1 What if…Choice of the functional form

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1 What if…Choice of the functional form 1 What if…Choice of the functional form

2 Exponential function Exponential f unctions are functions which can be represented by graphs similar to2 Exponential function Exponential f unctions are functions which can be represented by graphs similar to the graph on the right

3 Yellow = 4 xx Green = e xx Black = 3 xx Red = 23 Yellow = 4 xx Green = e xx Black = 3 xx Red = 2 xx

4 As you could see in the graph, the larger the base, the faster the function4 As you could see in the graph, the larger the base, the faster the function increased If we place a negative sign in front of the x, the graphs will be reflected(flipped) across the y-axis

5 Yellow = 4 -x-x Green = e -x-x Black = 3 -x-x Red = 25 Yellow = 4 -x-x Green = e -x-x Black = 3 -x-x Red = 2 -x-x

6 Exponential functions decrease if 0  b 1  1 and increase if b 16 Exponential functions decrease if 0 < b 1

7 Power function 7 Power function

8 Logarithmic function number of methylene groups, n 0 100 200 300 400 500 Tf(exp)/ K8 Logarithmic function number of methylene groups, n 0 100 200 300 400 500 Tf(exp)/ K

9 Hyperbolic function  y x 0 y=1/x 9 Hyperbolic function y x 0 y=1/x

10 Quadratic function y=(662, 678)+(-88, 916)*x+(3, 08855)*x^2 10111213141516171819 x 20 30 40 50 60 70 8010 Quadratic function y=(662, 678)+(-88, 916)*x+(3, 08855)*x^2 10111213141516171819 x 20 30 40 50 60 70 80 90 100 Y

11 Logistic function 11 Logistic function

12 General information NON LINEAR MODELS OFTEN ARE USED FOR SITUATION IN WHICH THE RATE OF12 General information NON LINEAR MODELS OFTEN ARE USED FOR SITUATION IN WHICH THE RATE OF INCREASE OR DECREASE IN THE DEPENDENT VARIABLE (WHEN PLOTTED AGAINST A PARTICULAR INDEPENDENT VARIABLE) IS NOT CONSTANT.

13 General information SOME OF THESE MODELS REQUIRED A TRANSFORMATION TO THE INDEPENDENT VARIABLE. 13 General information SOME OF THESE MODELS REQUIRED A TRANSFORMATION TO THE INDEPENDENT VARIABLE.

14 Transformation Logarithms Substitution Data transformations can be used to convert an equation into a linear14 Transformation Logarithms Substitution Data transformations can be used to convert an equation into a linear form

15 Exponential function 15 Exponential function

16 Power function  16 Power function

17 Quadratic function 17 Quadratic function

18 Polynomial function 18 Polynomial function

19 Hyperbolic function 19 Hyperbolic function

20 Logarithmic function 20 Logarithmic function

21 Logistic function 21 Logistic function

22 Linear function b 1 = 2, 50 X Y 1, 2 3, 08 1, 422 Linear function b 1 = 2, 50 X Y 1, 2 3, 08 1, 4 3, 58 1, 6 4, 08 1, 8 4, 58 XY — 0, 20, 5 Y/ X — 2,

23 Exponential function. XY 1, 24, 963 1, 46, 482 1, 811, 086 354, 872 XY23 Exponential function. XY 1, 24, 963 1, 46, 482 1, 811, 086 354, 872 XY — 0, 21, 519 0, 44, 604 1, 243, 786 Y/ X — 7, 595 11, 51 36, 4883 log y 0, 6957 0, 8117 1, 0448 1, 7394 log y/x 0, 580 b 1 =10 0, 580 =3,

24 Power function XY 1, 42, 027 24, 28 529, 37 9100, 9 XY -- 0,24 Power function XY 1, 42, 027 24, 28 529, 37 9100, 9 XY — 0, 62, 253 325, 085 471, 535 Y/ X — 3, 755 8, 36167 17, 88375 log y 0, 3069 0, 6314 1, 4678 2, 0039 log y/x 0, 219 0, 316 0, 294 0, 223 log x 0, 146 0, 301 0, 699 0, 954 log y/log x 2, 10 b 1 = 2,

25 EXPONENTIAL POWER Independent variable is a power exponent  Independent variable is a power base25 EXPONENTIAL POWER Independent variable is a power exponent Independent variable is a power base Form of model : x bby 10ˆ1 0ˆ b xby Interpretation of the coefficients b 0 — is the value of Y if independent variable is equal to zero. b 0 — is the value of Y if independent variable is equal to one b 1 — is the growth rate Y. If the independent variable increases 1 unit, the dependent variable will change (increase, if b 1 >1, or decrease, if b 1 0, or decrease, if b 1 <0) b 1 %, on average. Comparison

26 Comparison EXPONENTIAL POWER Linear transformation - logarithms Linear form 10 loglogˆlogbxbyxbbyloglogˆlog 10 Parameters estimation –26 Comparison EXPONENTIAL POWER Linear transformation — logarithms Linear form 10 loglogˆlogbxbyxbbyloglogˆlog 10 Parameters estimation – OLS: YXXX b b. TT log)( log 1 1 0 YXXX b b. TTloglog)log(log 1 1 0 Matrix and vector : 2 xx xn XX T yx y YX T log log 2 )(loglog xx xn XX T yx y YXT loglog

27 Comparison EXPONENTIAL POWER After log b 0  and log b 1 are estimated we27 Comparison EXPONENTIAL POWER After log b 0 and log b 1 are estimated we should check goodness of fit (standard error of the estimate, indetermination coefficient, test parameters individually and check residuals’ characteristics – at least linearity) for the linear form. 10 loglogˆlogbxbyxbbyloglogˆlog 10 1 )ˆlog(log 2 kn yy S ii e To interpret the results, antilog b 0 and b 1 should be calculated To interpret the results, antilog b 0 should be calculated




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