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Data Analysis
Thomas Hughes
Data Science – ITEC/CSCI/ERTH-4350/6350
Week 4, September 22, 2015
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Contents
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Preparing for data analysis
Completing and presenting results
Statistics,
Distributions
Filtering, etc.
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Types of data
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Data types
• Time-based, space-based, image-based, …
• Encoded in different formats
• May need to manipulate the data, e.g.
– In our Data Mining tutorial and conversion to
ARFF
– Coordinates
– Units
– Higher order, e.g. derivative, average
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Induction or deduction?
• Induction: The development of theories from
observation
– Qualitative – usually information-based
• Deduction: The testing/application of theories
– Quantitative – usually numeric, data-based
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Accurate vs. Precise
http://climatica.org.uk/climate-science-information/uncertainty
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‘Signal to noise’
• Understanding accuracy and precision
– Accuracy
– Precision
• Affects choices of analysis
• Affects interpretations (g-i-g-o)
• Leads to data quality and assurance
specification
• Signal and noise are context dependent
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Other considerations
• Continuous or discrete
• Underlying reference system
• Oh yeah: metadata standards and
conventions
• The underlying data structures are important
at this stage but there is a tendency to read in
partial data
– Why is this a problem?
– How to ameliorate any problems?
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Outlier
• An extreme, or atypical, data value(s) in a
sample.
• They should be considered carefully, before
exclusion from analysis.
• For example, data values maybe recorded
erroneously, and hence they may be
corrected.
• However, in other cases they may just be
surprisingly different, but not necessarily
'wrong'.
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Special values in data
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Fill value
Error value
Missing value
Not-a-number
Infinity
Default
Null
Rational numbers
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Gaussian Distributions
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Spatial example
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Spatial roughness…
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Statistics
• We will most often use a Gaussian
distribution (aka normal distribution, or bellcurve) to describe the statistical properties of
a group of measurements.
• The variation in the measurements taken
over a finite spatial region may be caused by
intrinsic spatial variation in the measurement,
by uncertainties in the measuring method or
equipment, by operator error, ...
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Mean and standard deviation
• The mean, m, of n values of the
measurement of a property z (the average).
– m = [ SUM {i=1,n} zi ] / n
• The standard deviation s of the
measurements is an indication of the amount
of spread in the measurements with respect
to the mean.
– s2 = [ SUM {i=1,n} ( zi - m )2 ] /n
• The quantity s2 is known as the variance of
the measurements.
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Width of distribution
• If the data are truly distributed in a Gaussian
fashion, 65% of all the measurements fall
within one s of the mean: i.e. the condition
–s-m<z<s+m
• is true about 2/3 of the time.
• Accordingly, the more spread the
measurements are away from the mean, the
larger s will be.
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Measurement description
– by its mean and standard deviation.
• Often a measurement at a sampling point is made
several times and these measurements are
grouped into a single one, giving the statistics.
• If only a single measurement is made (due to cost
or time), then we need to estimate the standard
deviation in some way, perhaps by the known
characteristics of our measuring device.
• An estimate of the standard deviation of a
measurement is more important than the
measurement itself.
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Weighting
• In interpolation, the data are often weighted
by the inverse of the variance ( w = s-2 ) when
used in modeling or interpolations. In this
way, we place more confidence in the betterdetermined values.
• In classifying the data into groups, we can do
so according to either the mean or the scatter
or both.
• Excel has the built-in functions AVERAGE
and STDEV to calculate the mean and
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standard deviation for a group of values.
More on interpolation
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Global/ Local Methods
• Global methods ~ in which all the known data
are considered
• Local methods ~ in which only nearby data
are used.
• Local methods and most often the global
methods also rely on the premise that nearby
points are more similar than distant points.
• Inverse Distance Weighting (IDW) is an
example of a global method.
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More…
• Local methods include bilinear interpolation
and planar interpolation within triangles
delineated by 3 known points.
• Global Surface Trends: Fitting some form of a
polynomial to data to predict values at unsampled points.
• Such fitting is done by regression – estimates
of coefficients by least-squares fit to data.
– Produces a continuous field
– Continuous first derivatives
– Values NOT reproduced exactly at observation points
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Geospatial means x and y
• In two spatial dimensions (map view x-y
coordinates) the polynomials take the form:
– f(x, y) = SUM r+s <= p ( brs xr ys )
• where b represents a series of coefficients
and p is the order of the polynomial trend
surface.
