Online NON-Linear Regression Homework Help

Welcome to the sample assignment from StatisticsHomeworkHelper.com, where we make advanced statistical concepts accessible through practical online guidance. In this sample, we dive into non-linear regression, a crucial technique for analyzing complex relationships in data. By working through real-life examples, you’ll learn how to apply non-linear regression to uncover intricate patterns and make accurate predictions. This example will bolster your understanding of non-linear regression, ensuring you can confidently tackle your homework and master this vital statistical method, all with the convenience of online support.

1. [phone] https://www.statisticshomeworkhelper.com / [email protected] NON-Linear Regression Homework Help

2. https://www.statisticshomeworkhelper.com / Introduction Welcome to a sample assignment from StatisticsHomeworkHelper.com, where we simplify advanced statistical concepts through clear, practical examples. In this sample, we explore non-linear regression, a powerful tool for modeling complex relationships in data. By examining real-world applications and interpreting non-linear patterns, you’ll gain insights into how non-linear regression can be used to make accurate predictions and understand intricate data trends. This example will strengthen your ability to apply non-linear regression techniques, ensuring a comprehensive grasp of this essential statistical method.

3. https://www.statisticshomeworkhelper.com / 1 . When using the transformed data model to find the constants of the regression model y  ae bx to best fit  x 1 , y 1  ,  x 2 , y 2  ,........  x n , y n  , the sum of the square of the residuals that is minimized is

4. https://www.statisticshomeworkhelper.com / Solution The correct answer is (B). Taking the natural log of both sides of the regression model y  ae bx gives ln  y   ln  a   bx The residual at each data point x i is E i  ln  y i   ln  a   bx i The sum of the square of the residuals for the transformed data is n

5. https://www.statisticshomeworkhelper.com / 2. It is suspected from theoretical considerations that the rate of water flow from a firehouse is proportional to some power of the nozzle pressure. Assume pressure data is more accurate. You are transforming the data. Flow rate, F (gallons/min) 96 129 135 145 168 235 Pressure, p (psi) 11 17 20 25 40 55 The exponent of the nozzle pressure in the regression model F  ap b most nearly is 0.49721 0.55625 0.57821 0.67876 Solution The correct answer is (A).

6. https://www.statisticshomeworkhelper.com / The transforming of the above data is done as follows. F  ap b ln ( F )  ln ( a )  b ln ( p ) z  a  bx where z  ln  F  x  ln( p ) a  ln  a  implying a  e a There is a linear relationship between z and x . Linear regression constants are given by n  x i z i   x i  z i

7. https://www.statisticshomeworkhelper.com / Since n  6  x i z i  ln(11)  ln(96)  ln(17)  ln(129)  ln(20)  ln(135) i  1  ln(25)  ln(145)  ln(40)  ln(168)  ln(55)  ln(235)  96.208

8. https://www.statisticshomeworkhelper.com /  x i  ln(11)  ln(17)  ln(20)  ln(25)  ln(40)  ln(55)  19.142 i  1  z i  ln(96)  ln(129)  ln(135)  ln(145)  ln(168)  ln(235)  29.890 i  1 6 2 2 2 2 2 2 2  i i  1 x  (ln(11))  (ln(17))  (ln(20))  (ln(25))  (ln(40))  (ln(55))  62.779 then b  6  96.208  19.142  29.890 6  62.779  19.142 2  577.25  572.15 376.67  366.41  0.49721 Can you now find what a is?

9. 2 1  k  3. The transformed data model for the stress- strain curve   k  e for concrete in compression, where  is the stress and  is the strain, is (A) ln    ln  k 1   ln    k 2  1 2 (B) ln     ln  k   k  1 2      (C) ln     ln  k   k  (D) ln    ln( k 1  )  k 2  Solution The correct answer is (B)   k  e  k 2  1 The model can be rewritten as

10. https://www.statisticshomeworkhelper.com /   2 1  k   k e To transform the data, we take the natural log of both sides 4. In nonlinear regression, finding the constants of the model requires solving simultaneous nonlinear equations. However in the exponential model y  ae bx that is best fit to  x 1 , y 1  ,  x 2 , y 2  ,........,  x n , y n  , the value of b can be found as a solution of a single nonlinear equation. That nonlinear equation is given by

11. https://www.statisticshomeworkhelper.com / Solution The correct answer is (B).

12. https://www.statisticshomeworkhelper.com / 1 1 2 2 n n Given  x , y  ,  x , y  ,........,  x , y  bx , best fit y  ae to the data. The variables a and b are the constants of the exponential model. The residual at each data point x i is E i  y i  ae bx i The sum of the square of the residuals is n (1) r  i S  E i  1 2 i i  1    y  ae bx  2 n i (2) To find the constants a and b of the exponential model, we find where S r is a local minimum or maximum by differentiating with respect to a and b and equating the resulting equations to zero.

