**3. Methodology**

#### **3.1 Statistical analysis**

Water quality parameters have been analyzed using the numerical index, multivariate statistics, and earth observation datasets [25]. The average value, positional average, and the maximum frequency values in the series datasets are estimated with mean, median, and mode correspondingly. Variability of the sample datasets is measured with standard deviation, and peakedness is estimated by kurtosis. The symmetry between data points is estimated with skewness approaches. Coefficient of variation gives the extent of variability of data in a sample.

#### *3.1.1 Regression analysis*

This analysis examines the influence of one or more independent variables on a dependent variable. The regression equation with dependent variable Y and independent variable X is represented as:

$$Y = m\_{\mathbb{P}^\infty} X + C$$

where C is a constant of integration [5]:

$$\begin{aligned} \text{Regression coefficient} &= m\_{\text{px}} = r \ast \left(\sigma\_{\prime}\prime\_{\sigma\_{x}}\right) \\ \text{Correlation coefficient} &= r = \frac{\left[\Psi(XY) - \Psi(X)E(Y)\right]}{\left[\left(\Psi(X^{2}) - \Psi(X)^{2}\right)\left(\Psi(Y^{2}) - \Psi(Y)^{2}\right)\right]^{1/2}} \\ &= \left[\frac{COV(X,Y)}{\sigma\_{Y}\sigma\_{X}}\right] \end{aligned}$$

where σ<sup>y</sup> and σ<sup>x</sup> are standard deviation of variables Y and X, respectively, and Ψ(X), Ψ(Y), and Ψ(XY) are the expected value of variables X, Y, and XY, respectively.

#### **3.2 Fractal dimension (FD)**

The term fractal comes from the Latin word fractus means "fraction or broken"; the basic concept lies in the fact that fractals have a large degree of *Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality DOI: http://dx.doi.org/10.5772/intechopen.92865*

self-similarity within themselves which was coined by Benot Mandelbrot in 1975. Fractals are characterized by self-similarity property having similar characteristics when analyzed over a large range of scales, and individually a single entity will have similar characteristics to that of the whole fractal [26, 27]. Fractal dimension estimation from a fractal set has various methods due its simplicity and automatic computability. The box counting is one of the major categories of fractal analysis and the most used technique to analyze image features such as texture segmentation, shape classification, and graphic analysis in many fields [28, 29]. The variance and spectral methods are two other major categories of fractal dimension analysis of a time series that recognize the determinism and randomness in data [30]. To study the naturally complex features such as coastlines, river boundaries, mountains, and clouds, the fractal dimension analysis has also provided a mathematical model as a fractal geometry [31, 32]. The glacial and fluvial morphologies are distinguished by using an automated approach (i.e., multifractal). In previous study, a multifractal detrended fluctuation analysis (MFDFA) has been carried out to estimate the variation of elevation profile of glacial and fluvial landscapes [33]. It has been observed that glacial landscapes reveal more complex structure than that of the fluvial landscapes as indicated by fractal parameters, such as degree of multifractality, asymmetry index, etc. The basic definition of fractal dimension is the Hausdorff dimension; however, box counting or box dimension is another popular definition which is easy to calculate.

#### **3.3 The Hurst exponent**

quality parameters used for the analysis are obtained through the website of the

In this study, we compute the Hurst exponent, fractal dimensions, and the predictability index (PI) of water quality parameters such as (1) turbidity, (2) chloride, (3) ferrous, (4) nitrate, (5) pH, (6) calcium, (7) magnesium, (8) fluoride, (9) total dissolved salts **(**TDS), (10) alkalinity, (11) hardness, and (12) sulfate, at high-altitude Tehri stations in the Himalayan foothills using the fractal theory. **Figure 2** shows the box plot of all the aforementioned 12 water parameters obtained for the study site. The irregular pattern in the WQI can be used in prediction purposes by analyzing its dynamic flow (i.e., chaotic, random, or deterministic structural pattern). Proper identification, classification, and mapping of water parameters of high-intensive and complex nature require frequent moni-

