1. (for Prof. Oleg Shpyrko)
PHYSICS 2CL – SPRING Physics Laboratory: Electricity and Magnetism, Waves and Optics
Prof. Leonid Butov
(for Prof. Oleg Shpyrko)
Mayer Hall Addition (MHA) 3681, ext
Office Hours: Mondays, 3PM-4PM.
Lecture: Mondays, 2:00 p.m. – 2:50 p.m., York Hall 2722
Course materials via webct.ucsd.edu
(including these lecture slides, manual, schedules etc.)

2. Today’s Plan:
Chi-Squared, least-squared fitting Next week: Review Lecture (Prof. Shpyrko is back)

3. Long-term course schedule
Week
Lecture Topic
Experiment 1
Mar. 30
  Introduction
NO LABS 2
Apr. 6
  Error propagation; Oscilloscope; RC circuits 3
Apr. 13
  Normal distribution; RLC circuits 4
Apr. 20
  Statistical analysis, t-values; 5
Apr. 27
  Resonant circuits 6
May 4
  Review of Expts. 4, 5, 6 and 7
4, 5, 6 or 7 7
May 11
Least squares fitting, c2 test 8
May 18
  Review Lecture 9
May 25
No Lecture (UCSD Holiday: Memorial Day)
No LABS,
Formal Reports Due 10
June 1
Final Exam
Schedule available on WebCT

4. Labs Done This Quarter 0. Using lab hardware & software
Analog Electronic Circuits (resistors/capacitors)
Oscillations and Resonant Circuits (1/2)
Resonant circuits (2/2)
Refraction & Interference with Microwaves
Magnetic Fields
LASER diffraction and interference
Lenses and the human eye
This week’s
lab(s),
3 out of 4

5. LEAST SQUARES FITTING (Ch.8)
Purpose:
1) Agreement with theory?
2) Parameters
y(x) = Bx

6. LINEAR FIT x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 y(x) = A +Bx :
LINEAR FIT
y(x) = A +Bx :
A – intercept with y axis
B – slope x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6
q where B=tan q A

7. ? LINEAR FIT x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 y(x) = A +Bx y=-2+2x
LINEAR FIT ?
y(x) = A +Bx x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6
y=-2+2x
y=9+0.8x

8. LINEAR FIT x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6 y(x) = A +Bx y=-2+2x
LINEAR FIT
y(x) = A +Bx x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6
y=-2+2x
y=9+0.8x
Assumptions:
dxj << dyj ; dxj = 0
yj – normally distributed
sj: same for all yj

9. S LINEAR FIT: y(x) = A + Bx Yfit(x) 2 Quality [yj-yfitj] of the fit
Method of linear regression, aka the least-squares fit….
Yfit(x)
[yj-yfitj] S 2
Quality
of the fit
y4-yfit4
y3-yfit3

10. S LINEAR FIT: y(x) = A + Bx true value 2 minimize [yj-(A+Bxj)]
Method of linear regression, aka the least-squares fit….
true value
[yj-(A+Bxj)] S 2
minimize
y4-(A+Bx4)
y3-(A+Bx3)

11. What about error bars?
Not all data points are created equal!

12. Weight-adjusted average:
Reminder:
Typically the average
value of x is given as:
Sometimes we want to weigh data points with some
“weight factors” w1, w2 etc:
You already KNOW this – e. g. your grade:
Weights: 20 for Final Exam, 20 for Formal Report, and
12 for each of 5 labs – lowest score gets dropped)

13. More precise data points should carry more weight!
Idea: weigh the points with the ~ inverse of their error bar

14. Weight-adjusted average:
How do we average values with different uncertainties?
Student A measured resistance 100±1 W (x1=100 W, s1=1 W)
Student B measured resistance 105±5 W (x2=105 W, s2=5 W)
Or in this case calculate for i=1, 2:
with “statistical” weights:
BOTTOM LINE: More precise measurements get weighed more heavily!

15. c2 TEST for FIT (Ch.12) How good is the agreement
between theory and data?

16. d = N - c c2 TEST for FIT (Ch.12) # of degrees of freedom # of data
points
# of parameters
calculated from data
# of constraints
(Example: You can always draw a perfect line through 2 points)

17. LEAST SQUARES FITTING true value xj yj y=f(x) y4-(A+Bx4) y3-(A+Bx3) …
y(x)=A+Bx+Cx2+exp(-Dx)+ln(Ex)+…
y4-(A+Bx4)
y3-(A+Bx3) 1.
2. Minimize c2: …
3.  A in terms of xj yj ; B in terms of xj yj , …
4. Calculate c2
5. Calculate
6. Determine probability for

18. Usually computer program (for example Origin) can minimize
as a function of fitting parameters (multi-dimensional landscape)
by method of steepest descent.
Think about rolling a bowling ball in some energy landscape
until it settles at the lowest point
Best fit (lowest c2)
Sometimes the fit
gets “stuck” in a local minimum like this one.
Solution? Give it a “kick” by resetting one of the fitting parameters and trying again
Fitting Parameter Space

19. Example: fitting datapoints to y=A*cos(Bx)
“Perfect” Fit

20. Example: fitting datapoints to y=A*cos(Bx)
“Stuck” in local
minima of c2landscape fit

21. Next on PHYS 2CL:
Monday, May 18,  Review Lecture