4.3 REGRESSION CURVE FITTING Using this feature the user can fit an analytic function to a data. The program allows four types of functions: 1. Polynomial : 1st to the 8th order polynom (A+B*X+C*X^2+..... ..+H*X^8). The data should be in the form of two columns, the first column is the dependent variable and the second is the independent variable. 2. Log/power : Product of powers (8 max) (A*B^X*C^Y*.....*H^U), the data can be in the form of 2 to 8 columns. The first column is the independent variable. 3. Exponent : A*EXP(X), and the data should be in the form of two columns, the first column is the dependent variable and second is the independent variable. 4. Multi_linear: Linear function of 8 variables maximum, (A+B*Y+C*Y+.......+H*U) the data can be in the form of 2 to 8 columns. The first column is the dependent variable. Where: A,B,C...H - Coefficients to be found X,Y,....U - Independent variables 4.3A urve_fit Choose the urve_fit menu item to get the next menu: Import Poly. Log/power Exp. Multi_linear Data_save Graph_save Help 4.3B mport Choose the mport menu item to get the next prompt: Input the file name - ISRA6 Type the file name and press ENTER to view the next screen : ____________________________________________________________________________ Import Poly. Log/power Exp. Multi_linear Data_save Graph_save Help Import data file (list) of numbers [ESC] - Previous menu A B C D E 1 ==================================================================== 2 Units | Output/Input | Memory | 3 ==================================================================== 4 None | 0.00 | 0.00| 5 ==================================================================== GA GB GC GD GE 3 8.154845 1 4 22.16716 2 5 60.25661 3 6 163.7944 4 7 445.2394 5 8 1210.286 6 9 3289.899 7 10 8942.873 8 11 24309.25 9 12 66079.39 10 13 14 15 16 ____________________________________________________________________________ NOTE: The imported data file represents an exponent function. 4.3C

olynomial Press

to get the next prompt: Polynom order (8 max) - 3 The user can choose 1st to 8th order polynom. In this example we will try to fit a 3rd order polynom to the data. The program will display the regression curve fitting graphs (the data and the fitted curve (see POLYNOM3.PIC). If you print this graph using Lotus PRINTGRAPH you will find that we need to use higher order polynom. Type 3 and press ENTER to view the graph. Press [ESC] to see the next screen: ____________________________________________________________________________ Import Poly. Log/power Exp. Multi_linear Data_save Graph_save Help Import data file (list) of numbers [ESC] - Previous menu A B C D E 1 ==================================================================== 2 Units | Output/Input | Memory | 3 ==================================================================== 4 None | 0.00 | 0.00| 5 ==================================================================== FQ FR FS FT FU FV FW 3 Regression Output: 4 Constant -15553.4 5 Std Err of Y Est 4428.812 6 R Squared 0.970260 7 No. of Observations 10 8 Degrees of Freedom 6 9 10 X Coefficient(s) 11 12 17328.98 -4880.03 391.9952 13 14 Std Err of Coef. 15 16 6442.657 1328.901 79.68790 ____________________________________________________________________________ This is the standard Lotus regression output table. The fitted curve can be expressed analytically as: Y = -15553.4 +17328.98*X -4880.03*X^2 + 391.9952*X^3 4.3D og/power This option enables the use of a product of powers x y u like: A * B * C ... G as the fitted function. Where x,y and u are to be determined and A,B,C and G are known. The maximum number of products is 8, but the user needs to use only the number of data columns less 1. For example if the data file includes 7 columns the first column is the dependent function and the rest 6 are for the independent variables, therefore the user can fit a function which is a product of 6 powers maximum. In our example we have only two columns therefore we need to choose 1 power only. Press to get the next prompt: Number of powers (Number of columns LESS ONE!, 8 max) - 1 Since in this example we have only two columns type 1 and press ENTER. The program displays