October 2017, 13(4): 1701-1721. doi: 10.3934/jimo.2017014

Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization

1. 

Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

2. 

Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

* Corresponding author: Roya Soltani

Received  August 2015 Published  December 2016

This study makes use of the artificial intelligence approaches combined with some nonlinear optimization techniques for optimization of a well-known problem in financial engineering called yield curve. Yield curve estimation plays an important role on making strategic investment decisions. In this paper, we use two well-known parsimonious estimation models, Nelson-Siegel and Extended Nelson-Siegel, for the yield curve estimation. The proposed models of this paper are formulated as continuous nonlinear optimization problems. The resulted models are then solved using some nonlinear optimization and meta-heuristic approaches. The optimization techniques include hybrid GPSO parallel trust region-dog leg, Hybrid GPSO parallel trust region-nearly exact, Hybrid GPSO parallel Levenberg-Marquardt and Hybrid genetic electromagnetism like algorithm. The proposed models of this paper are examined using some real-world data from the bank of England and the results are analyzed.

Citation: Roya Soltani, Seyed Jafar Sadjadi, Mona Rahnama. Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1701-1721. doi: 10.3934/jimo.2017014
References:
[1]

I. Baki, Yield curve estimation by spline-based model, A thesis submitted to the Graduate school of Applied Mathematics of The Middle East Technical University, 2006.

[2]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3$^{rd}$ edition, John Wiley and Sons, 2006. doi: 10.1002/0471787779.

[3]

S. I. Birbil and S. C. Fang, An electromagnetism-like mechanism for global optimization, Journal of Global Optimization, 25 (2003), 263-282. doi: 10.1023/A:1022452626305.

[4]

R. R. Bliss, Testing term structure estimation methods, Working Paper WP 96-12a, Federal Reserve Bank of Atlanta, 1996.

[5]

D. J. Bolder and D. Streliski, Yield Curve Modeling at the Bank of Canada, Technical Report TR 84, Bank of Canada, 1999.

[6]

M. Clerc and J. Kennedy, The particle swarm explosion stability and convergence in a multi dimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2001), 58-73.

[7]

A. Csajbok, Zero-coupon yield curve estimation from a central bank perspective, Working Paper WP 1998-2, National Bank of Hungary, 1998.

[8]

F. X. Diebold and C. Li, Forecasting the term structure of government bond yields, Journal of Econometrics, 130 (2006), 337-364. doi: 10.1016/j.jeconom.2005.03.005.

[9]

R. C. Eberhart and Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the Congress on Evolutionary Computation, San Diego. CA, (2000), 84-88. doi: 10.1109/CEC.2000.870279.

[10]

A. HanjooriM. Amiri and A. Alimi, Forecasting stock price using grey-fuzzy technique and portfolio optimization by invasive weed optimization algorithm, Decision Science Letters, 2 (2013), 175-184.

[11]

M. Ioannides, A comparison of yield curve estimation techniques using UK data, Journal of Banking and Finance, 27 (2003), 1-26. doi: 10.1016/S0378-4266(01)00217-5.

[12]

J. Kennedy and R. C. Eberhart, Particle swarm optimization, Proceeding of IEEE International Conference on Neural Networks, (1995), 1942-1948. doi: 10.1109/ICNN.1995.488968.

[13]

P. Manousopoulos and M. Michalopoulos, A comparison of yield curve estimation methods: The Greek case, Journal of Financial Decision Making, 1 (2005), 33-46.

[14]

P. Manousopoulos and M. Michalopoulos, Yield curve construction as a non-linear optimization problem, Proceedings of the 18th Hellenic Conference on Operations Research, 2006.

[15]

P. Manousopoulos and M. Michalopoulos, Comparison of non-linear optimization algorithms for yield curve estimation, European Journal of Operational Research, 192 (2009), 594-602. doi: 10.1016/j.ejor.2007.09.017.

[16]

J. H. McCulloch, Measuring the term structure of interest rates, Journal of Business, 44 (1971), 19-31. doi: 10.1086/295329.

[17]

J. H. McCulloch, The tax-adjusted yield curve, The Journal of Finance, 30 (1975), 811-830. doi: 10.1111/j.1540-6261.1975.tb01852.x.

[18]

C. R. Nelson and A. F. Siegel, Parsimonious modeling of yield curves, The Journal of Business, 60 (1987), 473-489. doi: 10.1086/296409.