13. https://www.statisticshomeworkhelper.com / or   n i  1 2 bx n i  1 bx i i i  y e  a e  n n i  1 i  1 bx 2 bx  i i  i i i Equations (4a) and (4b) are simultaneous nonlinear equations with constants a and b . This is unlike linear regression where the equations to find the constants of the model are simultaneous but linear .

14. In general, iterative methods (such as the Gauss- Newton iteration method, Method of Steepest Descent, Marquardt's Method, Direct search, etc) must be used to find values of a and b . However, in this case, from Equation (4a), a can be written explicitly in terms of b as n n bx  i i y e  e 2 bx i a  i  1 i  1 This equation is still a nonlinear equation in terms of b , and can be solved best by numerical methods such as the bisection method or the secant method . n i  1 2 bx  i n bx  i  1 i bx n i  1  i i i x e  i  e 2 bx i i y x e  y e

15. https://www.statisticshomeworkhelper.com / You can now show that these values of of a and b, correspond to a local minimum, and since the above nonlinear equation has only one real solution, it corresponds to an absolute minimum. There is a functional relationship between the mass density p of air and the altitude h above the sea level. In the regression model   k 1 e 2 , the constant k 2 is found as k 2  0.1315 . Assuming the  k h mass density of air at the top of the atmosphere is 1/1000 th of the mass density of air at sea level. The altitude in kilometers of the top of the atmosphere most nearly is

16. 46.2 46.6 49.7 52.5 Solution The correct answer is (D). Note to the student: See the alternative answer given later as that is quite a bit shorter. Since k 2  0.1315 is given, the sum of the square of the residual is 2 1 n i  1  0.1315 h r i i S      k e  https://www.statisticshomeworkhelper.com /

17. First we need to find the value of the constant k 1 . i i 1 i  1  0.1315 h  0.1315 h i 1  e    k  S r   n 2    k e   1  i  1  2  0.1315 h n n i  1  0.1315 h  i i i  k e   e Thus, k  n  i i  e  0.263 h i i  1 i  1 n  0.1315 h  e 1 Since  1.15  0.95879  1.10  0.91928  1.05  0.84508  0.95  0.81026  3.7709 i  1  0.1315  1.60  0.95 e  0.1315  1.28  1.15 e  0.1315  0.32  1.10 e  0.1315  0.64  1.05 e  0.1315 h  i n  4 n i  e https://www.statisticshomeworkhelper.com /

18.  e  0.263 h i  e  0.263  0.32  e  0.263  0.64  e  0.263  1.28  e  0.263  1.60 i  1  0.91928  0.84508  0.71417  0.65652  3.1351 the value of the constant k 1 is https://www.statisticshomeworkhelper.com /

19. https://www.statisticshomeworkhelper.com / Alternative Answer: Note to the student: Do we really need to find k 1 for this problem?   k 1 e  0.1315 h  sea  level  k 1 e  0.1315   k 1   k e  0.1315 h top top 1

20. https://www.statisticshomeworkhelper.com / 6. A steel cylinder at 80° F of length 12" is placed in a commercially available liquid nitrogen bath (  315  F) . If the thermal expansion coefficient of steel behaves as a second order polynomial function of temperature and the polynomial is found by regressing the data below,

21. https://www.statisticshomeworkhelper.com / Temperature, T (°F) Thermal expansion Coefficient,  (  in/in/°F)  320 2.76  240 3.83 160 4.72  80 5.43 6.00 80 6.47 the reduction in the length of the cylinder in inches most nearly is 0.0219 0.0231 0.0235 0.0307 Solution The correct answer is (C). We are fitting the above data to the following polynomial.

22. https://www.statisticshomeworkhelper.com / 1 2   a  a T  a T 2 2 2 r i 1 i 2 i S      a  a T  a T  There is a quadratic relationship between the thermal expansion coefficient and the temperature, and the coefficients a , a 1 , and a 2 are found as follows 2 2 2 2   1   i  1 i 1 i 2 i  a  S r   n 2    a  a T  a T  i 2 i  1 i 1 i 2 i  T    a  S r   n 2    a  a T  a T  i 1 i  1 i 1 i 2 i   T    a  S r   n 2    a  a T  a T  which gives

23. https://www.statisticshomeworkhelper.com / Table 1 Summations for calculating constants of model.

24. https://www.statisticshomeworkhelper.com / Table 1 ( cont)            We have     2  1     7.0022  10  1   2.9210  10  5   1.9840  10 5  7.2000  10 2  7.2000  10 2 1.9840  10 5  5.0688  10 7 1.4541  10 10   a   5.0688  10 7 a   2.4744  10  3 1.9840  10 5   a  6 Solving the above system of simultaneous linear equations, we get

25. https://www.statisticshomeworkhelper.com /