Water quality parameters have been analyzed using the numerical index, multivariate statistics, and earth observation datasets [25]. The average value, positional average, and the maximum frequency values in the series datasets are estimated with mean, median, and mode correspondingly. Variability of the sample datasets is measured with standard deviation, and peakedness is estimated by kurtosis. The symmetry between data points is estimated with skewness approaches. Coefficient

This analysis examines the influence of one or more independent variables on a dependent variable. The regression equation with dependent variable Y and inde-

*Y* ¼ *myxX* þ *C*

Regression coefficient ¼ *myx* ¼ *r* ∗ *<sup>σ</sup><sup>y</sup>*

where σ<sup>y</sup> and σ<sup>x</sup> are standard deviation of variables Y and X, respectively, and

<sup>Ψ</sup> *<sup>X</sup>*<sup>2</sup> � � � <sup>Ψ</sup>ð Þ *<sup>X</sup>* <sup>2</sup> � �

Correlation coefficient <sup>¼</sup> *<sup>r</sup>* <sup>¼</sup> ½ � <sup>Ψ</sup>ð Þ� *XY* <sup>Ψ</sup>ð Þ *<sup>X</sup> E Y*ð Þ

<sup>¼</sup> *COV X*ð Þ , *<sup>Y</sup> σYσ<sup>X</sup>* � �

Ψ(X), Ψ(Y), and Ψ(XY) are the expected value of variables X, Y, and XY,

The term fractal comes from the Latin word fractus means "fraction or broken"; the basic concept lies in the fact that fractals have a large degree of

*=σx* � �

<sup>Ψ</sup> *<sup>Y</sup>*<sup>2</sup> � � � <sup>Ψ</sup>ð Þ *<sup>Y</sup>* <sup>2</sup> h i � � <sup>1</sup>*=*<sup>2</sup>

Ministry of Drinking Water & Sanitation, New Delhi, India.

toring of these datasets especially in the context of drinking water.

of variation gives the extent of variability of data in a sample.

**3. Methodology**

**3.1 Statistical analysis**

*Fractal Analysis - Selected Examples*

*3.1.1 Regression analysis*

respectively.

**24**

**3.2 Fractal dimension (FD)**

pendent variable X is represented as:

where C is a constant of integration [5]:

Hurst exponent (H) is used as a measure of long-term memory of time series and a real-valued time series defined as the exponent in the asymptotic scaling relation [30, 34]. The Hurst exponent and fractal dimension are also directly related to each other and indicate the roughness of a surface. The Hurst exponent's value lies in a time series as persistent (0.5 < H ≤ 1) or anti-persistent (0 ≤ H < 0.5), and when the data are not intercorrelated, then H = 0.5 which implies that the series is unpredictable. This approach is used in various complex engineering fields as it provides statistical self-similarity relationship.

In terms of asymptotic scaling relation, the Hurst exponent of real-valued time series is defined as:

$$\left\langle \frac{\mathcal{R}(n)}{\mathcal{S}(n)} \right\rangle = \mathsf{C}n^H, \mathsf{as}!n! \mathsf{approaches} \text{ to infinity} \tag{1}$$

where C is a constant, angular brackets h i ⋯ denote expected value, S(n) is the standard deviation of the first "n" data of the seriesf g *X*1, *X*2, ⋯, *Xn* , and R(n) is their range:

$$R(n) = \max\left\{X\_1, X\_2, \dots, X\_n\right\} - \min\left\{X\_1, X\_2, \dots, X\_n\right\}$$

The Hurst exponent H is calculated from rescaled range technique and can also be computed from wavelet method for the time seriesf g *X*1,*X*2, ⋯, *Xn* .