[19]

J. Nocedal and S. J. Write, Numerical Optimization, Springer Science and Business Media, 2006. doi: MR2244940.

[20]

M. Orouji, Oil price shocks and stock market returns, Accounting, 2 (2016), 103-108. doi: 10.5267/j.ac.2016.2.005.

[21]

M. Pooter, Examining the Nelson-Siegel Class of Term Structure Models, Tinbergen Institute, Discussion paper, 2007.

[22]

P. Soderlind and L. E. O. Svensson, New Techniques to Extract Market Expectations from Financial Instruments, Working Paper WP 5877, National Bureau of Economic Research, 1996. doi: 10.3386/w5877.

[23]

L. E. O. Svensson, Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, Working Paper WP 4871, National Bureau of EconomicResearch, (1994). doi: 10.3386/w4871.

[24]

L. E. O. Svensson, Estimating forward interest rates with the Extended Nelson-Siegel method, Sveriges Riksbank Quarterly Review, 3 (1995), 13-26.

show all references

References:
[1]

I. Baki, Yield curve estimation by spline-based model, A thesis submitted to the Graduate school of Applied Mathematics of The Middle East Technical University, 2006.

[2]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3$^{rd}$ edition, John Wiley and Sons, 2006. doi: 10.1002/0471787779.

[3]

S. I. Birbil and S. C. Fang, An electromagnetism-like mechanism for global optimization, Journal of Global Optimization, 25 (2003), 263-282. doi: 10.1023/A:1022452626305.

[4]

R. R. Bliss, Testing term structure estimation methods, Working Paper WP 96-12a, Federal Reserve Bank of Atlanta, 1996.

[5]

D. J. Bolder and D. Streliski, Yield Curve Modeling at the Bank of Canada, Technical Report TR 84, Bank of Canada, 1999.

[6]

M. Clerc and J. Kennedy, The particle swarm explosion stability and convergence in a multi dimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2001), 58-73.

[7]

A. Csajbok, Zero-coupon yield curve estimation from a central bank perspective, Working Paper WP 1998-2, National Bank of Hungary, 1998.

[8]

F. X. Diebold and C. Li, Forecasting the term structure of government bond yields, Journal of Econometrics, 130 (2006), 337-364. doi: 10.1016/j.jeconom.2005.03.005.

[9]

R. C. Eberhart and Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the Congress on Evolutionary Computation, San Diego. CA, (2000), 84-88. doi: 10.1109/CEC.2000.870279.

[10]

A. HanjooriM. Amiri and A. Alimi, Forecasting stock price using grey-fuzzy technique and portfolio optimization by invasive weed optimization algorithm, Decision Science Letters, 2 (2013), 175-184.

[11]

M. Ioannides, A comparison of yield curve estimation techniques using UK data, Journal of Banking and Finance, 27 (2003), 1-26. doi: 10.1016/S0378-4266(01)00217-5.

[12]

J. Kennedy and R. C. Eberhart, Particle swarm optimization, Proceeding of IEEE International Conference on Neural Networks, (1995), 1942-1948. doi: 10.1109/ICNN.1995.488968.

[13]

P. Manousopoulos and M. Michalopoulos, A comparison of yield curve estimation methods: The Greek case, Journal of Financial Decision Making, 1 (2005), 33-46.

[14]

P. Manousopoulos and M. Michalopoulos, Yield curve construction as a non-linear optimization problem, Proceedings of the 18th Hellenic Conference on Operations Research, 2006.

[15]

P. Manousopoulos and M. Michalopoulos, Comparison of non-linear optimization algorithms for yield curve estimation, European Journal of Operational Research, 192 (2009), 594-602. doi: 10.1016/j.ejor.2007.09.017.

[16]

J. H. McCulloch, Measuring the term structure of interest rates, Journal of Business, 44 (1971), 19-31. doi: 10.1086/295329.

[17]

J. H. McCulloch, The tax-adjusted yield curve, The Journal of Finance, 30 (1975), 811-830. doi: 10.1111/j.1540-6261.1975.tb01852.x.

[18]

C. R. Nelson and A. F. Siegel, Parsimonious modeling of yield curves, The Journal of Business, 60 (1987), 473-489. doi: 10.1086/296409.

