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**(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.)

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

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**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

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**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

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**LEAST SQUARES FITTING (Ch.8)**

Purpose: 1) Agreement with theory? 2) Parameters y(x) = Bx

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**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

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**? 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

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**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

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**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

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**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)

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

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**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)

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**More precise data points should carry more weight!**

Idea: weigh the points with the ~ inverse of their error bar

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**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!

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**c2 TEST for FIT (Ch.12) How good is the agreement**

between theory and data?

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**d = N - c c2 TEST for FIT (Ch.12) # of degrees of freedom # of data**

points # of parameters calculated from data # of constraints

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**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

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**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

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**Example: fitting datapoints to y=A*cos(Bx)**

“Perfect” Fit

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**Example: fitting datapoints to y=A*cos(Bx)**

“Stuck” in local minima of c2landscape fit

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

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