PHYSICS 2CL – SPRING 2009
Physics Laboratory: Electricity and
Magnetism, Waves and Optics
Prof. Leonid Butov
(for Prof. Oleg Shpyrko)
oshpyrko@physics.ucsd.edu
Mayer Hall Addition (MHA) 3681, ext. 4-3066
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.)
Today’s Plan:
Chi-Squared, least-squared fitting
Next week:
Review Lecture (Prof. Shpyrko is back)
Long-term course schedule
Week
Lecture Topic
1 Mar. 30 Introduction
2 Apr. 6
Error propagation; Oscilloscope;
RC circuits
Experiment
NO LABS
0
3 Apr. 13 Normal distribution; RLC circuits
1
4 Apr. 20 Statistical analysis, t-values;
2
5 Apr. 27 Resonant circuits
3
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
4, 5, 6 or 7
9 May 25 No Lecture (UCSD Holiday:
Memorial Day)
No LABS,
Formal Reports Due
10 June 1
Final Exam
4, 5, 6 or 7
NO LABS
Schedule available on WebCT
Labs Done This Quarter
This week’s
lab(s),
3 out of 4
0. Using lab hardware & software
1. Analog Electronic Circuits
(resistors/capacitors)
2. Oscillations and Resonant Circuits (1/2)
3. Resonant circuits (2/2)
4. Refraction & Interference with
Microwaves
5. Magnetic Fields
6. LASER diffraction and interference
7. Lenses and the human eye
LEAST SQUARES FITTING (Ch.8)
Purpose:
1) Agreement with theory?
30
2) Parameters
y = f(x)
20
10
y(x) = Bx
0
0
5
10
15
x
20
25
LINEAR FIT
y(x) = A +Bx :
A – intercept with y axis
B – slope
30
y(x)
20
q where B=tan q
10
A
0
0
5
10
15
x
20
25
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
LINEAR FIT
30
y(x) = A +Bx
y=-2+2x
y=9+0.8x
y(x)
20
10
0
0
5
10
15
x
20
25
x1
y1
x2
x3
x4
x5
x6
y2
y3
y4
y5
y6
LINEAR FIT
30
y(x) = A +Bx
y=-2+2x
y=9+0.8x
y(x)
20
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
Assumptions:
1) dxj << dyj ; dxj = 0
10
2) yj – normally distributed
3) sj: same for all yj
0
0
5
10
15
x
20
25
LINEAR FIT: y(x) = A + Bx
Method of linear regression, aka the least-squares fit….
Quality
of the fit
30
S
[yj-yfitj] 2
y(x)
20
y4-yfit4
y3-yfit3
10
0
0
5
10
15
x
20
25
LINEAR FIT: y(x) = A + Bx
Method of linear regression, aka the least-squares fit….
minimize
30
S
[yj-(A+Bxj)]2
20
y(x)
y4-(A+Bx4)
y3-(A+Bx3)
10
0
0
5
10
15
x
20
25
What about error bars?
Not all data points are created equal!
30
y(x)
20
10
0
0
5
10
15
x
20
25
Weight-adjusted average:
x

x
Reminder:
Typically the average
value of x is given as:
i
N
x1  x2  ...  x N

N
Sometimes we want to weigh data points with some
“weight factors” w1, w2 etc:
wx

x
w
i
i
i
w1 x1  w2 x2  ...  wN xN

w1  w2  ...  wN
You already KNOW this – e. g. your grade:
GRADE 
20%  Formal  12%   LABS  20%  FINAL
20%  12% * 5  20%
Weights: 20 for Final Exam, 20 for Formal Report, and
12 for each of 5 labs – lowest score gets dropped)
More precise data points should carry more weight!
Idea: weigh the points with the ~ inverse of their error bar
30
y(x)
20
10
0
0
5
10
15
x
20
25
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)
wx

x
w
i
i
i
w1 x1  w2 x2  ...  wN xN

w1  w2  ...  wN
Or in this case calculate for i=1, 2:
x
w1 x1  w2 x2
w1  w2
w1 
1
s
2
1
with “statistical” weights:
w2 
1
s 22
BOTTOM LINE: More precise
measurements get weighed
more heavily!
c2 TEST for FIT (Ch.12)
N
c 
2
y
j
s 2j
j 1
30
 f x j ))
2
y(x)
20
10
How good is the agreement
between theory and data?
0
0
5
10
15
x
20
25
c2 TEST for FIT (Ch.12)
N
c 
2
y
 f x ))
2
j
j
s
j 1
2
j

Ns y2
s
2
y
N
c~ 2 
30
c2
d
1
# of degrees of
freedom
20
y(x)
d=N-c
10
0
0
5
10
15
x
20
25
# of data
points
# of parameters
calculated from data
# of constraints
(Example: You can always draw a perfect line through 2 points)
LEAST SQUARES FITTING
xj yj y=f(x)
30
y(x)=A+Bx+Cx2+exp(-Dx)+ln(Ex)+…
y4-(A+Bx4)
y(x)
20
y3-(A+Bx3)
N
1.
c2  
y
j 1
10
2. Minimize
c2 :
c 2
0
A
j  f x j ))
2
s 2j
c 2
0 …
B
3.  A in terms of xj yj ; B in terms of xj yj , …
0
0
c
2
~
5 4. Calculate
10
15
20
25c 0 
5. Calculate
d
x
c~ 2  c~ 2
6. Determine probability for
c2
0
Usually computer program (for example Origin) can minimize c
as a function of fitting parameters (multi-dimensional landscape)
by method of steepest descent.
2
c
2
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
Example: fitting datapoints to y=A*cos(Bx)
“Perfect” Fit
Example: fitting datapoints to y=A*cos(Bx)
“Stuck” in local
minima of c2landscape fit
Next on PHYS 2CL:
Monday, May 18, Review Lecture