noise & uncertainty
DESCRIPTION
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
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…• Gaussian Distribution = continuous probability distribution which describes
most statistical data well N(,)
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mean: P(x)⋅ x dx = μ−∞
∞∫variance : P(x)⋅ (x − μ)2 dx = μ
−∞
∞∫ =σ 2
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
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P x;n,ρ( )x=0
n
∑ =1
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P x;n,ρ( )x=0
n
∑ ⋅ x =K = np
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n →∞ ⇒ Normaldistributionn →∞ and np = const ⇒ Poissonian distribution
Gaussian Distribution
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G(x) =1
2πσ 2exp −
x − μ( )2
2σ 2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Uncertainty of measurement expressed in terms of σ
Gaussian Distribution : FWHM
+t
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G(μ + t) =12G(μ) → t =1.177σ
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2.355σ
Central Limit Theorem• Sufficiently large number of independent random variables can be
approximated by a Gaussian Distribution.
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
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
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.
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
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
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
Examples
2. mB=10.0±0.2 and mV=9.0±0.1 What is the uncertainty of mB-mV?
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?
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