Theor Appl Climatol DOI 10.1007/s00704-015-1401-6

ORIGINAL PAPER

Memory persistency and nonlinearity in daily mean dew point across India Rajdeep Ray & Mofazzal Hossain Khondekar & Koushik Ghosh & Anup Kumar Bhattacharjee

Received: 12 September 2014 / Accepted: 28 January 2015 # Springer-Verlag Wien 2015

Abstract Enterprising endeavour has been taken in this work to realize and estimate the persistence in memory of the daily mean dew point time series obtained from seven different weather stations viz. Kolkata, Chennai (Madras), New Delhi, Mumbai (Bombay), Bhopal, Agartala and Ahmedabad representing different geographical zones in India. Hurst exponent values reveal an anti-persistent behaviour of these dew point series. To affirm the Hurst exponent values, five different scaling methods have been used and the corresponding results are compared to synthesize a finer and reliable conclusion out of it. The present analysis also bespeaks that the variation in daily mean dew point is governed by a non-stationary process with stationary increments. The delay vector variance (DVV) method has been exploited to investigate nonlinearity, and the present calculation confirms the presence of deterministic nonlinear profile in the daily mean dew point time series of the seven stations. R. Ray Department of Electronics and Communication Engineering, Dr. B. C. Roy Engineering College, Durgapur 713206, India e-mail: [email protected] M. H. Khondekar (*) Department of Applied Electronics and Instrumentation Engineering, Dr. B. C. Roy Engineering College, Durgapur 713206, India e-mail: [email protected] M. H. Khondekar e-mail: [email protected] K. Ghosh Department of Mathematics, University Institute of Technology, University of Burdwan, Burdwan 713104, India e-mail: [email protected] A. K. Bhattacharjee Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur 713209, India e-mail: [email protected]

1 Introduction With the advent of observation techniques, satellite communication and high performance computer technologies, the field of atmospheric sciences, meteorology and weather prediction made revolutionary and remarkable advancements. Significant changes in important climate attributes are evident in the daily measurements made over the past few decades by thousands of dedicated weather observers worldwide. Some of these changing climate attributes have statistical significance in that their pattern, persistence and amplitude. There are three important features of the environment that are changing significantly and with consequence: temperature, water vapour and precipitation. In India, the need for more accurate forecasting and informative and elaborate climate database has been enhanced on account of rapid economic growth of the country (Vision Document on Atmospheric Sciences, prepared for The Ministry of Earth Sciences, 2010). Observations of dew point at a number of stations are being made near the earth’s surface by trained observers, automatic weather stations or buoys (National Climatic Data Center). These observatories over some periods of time and at different altitudes provide discrete time/spatial signals (Mathur and Paras 2012). As the dew point variation is related to the fluctuation of the humidity and water vapour, its statistical analysis may retrieve useful insight information about the Indian climate. In this paper, an initiative has been taken to understand dynamical feature of the variations in dew point at different geographical location in India. To serve this purpose, the daily mean dew point from 9 October 1996 to 1 February 2013 recorded at seven different stations viz. Kolkata, Chennai, New Delhi, Mumbai, Bhopal, Agartala and Ahmedabad representing different geographical zones of India is taken as the signals under investigation. Figure 1 shows the positional importance of all the stations within the country.

R. Ray et al.

Fig. 1 Seven stations of interest

To make the data convenient for the analysis proper interpolation (Baxter 1992), smoothing (Ostertagova and Ostertag 2012) and de-noising (Briggs and Levine 1997); (Donoho 1995) of these time series have been performed with utmost care. Scaling analysis on a time series is performed in order to understand the strength of long-range correlations in the fluctuation behaviour of that time series. Quantitatively, the selfaffinity of fluctuation time series is characterized by the scaling behaviour. Hurst exponent works as a very efficient tool to identify this scaling behaviour. The Hurst exponents (Hurst et al. 1965); (Bezsudnov and Snarskii 2014) using five different methods (Wavelet variance analysis (Gopir et al. 2010), Higuchi (Higuchi 1988), detrended fluctuation analysis (DFA) (Peng et al. 1994); (Shadkhoo et al. 2009); (Rivera-Castro et al. 2012); (Zebendea et al. 2011), visibility graph analysis (Lacasa et al. 2009); (Bezsudnov and Snarskii 2014); (Zhuang et al. 2014) and general method of estimating Hurst exponent (Matteo et al. 2003) have been calculated to decide the statistical nature of the signals with respect to different scales and to identify whether the signal is fractional Brownian motion (Changa and Geman 2013); (Florindo and Bruno 2012) or fractional Gaussian noise (Mandelbrot and VanNess 1968) nay whether the signal is stationary or non-stationary. Real wavelet functions can only return information on amplitude. As a result, real wavelet functions are practically employed to find peaks or discontinuities. On the contrary, complex wavelet functions basically return information both on amplitude and phase. Therefore, these are better adapted to capture oscillatory behaviour. In the present work, we have applied continuous wavelet transform involving complex coefficient for scaling analysis. Higuchi method relies on finding fractal lengths to determine the fractal dimension and on that basis finally the Hurst exponent of the present signals. DFA is basically used to remove the trend in a time series. Trend basically distorts or obscures the relationship of interest. By removing the trend, true oscillatory behaviour of a time series can be identified effectively. Visibility graph analysis (VGA) works to convert time series into complex networks. The converted networks exhibit significant properties of time series within the structure of network and help us to go for scaling analysis of the time series. General method of estimating