[19]

J. Nocedal and S. J. Write, Numerical Optimization, Springer Science and Business Media, 2006. doi: MR2244940.

[20]

M. Orouji, Oil price shocks and stock market returns, Accounting, 2 (2016), 103-108. doi: 10.5267/j.ac.2016.2.005.

[21]

M. Pooter, Examining the Nelson-Siegel Class of Term Structure Models, Tinbergen Institute, Discussion paper, 2007.

[22]

P. Soderlind and L. E. O. Svensson, New Techniques to Extract Market Expectations from Financial Instruments, Working Paper WP 5877, National Bureau of Economic Research, 1996. doi: 10.3386/w5877.

[23]

L. E. O. Svensson, Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, Working Paper WP 4871, National Bureau of EconomicResearch, (1994). doi: 10.3386/w4871.

[24]

L. E. O. Svensson, Estimating forward interest rates with the Extended Nelson-Siegel method, Sveriges Riksbank Quarterly Review, 3 (1995), 13-26.

Figure 1.  PSO solution space
Figure 2.  EM solution space
Figure 3.  Parallel trust regions
Figure 4.  Dogleg trajectory
Figure 5.  Interval plot for Extended Nelson-Siegel model
Figure 6.  Interval plot for Nelson-Siegel model
Figure 7.  Comparing the proposed methods regarding 12 sets of data taken from bank of England
Figure 8.  Comparing the proposed methods regarding 12 sets of data taken from bank of England
Figure 9.  Fitted models versus market data set 1
Figure 10.  Fitted Extended Nelson-Siegel models resulted from HGPSOPLM versus market data
Table 1.  Fitness Error
Methods
ModelsNelson-SiegelExtended Nelson-Siegel
AverageStandard deviationAverage Time (in minutes)Average deviationStandardAverage Time (in minutes)
HGEM0.1030290.0205781.6076080.0073070.0028065822.761143
HGPSO0.1039760.019731.050980.0120250.0066134851.89916
HGPSOPTR_NE0.1022080.02357729.18530.0048666670.00257340736.364
HGPSOPTR_DL0.1021670.02367931.36810.0051083330.00287764437.85
HGPSOPLM0.1025330.00636627.42750.0047833330.00264878535.74
Methods
ModelsNelson-SiegelExtended Nelson-Siegel
AverageStandard deviationAverage Time (in minutes)Average deviationStandardAverage Time (in minutes)
HGEM0.1030290.0205781.6076080.0073070.0028065822.761143
HGPSO0.1039760.019731.050980.0120250.0066134851.89916
HGPSOPTR_NE0.1022080.02357729.18530.0048666670.00257340736.364
HGPSOPTR_DL0.1021670.02367931.36810.0051083330.00287764437.85
HGPSOPLM0.1025330.00636627.42750.0047833330.00264878535.74
Table 2.  Parameter values for Extended Nelson-Siegel model
Variables
Methods $\beta_{0}$ $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\tau_{1}$ $\tau_{2}$
AverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviation
HGEM9.4960.384-8.9820.378-6.6560.142-12.3871.0312.4820.06123.9480.263
HGPSO9.4700.350-8.9780.381-6.5850.035-11.7860.4712.4800.07824.2390.144
HGPSOPTR-NE9.7961.007-9.3241.035-6.6880.549-13.3332.8832.5710.16224.6191.110
HGPSOPTR-DL9.9170.849-9.4480.866-6.7380.531-13.6662.4422.5860.11724.4621.286
HGPSOPLM9.9570.850-9.4910.873-6.7250.574-13.7962.4692.6040.11124.6101.119
Variables
Methods $\beta_{0}$ $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\tau_{1}$ $\tau_{2}$
AverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviation
HGEM9.4960.384-8.9820.378-6.6560.142-12.3871.0312.4820.06123.9480.263
HGPSO9.4700.350-8.9780.381-6.5850.035-11.7860.4712.4800.07824.2390.144