#### *3.3.1 Estimate of the Hurst exponent: Wavelet approach*

If f(t) is a self-affine random process, "t" a position parameter (time or distance), a > 0 is a scale (dilatation) parameter, w(t) is a mother wavelet, and

*Fractal Analysis - Selected Examples*

$$w\_{t,a}(t') = \frac{1}{\sqrt{a}}w\left(\frac{t'-t}{a}\right)$$

is its shifted, dilatated, and scaled version, then the continuous wavelet transform of f(t) is defined as:

$$W(t,a) = \frac{1}{\sqrt{a}} \int\_{-\infty}^{\infty} w\_{t,a}(t') f(t') dt' \tag{2}$$

Sr. No Parameters Mean Median Mode SD Skewness Kurtosis Coefficient of

1 Turbidity 1.619 1.135 1.000 1.504 1.747 3.134 0.929 2 Chloride 3.114 0.000 3.983 5.009 1.528 1.387 1.609 3 Ferrous 0.266 0.190 0.000 1.031 11.713 150.451 3.881 4 Nitrate 2.663 2.200 0.000 2.097 1.011 1.165 0.787 5 pH 7.189 7.200 0.251 0.691 7.377 74.926 0.096 6 Calcium 40.656 33.100 0.000 39.952 2.841 11.549 0.983 7 Magnesium 30.725 22.000 0.000 50.368 9.567 127.310 1.639 8 Fluoride 0.283 0.240 0.000 0.266 0.768 0.157 0.941 9 TDS 107.008 80.150 55.000 116.571 1.781 3.376 1.089 10 Alkalinity 115.239 110.000 1.272 63.728 0.331 0.683 0.553 11 Hardness 136.661 144.500 95.000 102.589 1.494 8.679 0.751 12 Sulfate 14.564 8.800 0.000 17.161 2.400 7.224 1.178

*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality*

*DOI: http://dx.doi.org/10.5772/intechopen.92865*

*Statistical analysis of groundwater parameters at Tehri District, Uttarakhand.*

**Regression equation r<sup>2</sup> H D**

Ferrous y = 0.12408\*x + 0.065435 0.032746 1.5146 0.48536 2.0293 Nitrate y = 0.22208\*x + 2.3121 0.02548 0.84779 1.1522 0.69558 pH y = 0.010597\*x + 7.1721 0.000532 1.2259 0.77415 1.4517 Calcium y = 3.6487\*x + 34.8722 0.018921 1.2706 0.72942 1.5412 Magnesium y = 2.8703\*x + 26.1718 0.007351 1.2307 0.76928 1.4614 Fluoride y = 0.054962\*x + 0.19432 0.097003 0.93987 1.0601 0.87973 TDS y = 15.5427\*x + 81.9829 0.040221 0.85579 1.1442 0.71158 Alkalinity y = 13.4538\*x + 93.5935 0.10097 0.82012 1.1799 0.64025 Hardness y = 12.7963\*x + 116.0534 0.035196 1.2005 0.7995 1.401 Sulfate y = 0.8647\*x + 13.2096 0.005754 0.90684 1.0932 0.81367

Turbidity Chloride y = 0.39645\*x + 3.7212 0.014385 0.83198 1.168 0.66396

Chloride Turbidity y = 0.036284\*x + 1.7326 0.014385 0.99887 1.0011 0.99774

Ferrous y = 0.0073198\*x + 0.2440 0.001245 1.6072 0.39276 2.2145 Nitrate y = 0.094225\*x + 2.9621 0.050117 0.98909 1.0109 0.97818 pH y = 0.017067\*x + 7.1368 0.015072 1.1173 0.88266 1.2347 Calcium y = 0.18703\*x + 41.362 0.000543 1.3236 0.6764 1.6472 Magnesium y = 1.4045\*x + 35.148 0.019231 1.3891 0.61086 1.7783 Fluoride y = 0.0032535\*x + 0.2733 0.003714 1.0405 0.95945 1.0811 TDS y = 5.7221\*x + 89.5595 0.059563 0.9857 1.0143 0.9714 Alkalinity y = 3.5154\*x + 104.5777 0.075324 0.92132 1.0787 0.84263 Hardness y = 3.3113\*x + 126.6005 0.02575 1.3457 0.65428 1.6914 Sulfate y = 1.5794\*x + 9.7489 0.20972 1.0341 0.96585 1.0683