Hurst exponent basically employs dispersion (standard deviation) to evaluate the Hurst exponent by means of the exponent of the corresponding power law. In the present work, all these five methods are used independently to have a large and reliable perspective of the scaling behaviour of the present signals. Delay vector variance (DVV) (Guatama et al. 2004); (Hossain et al. 2012); (Ahmed 2014) has been applied on the data to investigate the presence of nonlinear dynamics in the time series. The method is based upon the examination of local predictability of signals. Additionally, it spans the complete range of local linear models due to the standardization to the distribution of pair-wise distances between delay vectors.

2 Pre-processing of the data To proceed further for investigation, there is a need to preprocess the signals. 2.1 Interpolation As there are some missing data within the dew point time series, radial basis function interpolation method (Baxter 1992); (Buhmann 2000) has been used to interpolate the missing data. As per the ability to fit data at the gaps and altogether to produce a smooth curve, there are different forms of functions like Gaussian, multi-quadrics, linear, cubic, thin plate (Buhmann 1990); (Buhmann and Powell 1990), etc. All of the radial basis function methods are robust interpolators (Comac 1995), so they attempt to honour the trend and characteristics of the data present in the time series while interpolating the missing values. A radial basis function (RBF) is a function based on a scalar radius given by: φðrÞ ¼ φðjx−xi jÞ

ð1Þ

where φ(r), r, xi are radial basis function, scalar radius and new data point, respectively. 2.2 Simple exponential smoothing Due to the distribution of errors embedded in the given time series data, some sort of randomness may arise in it. In those cases, the methods of smoothing/averaging can be very useful techniques to reduce the bulk of errors accumulated at different locations in the time series data. Here, simple exponential smoothing technique has been used for the smoothing purpose (Ostertagova and Ostertag 2012). Exponential smoothing assigns exponentially decreasing weights as the observation get older. In other words, recent observations are given relatively more weight in forecasting than the older observations. The raw data sequence is represented by {xi}, and the output of the

Memory persistency and nonlinearity in daily mean dew point

exponential smoothing algorithm is {st}, which may be regarded as a best estimate of what the next value of x will be. When the sequence of observations begins at time t=0, the simplest form of exponential smoothing is given by s 0 ¼ x0 st ¼ αxt−1 þ ð1−αÞst−1

ð2Þ

where α is smoothing constant in the interval (0, 1) (Brown and Meyer 1961). For effective utility of this smoothing and to maintain positional importance, α should be chosen at the right hand neighbourhood of 0.5. For the present calculation, we have takenα=0.51. 2.3 De-noising the data using discrete wavelet transform In practical cases, all the observed data involve some amount of circumstantial errors which may creep in due to change in environment or systemic error which is due to factors inherent in the manufacture of the measuring instrument arising out of tolerances in the components of the instruments. Analysis of such data in presence of error may often fail to give accurate information. So, initially, there is a need to remove these errors up to a satisfactory level and to serve this purpose, different methods of de-noising are used. In the present work, for the de-noising the data, the discrete wavelet transform (Chui et al. 1994); (Daubechies 1992) has been exploited. The reason is its orthogonality property can fulfil the purpose satisfactorily over a discrete time signal compared to continuous wavelet transform (Polikar 1999). Let the data be written as the real-valued matrix G. It is possible to construct an orthogonal operator ψ, called the mother wavelet, such that the discrete wavelet transform (WT) is given by W ¼ ψG