• The summation is over all possible positive
integers r and s such that their sum is less
than or equal to the polynomial order p.
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p=1 / p=2
• For example, if p =1, then
– f(x, y) = b00 + b10 x + b01 y
– which is the equation of a plane.
• If p = 2, then
– f(x, y) = b00 + b10 x + b01 y + b11 x y + b20 x2 + b02
y2
•
For a polynomial order p the number of coefficients is (p+1)(p+2)/2.
In trend analysis or smoothing, these polynomials are estimated by
regression.
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Regression
• Is the process of finding the coefficients that
produce the best-fit to the observed values.
• Best-fit is generally described as minimizing
the squares of the misfits at each point, that
is,
– SUM {i=1,n} [ fi(x, y) – zi(x, y) ]2
• i.e. it is minimized by the choice of
coefficients (this minimization is commonly
called least-squares).
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Coefficients
• To estimate the coefficients we need at least
as many or preferably more observations as
coefficients. Otherwise? Underdetermined!
• Once we estimate the coefficients, the
surface trend is defined everywhere.
• NB. The Excel function LINEST can be used
to solve for the coefficients.
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Choices…
• The choice of how many coefficients to use
(the order of the polynomial) depends on how
smooth you think the variations in the
property is, and on how well the data are fit
by lower order polynomials.
• In general, adding coefficients always
improves the fit to the data to the extreme
that if the number of coefficients equals the
number of observations, the data can be fit
perfectly.
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• But this assumes that the data are perfect.
Multi-variate analysis
• Multivariate analysis is the procedure to use if
we want to see if there is a correlation
between any pair of attributes in our data.
• As earlier, you perform a linear regression to
find the correlations.
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Example – gis/data/MULTIVARIATE.xls
City
A
B
C
D
E
F
G
%_college
10.81
11.12
11.11
13.67
14.57
13.09
12.07
Income
32402
24013
45765
48231
24756
24474
43989
Population Homeowners
45902
17427
23853
20226
21170
9016
39707
12052
60872
39397
54764
5028
77830
41578
area
171456
100132
85838
129907
60162
99641
90
158714
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Multivariate analysis is the procedure to use if we want to see if
there is a correlation between any pair of attributes in our data.
As earlier, we will perform a linear regression to find the
correlations.
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Analysis – i.e. Science question
• We want to see if there is a correlation
between the percent of the college-educated
population and the mean Income, the overall
population, the percentage of people who
own their own homes, and the population
density.
• To do so we solve the set of 7 linear
equations of the form:
• %_college = a x Income + b x Population + c
x Homeowners/Population + d x
Population/area + e
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• We solve for for the coefficients a through e.
• This is done with Excel with the LINEST
function, giving the result:
Pop_density Homeowners Population
Incomes
constant
d
c
b
a
e
5.559033
-1.4858663
-1.73E-05
3.47E-05
10.15676 Coefficients
2.811892
2.26476261
3.57E-05
5.64E-05
2.895513 uncertainties
– Revealing that population density correlates with
college-educated percentage at a significant
level.
– => college-educated people prefer to live in
densely populated cities.
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Bi-linear Interpolation
• In two-dimensions we can interpolate
between points in a regular or nearly regular
grid.
• This interpolation is between 4 points, and
hence it is a local method.
– Produces a continuous field
– Discontinuous first derivative
– Values reproduced exactly at grid points
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Example
x0,y0
t = [ x0 – x1 ] / [ x2 - x1 ]
and
u = [ y0 – y1 ] / [ y4 - y1 ]
• The red squares represent 4 known values of z(x, y)
and our goal is to estimate the value of z at the new
point (blue circle) at (x0, y0).
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Calculating…
• Let
• t = [ x0 – x1 ] / [ x2 - x1 ] and
• u = [ y0 – y 1 ] / [ y 4 - y 1 ]
i.e. the fractional distances the new point is
along the grid axes in x and y, respectively,
where the subscripts refer to the known
points as numbered above.
Then
• z (x0 , y0 ) = (1-t) (1-u) z1 + t (1-u) z2 + t u z3 +
(1-t ) u z4
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Bilinear interpolation for a central point
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Bilinear interpolation of 4 unequal corner points.
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Lines connecting grid points are straight but diagonals are curved.
Bilinear interpolation -> a curvature of the surface within the grid.
Other interpolation
• Delaunay triangles: sampled points are
vertices of triangles within which values form
a plane.
