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Finding Regression Coefficient
To Predict Radiation
Contents:
• Finding of regression coefficient using
•
•

Advance Mathe...
Advance Mathematics for Planning

OBJECTIVE
*The objective of this analysis is to predict the radiation level at 6th locat...
Date

D-1

9/2/2013 1.318
9/3/2013
1.3
9/4/2013 1.324
…..
…..
11/8/2013 1.294
11/11/2013 1.32

D-14

D-15

D-20

D-24

2.1...
The regression equation is
*There is no constant term in the equation.
model_radiation=lm(D.1~D.14+D.15+D.20+D.24+D.26+0,d...
Advance Mathematics for Planning

Regression analysis using
Genetic algorithm
About genetic algorithm
1.
2.
3.

4.
5.

Sta...
Advance Mathematics for Planning

Regression analysis using
Genetic algorithm
Solution using genetic algorithm
Two fitness...
Solution using genetic algorithm
CASE-1: Maximizing R2 value
=> α=[ 0.4702 0.9999 0.0861 0.0232 0.0957] and R2=0.4688
CASE...
Final adopted values

Radiation at location D-1: Actual Vs Predicted

11/11/2013

11/9/2013

11/7/2013

11/5/2013

11/3/20...
Advance Mathematics for Planning

Summary
• Regression coefficients was calculated using two method.

