Time Series Analysis of Water Quality of Ramgarh Lake of Rajasthan

Published in:Rajesh Singh, F. Smarandache (Editors) STUDIES IN SAMPLING TECHNIQUES AND TIME SERIES ANALYSIS Zip Publishing, Columbus, USA, 2011ISBN 978-1-59973-159-9pp. 5 - 23

Rajesh Singh, Shweta Maurya Department of Statistics, Banaras Hindu University

Varanasi-221005, INDIA

Ashish K. Singh Raj Kumar Goel Institute of Technology, Ghaziabad, India.

Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup, USA

5

Abstract

In this chapter an attempt has been made to study the effect of certain pollutants

on water of Ramgarh Lake of Rajasthan. Time series analysis of the observed data has

been done using trend, single exponential smoothing and double exponential smoothing

methods. Keywords: Pollutants, trend, single, double exponential smoothing, time series.

1. IntroductionSeventy percent of the earth’s surface is covered by water. Water is undoubtedly

the most precious natural resource that exists on our planet. Water is an important

component of the eco-system and any imbalance created either in terms of amount which

it is represent or impurities added to it, Can harm the whole eco-system. Water pollution

occurs when a body of water is adversely affected due to the addition of large amount of

pollutant materials to the water. When it is unfit for its intended use, water is considered

polluted. There are various sources of water pollution (for detail refer to Jain (2011))

Some of the important water quality factors are:

1) Dissolved oxygen (D.O.)

2) Biological oxygen demand (B.O.D.)

3) Nitrate

6

4) Coliform

5) P.H.

Chemical analysis of any sample of water gives us a complete picture of its

physical and chemical constituents. This will give us only certain numerical value but

for estimating exact quality of water a time series system has been developed known

as water quality trend, which gives us the idea of whole system for a long time (see

Jain(2011)).

In this chapter we are calculating the trend values for five water parameters of

Ramgarh lake in Rajasthan for the year 1995-2006 and for three parameters of Mahi river

for the year 1997-2008.these methods viz. trend analysis, single smoothing are used to

analyze the data.

2. Methodology After ensuring the presence of trend in the data, smoothing of the data is the next

requirement for time series analysis. For smoothing the common techniques discussed by

Gardner(1985) are trend, simple exponential smoothing (SES), double exponential

smoothing (DES), triple exponential smoothing (TES) and adaptive response rate simple

exponential smoothing (ARRSES). Jain (2011) used trend method to analyze the data.

We have extended the work of Jain (2011) and analyzed the data using SES and DES and

compared these with the help of the available information. The methods are described

below:

1. Fitting Of Straight line

The equation of the straight line is-

Ut=a+bt

where,

ut=observed value of the data, a=intercept value, b=slope of the straight line and

t=time (in years)

Calculation for a and b:

7

The normal equations for calculating a and b are

∑Ut=n a+b∑t

∑t Ut=at+b∑t**2

2. Single Exponential Smoothing

The basic equation of exponential smoothing is

St = α yt-1+ (1-α) St-1 , 0 ≤ α ≤1

and parameter α is called the smoothing constant.

Here, Si stands for smoothed observation or EWNA and y stands for the original

observation. The subscripts refer to the time periods 1,2,3…..,n.

The smoothed series starts with the smoothed version of the second observation.

3. Double Exponential Smoothing

Single smoothing does not excel in following the data when there is a trend. This

situation can be improved by the introduction of a second equation with a second

constant γ, which must be chosen in conjunction with α.

St=α yt+ (1-α) (St-1+bt-1) , 0≤α≤1

bt= γ (St-St-1) + (1- γ ) bt-1, 0≤ γ ≤1.

For forecasting using single and double exponential smoothing following method is used-

Forecasting with single exponential smoothing

St=αyt-1+(1-α)St-1 0≤ α ≤1

The new forecast is the old one plus an adjustment for the error that occurred in the last

forecast.

8

Boot strapping of forecasts

St+1=α yorigin +(1-α) St

This formula works when last data points and no actual observation are available.

Forecasting with double exponential smoothing

The one period-ahead forecast is given by:

Ft+1=St+bt

The m-periods-ahead forecast is given by:

Ft+m=St+mbt

( for detail of these methods refer to Gardner (1985)).

4. Results and Discussion

Table 1 shows the data on Dissolved Oxygen (D.O.) for Ramgardh Lake for the

years 1995-2006 and fitted values using trend, single and double exponential smoothing.

