session: modeling, simulation and optimization

8/14/2019 Session: Modeling, Simulation and Optimization

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Session: Modeling, Simulation and Optimization

Applications and algorithms of non-

linear regression using least squares

Approximation based on Case Studies

S. Vignesh, T.K. Premannanth

Department of Chemical Engineering,

St. Josephs College of Engineering,

Chennai 600119.

By


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Abstract

Three data sets from literature were taken to investigate the importance of method of

least squares in approximation methods in engineering. The case studies such as

heat capacity data to quadratic equation in temperature, vapour liquid equilibrium

data to Wilson equation and fitting Gilliland Sherwood data were taken Multi Non

Linear Regression (MNLR) was successfully obtained using least square

approximation. The regression model obtained was subjected to Distributed Errors

Which was characterized by decrease of some global error measure with respect to

the whole approximation interval as the order of approximation increases. The best

Multiple Non Linear Regression model was evaluated with the small value of error.

Graphs to evaluate Goodness of Fit were drawn for the three data sets which werefound to be good.


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Introduction

The so-called method of least squares is a very important approximationmethod in engineering.

This method is perhaps the best known distributed error approximationmethod. Least square approximation is valuable in problems such asfitting equations to discrete data points and in analyzing measurementerrors.

The subject of least square analyses also plays central role in theapplication of the theory of statistics, which treats problems involvingrandom.

The subject of random variables and statistics is beyond the scope of ourpresentation, we will therefore use least squares in our casestudies.

Least squares are also useful for continuous approximations, such asdeveloping simple approximation to known functions.

In least squares, distributed error methods are characterized by adecrease of some global error measure with respect to the wholeapproximation interval as the order of approximation increases.


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Linear Regression Linear regression is solved by the method of least squares and the

error percentage was found out and Goodness of Fit graph wasplotted. The Least Square equations are as follows.

General equation:

y = a0 + a1x (2.1) where,

a0 = (yi / n) - a1 (xi/n) (2.2)

a1 = [xi yi (xi yi)/ n] / xi^2 (xi )^2 / n (2.3)

Where a0 and a1 are constants that are determined.


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Polynomial Regression

Non Linear Regression is solved by polynomial Regressionmethod (Second order). The Polynomial Regression equations areas follows:

General equation:

y = a0 + a1x + a2x^2 + ..+ anx^n (2.4)

(2.5)

On solving the above matrix, we can get the values of a0, a1 & a2.


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Case Studies

Three case studies has been

analyzed, they are:

Heat capacity data to

quadratic equation intemperature.

Vapour liquid equilibrium datato Wilson equation.

Fitting Gilliland-Sherwood

data.


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Case I

Heat Capacity Data to theQuadratic Equation

Here we have analyzed the heat capacity data for liquidmethylcyclohexane (C7H14) using the equation:-

Cp = a0 + a1T

Where Cp is the Heat capacity, T is the absolutetemperature and a0 and a1 are parameters which we havefound out by Linear Least Squares.


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T, K CP, KJ/KgK

150 1. 426

160 1. 447

170 1. 469

180 1. 492

190 1. 516

200 1. 541

210 1.567

220 1. 596

230 1. 627

240 1. 661

250 1. 696

260 1. 732

270 1. 770

280 1. 808

290 1. 848

300 1. 888

Data: TABLE I (a)


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On solving these datas by Least Square method, we got thevalues of a0 and a1 and it was found to be0.96 and 0.00297respectively.

On substituting the values of a0 and a1 in the equation 2.1, we gotthe predicted value, which seems to benear when compared to

the experimental values.

y = 0.96 + 0.00297x

On substituting the temperature values on the above equation, weget the predicted values


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PredictedValues Experimental values

1.405 1.426

1.434 1.447

1.457 1.469

1.489 1.492

1.509 1.516

1.538 1.541

1.558 1.567

1.587 1.596

1.611 1.627

1.658 1.661

1.684 1.696

1.725 1.732

1.768 1.770

1.795 1.808

1.821 1.848

1.867 1.888

TABLE I (b)


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Goodness of fit

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

EXP

1

EXP

2

EXP

3

EXP

4

EXP

5

EXP

6

EXP

7

EXP

8

EXP

9

EXP

10

EXP

11

EXP

12

EXP

13

EXP

14

EXP

15

EXPERIMENT

ALPREDICTED

The above graph shows the Goodness of Fit for the Heat Capacity Dataof Methylcyclohexane. The experimental andpredicted values showedabove shows the minimum percentage of error. The error percentagewould approximately lies between 2-5%.

