Environmental Data Analysis with MatLab
2nd Edition
Lecture 21:
Interpolation
SYLLABUS
Lecture 01
Lecture 02
Lecture 03
Lecture 04
Lecture 05
Lecture 06
Lecture 07
Lecture 08
Lecture 09
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Lecture 25
Lecture 26
Using MatLab
Looking At Data
Probability and Measurement Error
Multivariate Distributions
Linear Models
The Principle of Least Squares
Prior Information
Solving Generalized Least Squares Problems
Fourier Series
Complex Fourier Series
Lessons Learned from the Fourier Transform
Power Spectra
Filter Theory
Applications of Filters
Factor Analysis
Orthogonal functions
Covariance and Autocorrelation
Cross-correlation
Smoothing, Correlation and Spectra
Coherence; Tapering and Spectral Analysis
Interpolation
Linear Approximations and Non Linear Least Squares
Adaptable Approximations with Neural Networks
Hypothesis testing
Hypothesis Testing continued; F-Tests
Confidence Limits of Spectra, Bootstraps
Goals of the lecture
to introduce
Interpolation
the process of filling in missing data points
A(t)
Scenario 1: data are collected at irregular time intervals, but
you want to compute power spectral density, which requires
evenly sampled data.
1
time
2
?
psd
0
frequency
A(t)
Scenario 2: two datasets are collected with different
sampling intervals, but you want to combine them into a
scatter plot
2
1
time
2
B(t)
1
?
B
0
A
in both scenarios
the times that the data are collected at are
inconvenient
we encountered a problem similar to
this one back in Lecture 8,
where we used
prior information
to fill in data gaps
dobs(t)
observed data with missing points
0
1
time
2
dest(t)
estimated data with missing points filled in
0
1
time
2
find diest so that
est
di
≈
obs
di
at the observation points
and
est
di
roughness of
≈0
everywhere
the solution is inexact
est
di
≠
obs
di
everywhere
and
roughness of
≠0
everywhere
est
di
but the inexactness isn’t a problem
because
both
observations
and
prior information
have error
now we examine an alternative
approach
traditional interpolation
similar, but subtly different
find d(t) so that
d(ti) =
obs
di
at the observation points
and
roughness of d(t) = 0
in between the observation points
find d(t) so that
d(ti) =
exact
obs
di
at the observation points
and
exact
roughness of d(t) = 0
in between the observation points
find d(t) so that
d(ti) =
obs
di
“interpolant”
at the observation points
and
roughness of d(t) = 0
in between the observation points
advantage
interpolant d(t) is an analytic function that is known
everywhere
disadvantage
the observation points are singled out as special
advantage
interpolant d(t) is an analytic function that is known
everywhere
can evaluate d(t) at any time, t
can differentiate d(t), integrate it, etc.
disadvantage
the observation points are singled out as special
d(t) behaves differently at the observation
points than between them
the interpolation problem
find an interpolant
d(t)
that goes through all the data points
and
“does something sensible”
or
“satisfies some prior information”
between them
some obvious ideas don’t work at all
an (N-1) order polynomial can easily be constructed to
that it passes through N points
so use a polynomial for d(t)
example
d(t)
time, t
example
d(t)
what happened here?
time, t
and here?