ð3Þ

where W is the wavelet transform matrix of G whose elements wi,j are called wavelet coefficients (Donoho and Johnstone 1994). In other words, we project G onto the orthonormal basis ψ yielding a transformed field W. If ψ is well chosen, the transformed field or matrix W can be sparse; that is, many, even the vast majority, of elements may be at or near zero, with only a few elements being relatively large. The proper choice of mother wavelet and elimination of significant elements are the two key tasks in DWT de-noising. These insignificant or zero elements can be eliminated by thresholding (Donoho 1992). 2.3.1 Choice of the mother wavelet There exist many different mother wavelets (orthogonal bases) (Goel and Vidakovic 1995); (Katul and Vidakovic 1995). To choose the best mother wavelet for a particular signal, we

adopt the procedure based on minimizing the entropy of the wavelet transformed matrix (Goel and Vidakovic 1995); (Katul and Vidakovic 1995). The idea is that the WT disbalances the energy of the signal and that the most disbalanced transform is best. This approach makes intuitive sense in that the minimum entropy will be with the transform that produces the greatest ratio of few large coefficients to many small coefficients. We start with a library of mother wavelets, compute an entropy score for each transform, and pick the mother wavelet that produces the best score. We use the library of mother wavelets based on the increasing coefficients of Daubechies, Coiflet and Symmlet wavelets along with the Haar wavelet (Daubechies 1992). The measure found to be the most resilient is the Shannon entropy measure (Goel and Vidakovic 1995) given by φ ðW Þ ¼ −

X i; j

w0 i; j logw0 i0 j ;

ð4Þ

where w′ i,j are the non-negative normalized wavelet coefficients, that is, w′ i,j =|wi,j|/∑|wi,j| and 0log0=0 by definition. The best WT will minimize φ(W) from the library of possible mother wavelets. The best possible choices of mother wavelets for dew point data of different stations have been shown in Fig. 2 2.3.2 Wavelet thresholding Thresholding (Donoho and Johnstone 1994) refers to the process of shrinking the coefficients of W that is, setting to zero or shrinking toward certain coefficients, in an effort to remove insignificant information. Generally, there are two manners of thresholding, hard and soft. Hard thresholding is of the form wi; j ¼

0; wi; j ;

jwi; j j jwi; j j

< λ ≥ λ

ð5Þ

where all |wi,j|

ORIGINAL PAPER

Memory persistency and nonlinearity in daily mean dew point across India Rajdeep Ray & Mofazzal Hossain Khondekar & Koushik Ghosh & Anup Kumar Bhattacharjee

Received: 12 September 2014 / Accepted: 28 January 2015 # Springer-Verlag Wien 2015

Abstract Enterprising endeavour has been taken in this work to realize and estimate the persistence in memory of the daily mean dew point time series obtained from seven different weather stations viz. Kolkata, Chennai (Madras), New Delhi, Mumbai (Bombay), Bhopal, Agartala and Ahmedabad representing different geographical zones in India. Hurst exponent values reveal an anti-persistent behaviour of these dew point series. To affirm the Hurst exponent values, five different scaling methods have been used and the corresponding results are compared to synthesize a finer and reliable conclusion out of it. The present analysis also bespeaks that the variation in daily mean dew point is governed by a non-stationary process with stationary increments. The delay vector variance (DVV) method has been exploited to investigate nonlinearity, and the present calculation confirms the presence of deterministic nonlinear profile in the daily mean dew point time series of the seven stations. R. Ray Department of Electronics and Communication Engineering, Dr. B. C. Roy Engineering College, Durgapur 713206, India e-mail: [email protected] M. H. Khondekar (*) Department of Applied Electronics and Instrumentation Engineering, Dr. B. C. Roy Engineering College, Durgapur 713206, India e-mail: [email protected] M. H. Khondekar e-mail: [email protected] K. Ghosh Department of Mathematics, University Institute of Technology, University of Burdwan, Burdwan 713104, India e-mail: [email protected] A. K. Bhattacharjee Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur 713209, India e-mail: [email protected]