Table 1: Data on Dissolved Oxygen (D.O.) for Ramgardh Lake for the years 1995-2006 and fitted values using trend, single and double exponential smoothing

Observed Data

Trend values

Single exponential smoothing(alpha=0.1)

Double exponential smoothing(alpha=0.9

and gamma=0.1) 1995 5.12 5.3372 5.12 1996 5.75 5.3036 5.12 5.707 1997 5.26 5.27 5.183 5.32857 1998 5.72 5.2364 5.1907 5.698556 1999 4.64 5.2028 5.24363 4.765484 2000 5.14 5.1692 5.183267 5.110884 2003 4.23 5.1356 5.17894 4.329044 2004 6.026 5.102 5.084046 5.858346 2005 4.46 5.0684 5.178242 4.616965 2006 5.52 5.0348 5.106417 5.4327 Total 51.866 51.86 46.46824 51.96755

Ut = 5

with

of α a

the d

comb

tried

Figur

neces

adequ

levels

www

above

Tim

ese

riesv

alue

s

For trend,

5.1866-0.01

mean square

are tried and

ata, Holt’s d

binations of

and MSE =

re 1- Gr

expon

the ye

Adequate

ssary elemen

uate oxygen

s in water d

w.state.ky.us)

Form Tab

e the require

0

1

2

3

4

5

6

7

1995 1

Tim

e se

ries v

alue

s

, the fitted li

69 * t

ed error (MS

d minimum

double expon

α and γ bot

0.0094 was

raph of obse

nential smoo

ears 1995-20

e dissolved

nt to all fo

n levels in or

drop below

).

ble 1 and Fi

ed standard 5

1996 1997 199

ne is

SE) 0.3077.

MSE = 0.39

nential smoo

th ranging b

least for α =

erved data an

othing of Di

006

oxygen is n

orms of life

rder to prov

5.0 mg/l, a

igure 1, we o

5.0 mg/l, exc

98 1999 2000

Year

Disso

9

For single e

915 was obt

othing was fo

etween 0.1

= 0.9 and γ =

nd fitted val

issolved Ox

necessary fo

. Natural st

vide for aero

aquatic life

observe that

cept for the t

0 2003 2004 2

olved oxy

exponential s

tained for α

found to be m

and 0.9 with

= 0.1.

lues using t

xygen (D.O.)

or good wat

tream purifi

obic life form

is put unde

t level of D.

two years 19

2005 2006

ygen

smoothing v

= 0.1. For

most appropr

h increment

trend, single

) for Ramga

ter quality.

ication proc

ms. As disso

r stress ( fo

O. in Ramga

999 and 2005

Observed

Trend va

single exsmoothin

Double esmoothingamma=

various value

smoothing o

riate. Variou

s of 0.1 wer

e and doubl

ardh Lake fo

Oxygen is

esses requir

olved oxyge

or details se

arh Lake wa

5.

d Data

alues

xponential ng(alpha=0.1)

exponential ng(alpha=0.9 &

=0.1)

es

of

us

re

le

or

a

re

en

ee

as

&

Ta

Figur

Tim

ese

riesv

alue

s

able 2: Com

Per

23456789

11111111112

re 2: Graph o

0

2

4

6

8

1 2

Tim

e se

ries v

alue

s

mparison of f

riod Obs

D1 52 53 54 55 46 57 48 6.9 4

10 511 12 13 14 15 16 17 18 19 20

of forecasts

3 4 5

Co

Observed D

forecasts

served Data Fo5.12 5.75 5.26 5.72 4.64 5.14 4.23 .026

4.46 5.52

6 7 8

ompariso

Data Fo

10

orecast(sing

5.7437 5.25923

5.7147074.6460363

5.140432674.239489406.016580464.467182415.515864175.147775735.184998155.218498345.248648505.275783655.300205295.322184765.341966285.359769655.37579269

9 10 11

Period

on of fo

orecast(single)

gle) Foreca

65.7

6.513 5.017 5.81

03 4.363 7.51

6 4.6275 6.6831 558 542 508 557 91 662 686 657 692

12 13 14 1

recasts

) Fore

ast(double) 5.32

.23084 7869651

10832048 13406419 15207177 2655631

11558059 21632535 86376834 5.6266 5.7442 5.8618 5.9794 6.097

6.2146 6.3322 6.4498 6.5674 6.685

15 16 17 18

cast(double)

8 19 20

11

Table 3 shows the data on Nitrate for Ramgardh Lake for the years 1995-2006

and fitted values using trend, single and double exponential smoothing.