**Table 1.**

**27**

**Y Parameters-**

**X**

variation

**(Fractal)**

**PI**

If the time series f(t) is self-affine, the variance of *W t*ð Þ , *a* will scale with the dilatation parameter asymptotically as:

$$V(\mathfrak{a}) = \left\langle \mathcal{W}^2 \right\rangle - \left\langle \mathcal{W} \right\rangle^2 \mathfrak{a} \mathfrak{a}^\delta \tag{3}$$

When the exponent "δ" is between �1 and 3 (i.e., �1 ≤ δ ≤ 3), the Hurst exponent is defined as:

$$H\_w = \begin{cases} \frac{\delta + 1}{2} & \text{if } \quad -1 \le \delta < 1 \quad (\text{FGN})\\ \frac{\delta - 1}{2} & \text{if } \quad 1 \le \delta \le 3 \quad (\text{FBM}) \end{cases} \tag{4}$$

where FGN is the fractal Gaussian noise and FBM is the fractional Brownian motion. The Hurst exponent is linked with fractal dimension (D) and defined as:

$$\mathbf{H} = \mathbf{2} - \mathbf{D} \tag{5}$$

Now the climate predictability index is given as:

$$PI = \mathcal{Z}|D - \mathbf{1.5}| = \mathcal{Z}|\mathbf{0.5} - H|\tag{6}$$

If PI is close to zero, climate is unpredictable. The closer the PI to 1, the more predictable the climate is.

#### **4. Results and discussion**

To distinguish the fresh and contaminated water and establish relationship between the parameters have become a major concern for environmentalists and health workers. And due to increased levels pollutants, it is very challenging for municipal authorities to make availability of clean drinking water especially in developing countries. The statistical relationship of water models depends on the dynamics of climatic as well as soil parameters and thermodynamic processes among the surface water parameters. The established statistical relationship among the various water quality parameters is shown in **Table 1**, which suggested that the variation among these parameters occurs due to variability in the originating environment and is affected by terrain conditions by which it flows down. In dynamic systems, this kind of response generates irregularity, which may show a random pattern of certain type. **Figure 1** shows the box plot of the 12 water quality parameters over the study area. **Table 2** shows the Hurst exponent (H) estimated through standard wavelet techniques and compared with regression equation, and coefficient of each water parameters shows whether they have Brownian time series (or true random walk) behavior with the other related parameters or not. The summary

Sr. No Parameters Mean Median Mode SD Skewness Kurtosis Coefficient of variation 1 Turbidity 1.619 1.135 1.000 1.504 1.747 3.134 0.929 2 Chloride 3.114 0.000 3.983 5.009 1.528 1.387 1.609 3 Ferrous 0.266 0.190 0.000 1.031 11.713 150.451 3.881 4 Nitrate 2.663 2.200 0.000 2.097 1.011 1.165 0.787 5 pH 7.189 7.200 0.251 0.691 7.377 74.926 0.096 6 Calcium 40.656 33.100 0.000 39.952 2.841 11.549 0.983 7 Magnesium 30.725 22.000 0.000 50.368 9.567 127.310 1.639 8 Fluoride 0.283 0.240 0.000 0.266 0.768 0.157 0.941 9 TDS 107.008 80.150 55.000 116.571 1.781 3.376 1.089 10 Alkalinity 115.239 110.000 1.272 63.728 0.331 0.683 0.553 11 Hardness 136.661 144.500 95.000 102.589 1.494 8.679 0.751 12 Sulfate 14.564 8.800 0.000 17.161 2.400 7.224 1.178

*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality DOI: http://dx.doi.org/10.5772/intechopen.92865*

**Table 1.**

*wt*,*<sup>a</sup> t*

*W t*ð Þ¼ , *a*

*δ* þ 1

8 ><

>:

Now the climate predictability index is given as:

*δ* � 1

transform of f(t) is defined as:

*Fractal Analysis - Selected Examples*

exponent is defined as:

predictable the climate is.

