QMDA Review Session. Things you should remember 1. Probability & Statistics.

QMDA

Review Session

Things you should remember

1. Probability & Statistics

the Gaussian or normal distribution

p(x) = exp{ - (x-x)2 / 22 ) 1(2)

expected value

variance

x

p(x)

x x+2x-2

95%

Expectation =

Median =

Mode = x

95% of probability within 2 of the expected value

Properties of the normal distribution

Multivariate Distributions

The Covariance Matrix, C, is very important

Cij

the diagonal elements give the variance of each x i

xi2 = Cii

The off-diagonal elemements of C indicate whether pairs of x’s are correlated. E.g.

C12

x2

x1

x1

x2

C12<0negative correlationx2

x1

x1

x2

C12>0positive correlation

the multivariate normal distribution

p(x) = (2)-N/2 |Cx|-1/2 exp{ -1/2 (x-x)T Cx-1 (x-x) }

has expectation x

covariance Cx

And is normalized to unit area

if y is linearly related to x, y=Mx then

y=Mx (rule for means)

Cy = M Cx MT

(rule for propagating error)

These rules work regardless of the distribution of x

2. Least Squares

Simple Least SquaresLinear relationship between data, d, and model, m

d = Gm

Minimize prediction error E=eTe with e=dobs-Gm

mest = [GTG]-1GTd

If data are uncorrelated with variance, d2, then

Cm = d2 [GTG]-1

Least Squares with prior constraints

Given uncorrelated with variance, d2, that satisfy a

linear relationship d = Gm

And prior information with variance, m2, that satisfy a

linear relationship h = Dm

The best estimate for the model parameters, mest, solves

G

D

d

h m =

Previously, we discussed only the special case h=0

With = m/d.

Newton’s Method for Non-Linear Least-Squares Problems

Given data that satisfies a non-linear relationship d = g(m)

Guess a solution m(k) with k=0 and linearize around it:

m = m-m(k) and d = d-g(m(k)) and d=Gm

With Gij = gi/mj evaluated at m(k)

Then iterate, m(k+1) = m(k) + m withm=[GTG]-1GTd

hoping for convergence

3. Boot-straps

Investigate the statistics of y by

creating many datasets y’and examining their statistics

each y’ is created throughrandom sampling with replacement

of the original dataset y

y1

y2

y3

y4

y5

y6

y7

…

yN

y’1

y’2

y’3

y’4

y’5

y’6

y’7

…

y’N

4

3

7

11

4

1

9

…

6

N o

rigi

nal d

ata

Ran

dom

inte

gers

in

the

rang

e 1-

N

N r

esam

pled

dat

aN1

i y’i

Compute estimate

Now repeat a gazillion times and examine the resulting distribution of estimates

Example: statistics of the mean of y, given N data

4. Interpolation and Splines

linear splines

xxi xi+1

y iy i+

1y

in this intervaly(x) = yi + (yi+1-yi)(x-xi)/(xi+1-xi)

1st derivative discontinuous here

cubic splines

xxi xi+1

y iy i+

1

y

cubic a+bx+cx2+dx3 in this interval

a different cubic in this interval

1st and 2nd derivative continuous here

5. Hypothesis Testing

The Null Hypothesisalways a variant of this theme:

the results of an experiment differs from the expected value only because

of random variation

Test of Significance of Resultssay to 95% significance

The Null Hypothesis would generate the observed result less than 5% of

the time

Four important distributions

Normal distribution

Chi-squared distribution

Student’s t-distribution

F-distribution

Distribution of 2 = i=1Nxi

2

Distribution of xi

Distribution of t = x0 / { N-1 i=1Nxi

2 }

Distribution of F = { N-1i=1N xi

2} / { M-1i=1M xN+i

2 }

5 testsmobs = mprior when mprior and prior are known

normal distribution

obs = prior when mprior and prior are knownchi-squared distribution

mobs = mprior when mprior is known but prior is unknownt distribution

1obs =

obs when m1prior and m2

prior are knownF distribution

m1obs = m

obs when 1prior and

prior are unknownmodified t distribution

6. filters

g(t) = -

t f(t-) h() dgk = t p=-

k fk-p hp

g(t) = 0

f() h(t-) dgk = t p=0

fp hk-p

or alternatively

Filtering operation g(t)=f(t)*h(t)