Table 3: Data on Nitrate for Ramgardh Lake for the years 1995-2006 and fitted values using trend, single and double exponential smoothing

Year(x) Observed Data

Trend values

Single exponential smoothing(alpha=0.1)

Double exponential smoothing(alpha=0.9 and gamma=0.1)

1995 0.32 0.588 0.32 1996 1.28 0.546 0.32 1.176 1997 0.38 0.504 0.416 0.46096 1998 0.08 0.462 0.4124 0.104883 1999 0.04 0.42 0.37916 0.028797 2000 0.92 0.378 0.345244 0.815205 2003 0.24 0.336 0.40272 0.300708 2004 0.25 0.294 0.386448 0.247331 2005 0.186 0.252 0.372803 0.184874 2006 0.3 0.21 0.354123 0.281431 Total 3.996 3.99 3.388897 3.920189

For trend, the fitted line is

Ut = 0.3996 - 0.0216 * t

with MSE= 0.3078. For single exponential smoothing various values of α are tried and

minimum MSE = 0.3412 was obtained for α = 0.1. For smoothing of the data, Holt’s

double exponential smoothing was found to be most appropriate. Various combinations

of α and γ both ranging between 0.1 and 0.9 with increments of 0.1 were tried and MSE =

0.0033 was least for α = 0.9 and γ = 0.1.

Nitrites can produce a serious condition in fish called "brown blood disease."

Nitrites also react directly with hemoglobin in human blood and other warm-blooded

animals to produce methemoglobin. Methemoglobin destroys the ability of red blood

cells to transport oxygen. This condition is especially serious in babies under three

months of age. It causes a condition known as methemoglobinemia or "blue baby"

disease. Water with nitrite levels exceeding 1.0 mg/l should not be used for feeding

babie

have

was b

Figur

Nitrit

anim

cells

mont

disea

babie

have

was b

time

serie

sval

ues

es. Nitrite/ni

no effect on

Form Ta

below the sta

re 3: Graphexpon

Nitrites c

tes also reac

als to produ

to transpor

ths of age.

ase. Water w

es. Nitrite/ni

no effect on

Form Ta

below the sta

00.20.40.60.8

11.21.4

1995

time

serie

s val

ues

itrogen level

n warm wate

able 3 and F

andard 1.0 m

h of observnential smoo

can produce

ct directly w

uce methem

rt oxygen. T

It causes a

with nitrite

itrogen level

n warm wate

able 3 and F

andard 1.0 m

1996 1997 19

ls below 90

r fish (for de

Figure 3, we

mg/l, except

ved data andothing of Ni

a serious c

with hemogl

moglobin. M

This conditi

a condition

levels excee

ls below 90

r fish ( for d

Figure 3, we

mg/l, except

998 1999 2000

year

12

mg/l and n

etails see ww

e observe th

for the year

d fitted valitrate for Ram

condition in

lobin in hum

ethemoglobi

ion is espec

known as m

eding 1.0 m

mg/l and n

details see ww

e observe tha

for the year

0 2003 2004 2

Nitrate

nitrate levels

ww.state.ky.u

hat level of N

1996.

ues using tmgardh lake

n fish called

man blood a

in destroys

cially seriou

methemoglo

mg/l should

nitrate levels

ww.state.ky

at level of N

1996.

2005 2006

below 0.5

us).

Nitrate in R

trend, singlee for the year

d "brown blo

and other w

the ability

us in babies

obinemia or

not be used

below 0.5

.us).

Nitrate in R

Observe

Trend va

single exsmoothi

Double esmoothigamma=

mg/l seem t

Ramgarh Lak

e and doublrs 1995-2006

ood disease

warm-bloode

of red bloo

s under thre

"blue baby

d for feedin

mg/l seem t

Ramgarh Lak

ed Data

alues

xponential ing(alpha=0.1)

exponential ing(alpha=0.9 &=0.1)

to

ke

le 6

."

ed

od

ee

y"

ng

to

ke

&

T

Figur

Tim

ese

riesv

alue

s

Table 4: C

P

re 4 : Graph

-0.5

0

0.5

1

1.5

1Tim

e se

ries v

alue

s

Comparison o

eriod Obs

D1 02 13 04 05 06 07 08 09 0.