**4. Results and discussion**

**26**

dilatation parameter asymptotically as:

*Hw* ¼

<sup>0</sup> ð Þ¼ <sup>1</sup>

is its shifted, dilatated, and scaled version, then the continuous wavelet

1 ffiffiffi *a* p

ffiffiffi *<sup>a</sup>* <sup>p</sup> *<sup>w</sup>*

∞ð

�∞

If the time series f(t) is self-affine, the variance of *W t*ð Þ , *a* will scale with the

*V a*ð Þ¼ *<sup>W</sup>*<sup>2</sup> � � � h i *<sup>W</sup>* <sup>2</sup>

When the exponent "δ" is between �1 and 3 (i.e., �1 ≤ δ ≤ 3), the Hurst

where FGN is the fractal Gaussian noise and FBM is the fractional Brownian motion. The Hurst exponent is linked with fractal dimension (D) and defined as:

If PI is close to zero, climate is unpredictable. The closer the PI to 1, the more

To distinguish the fresh and contaminated water and establish relationship between the parameters have become a major concern for environmentalists and health workers. And due to increased levels pollutants, it is very challenging for municipal authorities to make availability of clean drinking water especially in developing countries. The statistical relationship of water models depends on the dynamics of climatic as well as soil parameters and thermodynamic processes among the surface water parameters. The established statistical relationship among the various water quality parameters is shown in **Table 1**, which suggested that the variation among these parameters occurs due to variability in the originating environment and is affected by terrain conditions by which it flows down. In dynamic systems, this kind of response generates irregularity, which may show a random pattern of certain type. **Figure 1** shows the box plot of the 12 water quality parameters over the study area. **Table 2** shows the Hurst exponent (H) estimated through standard wavelet techniques and compared with regression equation, and coefficient of each water parameters shows whether they have Brownian time series (or true random walk) behavior with the other related parameters or not. The summary

*t* <sup>0</sup> � *t a* � �

*wt*,*<sup>a</sup> t*

<sup>2</sup> *if* �1≤*δ*<1 FGN ð Þ

<sup>2</sup> *if* <sup>1</sup>≤*δ*≤3 FBM ð Þ

<sup>0</sup> ð Þ*f t*<sup>0</sup> ð Þ*dt*<sup>0</sup> (2)

∝*a<sup>δ</sup>* (3)

H ¼ 2 � D (5)

*PI* ¼ 2j j *D* � 1*:*5 ¼ 2 0j j *:*5 � *H* (6)

(4)

*Statistical analysis of groundwater parameters at Tehri District, Uttarakhand.*



**Y Parameters-**

**29**

**X**

*DOI: http://dx.doi.org/10.5772/intechopen.92865*

**Regression equation r<sup>2</sup> H D**

TDS y = 0.038294\*x + 108.737 0.000172 0.56384 1.4362 0.12768 Alkalinity y = 0.62046\*x + 90.0942 0.1511 0.53726 1.4627 0.074521 Hardness y = 0.67582\*x + 109.2303 0.069075 0.82607 1.1739 0.65214 Sulfate y = 0.089831\*x + 10.9473 0.043691 0.67248 1.3275 0.34495