Fig I (c)


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Case II

Vapour Liquid Equilibrium toWilson Equation

Vapour Liquid equilibrium data were taken from Heptane-Toluene binary system at 1 atm pressure.

Here we fitted activity coefficient data to Wilson Equation As it requiredNon-Linear regression, we have used

polynomial method and we havegot the predicted valueswhich are nearer to the experimental.

Here the equation we use is:

y = a0 + a1x + a2x^2 + ..+ anx^n


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Data:

xi yi1.000 0.0000

0.790 0.1259

0.596 0.1509

0.480 0.1392

0.390 0.1250

0.293 0.1111

0.220 0.0950

0.150 0.0707

0.065 0.0290

0.000 0.0000

TABLE II (a)


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On solving these data's by Polynomial method, we got the valuesof a0, a1 and a2 and it was found to be -0.00425, 0.575, -0.5568respectively.

On substituting these values on the equation 2.4, we got thepredicted values which were very nearer to the experimental

values.

y = -0.00425 + 0.575x -0.5568x^2

On substituting the xi values, we got the predicted values which

are tabulated as follows.


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Predicted Experimental

0.0012 0.000

0.1154 0.1259

0.1495 0.1509

0.1435 0.1392

0.1343 0.1250

0.1102 0.1111

0.0925 0.0950

0.0695 0.0707

0.0278 0.0290

0.0000 0.0000

TABLE II (b)


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Goodnessof fit

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

EXP 1EXP 2 EXP 3 EXP 4 EXP 5 EXP 6 EXP 7 EXP 8EXP 9

EXPERIMENTAL

PREDICTED

The above graph shows the Goodness of Fit for the Vapour LiquidEquilibrium data. The experimental andpredicted lines showed aboveshows the minimum percentage of error. The error percentage wasapproximately lies between 6 7%.

Fig II (c)


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Case III

Mass Transfer Data of Gilliland-Sherwood Equation

The Mass Transfer Data's were taken and analyzed by Gilliland-

Sherwood Equation As it required Non-Linear regression, we used the polynomial method

and we have got the predicted values which are nearer to theexperimental values.

Here the equation we use is:

y = a0 + a1x + a2x^2 + ..+ anx^n


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Data:

xi yi

43.7 0.60

24.2 1.80

51.6 1.87

32.3 1.86

26.1 2.16

92.8 2.17

TABLE III (a)


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On solving these datas by polynomial method, we got the valuesof a0, a1 and a2 and it was found to be 16.11, -0.7588 and 0.0053respectively.

On substituting these values on the equation 2.4, we got thepredicted values which were very nearer to the experimental

values.

y = 16.11 0.7588x + 0.0053x^2

On substituting the xi values, we got the predicted values whichare tabulated below.


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PREDICTED EXPERIMENTAL

0.48 0.60

1.62 1.80

1.69 1.87

1.70 1.86

1.92 2.16

1.95 2.17

TABLE III (b)


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Goodness of fit

0

0.5

1

1.5

2

2.5

EXP 1 EXP 2 EXP 3 EXP 4 EXP 5 EXP 6

EXPERIMENT

ALPREDICTED

The above graph shows the Goodness of Fit for the Mass TransferData from Gilliland-Sherwood equation. The experimental andpredicted lines showed above shows the minimum percentage oferror. The error percentage was approximately lies between 20 25%. The error was little high since the data was Non-Linear.

Fig III (c)


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Results and Discussion

The three case studies analyzed using least squares andpolynomial regression yield good results.

The goodness of fit graphs was plotted and error % wascalculated for all the three datas.

The error percentage was found to be approximately2to5% in the first case study, similarly 7to10% in thesecond and 20to25% in the third respectively.