HGPSOPTR-NE9.7961.007-9.3241.035-6.6880.549-13.3332.8832.5710.16224.6191.110
HGPSOPTR-DL9.9170.849-9.4480.866-6.7380.531-13.6662.4422.5860.11724.4621.286
HGPSOPLM9.9570.850-9.4910.873-6.7250.574-13.7962.4692.6040.11124.6101.119
Table 3.  Parameter values for Nelson-Siegel model
Variables
Methods $\beta_{0}$ $\beta_{1}$ $\beta_{2}$ $\tau_{1}$
AverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviation
HGEM5.28810.0004-3.80430.0053-5.47050.00931.51920.0013
HGPSO5.24670.0919-4.52820.0732-5.22690.54621.65150.2967
HGPSOPTR-NE5.03210.0581-4.34500.0458-4.14690.04222.24020.0073
HGPSOPTR-DL5.00280.9806-4.31480.8265-4.08924.90732.2732.6319
HGPSOPLM5.08320.0581-4.39760.0458-4.25170.04222.18240.0072
Variables
Methods $\beta_{0}$ $\beta_{1}$ $\beta_{2}$ $\tau_{1}$
AverageStandard deviationAverageStandard deviationAverageStandard deviationAverageStandard deviation
HGEM5.28810.0004-3.80430.0053-5.47050.00931.51920.0013
HGPSO5.24670.0919-4.52820.0732-5.22690.54621.65150.2967
HGPSOPTR-NE5.03210.0581-4.34500.0458-4.14690.04222.24020.0073
HGPSOPTR-DL5.00280.9806-4.31480.8265-4.08924.90732.2732.6319
HGPSOPLM5.08320.0581-4.39760.0458-4.25170.04222.18240.0072
Table 4.  Extended Nelson-Siegel Eigenvalues and Gradient norms
MethodDataEigenvaluesGnorm
Set1219.7177.2452.5300.04100.00090.0004
Set2220.7527.1112.7480.0420.00010.0010.011
Set3214.8486.9422.7330.04100.00090.006
Set4188.9255.9993.5570.0460.0010.00060.026
Set5198.8356.2543.1340.072-0.0050.0030.861
Set6184.0565.8632.9290.0390.00020.00060.003
HGPSOPTR-NESet7220.4116.6173.2330.04300.0020.041
Set8201.8486.4902.7540.03900.00070.002
Set9201.7826.4502.7120.047-0.0020.0020.299
Set10192.9706.1122.9710.0400.00070.00010.002
Set11192.5746.0873.0860.043-0.0010.0010.374
Set12206.7786.6372.8960.0420.00010.00090.004
Average203.6246.4842.9400.045-0.00050.00120.136
Set1215.7817.0932.6240.04200.00090.009
Set2217.8997.0742.6970.04200.00090.0008
Set3210.8606.8062.8260.04200.00090.0137
Set4208.0926.8042.7280.04100.00080.009
Set5198.8356.2543.1340.072-0.0050.0030.861
Set6180.2925.6333.2360.0420.00040.00070.009
HGPSOPTR-DLSet7220.4116.6173.2330.04300.0020.041
Set8216.1296.7063.0190.0410.00010.0020.014
Set9194.5996.1782.8950.0390.00010.00070.007
Set10195.8456.2642.8110.03900.00070.0121
Set11195.0956.1532.9950.04000.00070.0002
Set12207.9496.6822.8810.04200.00090.002
Average205.1496.5222.9230.044-0.00040.0010.0815
Set1225.2727.3322.5880.04200.0010.003
Set2217.6867.0652.7090.04200.00090.005
Set3212.3556.8632.7810.04200.00090.001
Set4210.6916.9012.6840.04200.00080.004
Set5198.8356.2543.1340.072-0.0050.0030.861
Set6184.1615.8652.9330.0390.00020.00060.008
HGPSOPLMSet7220.4116.6173.2330.04300.0020.041
Set8201.9166.4942.7510.03900.00070.007
Set9195.2456.2112.8680.03900.00070.003
Set10192.976.1122.9710.040.00010.00070.002
Set11192.5756.0883.0860.043-0.0010.0010.374
Set12206.7346.6362.8950.04200.00090.002
Average204.9046.5362.8860.044-0.00050.0010.109
MethodDataEigenvaluesGnorm
Set1219.7177.2452.5300.04100.00090.0004
Set2220.7527.1112.7480.0420.00010.0010.011
Set3214.8486.9422.7330.04100.00090.006
Set4188.9255.9993.5570.0460.0010.00060.026
Set5198.8356.2543.1340.072-0.0050.0030.861
Set6184.0565.8632.9290.0390.00020.00060.003
HGPSOPTR-NESet7220.4116.6173.2330.04300.0020.041
Set8201.8486.4902.7540.03900.00070.002