10 011 12 13 14 15 16 17 18 19 20

h of forecast

2 3 4

C

Observed

of forecasts

served Data For0.32 1.28 0.38 0.08 0.04 0.92 0.24 0.25 .186 0.3

ts

5 6 7

Comparis

Data

13

recast(singl0.416

0.4124 0.37916

0.345244 0.40272

0.386448 0.372803 0.354123 0.34871

0.348711 0.34384

0.339456 0.33551

0.331959 0.328763 0.325887 0.323298 0.320968 0.318872 0.316984

8 9 10 1perio

son of fo

Forecast(singl

e) Forecas0

1.10.32-0.0-0.10.840.220.1

0.110.240.20.20.10.10.0

0.00.0-0.0-0.0

-0

11 12 13 14d

orecasts

le) For

st(double) 0.24 1896 28832 07203 12795 47085 23314 7474

14309 44291

24426 20712

6998 3284

0957 05856 02142 01572 05286 0.09

4 15 16 17

s

recast(double)

7 18 19 20

)

14

Table 5 shows the data on B.O.D. for Ramgardh Lake for the years 1995-2006

and fitted values using trend, single and double exponential smoothing.

Table 5: Data on Biological oxygen demand (B.O.D.) for Ramgardh Lake for the years

1995-2006 and fitted values using trend, single and double exponential smoothing

Year(x) Observed

Data Trend values

single exponential smoothing(alpha=0.1)

Double exponential smoothing(alpha=0.9

& gamma=0.1) 1995 1.96 5.122 1.96 1996 14.09 4.762 1.96 12.87167 1997 1.84 4.402 3.173 3.047483 1998 1.8 4.042 3.0397 1.920392 1999 1.46 3.682 2.91573 1.490847 2000 2.78 3.322 2.770157 2.633116 2003 2.76 2.962 2.771141 2.742563 2004 1.976 2.602 2.770027 2.049477 2005 2.78 2.242 2.690624 2.697155 2006 3.58 1.882 2.699562 3.489379 Total 35.026 35.02 24.78994 34.90208

Biochemical oxygen demand is a measure of the quantity of oxygen used by

microorganisms (e.g., aerobic bacteria) in the oxidation of organic matter. Natural

sources of organic matter include plant decay and leaf fall. However, plant growth and

decay may be unnaturally accelerated when nutrients and sunlight are overly abundant

due to human influence. Urban runoff carries pet wastes from streets and sidewalks;

nutrients from lawn fertilizers; leaves, grass clippings, and chapter from residential areas,

which increase oxygen demand. Oxygen consumed in the decomposition process robs

other aquatic organisms of the oxygen they need to live. Organisms that are more tolerant

of lower dissolved oxygen levels may replace a diversity of natural water systems contain

bacteria, which need oxygen (aerobic) to survive. Most of them feed on dead algae and

other dead organisms and are part of the decomposition cycle. Algae and other producers

in the water take up inorganic nutrients and use them in the process of building up their

organic tissues (for details refer to www.freedrinkingwater.com).

For tr

with

minim

doub

of α a

0.300

Figur

Tim

ese

riesv

alue

srend, the fitt

Ut = 3.502

MSE= 11.74

mum MSE =

le exponenti

and γ both ra

00 was least

re 5: Grapexpon

02468

10121416

1995

Tim

e se

ries v

alue

sted line is

26+0.18094

4366. For si

= 17.1093 w

ial smoothin

anging betw

for α = 0.

ph of observnential smoo

1996 1997 19

Bio

* t

ngle expone

was obtained

ng was foun

een 0.1 and

.9 and γ = 0.

ved data anothing of B.O

998 1999 2000

Year

ological

15

ential smooth

d for α = 0.1

nd to be mos

0.9 with inc

1.