1 Introduction With the advent of observation techniques, satellite communication and high performance computer technologies, the field of atmospheric sciences, meteorology and weather prediction made revolutionary and remarkable advancements. Significant changes in important climate attributes are evident in the daily measurements made over the past few decades by thousands of dedicated weather observers worldwide. Some of these changing climate attributes have statistical significance in that their pattern, persistence and amplitude. There are three important features of the environment that are changing significantly and with consequence: temperature, water vapour and precipitation. In India, the need for more accurate forecasting and informative and elaborate climate database has been enhanced on account of rapid economic growth of the country (Vision Document on Atmospheric Sciences, prepared for The Ministry of Earth Sciences, 2010). Observations of dew point at a number of stations are being made near the earth’s surface by trained observers, automatic weather stations or buoys (National Climatic Data Center). These observatories over some periods of time and at different altitudes provide discrete time/spatial signals (Mathur and Paras 2012). As the dew point variation is related to the fluctuation of the humidity and water vapour, its statistical analysis may retrieve useful insight information about the Indian climate. In this paper, an initiative has been taken to understand dynamical feature of the variations in dew point at different geographical location in India. To serve this purpose, the daily mean dew point from 9 October 1996 to 1 February 2013 recorded at seven different stations viz. Kolkata, Chennai, New Delhi, Mumbai, Bhopal, Agartala and Ahmedabad representing different geographical zones of India is taken as the signals under investigation. Figure 1 shows the positional importance of all the stations within the country.

R. Ray et al.

Fig. 1 Seven stations of interest

To make the data convenient for the analysis proper interpolation (Baxter 1992), smoothing (Ostertagova and Ostertag 2012) and de-noising (Briggs and Levine 1997); (Donoho 1995) of these time series have been performed with utmost care. Scaling analysis on a time series is performed in order to understand the strength of long-range correlations in the fluctuation behaviour of that time series. Quantitatively, the selfaffinity of fluctuation time series is characterized by the scaling behaviour. Hurst exponent works as a very efficient tool to identify this scaling behaviour. The Hurst exponents (Hurst et al. 1965); (Bezsudnov and Snarskii 2014) using five different methods (Wavelet variance analysis (Gopir et al. 2010), Higuchi (Higuchi 1988), detrended fluctuation analysis (DFA) (Peng et al. 1994); (Shadkhoo et al. 2009); (Rivera-Castro et al. 2012); (Zebendea et al. 2011), visibility graph analysis (Lacasa et al. 2009); (Bezsudnov and Snarskii 2014); (Zhuang et al. 2014) and general method of estimating Hurst exponent (Matteo et al. 2003) have been calculated to decide the statistical nature of the signals with respect to different scales and to identify whether the signal is fractional Brownian motion (Changa and Geman 2013); (Florindo and Bruno 2012) or fractional Gaussian noise (Mandelbrot and VanNess 1968) nay whether the signal is stationary or non-stationary. Real wavelet functions can only return information on amplitude. As a result, real wavelet functions are practically employed to find peaks or discontinuities. On the contrary, complex wavelet functions basically return information both on amplitude and phase. Therefore, these are better adapted to capture oscillatory behaviour. In the present work, we have applied continuous wavelet transform involving complex coefficient for scaling analysis. Higuchi method relies on finding fractal lengths to determine the fractal dimension and on that basis finally the Hurst exponent of the present signals. DFA is basically used to remove the trend in a time series. Trend basically distorts or obscures the relationship of interest. By removing the trend, true oscillatory behaviour of a time series can be identified effectively. Visibility graph analysis (VGA) works to convert time series into complex networks. The converted networks exhibit significant properties of time series within the structure of network and help us to go for scaling analysis of the time series. General method of estimating

Hurst exponent basically employs dispersion (standard deviation) to evaluate the Hurst exponent by means of the exponent of the corresponding power law. In the present work, all these five methods are used independently to have a large and reliable perspective of the scaling behaviour of the present signals. Delay vector variance (DVV) (Guatama et al. 2004); (Hossain et al. 2012); (Ahmed 2014) has been applied on the data to investigate the presence of nonlinear dynamics in the time series. The method is based upon the examination of local predictability of signals. Additionally, it spans the complete range of local linear models due to the standardization to the distribution of pair-wise distances between delay vectors.

2 Pre-processing of the data To proceed further for investigation, there is a need to preprocess the signals. 2.1 Interpolation As there are some missing data within the dew point time series, radial basis function interpolation method (Baxter 1992); (Buhmann 2000) has been used to interpolate the missing data. As per the ability to fit data at the gaps and altogether to produce a smooth curve, there are different forms of functions like Gaussian, multi-quadrics, linear, cubic, thin plate (Buhmann 1990); (Buhmann and Powell 1990), etc. All of the radial basis function methods are robust interpolators (Comac 1995), so they attempt to honour the trend and characteristics of the data present in the time series while interpolating the missing values. A radial basis function (RBF) is a function based on a scalar radius given by: φðrÞ ¼ φðjx−xi jÞ