Chloride y = 0.013693\*x + 3.5007 0.019231 0.5427 1.4573 0.0854 Ferrous y = 0.0005026\*x + 0.2510 0.000602 1.121 0.87899 1.242 Nitrate y = 0.012498\*x + 2.2868 0.090436 0.70056 1.2994 0.40112 pH y = 0.0008809\*x + 7.1622 0.004119 1.01946 1.0054 0.98923 Calcium y = 0.43765\*x + 27.2961 0.30508 0.88407 1.1159 0.76813 Fluoride y = 0.0011794\*x + 0.2470 0.050062 0.69593 1.3041 0.39186 TDS y = 0.16172\*x + 112.1606 0.00488 0.52529 1.4747 0.050579 Alkalinity y = 0.16349\*x + 110.361 0.01671 0.5019 1.5041 0.008196 Hardness y = 0.94027\*x + 107.8115 0.21297 0.8051 1.1949 0.61019 Sulfate y = 0.009879\*x + 14.3066 0.000842 0.62993 1.3701 0.25985

Magnesium Turbidity y = 0.0025611\*x + 1.5419 0.007351 0.57726 1.4227 0.15451

*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality*

Fluoride Turbidity y = 1.7649\*x + 1.1207 0.097003 0.82823 1.1718 0.65646

TDS Turbidity y = 0.0025877\*x + 1.3435 0.040221 1.0328 0.96716 1.0657

Alkalinity Turbidity y = 0.0075049\*x + 0.75479 0.10097 1.1155 0.88449 1.231

Chloride y = 0.021427\*x + 0.60594 0.075324 1.0438 0.95622 1.0876 Ferrous y = 0.0012524\*x + 0.12203 0.005981 1.6487 0.35134 2.2973

Chloride y = 0.010409\*x + 1.963 0.059563 1.0908 1.0091 0.98171 Ferrous y = 0.0001491\*x + 0.28255 0.000284 1.548 0.452 2.096 Nitrate y = 0.0039588\*x + 3.0963 0.048631 0.94004 1.06 0.88008 pH y = 5.9129e-05\*x + 7.183 9.95E-05 1.0928 0.9072 1.1856 Calcium y = 0.004486\*x + 41.267 0.000172 1.3135 0.68655 1.6269 Magnesium y = 0.030176\*x + 34.0583 0.00488 1.3516 0.64839 1.7032 Fluoride y = 0.0003996\*x + 0.32624 0.030805 1.0191 0.98091 1.0382 Alkalinity y = 0.20175\*x + 93.7774 0.13638 0.93687 1.0631 0.87375 Hardness y = 0.26534\*x + 108.3566 0.090894 1.3517 0.64832 1.7034 Sulfate y = 0.037752\*x + 10.565 0.06587 1.0279 0.97206 1.0559

Chloride y = 1.1415\*x + 2.7551 0.003714 0.76375 1.2362 0.5275 Ferrous y = 0.18644\*x + 0.21372 0.002302 1.6654 0.33457 2.3309 Nitrate y = 2.0745\*x + 2.0841 0.069242 1.0143 0.98566 1.0287 pH y = 0.20299\*x + 7.1318 0.006077 1.4292 0.5708 1.8584 Calcium y = 21.1354\*x + 34.7963 0.019771 1.2721 0.72787 1.5443 Magnesium y = 42.446\*x + 18.7946 0.050062 1.3075 0.6925 1.615 TDS y = 77.0811\*x + 129.021 0.030805 0.74411 1.2559 0.48822 Alkalinity y = 63.845\*x + 97.3061 0.070809 0.7059 1.2941 0.41179 Hardness y = 43.9117\*x + 124.3496 0.012907 1.0748 0.92515 1.1497 Sulfate y = 20.7924\*x + 8.7184 0.1036 0.89597 1.104 0.79193

**(Fractal)**

**PI**


*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality DOI: http://dx.doi.org/10.5772/intechopen.92865*