“convolution”

How to do convolution by handx=[x0, x1, x2, x3, x4, …]T and y=[y0, y1, y2, y3, y4, …]T

x0, x1, x2, x3, x4, …… y4, y3, y2, y1, y0

x0y0

Reverse on time-series, line them up as shown, and multiply rows. This is first element of x*y

[x*y]2=

x0, x1, x2, x3, x4, …… y4, y3, y2, y1, y0

x0y1+x1y0

Then slide, multiply rows and add to get the second element of x*y

And etc …

[x*y]1=

g0

g1

…gN

h0

h1

…hN

f0 0 0 0 0 0f1 f0 0 0 0 0…fN … f3 f2 f1 f0

= t

g = F h

Matrix formulations of g(t)=f(t)*h(t)

g0

g1

…gN

f0

f1

…fN

h0 0 0 0 0 0h1 h0 0 0 0 0…hN … h3 h2 h1 h0

= t

g = H f

and

X(0)X(1)X(2)…X(N)

f0

f1

…fN

A(0) A(1) A(2) …A(1) A(0) A(1) … A(2) A(1) A(0) ……A(N) A(N-1) A(N-2) …

=

Least-squares equation [HTH] f = HTg

g = H f

g0

g1

…gN

f0

f1

…fN

h0 0 0 0 0 0h1 h0 0 0 0 0…hN … h3 h2 h1 h0

= t

Autocorrelation of hCross-correlation of h and g

Ai and Xi

Auto-correlation of a time-series, T(t)

A() = -+

T(t) T(t-) dt

Ai = j Tj Tj-i

Cross-correlation of two time-series T(1)(t) and T(2)(t)

X() = -+

T(1)(t) T(2)(t-) dt

Xi = j T(1)j T(2)

j-i

7. fourier transforms and spectra

Integral transforms:

C() = -+

T(t) exp(-it) dt

T(t) = (1/2) -+

C() exp(it) d

Discrete transforms (DFT)

Ck = n=0N-1 Tn exp(-2ikn/N ) with k=0, …, N-1

Tn = N-1k=0N-1 Ck exp(+2ikn/N ) with n=0, …, N-1

Frequency step: t = 2/N

Maximum (Nyquist) Frequency max = 1/ (2t)

Aliasing and cyclicity

in a digital world n+N = n

andsince time and frequency play symmetrical roles in exp(-it)

tk+N = tk

One FFT that you should know:

FFT of a spike at t=0 is a constant

C() = -+

(t) exp(-it) dt = exp(0) = 1

Error Estimates for the DFTAssume uncorrelated, normally-distributed data, dn=Tn, with

variance d2

The matrix G in Gm=d is Gnk=N-1 exp(+2ikn/N )

The problem Gm=d is linear, so the unknowns, mk=Ck, (the coefficients of the complex exponentials) are also normally-distributed.

Since exponentials are orthogonal, GHG=N-1I is diagonaland Cm= d

2 [GHG]-1 = N-1d2I is diagonal, too

Apportioning variance equally between real and imaginary parts of Cm, each has variance 2= N-1d

2/2.The spectrum sm

2= Crm

2+ Cim

2 is the sum of two uncorrelated, normally distributed random variables and is thus

2-distributed.

The 95% value of 2 is about 5.9, so that to be significant, a

peak must exceed 5.9N-1d2/2

Convolution Theorem

transform[ f(t)*g(t) ] =

transform[g(t)] transform[f(t)]

Power spectrum of a stationary time-series

T(t) = stationary time series

C() = -T/2

+T/2 T(t) exp(-it) dt

S() = limT T-1 |C()|2

S() is called the power spectral density, the spectrum normalized by the length of the time series.

