20) for the first two seasons Jan-Feb and Mar-Apr- May. 110
Figure 16: Non-parametric point wise Hölder regularity estimation using DWT with two different wavelets (db2 and db20) for the last two seasons JJAS and OND. 4.3.4. Wavelet Based Multifractal Analysis Various phenomena (data) show fractal and multifractal behavior when plotted against time. Multifractals have infinite number of dimensions associated with them. Signals which are multifractals, are singular (i.e. changes abruptly) at almost every point. There exist three main multifractal spectra, viz. the Hausdroff, large deviation and Legendre spectra. Basically, any of these three spectra provides information as to which singularities occur in the signal, and which are dominant: a spectrum is a one dimensional curve where abscissa represents the Hölder exponents actually present in the signal, and ordinates are related to the amount of points where we encounter a given singularity. The Hausdroff spectrum gives geometrical information pertaining to the dimension of sets of points in a signal having a given Hölder exponent. This is the most precise spectrum from a mathematical point of view, but is also the most difficult one to estimate. 111
Large deviation spectrum yields statistical information, related to the probability of finding a point with a given Holder exponent in the signal. More precisely, it measures how this probability behaves with the change in resolution. Legendre spectrum is a concave approximation to the large deviation spectrum. Its main purpose is to yield much more robust estimates, though at the expense of a loss of information. It could be based on box method or CWT techniques. In the sequel we show some sample results for the spectra computed with the Legendre technique (see theorems 4.2.1 and 4.2.2.). Figures 17 to 24 show the results of the CWT (Morlet wavelet) based estimation of the Legendre spectrum which represents an approximation of the Hausdroff spectrum for the four different seasons. Figures 25 and 26 represent the results for the annual rainfall. Figure 17: CWT of Jan-Feb season rainfall using a Morlet wavelet of size 8 and 128 voices. Each of the figures 17, 19, 21, 23, 25 consists of two parts. The first part on the top of each of these figures represents the signal or raw data. The second part of these figures shows the analysed pattern with the application of Morlet wavelet of size 8 and 128 voices. Figures 18, 20, 22, 24 show the Legendre spectrum estimation by the 112
application of CWT (Morlet wavelet of size 8, 128 voices) of four different seasons. Figure 26 shows multifractal analysis of the annual rainfall data. Figure 18: Multifractal analysis of Jan-Feb rainfall data using CWT (Morlet wavelet of size 8 and 128 voices, LS regression and local maxima). Figure 19: CWT of March-April-May season rainfall using a Morlet wavelet of size 8 and 128 voices. 113
Figure 20: Multifractal analysis of March-April-May rainfall data using CWT (Morlet wavelet of size 8 and 128 voices, LS regression and local maxima). Figure 21: CWT of June-July-Aug-Sept season rainfall using a Morlet wavelet of size 8 and 128 voices. 114
Figure 22: Multifractal analysis of June-July-Aug-Sept rainfall data using CWT (Morlet wavelet size 8, 128 voices, LS regression and local maxima). Figure 23: CWT of Oct-Nov-Dec season rainfall using a Morlet wavelet of size 8 and 128 voices. 115
Figure 24: Multifractal analysis of Oct-Nov-Dec rainfall data using CWT (Morlet wavelet of size 8, 128 voices, LS regression and local maxima). Figure 25: CWT of the annual rainfall using a Morlet wavelet of size 8 and 128 voices. 116
Figure 26: Multifractal analysis of the annual rainfall data using CWT (Morlet wavelet of size 8, 128 voices, LS regression and local maxima). 4.4. Conclusion Wavelet analysis provided visualization of the real life data at different levels. Fig. 1 to 10 provides decomposition of seasonal and annual Indian rainfall data for the period 1813-1995 at different levels through applications of Daubechies 3 and Coifman 5 wavelets. Wavelet toolbox of MATLAB has been used. Multifractal analysis provides both a local and a global description of the singularities of a signal. The local description is obtained via the Hölder exponent where as the global one is contained in various multifractal spectra (Addison (2002), Feder (1988), Mallat (1999), Mandelbrot and Hudson (2004) and Turiel et al. (2006)). Table 1 presents fractal dimension of seasonal and annual rainfall data. Figs. 11 to 13 represent the pointwise regularity of Indian rainfall data for the period 1813-1995 for annual as well as seasonal data using the parametric approach codes using Fraclab. The results of the non-parametric approach using the DWT for four different wavelets are shown in Figs. 14 to 16. Figs. 17 to 26 represent singularity spectrum for the seasonal as well as annual rainfall estimated using Morlet wavelet. The results of this work provide a general view of the pattern of Indian rainfall data of different seasons along with the 117
annual pattern. Whether we can connect these results with predictability indices studied by Rangarajan and Sant (2004) is an open problem. By applying results of Table 1, we can estimate the Hurst exponent of the data under consideration using relation. On the basis of the values of H, persistency of the data can be predicted (We know a datum is persistent if ( ). ************* 118