nd fitted valO.D. for Ram

2003 2004 20

oxygen

hing various

1. For smoo

st appropriat

crements of 0

lues using tmgardh Lake

005 2006

demand

s values of α

othing of the

te. Various

0.1 were trie

trend, singlee for the yea

d Obse

Trend

singlesmoo.1)Doubsmoo.9 & g

α are tried an

e data, Holt’

combination

ed and MSE

e and doublars 1995-200

erved Data

d values

e exponential othing(alpha=0

ble exponentiaothing(alpha=0gamma=0.1)

nd

’s

ns

=

le 06

0

l 0

Tabl

Figur

Tim

ese

ries

valu

es

le 6: Com

P

re 6 : Graph

0

5

10

15

1 2

Tim

e se

ries

valu

es

mparison of

eriod Obs

D1 12 143 14 5 16 27 28 1.9 2

10 311 12 13 14 15 16 17 18 19 20

h of forecast

2 3 4 5

Com

Observed Dat

forecasts

served Data For1.96 4.09

1.84 1.8

1.46 2.78 2.76 .976

2.78 3.58

ts

6 7 8

mpariso

ta For

16

recast(singl

3.173 3.0397

2.91573 2.770157 2.771141 2.770027 2.690624 2.699562 2.787606

2.7871 2.86639

2.937751 3.001976 3.059778

3.1118 3.15862

3.200758 3.238683 3.272814

9 10 11 12

Period

n of fore

ecast(single)

e) Forecas1.9013.93.001.761.312.582.711.952.673.54

3.53.63.63.73.73.83.

3.94.04.0

2 13 14 15

ecasts

Foreca

st(double) 06667 91483 03915 68471 11164 85629 10768 51553 73792 47575 5474 6055 6636 7217 7798 8379 .896 9541 0122 0703

16 17 18 1

ast(double)

19 20

17

Table 7 shows the data on Total Colifirm for Ramgardh Lake for the years

1995-2006 and fitted values using trend, single and double exponential smoothing.

Table 7 : Data on Total Colifirm for Ramgardh lake for the years 1995-2006 and fitted

values using trend, single and double exponential smoothing

Year(x) Observed

Data Trend values

single exponential smoothing(alpha=0.6)

Double exponential smoothing(alpha=0.9

& gamma=0.1) 1995 1169 687.894 1169 1996 285 615.314 1169 339.2333 1997 840.75 542.734 638.6 751.5507 1998 144 470.154 759.89 173.7353 1999 65.33 397.574 390.356 42.47463 2000 121.5 324.994 195.3404 81.95854 2003 119 252.414 151.0362 87.21566 2004 720.6 179.834 131.8145 632.042 2005 86 107.254 485.0858 123.3548 2006 61.66 34.674 245.6343 47.21817 Total 3612.84 3612.84 4166.757 3447.783

Total coliform bacteria are a collection of relatively harmless microorganisms that

live in large numbers in the intestines of man and warm- and cold-blooded animals. They

aid in the digestion of food. A specific subgroup of this collection is the fecal coliform

bacteria, the most common member being Escherichia coli. These organisms may be

separated from the total coliform group by their ability to grow at elevated temperatures

and are associated only with the fecal material of warm-blooded animals. The presence of

fecal coliform bacteria in aquatic environments indicates that the water has been

contaminated with the fecal material of man or other animals. The presence of fecal

contamination is an indicator that a potential health risk exists for individuals exposed to

this water (for details see www.state.ky.us).

with

minim

doub

of α

MSE

Figur

For trend,

Ut = 361.2

MSE= 9989

mum MSE =

le exponenti

and γ both

E = 2432.458

re 7: Grapexpon1995-

0

200

400

600

800

1000

1200

1400

19

Tim

e se

ries v

alue

s

, the fitted li

284 - 36.2989

96.33. For si

= 205949.6 w

ial smoothin

h ranging be

8 was least fo

ph of observnential smoo-2006.

995199619971

ne is

9* t

ngle expone

was obtaine

ng was foun

etween 0.1 a

or α = 0.9 an

ved data anothing of T

199819992000

Year

Tota

18

ential smooth

d for α = 0.6

nd to be mos

and 0.9 wit

nd γ = 0.1.

nd fitted valTotal Colifor

020032004200

al colifor

hing various

6. For smoo

st appropriat

th increment

lues using trm for Ramg

052006

rm

s values of α

othing of the

te. Various

ts of 0.1 w

trend, singlegardh Lake

Observed Da

Trend values

single exponsmoothing(a

Double exposmoothing(agamma=0.1)

