CHEG REU Polymath Regression
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POLYMATH WORKSHOP - Regression and Data Analysis
Data Table
The data table is used for input, manipulation and storage of numerical data. The data are stored in a
columnwise fashion where every column is associated with a name (variable) and can be addressed
separately. The stored data can be regressed (meaning fitting a straight line, various curves and
equations to the data using multiple linear, polynomial and nonlinear regression techniques), analyzed
(meaning interpolated, differentiated, integrated and various statistics are calculated) and plotted.
Linear & Polynomial Regression
This part of the program will fit a polynomial of the form:
P(x) = a0 + a1*x + a2*x^2 + . . . + an*x^n
where a0, a1, ..., an are regression parameters to a set of N tabulated values of x (independent variable)
versus y (dependent variable). The highest degree allowed for a polynomial is N - 1 (thus n >= N - 1).
The program calculates the coefficients a0, a1, ..., an by minimizing the sum of squares of the deviations
between the calculated P(x) above and the corresponding value of y for each value of x.
Multiple Linear Regression
This part of the program will fit a linear function of the form:
y(x1, x2, ..., xn) = a0 + a1*x1 + a2*x2 + ... + an*xn
where a0, a1, ..., an are regression parameters, to a set of N tabulated values of x1, x2, ..., xn (independent
variables) versus y (dependent variable). Note that the number of data points must be greater than n+1
(thus N >= n+1). The program calculates the coefficients a0, a1, ..., an by minimizing the sum of squares
of the deviations between the calculated and the data for y.
Nonlinear Regression
This part of the program will fit a nonlinear function of the form:
y = f (x1, x2, …, xn, a0, a1, a2, …, am)
where a0, a1, …, an are regression parameters to a set of N tabulated values of x1, x2, …, xn (independent
variables) versus y (dependent variable). Note that the number of data points must be greater than m + 1
(thus N >= m + 1).
CHEG REU Polymath Regression
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POLYMATH Problem Data Set
The following table presents vapor pressure versus temperature for benzene.
For POLYMATH
Let TC = Temperature (˚C)
Let P = Pressure (mm Hg)
ENTER BENZENE DATA SET INTO POLYMATH
Linear Regression (Polynomial of degree 1)
P(x) = a0 + a1*x
General
P(TC) = a0 + a1*TC
Problem 1
(Linear Regression)
PROBLEM 1 - SOLVE FOR BENZENE DATA SET
Polynomial Regression (n is degree of polynomial)
P(x) = a0 + a1*x + a2*x^2 + . . . + an*x^n
P(TC) = a0 + a1*TC + a2*TC^2
General
Problem 2
(Second Degree Polynomial Regression)
PROBLEM 2 - SOLVE FOR BENZENE DATA SET
CHEG REU Polymath Regression
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Multiple Linear Regression
y(x1, x2, ..., xn) = a0 + a1*x1 + a2*x2 + ... + an*xn
General
log(P) = A + B/T + C*log(T) + D*T^2
Problem 3
(Riedel equation)
where T is the temperature in Kelvin and A, B, C and D are the parameters
For POLYMATH Variable Transformations
TC = Temperature (˚C)
P = Pressure (mm Hg)
TK = TC + 273.15
logP = log(P)
Trec = 1/TK
T2 = TK^2
LogT = log(TK)
logP = a0 + a1*Trec + a2*logT + a3*T2
PROBLEM 3 - SOLVE FOR BENZENE DATA SET
Nonlinear Regression
y = f (x1, x2, …, xn, a0, a1, a2, …, am)
logP = A + B/(TC + C)
General
Example Nonliner Regression Problem
(Antoine Equation)
For POLYMATH Variable Transformations
TC = Temperature (˚C)
P = Pressure (mm Hg)
logP = log(P)
Initial estimates for the parameters must also be provided.
For this example: Initial estimates are:
A = 6, B = -1000, and C = 200
PROBLEM 4 - DETERMINE ANTOINE EQUATION CONSTANTS FOR
BENZENE DATA SET