**Y Parameters-**

*Fractal Analysis - Selected Examples*

**28**

**X**

**Regression equation r<sup>2</sup> H D**

Chloride y = 0.1701\*x + 3.0332 0.001245 0.3567 1.6433 0.2866 Nitrate y = 0.21107\*x + 2.6158 0.010822 0.52792 1.4721 0.055837 pH y = 0.018421\*x + 7.1844 0.000756 0.80777 1.1922 0.61553 Calcium y = 4.7305\*x + 39.5253 0.014954 0.66518 1.3348 0.33036 Magnesium y = 1.1981\*x + 30.5048 0.000602 0.63682 1.3632 0.27364 Fluoride y = 0.012348\*x + 0.28012 0.002302 0.50357 1.4964 0.007137 TDS y = 1.9056\*x + 107.6835 0.000284 0.34176 1.6582 0.31647 Alkalinity y = 4.7751\*x + 114.1275 0.005981 0.32296 1.677 0.35407 Hardness y = 5.5291\*x + 135.3207 0.00309 0.50895 1.491 0.017901 Sulfate y = 2.2747\*x + 14.0048 0.018721 0.42956 1.5704 0.14088

Ferrous Turbidity y = 0.2639\*x + 1.5505 0.032746 0.40358 1.5964 0.19285

Nitrate Turbidity y = 0.11474\*x + 1.3143 0.02548 0.68125 1.3188 0.3625

pH Turbidity y = 0.050188\*x + 1.2601 0.000532 0.37731 1.6227 0.24538

Calcium Turbidity y = 0.0051856\*x + 1.4094 0.018921 0.65827 1.3417 0.31654

Chloride y = 0.0029044\*x + 3.197 0.000543 0.57117 1.4288 0.14234 Ferrous y = 0.0031612\*x + 0.13763 0.014954 1.2934 0.70663 1.5867 Nitrate y = 0.015704\*x + 2.0315 0.089649 0.7484 1.2516 0.4968 pH y = 0.0014165\*x + 7.1316 0.006686 1.111 0.88902 1.222 Magnesium y = 0.69708\*x + 2.3928 0.30508 0.97651 1.0235 0.95303 Fluoride y = 0.00093543\*x + 0.24526 0.019771 0.74791 1.2521 0.49582

Chloride y = 0.88314\*x + 3.2706 0.015072 0.28645 1.7135 0.4271 Ferrous y = 0.041021\*x + 0.02835 0.000756 0.9331 1.0669 0.8662 Nitrate y = 0.44965\*x + 0.56059 0.022056 0.52027 1.4797 0.040543 Calcium y = 4.7201\*x + 6.8518 0.006686 0.66003 1.34 0.32006 Magnesium y = 4.6756\*x + 2.7902 0.004119 0.65269 1.3473 0.30538 Fluoride y = 0.029938\*x + 0.068174 0.006077 0.5019 1.5008 0.001615 TDS y = 1.682\*x + 95.0834 9.95E-05 0.2787 1.7213 0.4426 Alkalinity y = 3.1687\*x + 92.6195 0.001183 0.2732 1.7268 0.4536 Hardness y = 2.6385\*x + 117.8255 0.000316 0.46067 1.5393 0.078657 Sulfate y = 2.1465\*x + 0.82098 0.007486 0.39596 1.604 0.20807

Chloride y = 0.53189\*x + 4.4998 0.050117 0.662 1.338 0.324 Ferrous y = 0.051273\*x + 0.12956 0.010822 1.5921 0.40791 2.1842 pH y = 0.049051\*x + 7.0583 0.022056 1.3583 0.64173 1.7165 Calcium y = 5.7087\*x + 25.5324 0.089649 1.1608 0.83923 1.3215 Magnesium y = 7.2362\*x + 11.4886 0.090436 1.2002 0.7998 1.4004 Fluoride y = 0.033377\*x + 0.19422 0.069242 0.92495 1.0751 0.84989 TDS y = 12.2844\*x + 140.0001 0.048631 0.62589 1.3741 0.25179 Alkalinity y = 0.37421\*x + 114.4004 0.000151 0.58893 1.4111 0.17786 Hardness y = 3.2928\*x + 127.996 0.004511 0.91075 1.0892 0.82151 Sulfate y = 0.41289\*x + 15.7144 0.002539 0.80191 1.1981 0.60382

**(Fractal)**

**PI**


suggests that the sample data points are close together. The positive skewness (1.747) of data points reveals that the curve is not symmetrical, and the kurtosis value 3.13 shows that the sample datasets are platykurtic. Turbidity has persistent behavior with chloride, nitrate, fluoride, TDS, alkalinity, and sulfate and antipersistent behavior with ferrous, PH, Ca, Mg, and hardness parameters.

