Case Study: Spline Approximation

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Case study background and problem formulations

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

PROBLEM1: problem_St_Pen Minimize st_pen(spline_sum) (function Standard Penalty applied to Spline Sum) Calculate:nmeanabs_pen(spline_sum) (function Mean Absolute Penalty applied to Spline Sum)nspline_sum (function Spline Sum) ——————————————————————–———————————————— st_pen = Standard Penaltynmeanabs_pen = Mean Absolute Penaltynspline_sum = Spline Sum calculates spline value depending upon regression variables for every scenario ——————————————————————–———————————————— ——————————————————————————— Third Degree Polynomial Spline Consisting of 5 Piecies ———————————————————————————

Dataset	20	4371	0.18954	0.12
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Run-File	Problem Statement		Data	Solution
Matlab Toolbox			Data
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

——————————————————————————— Third Degree Polynomial Spline Consisting of 30 Piecies ———————————————————————————

Dataset	120	4371	0.1888	3.82
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Run-File	Problem Statement		Data	Solution
Matlab Toolbox			Data
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

PROBLEM2: problem_Meanabs_Pen Minimize meanabs_pen(spline_sum) (function Mean Absolute Penalty applied to Spline Sum) Calculate:nst_pen(spline_sum) (function Standard Penalty applied to Spline Sum)nspline_sum (function Spline Sum) ——————————————————————–———————————————— meanabs_pen = Mean Absolute Penaltynst_pen = Standard Penaltynspline_sum = Spline Sum calculates spline value depending upon regression variables for every scenarion ——————————————————————————— Third Degree Polynomial Spline Consisting of 5 Piecies ———————————————————————————

Dataset	20	4371	0.13590	0.06
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Run-File	Problem Statement		Data	Solution
Matlab Toolbox			Data
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

———————————————————————————— Third Degree Polynomial Spline Consisting of 30 Piecies ————————————————————————————

Dataset	120	4371	0.13496	2.67
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Run-File	Problem Statement		Data	Solution
Matlab Toolbox			Data
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

——————————————————————————— Subroutine for spline transformation with figures ———————————————————————————

Dataset	120	4371	0.13496	2.67
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

PROBLEM3: problem_Logexp_Sum Maximize logexp_sum(spline_sum) (function Logarithms Exponents Sum applied to Spline Sum) Calculate:nlogexp_sum(spline_sum) (function Logarithms Exponents Sum applied to Spline Sum)nlogistic(spline_sum) (function Logistic applied to Spline Sum) ——————————————————————–———————————————— logexp_sum = Logarithms Exponents Sumnlogistic = Logistic calculate values of logistic function of spline approximation for every scenarionspline_sum = Spline Sum calculates spline value depending upon regression variables for every scenario ——————————————————————–———————————————— ——————————————————————————— Third Degree Polynomial Spline Consisting of 5 Piecies ———————————————————————————

Dataset	20	14920	-0.68571	0.53
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Run-File	Problem Statement		Data	Solution
Matlab Toolbox			Data
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

Dataset	120	14920	-0.68481	10.45
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Run-File	Problem Statement		Data	Solution
Matlab Toolbox			Data
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

Dataset	120	4371	0.13496	2.67
	# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.14GHz (sec)
Environments
Matlab Subroutines	Matlab Code		Data
R	R Code		Data

CASE STUDY SUMMARY

Splines are calibrated to approximate one dimension observation data. Input data for building a spline are vectors containing data of independent and dependent variables and parameters defining number of knots and smoothing degree of the spline. The splines are calibrated by minimizing various error functions, such as mean square error, mean absolute error, and maximum likelihood logistic regression function (PSG functions: st_pen, meanabs_pen, and logexp_sum, accordingly).