Set9201.7826.4502.7120.047-0.0020.0020.299
Set10192.9706.1122.9710.0400.00070.00010.002
Set11192.5746.0873.0860.043-0.0010.0010.374
Set12206.7786.6372.8960.0420.00010.00090.004
Average203.6246.4842.9400.045-0.00050.00120.136
Set1215.7817.0932.6240.04200.00090.009
Set2217.8997.0742.6970.04200.00090.0008
Set3210.8606.8062.8260.04200.00090.0137
Set4208.0926.8042.7280.04100.00080.009
Set5198.8356.2543.1340.072-0.0050.0030.861
Set6180.2925.6333.2360.0420.00040.00070.009
HGPSOPTR-DLSet7220.4116.6173.2330.04300.0020.041
Set8216.1296.7063.0190.0410.00010.0020.014
Set9194.5996.1782.8950.0390.00010.00070.007
Set10195.8456.2642.8110.03900.00070.0121
Set11195.0956.1532.9950.04000.00070.0002
Set12207.9496.6822.8810.04200.00090.002
Average205.1496.5222.9230.044-0.00040.0010.0815
Set1225.2727.3322.5880.04200.0010.003
Set2217.6867.0652.7090.04200.00090.005
Set3212.3556.8632.7810.04200.00090.001
Set4210.6916.9012.6840.04200.00080.004
Set5198.8356.2543.1340.072-0.0050.0030.861
Set6184.1615.8652.9330.0390.00020.00060.008
HGPSOPLMSet7220.4116.6173.2330.04300.0020.041
Set8201.9166.4942.7510.03900.00070.007
Set9195.2456.2112.8680.03900.00070.003
Set10192.976.1122.9710.040.00010.00070.002
Set11192.5756.0883.0860.043-0.0010.0010.374
Set12206.7346.6362.8950.04200.00090.002
Average204.9046.5362.8860.044-0.00050.0010.109
Table 5.  Nelson-Siegel Eigenvalues and Gradient norms
MethodDataEigenvaluesGnorm
Set 1166.47716.66453.3590.04990.0032
Set 2168.94736.82363.35930.04980.0069
Set 3167.37366.80913.35410.04890.0025
Set 4166.06046.53573.36990.04940.0033
Set 5161.83735.99443.40180.04820.0056
Set 6144.29633.04160.32470.00130.005
HGPSOPTR-NESet 7162.19935.9413.40660.04560.0081
Set 8164.16126.24433.39040.04770.0105
Set 9162.87066.11143.39420.04680.0054
Set 10162.22955.99823.40650.04730.0022
Set 11163.71366.21453.38850.04720.0009
Set 12167.11936.62723.37020.04950.0005
Average163.10716.08383.12710.04430.0045
Set1166.47716.66453.3590.04990.0032
Set2168.94736.82363.35930.04980.0069
Set3167.37366.80913.35410.04890.0025
Set4166.06046.53573.36990.04940.0033
Set5161.83735.99443.40180.04820.0056
Set6145.17372.98420.00110.31130.0065
HGPSOPTR-DLSet7162.19935.9413.40660.04560.0081
Set8164.16126.24433.39040.04770.0105
Set9162.86996.11173.39420.04680.0035
Set10162.22955.99823.40650.04730.0022
Set11162.77225.75163.40690.021.969
Set12167.11936.62723.37020.04950.0005
Average163.10176.04053.10170.06790.1685
Set1166.47716.66453.3590.04990.0032
Set2168.94736.82363.35930.04980.0069
Set3168.3686.80673.35430.04880.0066
Set4166.06976.53613.36990.04930.0069
Set5161.83735.99443.40180.04820.0056
Set6142.60023.14830.0020.35150.0065
HGPSOPLMSet7162.19935.9413.40660.04560.0081
Set8164.16126.24433.39040.04770.0105
Set9162.86996.11173.39420.04680.0035
Set10162.22955.99823.40650.04730.0022
Set11163.71366.21453.38850.04720.0009
Set12167.11936.62723.37020.04950.0005
Average163.04946.09253.10020.07350.0051
MethodDataEigenvaluesGnorm
Set 1166.47716.66453.3590.04990.0032
Set 2168.94736.82363.35930.04980.0069
Set 3167.37366.80913.35410.04890.0025
Set 4166.06046.53573.36990.04940.0033
Set 5161.83735.99443.40180.04820.0056
Set 6144.29633.04160.32470.00130.005
HGPSOPTR-NESet 7162.19935.9413.40660.04560.0081
Set 8164.16126.24433.39040.04770.0105
Set 9162.87066.11143.39420.04680.0054
Set 10162.22955.99823.40650.04730.0022
