Simple Null Hypothesis

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Types of Null Hypothesis
Simple Null Hypothesis
Data are a random realization of a unique process
Data are generated by a Gaussian process with
zero mean and unit variance
Example
Composite NH
Data are only a member of a family of processes
Data are generated by a Gaussian process with
unspecified mean and variance
Example
Let ? be the space of processes under
consideration, and let ?? ? ? be the set of
processes which are consistent with the null
hypothesis.
The null hypothesis declares that the process F
that generated the data is an element of the set
??. If this set consists of a single member, the
null hypothesis is simple. Otherwise, the null
hypothesis is composite. (i.e. the data were
generated by some process F ? ?? but it doesnt
say which one!)
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Types of Errors
Type I Error
The hypothesis is rejected but it is actually
true (false positive)
Type II Error
The hypothesis is accepted but it is actually not
true (false negative)
Given that the NH is actually true, the
probability of making a Type I error is the
significance level (a) of the test. We want
this level (or size) to be as low as possible (at
least lt 0.05)
Given that the NH is actually false, the
probability of making a Type II error is related
with the power (b) of the test. We want this
probability or power (1-b) to be as high as
possible
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Choice of Discriminating Statistic (T)
When the NH is composite, T should be pivotal
What is a pivotal statistic? The distribution of
T is the same for all members F of the family ??
of processes consistent with the chosen NH.
Why pivotal T? If T is pivotal, it does not
matter which F ? ?? is used as the basis for
generating realizations, all serve equal purpose.
But in practice, most of the discriminating
statistic to investigate low dimensional dynamical
nonlinearity are non-pivotal. And for that
purpose, the method of constrained realization is
beneficial than typical realization.
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There are infinite possibilities of
discriminating statistic to be used! But what
should be the guidelines?
The primary requirement Discriminating
statistic should not be directly derived from the
second order properties of the data like
histogram, autocorrelation, power spectrum etc.
Why so? The above properties are linear
statistic, and the surrogates, by definition,
Preserve the linear properties of the data.

• Choose a statistic which
• 1. Provides the maximum statistical power,
• 2. Is possibly related addressing some
  deterministic structure (e.g., continuity,
  smoothness, differentiability etc.)
• 3. Addresses the predictability properties of
  the process.
• 4. Checks the temporal asymmetry of the data.

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Typical vs. Constrained - Realizations
Data Chaotic Henon Model
Typical realizations AR Surrogates with model
order 6 (AIC criterion)
Discriminating statistic
A(1) autocorrelation at lag 1
FT-surrogates correctly rejects NH but not
AR-surrogates. AR-surrogates produce more
variance that is due to the nuisance parameter
A(1) from one realization to the next. But
FT-surrogates have much smaller spread.
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Size Power of the Test
Power and size are estimates as the fraction of
trials out of 1000 for which the null hypothesis
is rejected at the a 0.05 level.
FT-surrogate
FT-surrogate
AR-surrogate
AR-surrogate
Constrained realization is more accurate and
more powerful than typical realization.
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Another Example High Dimensional Chaotic Time
Series
Data Sum of five realizations of Henon map
Additive Gaussian noise AR-Surrogates Model
order 8 Discriminating statistic
FT-surrogates detect the nonlinearitybut
AR-surrogates do not.
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Discriminating statistic vs. Nuisance parameters
x AR-Surr. FT-Surr.
T is nonpivotal, i.e., discriminating statistic
depends on the nuisance parameters, (Here the
first two lags of autocorrelation)
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Local Flow
Kaplan Glass (1992) Phys. Rev. Lett.
For chaos, the direction of motion is a function
of position in phase space. And determinism
implies that points that are now close in phase
space do not diverge very much over
sufficiently short times. But for longer times,
sensitive dependence to initial conditions
starts to separate nearby trajectories.

• Reconstruct the state space with suitable
  embedding parameters.
• Coarse-grain partition the state space into
  boxes.
• Each grid box j contains nj points with time
  indices of tj,k for k1,2, , nj
• For each point, the displacement for a given
  translation horizon tp, was estimated Dxj,k
  x(tj,ktp) x(tj,k)
• The average displacement of the points in each
  box yields a family of displacement magnitudes
  Lnd for each embedding dimension d and value of
  nnj.
• The expectation value of Lnd for a random system
  is cd/nj0.5 where cd(2/d)0.5G((d1)/2)/G(d/2)
• The average displacement for all values of n at a
  given embedding dimension is

