1. R fitting to functions other than lines

2. Power-Law from log-log linear model

3. Recall the data used in an earlier lab that varied the length of a pendulum and measured its period (the time to swing back and forth)
Length (cm)
Period (s)
108
2.080 89
1.888 58
1.534 44
1.3352 25
1.0199

4. The Excel fit to a power-law yielded

5. Entering the same data into R looks like
# copy and paste
length = c(108, 89,58,44,25)
period = c(2.080, 1.888, 1.534, , )

6. Plotting the data in R looks like

7. Log-log linear model
That number looks familiar from the Excel Power-law fit. It was the power.

8. Can’t take log of 0 or negative numbers
One cannot take the log of zero or a negative number
Recall on the test we were plotting the kutosis of dice simulations versus how many dice were rolled. But the kurtosis was negative. Thus we had to take the absolute value and plot and fit that.
Behind the scenes in Excel there are log’s being taken, and it cannot handle the negative numbers

9. Isolating the power

10. Breaking it down pend_fit is the name we gave to the model
The model has coefficients (what we usually call slope and intercept)
Our x’s are log(pendulum$length) so we are getting the coefficient associated with the x – what we would normally call the slope
Originally it was the power

11. Breaking it down y = A x^b log(y) = log(A x^b)
log(y) = log(A) +log(x^b)
log(y) = log(A) + b log(x)
We see that the slope in log(y) versus log(x) is b – what was originally the power in the power law
By log() in R we mean the natural log ln()
intercept = log(A)
exp(intercept) = A
The exponential function and natural log functions are inverses

12. Obtaining the Coefficient

13. Displaying fit equation

14. Displaying the fit curve

15. Exponential Fit from log linear model

16. Data from on how air pressure depends on altitude
Altitude (miles)
Pressure (psi)
14.696
11.022
7.348
1.4696

17. Plot with Exponential fit from Excel

18. Enter the data into R #copy and paste
altitude = c(0, , , , , ,
, , )
pressure = c(14.696, , 7.348, , ,
, , , )

19. Plotting the data in R

20. This time take log(y) but not log(x) and obtain a linear model
Once again the “slope” number looks familiar when compared to what we got from Excel

21. Cannot take log(0)
When we used this data on the first test, some students got their x and y axes confused.
In this data x=0 is perfectly fine, but y=0 is not.
That is why for those with the wrong axes, the exponential fit was unavailable in Excel

22. Isolating the factor

23. Determining the coefficient

24. Displaying the equation

25. Displaying the curve