Relationship of power spectral density to DFT

To compute the Fourier transform, C(), you multiply the DFT coefficients, Ck, by t.

So to get power spectal densityT-1 |C()|2 =

(Nt)-1 |t Ck|2 =

(t/N) |Ck|2

You multiply the DFT spectrum, |Ck|2, by t/N.

Windowed Timeseries

Fourier transform of long time-series

convolved with the Fourier Transform of the windowing function

is Fouier transform of windowed time-series

Window FunctionsBoxcar

its Fourier transform is a sinc functionwhich has a narrow central peakbut large side lobes

Hanning (Cosine) taperits Fourier transform

has a somewhat wider central peakbut now side lobes

8. EOF’s and factor analysis

SamplesNM

(f1 in s1) (f2 in s1) (f3 in s1)

(f1 in s2) (f2 in s2) (f3 in s2)

(f1 in s3) (f2 in s3) (f3 in s3)

…

(f1 in sN) (f2 in sN) (f3 in sN)

(A in s1) (B in s1) (C in s1)



…

(A in sN) (B in sN) (C in sN)

=

(A in f1) (B in f1) (C in f1)



S = C F

Coefficients NM

Factors MM

Representation of samples as a linear mixing of factors

SamplesNM

(f1 in s1) (f2 in s1)

(f1 in s2) (f2 in s2)

(f1 in s3) (f2 in s3)

…

(f1 in sN) (f2 in sN)




…

(A in sN) (B in sN) (C in sN)

=



S C’ F’

selectedcoefficients

Np

selectedfactors pM

ignore f3

igno

re f

3

data approximated with only most important factors

p most important factors = those with the biggest coefficients

Singular Value Decomposition (SVD)

Any NM matrix S and be written as the product of three matrices

S = U VT

where U is NN and satisfies UTU = UUT

V is MM and satisfies VTV = VVT

and is an NM diagonal matrix of singular values, i

SVD decomposition of S

S = U VT

write as

S = U VT = [U ] [VT] = C F

So the coefficients are C = U

and the factors are

F = VT

The factors with the biggest i’s are the most important

Transformations of FactorsIf you chose the p most important factors, they define both a

subspace in which the samples must lie, and a set of coordinate axes of that subspace. The choice of axes is not unique, and could be changed through a transformation, T

Fnew = T Fold

A requirement is that T-1 exists, else Fnew will not span the same subspace as Fold

S = C F = C I F = (C T-1) (T F)= Cnew Fnew

So you could try to implement the desirable factors by designing an appropriate transformation matrix, T

9. Metropolis Algorithm and Simulated Annealing

Metropolis Algorithm

a method to generate a vector x of realizations of the distribution p(x)

The process is iterativestart with an x, say x(i)

then randomly generate another x in its neighborhood, say x(i+1), using a distribution

Q(x(i+1)|x(i))

then test whether you will accept the new x(i+1)

if it passes, you append x(i+1) to the vector x that you are accumulating

if it fails, then you append x(i)

a reasonable choice for Q(x(i+1)|x(i)) normal distribution with mean=x(i) and x

2 that quantifies the sense of neighborhood

The acceptance test is as followsfirst compute the quantify:

If a>1 always accept x(i+1)

If a<1 accept x(i+1) with a probability of a and accept x(i) with a probability of 1-a

p(x(i+1)) Q(x(i)|x(i+1))

p(x(i)) Q(x(i+1)|x(i))a =

Simulated Annealing

Application of Metropolis to Non-linear optimization

find m that minimizes E(m)=eTewhere e = dobs-g(m)

Based on using the Boltzman distribution for p(x) in the Metropolis

Algorithm

p(x) = exp{-E(m)/T}

where temperature, T, is slowly decreased during the iterations

10. Some final words

Start Simple !

Examine a small subset of your data and looking them over carefully

Build processing scripts incrementally, checking intermediated results at each stage

Make lots of plots and look them over carefully

Do reality checks

QMDA Review Session. Things you should remember 1. Probability & Statistics.

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