For a deterministic system, L ? 1 While for a
random walk, L ? 0
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Local Dispersion Method
Wayland et al. (1993) Phys. Rev. Lett
Choose a reference vector, x0, and find the k
nearest neighbors, x1, , xk. Let y0, ., yk
represent the image of these vectors. The
translation differences between the vectors and
their images will be vj yj - xj Compute the
average
Compute the translation error
The translation errors measures the normalized
spread of the displacements around each reference
vector, relative to the average displacement.
Choose Nref number of reference vectors and
compute the translation error for each. Find the
average (or median) of these value which provides
a robust measure ofthe underlying deterministic
structure. For deterministic series, vj will be
nearly equal and etrans is small. For stochastic
series, etrans is large.
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Simple Nonlinear Prediction
Choose a zeroth-order nonlinear predictor.
Fitting local constant maps to the attractor.
For each reference vector xr,find a fixed k
number of neighbors xj, j 1, , k.
Then, the predicted value with prediction horizon
tp will be
The prediction error,
If we use the mean as prediction, the error will
be
The, normalized prediction error
For deterministic series, NPE ? 0 For
stochastic series, NPE ? 1
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Kaplans d/e
Kaplan (1994) Physica D
This method examines the trajectories of points
in the embedding space for exceptional events
that track each other closely when
translated. In other words, if two points are
close together on a trajectory, the images of the
points at some short time later are more likely
to be close together if the system is
deterministic than if it is not.
Let the distance between two pairs of vectors
(as usual you can exclude pairs where (i-j) gt
tc, correlation time
The distance between the images of the vectors
after k time steps
The average separation
The stochastic systems are expected to have
values of ei,j that are independent of previous
separations di,j ? Er vs. r will tend to be a
line with zero slope For deterministic systems,
Er is small for small r, increasing r to a
maximum that is dependent on the size of the
trajectory.
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Examples Simulated Time Series
Chang et al. (1995) Chaos

• Henon Map
• Lorenz Flow
• Mackey-Glass Equation

Add in-band or colored noise to each
deterministic time series.
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Discriminating Statistic

• Local Flow The value of L should be greater than
  surrogates.
• Local Dispersion The value of ltetransgt should be
  smaller than surrogates.
• d-e Profile e values for small d should be
  smaller than surrogates.
• Local Prediction The values of NPE should be
  smaller than surrogates.

Choice of Embedding Parameters

• Two suggestions
• Ensure proper embedding procedure to get a valid
  d, t
• Choose those parameters which has a limited
  spread of values for different realizations of
  the same process even if at these parameters the
  embedding is not strictly valid

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data
Example Lorenz Flow 25 Noise
surrogates
tp
tp
tp
d
d
d
tp
tp
tp
d
d
d
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Example Henon Map
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Example Lorenz Flow
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Example Mackey-Glass System
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Measure of Smoothness

• Find directional vector Xtxt1-xt
• Find the angle between successive directional
  vectors
• Compute the index
• The lower the s, the higher the degree of
  smoothness

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Example Chaotic Rössler Equation
State space plot
Angles between Successive vectors
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Example Chaotic Rössler Equation
Histogram
Original
s 0.01
Angle variation
Power spectrum
Histogram
Angle variation
Surrogate
s 0.39
Power spectrum
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Palus (1995) Physica D
Redundancy

• It measures how much repetition of information
  occurs in the measurement.
• For low-dimensional deterministic dynamics,
  repeatable and distinct patterns
• produces redundancy because the measurements made
  in the past provide
• information about the present/future states.
• For random or stochastic dynamics, repetition
  happens only out of chance!
• Subsequent measurements provide fresh or new
  information.

If pr is the probability that the time series
will take values between x and xr,then,
H(1,r) is the average number of bits needed to
describe a single value of the time series with
a precision r ? Shannon Entropy Let H(d,r) be
the average number of bits needed to describe a
sequence of d values (i.e. the number of bits in
an d-dimensional embedding
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In general, because describing a sequence of d
values cannot take more bits than describing d
individual values. This redundancy R(d,r) d
. H(1,r) H(d,r) The block-entropy can also be
defined by correlation sum C(r) H2(d,r)
-logC(r) And redundancy R2(d,r)
d.H2(1,r) H2(d,r) In practice, the following
formula is used to estimate redundancy
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Approximate Entropy
Pincus (1991) PNAS

• Construct multi-dimensional state space from each
  signal xt xtxt xt-t xt-2t xt-(d-1)t
• Find,
• Compute,
• Approximate entropy,
• The lower the value of ApEn, the higher the
  degrees of recurrence
• Practical choice d 2, r 0.15std (or three
  times of the mean noise level)

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1. ApEn is a nonnegative and biased quantity.
Thus, two ApEn values should always be
compared with same parameter settings. 2. For a
stochastic process, K-S entropy is infinite, but
ApEn is finite. 3. For any two systems, the
following holds (a) If h1(A) lt h1(B), then
ApEn(d,r,N)(A) lt ApEn(d,r,N)(B) (b) If
ApEn(d,r,N)(A) lt ApEn(d,r,n)(B), then
ApEn(d1,r1,N)(A) lt ApEn(d1,r1,N)(B)
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Barahona Poon (1996) Nature
Polynomial Based Prediction
Volterra based approach but with closed loop (or
feedback)
Model with degree d and memory ?
Functional basis zm(n) is composed of all the
distinct combinations of the embedding space
coordinates (xn-1, xn-2, , xn-?) upto a degree d
with a totaldimension M (kd)! / (d! k!)
The short-term prediction power of the model is
then measured by one-step ahead prediction error
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The best model (?opt, dopt) can be found by
satisfying the following condition
where r ? 1,M is the number of polynomial terms
of truncated Volterra expasionfrom a certain
parameter set (?,d).