ð1Þ

where φ(r), r, xi are radial basis function, scalar radius and new data point, respectively. 2.2 Simple exponential smoothing Due to the distribution of errors embedded in the given time series data, some sort of randomness may arise in it. In those cases, the methods of smoothing/averaging can be very useful techniques to reduce the bulk of errors accumulated at different locations in the time series data. Here, simple exponential smoothing technique has been used for the smoothing purpose (Ostertagova and Ostertag 2012). Exponential smoothing assigns exponentially decreasing weights as the observation get older. In other words, recent observations are given relatively more weight in forecasting than the older observations. The raw data sequence is represented by {xi}, and the output of the

Memory persistency and nonlinearity in daily mean dew point

exponential smoothing algorithm is {st}, which may be regarded as a best estimate of what the next value of x will be. When the sequence of observations begins at time t=0, the simplest form of exponential smoothing is given by s 0 ¼ x0 st ¼ αxt−1 þ ð1−αÞst−1

ð2Þ

where α is smoothing constant in the interval (0, 1) (Brown and Meyer 1961). For effective utility of this smoothing and to maintain positional importance, α should be chosen at the right hand neighbourhood of 0.5. For the present calculation, we have takenα=0.51. 2.3 De-noising the data using discrete wavelet transform In practical cases, all the observed data involve some amount of circumstantial errors which may creep in due to change in environment or systemic error which is due to factors inherent in the manufacture of the measuring instrument arising out of tolerances in the components of the instruments. Analysis of such data in presence of error may often fail to give accurate information. So, initially, there is a need to remove these errors up to a satisfactory level and to serve this purpose, different methods of de-noising are used. In the present work, for the de-noising the data, the discrete wavelet transform (Chui et al. 1994); (Daubechies 1992) has been exploited. The reason is its orthogonality property can fulfil the purpose satisfactorily over a discrete time signal compared to continuous wavelet transform (Polikar 1999). Let the data be written as the real-valued matrix G. It is possible to construct an orthogonal operator ψ, called the mother wavelet, such that the discrete wavelet transform (WT) is given by W ¼ ψG

ð3Þ

where W is the wavelet transform matrix of G whose elements wi,j are called wavelet coefficients (Donoho and Johnstone 1994). In other words, we project G onto the orthonormal basis ψ yielding a transformed field W. If ψ is well chosen, the transformed field or matrix W can be sparse; that is, many, even the vast majority, of elements may be at or near zero, with only a few elements being relatively large. The proper choice of mother wavelet and elimination of significant elements are the two key tasks in DWT de-noising. These insignificant or zero elements can be eliminated by thresholding (Donoho 1992). 2.3.1 Choice of the mother wavelet There exist many different mother wavelets (orthogonal bases) (Goel and Vidakovic 1995); (Katul and Vidakovic 1995). To choose the best mother wavelet for a particular signal, we

adopt the procedure based on minimizing the entropy of the wavelet transformed matrix (Goel and Vidakovic 1995); (Katul and Vidakovic 1995). The idea is that the WT disbalances the energy of the signal and that the most disbalanced transform is best. This approach makes intuitive sense in that the minimum entropy will be with the transform that produces the greatest ratio of few large coefficients to many small coefficients. We start with a library of mother wavelets, compute an entropy score for each transform, and pick the mother wavelet that produces the best score. We use the library of mother wavelets based on the increasing coefficients of Daubechies, Coiflet and Symmlet wavelets along with the Haar wavelet (Daubechies 1992). The measure found to be the most resilient is the Shannon entropy measure (Goel and Vidakovic 1995) given by φ ðW Þ ¼ −

X i; j

w0 i; j logw0 i0 j ;

ð4Þ

where w′ i,j are the non-negative normalized wavelet coefficients, that is, w′ i,j =|wi,j|/∑|wi,j| and 0log0=0 by definition. The best WT will minimize φ(W) from the library of possible mother wavelets. The best possible choices of mother wavelets for dew point data of different stations have been shown in Fig. 2 2.3.2 Wavelet thresholding Thresholding (Donoho and Johnstone 1994) refers to the process of shrinking the coefficients of W that is, setting to zero or shrinking toward certain coefficients, in an effort to remove insignificant information. Generally, there are two manners of thresholding, hard and soft. Hard thresholding is of the form wi; j ¼

0; wi; j ;

jwi; j j jwi; j j

< λ ≥ λ

ð5Þ

where all |wi,j|