α are tried an

e data, Holt’

combination

ere tried an

e and doublfor the year

ata

s

nential alpha=0.6)

onential alpha=0.9 & )

nd

’s

ns

nd

le rs

19

Table 8: Comparison of forecasts

Period Observed

Data Forecast(single) Forecast(double) 1 1169 827.3333 2 285 638.6 -51.2433 3 840.75 759.89 441.3534 4 144 390.356 -163.224 5 65.33 195.3404 -273.915 6 121.5 151.0362 -198.843 7 119 131.8145 -164.98 8 720.6 485.0858 459.5482 9 86 245.6343 -82.7583 10 61.66 135.2497 -145.897 11 135.248 -145.897 12 91.0952 -339.012 13 73.43408 -532.127 14 66.36963 -725.242 15 63.54385 -918.357 16 62.41354 -1111.47 17 61.96142 -1304.59 18 61.78057 -1497.7 19 61.70823 -1690.82 20 61.67929 -1883.93

Figure 8: Comparison of forecasts

Tablevalue Tabl

Y

conce

range

e 9 shows thes using tren

e 9: Data ontrend, s

Year(x) Ob

1995 1996 1997 1998 1999 2000 2003 2004 2005 2006 Total 7

pH is a

entration of

e of 6.0 to 9.

-3000

-2000

-1000

0

1000

2000

Tim

e se

ries v

alue

s

he data on pHd, single and

n pH for Ramsingle and do

bserved Data

Tv

7.64 7.48 7.87 8.05 8.44 8.3

7.25 8.28 7.87

8.006 79.186

measure of

the hydroge

.0 appears to

1 2 3 4

Co

Obser

H for Ramgad double exp

mgardh Lakouble expon

Trend values

sismo

6.29 6.65 7.01 7.37 7.73 8.09 8.45 8.81 9.17 9.53 79.1

f the acidic

en ion [H+]

o provide pro

5 6 7

omparis

rved Data

20

ardh Lake forponential sm

ke for the yeanential smoot

ingle exponoothing(alp

7.64 7.608

7.66047.73832

7.8786567.9629257.82034

7.9122727.90381770.12473

or basic (

activity in

otection for t

8 9 10 11

Period

son of fo

Forecast(s

r the years 1moothing.

ars 1995-200thing

ential ha=0.2)

Dsm

2 6 5

4 2 7 3

alkaline) na

a solution d

the life of fr

1 12 13 14

d

orecasts

single)

995-2006 an

06 and fitted

Double expomoothing(al

& gamma7.64

7.50967.84498.042748.414178.327647.371508.19197.912928.003579.259

ature of a s

determines th

reshwater fis

15 16 17 1

Forecast(dou

nd fitted

d values usin

onential lpha=0.9 a=0.1)

67 63 46 77 45 03 54 23 57 14

solution. Th

he pH. A pH

sh and bottom

18 19 20

uble)

ng

he

H

m

dwell

syner

produ

Runo

amm

effect

effect

die (f

with

minim

doub

of α a

0.002

Figur

Tim

ese

riesv

alue

s

ling inverte

rgistic effect

uce effects

off from agr

onia, mercu

ts, if any, of

t at a pH of

for details se

For trend,

Ut = 7.918

MSE = 0.99

mum MSE =

le exponenti

and γ both ra

2735was leas

re 9: Grapexpon

0

2

4

6

8

10

12

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We observe from the calculations for the different parameters of pollutants that

double smoothing follows the data much closer than single smoothing. Furthermore, for

forecasting single smoothing cannot do better than projecting a straight horizontal line,

which is not very likely to occur in reality. So for forecasting purposes for our data

double exponential smoothing is more preferable.

Conclusion

From the above discussions, we observe that the various pollutants considered in

the article may have very hazardous effect on quality of water. Increase of pollutants in

water beyond a certain limit may be dangerous for aquatic animals. Also, according to

recent reports, most of the tap and well water in the India are not safe for drinking due to

presence of various pollutants in inappropriate percentage. Now, we have reached the

point where all sources of our drinking water, including municipal water systems, wells,

lakes, rivers, and even glaciers, contain some level of contamination. So, we need to

keep a routine check of the quality of water so that we can lead a healthy life.

References

Gardner, E. S. (1985) : Exponential smoothing- The state of the art. Jour. Of

Forecasting,4, 1-28.

Jain, Smita (2011): A Statistical study of effect of different pollutants on water (with

specific reference to Rajasthan). Unpublished thesis submitted to University of

Rajasthan, India.

www.state.ky.us

www.freedrinkingwater.com