solution
a low-order polynomial
has less potential for wild swings
so use many low-order polynomial
each valid in a small time interval
such a function is called a “spline”
simplest case
set of linear polynomials
each valid between two data points
“connect the data points with straight lines”
d
d(t)
ti
ti+1
t
advantages
conceptually very simple
always get what you expect
zero roughness between observations
disadvantage
d(t) has kinks at observation points
example
5
d(t) 0
-5
0
0.1
0.2
0.3
0.4
0.5
time, t
0.6
0.7
0.8
0.9
1
example
5
kink
d(t) 0
-5
0
0.1
0.2
0.3
0.4
0.5
time, t
0.6
0.7
0.8
0.9
1
in MatLab
observations
interpolated
observations
times of
interpolation
getting rid of the kinks
use cubic polynomials
Si(t) = c0 + c1 t + c2 t2 + c3 t3
each valid between two data points
cubic polynomial has 4 coefficients
two constrained by need to pass through two data
two to implement prior information
no kinks in d(t) or its first derivative
the trick
second derivative
of cubic
is linear
so use linear interpolation formula
for second derivative
2nd derivative
yi+1
yi
yi-1
ti-1
ti ti+1
t
2nd derivative
yi+1
yi
yi-1
ti-1
ti ti+1
t
the second derivative at the
observation points, denoted yi,
become an unknown in the problem
the second derivative is now integrated
twice to give the spline function
here ai and bi are two more unknowns that
arise from the integration constants
finally
one finds the y’s, a’s and b’s
so that the spline
1. goes through the observations
and
2. has a first derivative that is continuous
across the observation points
the solution involves solving a matrix
equation for the unknowns
(see text for details)
in MatLab
observations
interpolated
observations
times of
interpolation
example
d(t)
time, t
example
no kinks
d(t)
time, t
interpolation involves
prior information of smoothness
in generalized least-squares
the prior information of smoothness is quantified by
a roughness matrix, H
Hm
then we minimize the overall roughness, which is
to say the overall error in the prior information
(Hm)T (Hm)
note that
(Hm)T (Hm) = mT (HTH) m
but in generalized error also has the form
mT Cm-1 m
where Cm-1 is a covariance matrix
so in this case
Cm = (HTH)-1
so the prior information that the data are smooth
is equivalent to the requirement that they have a
specific covariance matrix
which for stationary time series is equivalent to saying
that they have a specific autocorrelation function
so an alternative, more flexible way of
interpolating data
is by specifying the autocorrelation function
that we want the results to have
this is called Kriging
(after Danie G Krige, its inventor)
Kriging
estimate data at arbitrary time t0
determine weights w
by
minimizing the variance of
with respect to wi
we’ll find that we don’t need to know d0true
only its autocorrelation
j
assuming
and
means approximately cancel
j
assuming
and
means approximately cancel
expand square
j
assuming
and
means approximately cancel
expand square
insert weighted average formula
j
assumming
and
means approximately cancel
expand square
insert weighted average formula
j
identify terms proportional to autocorrelation
assumming
and
now differentiate with respect to the weight, wk
which yields the matrix equation
Mw = v
now differentiate with respect to the weight, wk
which yields the matrix equation
Mw = v
note that the autocorrelation appears on both sides of the
equation, so that its overall normalization cancels out
all we need now do is specify an
autocorrelation function
for example
we could use the Normal function
the variance, L2, controls the width of the autocorrelation
and hence the smoothness of the interpolation
In MatLab
observations: tobs, dobs
interpolated values: test, dest
Normal autocorrelation function with variance L2
Example
A) Kriging
B) Generalized Least
Squares
2
2
1.5
1
1
d(t)
d(t 0
)
d(t)
0
d
d
0.5
-0.5
-1
-1
-1.5
-2
0
20
40
60
time,
t
x
80
100
-2
0
10
20
30
40
50
x
60
time, t
70
80
90
100
Interpolation in two-dimensions
construct an interpolant
d(x,y)
that goes through the observations
and
does something sensible in between
notion of bracketing observations
more complicated
d
y0
t
t0
1 dimensions
y0
x0
2 dimensions
x
notion of bracketing observations
more complicated
y
d
triangular
tile
ti
ti+1
t0
1 dimensions
t
y0
segment of
t-axis
x0
2 dimensions
x
Delaunay triangles
set of most equilateral triangles connecting data
points
B) Delaunay data
triangles
A) Observations
data
0
5
5
10
10
15
15
20
x 20
25
25
30
30
35
35
40
40
x
x
x
0
0
5
10
15
20
y
y
25
30
35
40
0
5
10
15
20
y
y
25
30
35
40
B) Delaunay data
triangles
A) Observations
data
0
5
5
10
10
15
15
20
x 20
25
25
30
30
35
35
40
40
x
x
x
0
0
5
10
15
20
y
y
25
30
35
40
0
5
10
15
20
y
25
30
35
y
triangle enclosing
a point of interest
40
C) Linear linear
Splines
interpolation
0
5
5
10
10
15
15
20
x 20
25
25
30
30
35
35
40
40
x
x
x
D) Cubic cubic
Splines
interpolation
0
0
5
10
15
20
y
y
25
30
35
40
0
5
10
15
y
20
y
25
30
35
40
In MatLab
linear splines
cubic splines