noise & uncertainty
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Noise & Uncertainty. ASTR 3010 Lecture 7 Chapter 2. Accuracy & Precision. Accuracy & Precision. True value. systematic error. Probability Distribution : P(x ). Uniform, Binomial, Maxwell , Lorenztian , etc … - PowerPoint PPT PresentationTRANSCRIPT
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Noise & Uncertainty
ASTR 3010
Lecture 7
Chapter 2
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Accuracy & Precision
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Accuracy & Precision
True value
systematic error
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Probability Distribution : P(x)• Uniform, Binomial, Maxwell, Lorenztian, etc…• Gaussian Distribution = continuous probability distribution which describes
most statistical data well N(,)
€
mean: P(x)⋅ x dx = μ−∞
∞∫variance : P(x)⋅ (x − μ)2 dx = μ
−∞
∞∫ =σ 2
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Binomial Distribution• Two outcomes : ‘success’ or ‘failure’
probability of x successes in n trials with the probability of a success at each trial being ρ
Normalized…
mean
when
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P x;n,ρ( ) =n!
x!(n − x)!ρ x (1− ρ )n−x
€
P x;n,ρ( )x=0
n
∑ =1
€
P x;n,ρ( )x=0
n
∑ ⋅ x =K = np
€
n →∞ ⇒ Normaldistributionn →∞ and np = const ⇒ Poissonian distribution
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Gaussian Distribution
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G(x) =1
2πσ 2exp −
x − μ( )2
2σ 2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Uncertainty of measurement expressed in terms of σ
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Gaussian Distribution : FWHM
+t
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G(μ + t) =12G(μ) → t =1.177σ
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2.355σ
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Central Limit Theorem• Sufficiently large number of independent random variables can be
approximated by a Gaussian Distribution.
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Poisson Distribution• Describes a population in counting experiments
number of events counted in a unit time.o Independent variable = non-negative integer numbero Discrete function with a single parameter μprobability of seeing x events when the average event rate is E.g., average number of raindrops per second for a storm = 3.25 drops/sec at time of t, the probability of measuring x raindrops = P(x, 3.25)
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PP (x;μ) =μ x
x!e−μ
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Poisson distribution
Mean and Variance
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x = xx=0
∞
∑ PP (x;μ) = xμ x
x!e−μ
⎛ ⎝ ⎜
⎞ ⎠ ⎟
x=0
∞
∑=K= μ
(x − μ)2 =K
= x 2 − μ 2
=K= μ €
x
x!x=0
∞
∑ = eμuse
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Signal to Noise Ratio• S/N = SNR = Measurement / Uncertainty• In astronomy (e.g., photon counting experiments), uncertainty = sqrt(measurement) Poisson statistics
Examples:• From a 10 minutes exposure, your object was detected at a signal strength of
100 counts. Assuming there is no other noise source, what is the S/N?
S = 100 N = sqrt(S) = 10S/N = 10 (or 10% precision measurement)
• For the same object, how long do you need to integrate photons to achieve 1% precision measurement?
For a 1% measurement, S/sqrt(S)=100 S=10,000. Since it took 10 minutes to accumulate 100 counts, it will take 1000 minutes to achieve S=10,000 counts.
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Weighted Mean• Suppose there are three different measurements for the distance to the
center of our Galaxy; 8.0±0.3, 7.8±0.7, and 8.25±0.20 kpc. What is the best combined estimate of the distance and its uncertainty?
wi = (11.1, 2.0, 25.0)xc = … = 8.15 kpcc= 0.16 kpc
So the best estimate is 8.15±0.16 kpc.
2
2
1 22
1
11
c
ii
n
ii
c
n
iiic wwxx
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Propagation of Uncertainty• You took two flux measurements of the same object.
F1 ±1, F2 ±2
Your average measurement is Favg=(F1+F2)/2 or the weighted mean. Then, what’s the uncertainty of the flux? we already know how to do this…
• You need to express above flux measurements in magnitude (m = 2.5log(F)). Then, what’s mavg and its uncertainty? F?m
• For a function of n variables, F=F(x1,x2,x3, …, xn),
22
23
2
3
22
2
2
21
2
1
2 ... nn
F xF
xF
xF
xF
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Examples
1. S=1/2bh, b=5.0±0.1 cm and h=10.0±0.3 cm. What is the uncertainty of S?
S
h
b
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Examples
2. mB=10.0±0.2 and mV=9.0±0.1 What is the uncertainty of mB-mV?
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Examples
3. M = m - 5logd + 5, and d = 1/π = 1000/πHIP
mV=9.0±0.1 mag and πHIP=5.0±1.0 mas.What is MV and its uncertainty?
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In summary…
Important Concepts• Accuracy vs. precision• Probability distributions and
confidence levels• Central Limit Theorem• Propagation of Errors• Weighted means
Important Terms• Gaussian distribution• Poisson distribution
Chapter/sections covered in this lecture : 2