*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality*

*DOI: http://dx.doi.org/10.5772/intechopen.92865*

Mean and mode values are in the order of 0.5, and thus the data show normal behavior even though the median is 0. High standard deviation (5.009) is observed between sample points. Skewness value (1.52) suggests that the curve is not symmetrical, and the kurtosis value (1.3) is less than 3. Chloride has the Brownian time series (true random walk) behavior with Fl, sulfate, and turbidity parameters. Thus, the curve is platykurtic. Chloride has persistent behavior with turbidity, nitrate, TDS, and alkalinity and anti-persistent behavior with Fe, PH, Ca, Mg, and hardness

Average, median, and mode values are approximately equal, and thus the data show normal behavior. Standard deviation value (1.031) exhibits that the sample points are close to each other. Skewness value (11.713) suggests that the curve is not symmetrical, and kurtosis value is very large; thus, the curve is not platykurtic. The sample dataset containing heavier outliers and Fe has Brownian time series (True random walk) behavior with nitrate, fluoride, and hardness parameters. It has persistent behavior with pH, Ca, and Mg and anti-persistent behavior with chloride,

Mean and median values and standard deviation are approximately equal; thus data exhibit normal behavior. This suggests that the sample data are close to each other. The skewness value (1.011) and kurtosis are less than 3; hence the curve is not symmetrical and platykurtic. Nitrate has persistent behavior with turbidity, chloride, fluoride, TDS, alkalinity, hardness, and sulfate and anti-persistent behav-

Average and median are almost same, i.e., 7.189 and 7.20, respectively, whereas the mode of pH is 0.25. These values are approximately equal and hence exhibit the normal behavior. Standard deviation (SD) is 0.691, and skewness is close to 0, and all values are also close to each other; thus pH is symmetrical. The curve is not platykurtic, as kurtosis is very large 74.92. It shows the Brownian time series behavior with fluoride (Fl) parameter; persistent behavior with Ca, Mg, Fe, and nitrate; and anti-persistent performance with different parameters, i.e., turbidity,

Mean, median, and mode values are not close to each other; thus the curve does not show normal behavior. High standard deviation (40) indicates that the Ca values are very much distributed from each other. It is positively skewed, and the

**4.2 Chloride**

parameters.

**4.4 Nitrate**

*4.4.1 pH*

*4.4.2 Calcium*

**31**

**4.3 Ferrous (Fe)**

TDS, alkalinity, and sulfate parameters.

ior with Ca, Mg, Fe, and PH parameters.

chloride, TDS, alkalinity, hardness, and sulfate.

#### **Table 2.**

*Regression equation, coefficient of correlation, Hurst exponent, fractal dimension, and probability index between water parameters at Tehri District.*

of the statistical and fractal analysis is shown in **Table 2**, and each WQI analysis is discussed subsequently.

#### **4.1 Turbidity**

The turbidity sample datasets exhibit normal behavior as the mean, median, and mode values are approximately equal. Standard deviation is found to be 1.5 and

suggests that the sample data points are close together. The positive skewness (1.747) of data points reveals that the curve is not symmetrical, and the kurtosis value 3.13 shows that the sample datasets are platykurtic. Turbidity has persistent behavior with chloride, nitrate, fluoride, TDS, alkalinity, and sulfate and antipersistent behavior with ferrous, PH, Ca, Mg, and hardness parameters.