The first two case studies error was obtained veryminimal and the best fit graph was plotted for theGilliland-Sherwood data it was little higher because thevalues are highly non-linear and it was difficult topolynomial apply regression to the result. Further studyhas to be made to minimize the error in the third casestudy.


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Discussion - Case I

In the first case study, the table I (a) shows the datataken for Performing regression.

Using those values and by manually calculating thenecessary terms, we have substituted those terms in the

equations 2.1, 2.2 and 2.3 and we have obtained thegeneral equation.

The table I (b) is the table containing the experimentaldata and predicted data. Using those values we haveplotted a graph I (c) which shows the goodness of fit.

From the graph we have concluded that the first casestudy has come out well with minimum error %. As wesee the graph we can see the two lines very closeindicating that the regression was successful with veryless error.


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Discussion - Case II

In the second case study, the table II (a) shows the data takenfor Performing regression.

Using those values and by manually calculating the necessaryterms, we have substituted those terms in the equations 2.4and 2.5 and we have obtained the general equation.

The table II (b) is the table containing the experimental dataand predicted data. Using those values we have plotted a graphII (c) which shows the goodness of fit.

From the graph we have concluded that the second case study

has come out well with minimum error %. As we see the graphwe can see the two lines very close indicating that theregression was successful with very less error.


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Discussion - Case III

In the third case study, the table III (a) shows the data taken forPerforming regression.

Using those values and by manually calculating the necessary terms, wehave substituted those terms in the equations 2.4 and 2.5 and we haveobtained the general equation.

The table III (b) is the table containing the experimental data and predicteddata. Using those values we have plotted a graph III (c) which shows thegoodness of fit.

From the graph we have concluded that the second case study has come outwell with minimum error %. As we see the graph we can see the two linesvery close indicating that the regression was successful with very less error.

The error % is high when compared to the first two cases because the datais non-linear.


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Algorithm

General algorithm for all the three cases: Step 1: The datas were taken from the case

studies. Step 2: Regression was applied to the data using

the formulas which are stated in the beginning. Step 3: The necessary values of Ao and A1 was

determined. Step 4: These values are substituted in the general

equations (2.1 for case study I and 2.4 forCase study II and III).

Step 5: Using that we have determined thepredicted values.

Step 6: A graph was drawn between theexperimental and the predicted data.

Step 7: The error percentage was calculated. Step 8: The graph plotted is the goodness of fit.


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Conclusion

The three sets of data's were analyzed and good results havebeen obtained in all three case studies.

The first 2 case studies have come perfectly with minimum errorand the goodness of fit graph is plotted and was found to begood.

For the third case study the error was little high when comparedto the other two, this is because the data is too non-linear.

Goodness of fit graph was plotted for the third case study andhas come out well.

For further minimizing the error for the Gilliland-Sherwood data,further studies have to be made.

On the whole the results i.e., the regression model obtained byusing Least Squares and Polynomial regression were successfulin prediction and the representation of the system was good.


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References1. Berezin I. S., and Zhidkov N. P., (1965), Computing Methods, Addision-Wesely,

Menlo Park, CA2. PERRY, R. H., Green, D. W., and Maloney, J. O. (1984), Chemical Engineers Hand

book3. Modeling and Analysis of Chemical Engineering Processes, Balu.K,

Padmanabhan.K, I.K. International Pvt. Ltd.4. Optimization Theory and Practice, Mohan C Joshi, Kannan M Moudgalya, Narosa

Publishing House.5. Neural Networks - A Comprehensive Foundation, Simon Haykin, Pearsoneducation second edition, 2004.

6. Neural Networks, Ananda Rao. M, Srinivas.J, Narosa Publishing House.7. Design and analysis of Experiments, Montgomery D C, 5th ed., John Wiley & sons,

New York, 2007.8. Experiment optimization in chemistry and chemical engineering, Akhnazarova S,

Kafarov V, MIR publishers, Moscow, 1982.

9. Experimental methods for engineers, Holman, McGraw Hill Publications.


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Our sincere thanks

Organizing committee - J.N.T.U College of Engineering,Anantapur

Judges and coordinator's for their best support.

Head of the departmentChemical Engineering

St. Josephs College of Engineering


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