the regression curve fitting graphs as Log-Log graph (the data and the fitted curves, see LOG1.PIC). If you print this graph using Lotus PRINTGRAPH you will find that the log/power option doesn't fit so well. Press [ESC] to get the next screen: ____________________________________________________________________________ Import Poly. Log/power Exp. Multi_linear Data_save Graph_save Help Import data file (list) of numbers [ESC] - Previous menu A B C D E 1 ==================================================================== 2 Units | Output/Input | Memory | 3 ==================================================================== 4 None | 0.00 | 0.00| 5 ==================================================================== FQ FR FS FT FU FV FW 3 Regression Output: 4 Constant 0.661512 5 Std Err of Y Est 0.986340 6 R Squared 0.905661 7 No. of Observations 10 8 Degrees of Freedom 8 9 10 X Coefficient(s) 11 12 3.930705 13 14 Std Err of Coef. 15 16 0.448525 ____________________________________________________________________________ In this screen it can be seen that R Squared is far from 1, as the R Squared gets closer to 1 the better the fit is. The fitted curve can be expressed as: LN(Y) = .661512 + 3.930705 * LN(X) and by representing the constant .661512 as LN(EXP(.661512)) we can write: Y = EXP(.661512) * X^3.930705 in case of more powers the function can be expressed as: Y = EXP(Constant) * X^coef1 * Y^coef2 *..... 4.3D xponent The program displays the regression curve fitting graphs as Log(Y) vs. X (the data and the fitted curves, see EXPONENT.PIC). If you print this graph using Lotus PRINTGRAPH you will find that it fits well. Press [ESC] to view the next screen: ____________________________________________________________________________ Import Poly. Log/power Exp. Multi_linear Data_save Graph_save Help Import data file (list) of numbers [ESC] - Previous menu A B C D E 1 ==================================================================== 2 Units | Output/Input | Memory | 3 ==================================================================== 4 None | 0.00 | 0.00| 5 ==================================================================== FQ FR FS FT FU FV FW 3 Regression Output: 4 Constant 1.098612 5 Std Err of Y Est 0.000000 6 R Squared 1 7 No. of Observations 10 8 Degrees of Freedom 8 9 10 X Coefficient(s) 11 12 1.000000 13 14 Std Err of Coef. 15 16 0.000000 ____________________________________________________________________________ In this screen it can be seen that R Squared=1 therefore the curve is well fitted. The fitted curve can be expressed as: LN(Y) = 1.098612 + 1 * X by representing the constant 1.098612 as LN(EXP(1.098612)) we can write: Y = EXP(1.098612) * EXP(X^1) and in general terms: Y = EXP(constant) * EXP(X) 4.3E ulti-linear Press to get the next prompt: Number of independent variables (Number of columns LESS ONE!,8max) - 1 In this example we will try to fit a 1 variable linear function to the data (because the imported file has only two columns). To see the results you can print the file (LINEAR.PIC). Type 1 and press ENTER to view the graph. Press [ESC] to get the next screen: ____________________________________________________________________________ Import Poly. Log/power Exp. Multi_linear Data_save Graph_save Help Import data file (list) of numbers [ESC] - Previous menu A B C D E 1 ==================================================================== 2 Units | Output/Input | Memory | 3 ==================================================================== 4 None | 0.00 | 0.00| 5 ==================================================================== FQ FR FS FT FU FV FW 3 Regression Output: 4 Constant -16853.7 5 Std Err of Y Est 15506.45 6 R Squared 0.513903 7 No. of Observations 10 8 Degrees of Freedom 8 9 10 X Coefficient(s) 11 12 4964.891 13 14 Std Err of Coef. 15 16 1707.204 ____________________________________________________________________________ In this screen it can be seen that R Squared is far from 1 (we tried to fit a linear function to data which describes an exponent), as R Squared get closer to 1 the better the fit is. The fitted curve can be expressed as: FUNC = Constant + coef1*X1 + coef2*X2 + .....