• The data analysis ...
Advance Mathematics for Planning

THANK

YOU

10
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  1. 1. Finding Regression Coefficient To Predict Radiation Contents: • Finding of regression coefficient using • • Advance Mathematics Statistical method Genetic algorithm (by minimizing error) References: 1. Course work in environmental geology 2. Course work in advance mathematics for planning 1 Nirmal Raj Joshi|13ME135|Structual Material Laboratory
  2. 2. Advance Mathematics for Planning OBJECTIVE *The objective of this analysis is to predict the radiation level at 6th location using known data of 5 locations. The data set contains radiation measured at 6 different spots. *Use various techniques to find the regression coefficients a) Linear regression b) genetic algorithm 2
  3. 3. Date D-1 9/2/2013 1.318 9/3/2013 1.3 9/4/2013 1.324 ….. ….. 11/8/2013 1.294 11/11/2013 1.32 D-14 D-15 D-20 D-24 2.132 2.132 2.158 ….. 2.116 2.084 1.48 1.46 1.378 ….. 1.438 1.384 3.144 3.468 3.32 ….. 3.356 3.156 2.128 2.13 1.924 ….. 2.036 1.936 2.858 2.89 2.874 ….. 2.724 2.904 6 D-26 Radiation level Advance Mathematics for Planning Data Radiation level at various locations 4 2 0 Time (days) D-1 Target data D-14 D-15 D-20 D-24 D-26 Input data 3
  4. 4. The regression equation is *There is no constant term in the equation. model_radiation=lm(D.1~D.14+D.15+D.20+D.24+D.26+0,data=rdata) summary(model_radiation) 1.4 R-Output Coefficients: Estimate Std. Error t value Pr(>|t|) D.14 0.226600 0.088572 2.558 0.0143 * D.15 0.407026 0.155673 2.615 0.0124 * D.20 0.025524 0.024257 1.052 0.2989 D.24 0.004203 0.081023 0.052 0.9589 D.26 0.050943 0.052996 0.961 0.3421 --Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.03299 on 41 degrees of freedom Multiple R-squared: 0.9994, Adjusted R-squared: 0.9993 F-statistic: 1.353e+04 on 5 and 41 DF, p-value: < 2.2e-16 Radiation at location D-1: Actual Vs Predicted Radiation level Advance Mathematics for Planning Regression analysis using normal statistical method 1.3 1.2 1.1 1 Time (days) D-1 ANOVA table Source of Variation Between Groups Within Groups SS 6.85E-07 0.135982 df 1 90 Total 0.135982 MS 6.85E-07 0.001511 F 0.000453 D-1 pred P-value 0.983061 91 *R2 value of this multiple regression is 0.9994 which indicated the linear model is quite reliable. *R2 value of PD-1 calculated and observed value is 0.4668. *F<Fcrit, hence the model is acceptable at 95% confidence level F crit 3.946876 4
  5. 5. Advance Mathematics for Planning Regression analysis using Genetic algorithm About genetic algorithm 1. 2. 3. 4. 5. Start with a randomly generated population of n l−bit chromosomes (candidate solutions to a problem). Calculate the fitness ƒ(x) of each chromosome x in the population. Repeat the following steps until n offspring have been created: a. Select a pair of parent chromosomes from the current population, the probability of selection being an increasing function of fitness. b. With probability pc (the "crossover probability" or "crossover rate"), cross over the pair at a randomly chosen point (chosen with uniform probability) to form two offspring. If no crossover takes place, form two offspring that are exact copies of their respective parents. c. Mutate the two offspring at each locus with probability pm (the mutation probability or mutation rate), and place the resulting chromosomes in the new population. Replace the current population with the new population. Go to step 2 until the fitness of successive population converges. Generate initial population Array of [P] values. Find fitness. Generate off-springs using elite population Set of new population Stop when convergence is met 5
  6. 6. Advance Mathematics for Planning Regression analysis using Genetic algorithm Solution using genetic algorithm Two fitness functions were used viz. (a) R2 and (b) E2=(Yactual-Ypred)2 separately. The program was run in MATLAB and results were obtained. For both run, the input parameters are: SN 1 Parameter Population type 2 Number of variables optimize Number of population Number of generation 3 4 5 Value Double numbers to 5 5000 100 or attainment of error in 1e-6 7 8 9 Allowable error consecutive population Population generation scheme Fitness scaling Selection function Reproduction scheme 10 Mutation function 6 precision Uniform Rank Roulette wheel Elite selection with crossover of 0.8 at single point Uniform 6
  7. 7. Solution using genetic algorithm CASE-1: Maximizing R2 value => α=[ 0.4702 0.9999 0.0861 0.0232 0.0957] and R2=0.4688 CASE-2: Minimizing E2=(Yactual-Ypred)2 => α=[0.1952 0.3847 0.0362 0.0502 0.0397] and R2=0.4600 Although R2 is low, the fitting is more realistic Radiation at location D-1: Actual Vs Predicted 3.500 3.000 2.500 Radiation Level Advance Mathematics for Planning Regression analysis using Genetic algorithm 2.000 1.500 1.000 0.500 0.000 Time (Days) D-1 pred with E2 D-1actual D-1pred with R2 7
  8. 8. Final adopted values Radiation at location D-1: Actual Vs Predicted 11/11/2013 11/9/2013 11/7/2013 11/5/2013 11/3/2013 11/1/2013 10/30/2013 10/28/2013 10/26/2013 10/24/2013 10/22/2013 10/20/2013 10/18/2013 10/16/2013 10/14/2013 10/12/2013 10/10/2013 10/8/2013 10/6/2013 10/4/2013 10/2/2013 9/30/2013 9/28/2013 9/26/2013 9/24/2013 9/22/2013 9/20/2013 9/18/2013 9/16/2013 9/14/2013 9/12/2013 9/10/2013 9/8/2013 9/6/2013 9/4/2013 1.400 1.350 1.300 1.250 1.200 1.150 1.100 1.050 1.000 9/2/2013 Radiation Level Advance Mathematics for Planning Regression analysis using Genetic algorithm Time (Days) D-1 pred with E2 Source of Variation Between Groups Within Groups SS 2.21E-05 0.135241 df 1 90 Total 0.135263 D-1actual MS 2.21E-05 0.001503 F 0.014681 P-value 0.90383 91 In the table, we see that F<Fcrit, thus it can be said that the distribution estimated by the regression coefficients α gives significantly correct value at 95% confidence level. F crit 3.946876 8
  9. 9. Advance Mathematics for Planning Summary • Regression coefficients was calculated using two method. • The data analysis tools should be selected wisely to get the correct results. For e.g. in case of GA, R2 value may not yield proper results. 9
  10. 10. Advance Mathematics for Planning THANK YOU 10
  • AbhinandanaRangasamy

    Dec. 27, 2017

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