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

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 local minima 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