Set 11163.71366.21453.38850.04720.0009
Set 12167.11936.62723.37020.04950.0005
Average163.10716.08383.12710.04430.0045
Set1166.47716.66453.3590.04990.0032
Set2168.94736.82363.35930.04980.0069
Set3167.37366.80913.35410.04890.0025
Set4166.06046.53573.36990.04940.0033
Set5161.83735.99443.40180.04820.0056
Set6145.17372.98420.00110.31130.0065
HGPSOPTR-DLSet7162.19935.9413.40660.04560.0081
Set8164.16126.24433.39040.04770.0105
Set9162.86996.11173.39420.04680.0035
Set10162.22955.99823.40650.04730.0022
Set11162.77225.75163.40690.021.969
Set12167.11936.62723.37020.04950.0005
Average163.10176.04053.10170.06790.1685
Set1166.47716.66453.3590.04990.0032
Set2168.94736.82363.35930.04980.0069
Set3168.3686.80673.35430.04880.0066
Set4166.06976.53613.36990.04930.0069
Set5161.83735.99443.40180.04820.0056
Set6142.60023.14830.0020.35150.0065
HGPSOPLMSet7162.19935.9413.40660.04560.0081
Set8164.16126.24433.39040.04770.0105
Set9162.86996.11173.39420.04680.0035
Set10162.22955.99823.40650.04730.0022
Set11163.71366.21453.38850.04720.0009
Set12167.11936.62723.37020.04950.0005
Average163.04946.09253.10020.07350.0051
Table 6.  Fitness Error for the Extended Nelson-Siegel model (Second quarter in 2015)
AverageStandard deviationAverage Time (in minutes)
HGEM0.0247590.0096743.0325
HGPSO0.0394970.0073241.1797
HGPSOPTR-NE0.0165410.01527135.3814
HGPSOPTR-DL0.0198750.0170039.1531
HGPSOPLM0.0157780.01423836.332
AverageStandard deviationAverage Time (in minutes)
HGEM0.0247590.0096743.0325
HGPSO0.0394970.0073241.1797
HGPSOPTR-NE0.0165410.01527135.3814
HGPSOPTR-DL0.0198750.0170039.1531
HGPSOPLM0.0157780.01423836.332
Table 7.  Comparison between LM results with respect to random, yesterday and PSO results as initial conditions
Random parameters as Initial conditionYesterday parameters estimation as Initial conditionPSO results as Initial condition
DateFinal best valueFinal best valueFinal best value
01/05/20080.5486-0.0368
02/05/20080.05400.43620.0081
06/05/20080.76400.52210.0489
07/05/20080.05340.45980.0084
08/05/20080.05360.62870.0144
09/05/20080.05500.65780.0166
12/05/20080.01260.72540.0672
13/05/20080.07050.64650.0064
14/05/20080.08250.49610.0037
15/05/20080.10400.44680.0346
16/05/20080.23650.71880.1200
19/05/20080.38370.43850.0742
20/05/20080.09140.49560.0023
21/05/20080.27510.36310.1220
22/05/20080.20990.29600.0838
23/05/20080.25040.44760.1214
27/05/20080.18630.39230.0768
28/05/20080.21890.33140.0869
29/05/20080.16620.16500.0880
30/05/20080.16370.16910.0846
Random parameters as Initial conditionYesterday parameters estimation as Initial conditionPSO results as Initial condition
DateFinal best valueFinal best valueFinal best value
01/05/20080.5486-0.0368
02/05/20080.05400.43620.0081
06/05/20080.76400.52210.0489
07/05/20080.05340.45980.0084
08/05/20080.05360.62870.0144
09/05/20080.05500.65780.0166
12/05/20080.01260.72540.0672
13/05/20080.07050.64650.0064
14/05/20080.08250.49610.0037
15/05/20080.10400.44680.0346
16/05/20080.23650.71880.1200
19/05/20080.38370.43850.0742
20/05/20080.09140.49560.0023
21/05/20080.27510.36310.1220
22/05/20080.20990.29600.0838
23/05/20080.25040.44760.1214
27/05/20080.18630.39230.0768
28/05/20080.21890.33140.0869
29/05/20080.16620.16500.0880
30/05/20080.16370.16910.0846
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