• In practice,
• One obtains the best linear model by searching
  for ?lin by minimizingC(r) with d 1.
• Repeat with increasing ? and d gt 1 to get the
  best nonlinear model.
• The nonlinear determinism is indicated if (i)
  dopt gt 1
• (ii) the best nonlinear model is better
  predictor that both the linear model and
  than the best linear and nonlinear models
  obtained from the surrogates.

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Example Chaotic ecological model with strong
periodic component
NH3 laser pulsations
Intensity of a star
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Practical Remarks
1. Inpulse Noise
The sudden spike or impulse has a damaging
effect on surrogates
Primarily, these impulses have a white power
spectrum after phaserandomization, such white
noisesspread into the time series.
Recipe Remove these spikes if they are not of
interest.
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2. Endpoint Mismatch
Any standard FFT program assumes that time series
to be periodic with the available data sequence
being just one period of the assumed infinite
time series. This means that the end points of
the time series are assumed to be equal. If the
endpoints of the time series are not equal, a
discontinuity results which Introduces spurious
high frequency power into the power spectrum.
Recipe Choose a subset of time series such that
the end points match.
Ex xn 1.9xn-1 - 0.9001xn-2 ?n
Effect of endpoint mismatch
After removal of endpoint mismatch
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3. Slow Drifts
If a trend is present in the signal,
phase-randomized surrogateappears quite
different from the original signal.
Recipe 1. Remove the trend. 2. Compute the
surrogate of the detrended series. 3. Adding
back the trend.
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4. Padding
A common approach while computing an FFT of a
time series is to pad the data sequence with
zeros so that the length of the series will be a
power of 2. But introducing such padded zeros
introduces more structure into the
surrogatewhich is not present in the original
data. This effect is likely to increase the
rateof false positive.
Recipe Use a Fourier Transform program which
can be applied on exact length of the data. or
Truncate the data sequence to be the right
length.
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Multivariate Surrogates
Prichard Theiler (1996) Physica D
Let, we have M simultaneously measured time
series, x1(t), x2(t), , xM(t)
Let X1(f), X2(f), , XM(f) be their Fourier
transforms.
The cross-correlation between jth and kth time
series is
For a linear Gaussian multidimensional process,
all relevant information about the process is
given by these cross-correlations.
Now, the cross-spectrum will be
A(f) is the Fourier amplitude, and f(f) is the
phase angle.
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We need to preserve (i) linear
auto-correlations (ii) linear
cross-correlations
Add same random phase sequence ?(f) to fj(f) for
j 1,, M
Then, surrogates can be obtained by
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Example Chromospheric Oscillations
Bhattacharya et al. (2001) Sol. Phys.
Surrogates w/o Spatial Correlations
Surrogates with Spatial Correlations
data
data
surrogates
surrogates
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Casdagli (1992) J. R. Stat. Soc.
DVS Algorithm
1, Normalize the time series to zero mean and
unit variance. 2. Divide the time series into
two parts (a) a training set or fitting set
to estimate the model
coefficients (b) a test set or
out-of sample set to
evaluate the model Nf is the number of points
in the fitting set Nt is the number of points in
the test set 3. Choose T (prediction horizon), d
(embedding dimension) 4. Choose a test delay
vector xi (i gt Nf) for a T-step ahead prediction
task. Compute the distances dij of the test
vector xi from the training vectors xj
for all j such that (d-1)t lt j lt i-T 5. Order
the distances dij and find k nearest neighbors
xj(1) through xj(k) of xj s

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6. Then T-step further state of xj was predicted
by using the following model
where the prediction coefficients a0, a1,, ad
by least-squares approach. This is equivalent to
an locally linear autoregressive model of order d
fitted to the k nearest neighbors to the test
point. Vary k in the range 2(d1) lt k lt Nf T
(d-1)t
7. Compute the prediction error
8. Repeat steps 4-7 as (jT) runs through the
test set, and compute the mean absolute
forecasting error
9. Vary d and plot Ed(k) as a function of the
number of nearest neighbors. Such a family of
curves is called a DVS plot.
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Example FIR Laser Data
Ed(k)
k
k
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Tokuda et al. (2002) Chaos
A Variant of DVS Plot
Here, one can estimate the in-sample prediction,
i,e, using the whole data set. Say, is the
predicted value of the original xt, and the
residual error The signal-to-noise ration
(SNR) will be
Here, noise refers to unpredictable components
within the data.
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When k is small, prediction accuracy is sensitive
to noise level ? low SNR With increasing k,
prediction performance gets better because
nonlinear structureis well modeled by local
linear predictor. With an intermediate value of
k, SNR reaches maximum. With further increasing
of k, the model gets closer to a global linear
predictor that may be a bad model for a
nonlinear process ? low SNR The degree of
nonlinearity is related with the difference
between optimally nonlinear prediction and
global linear prediction.
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Example Iregular Animal Vocalizations
Scream of Juvenile Macaque
DVS plot Juvenile Macaque
Call of Mother Macaque
DVS plot Mature Macaque
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Surrogate Analysis DVS plot
Juveniles scream
Mothers call