• Thiessen (Dirichlet / Voronoi) polygons: value
at unknown location equals value at nearest
known point.
• Splines: piece-wise polynomials estimated
using a few local points, go through all known
points.
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More …
• Bicubic interpolation
– Requires knowing z (x, y) and slopes dz/dx,
dz/dy, d2z/dxdy at all grid points.
• Points and derivatives reproduced exactly at grid
points
• Continuous first derivative
• Bicubic spline
– Similar to bicubic interpolation but splines are
used to get derivatives at grid points.
• Do some reading on these… will be important 37
for future assignments.
Spatial analysis of continuous fields
• Filtering (Smoothing = low-pass filter)
• High-pass filter is the image with the low-pass
(i.e. smoothing) removed
• One-dimension; V(i) = [ V(i-1) + 2 V(i) +
V(i+1) ] /4 another weighted average
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• Square window (convolution, moving window)
• New value for V is weighted average of points
within specified window.
– Vij = f [ SUM k=i-m, i+m SUM l=j-n, j+n Vkl wkl ] /
SUM wkl ,
– f = operator
– w = weight
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• Each cell can have same or different weight
but typically SUM wkl = 1. For equal
weighting, if n x m = 5 x 5 = 25, then each w
= 1/25.
• Or weighting can be specified for each cell.
For example for 3x3 the weight array might
be:
1/15
2/15
1/15
2/15
3/15
2/15
1/15
2/15
1/15
So Vij = [ Vi-1,j-1 + 2Vi,j-1 + Vi+1,j-1 + 2Vi-1,j + 3Vi,j + 2Vi+1,j
+Vi-1,j+1 +2Vi,j+1 +Vi+1,j+1 ] /15
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Low pass =smoothing
High pass – smoothing removed
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Low pass =smoothing
Modal filters
• The value or type at center cell is the most
common of surrounding cells.
• Example 3x3:
• AABCADCABB
• A B C A C B C B A C -> A A A C C C B B B
• BAACBCBBBA
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Or
• You can use the minimum, maximum, or
range. For example the minimum:
• AABCADCABB
• A B C A C B C B A C -> A A A A A A A A A
• BAACBCBBBA
– No powerpoint animation hell…
• Note - Because it requires sorting the values
in the window, it is a computationally
intensive task, the modal filter is considerably
less efficient than other smoothing filters.
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Median filter
• Median filters can be used to emphasize the longerrange variability in an image, effectively acting to
smooth the image.
• This can be useful for reducing the noise in an
image. The algorithm operates by calculating the
median value (middle value in a sorted list) in a
moving window centered on each grid cell.
• The median value is not influenced by anomalously
high or low values in the distribution to the extent
that the average is.
• As such, the median filter is far less sensitive to
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shot noise in an image than the mean filter.
Compare median, mean, mode
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Median filter
• Because it requires sorting the values in the window, a
computationally intensive task, the median filter is
considerably less efficient than other smoothing filters.
• This may pose a problem for large images or large
neighborhoods.
• Neighborhood size, or filter size, is determined by the userdefined x and y dimensions. These dimensions should be
odd, positive integer values, e.g. 3, 5, 7, 9...
• You may also define the neighborhood shape as either
squared or rounded.
• A rounded neighborhood approximates an ellipse; a rounded
neighborhood with equal x and y dimensions approximates a
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circle.
Slopes
• Slope is the first derivative of the surface;
aspect is the direction of the maximum
change in the surface.
• The second derivatives are called the profile
convexity and plan convexity.
• For surface the slope is that of a plane
tangent to the surface at a point.
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Gradient
• The gradient, which is a vector written as del
V, contains both the slope and aspect.
– del V = ( dV/dx, dV/dy )
• For discrete data we often use finite
differences to calculate the slope.
• In the plot above the first derivative at Vij
could be taken as the slope between points at
i-1 and i+1.
– d Vij / d x = ( Vi+1,j – Vi-1,j ) / (2 dx)
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Second derivative
• … is the slope of the slope. We take the
change in slope between i+1 and i, and
between i and i-1.
d2V / dx2 = [ ( Vi+1,j – Vi,j ) / dx - ( Vi,j – Vi-1,j ) / dx ] /
dx
• The slope, which is the magnitude of del V,
is:
| del V | = [ (d V / d x )2 + ( d V / d y )2 ]1/2
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End of Part I
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Summary
• Purpose of analysis should drive the type that
is conducted
• Many constraints due to prior management of
the data
• Become proficient in a variety of methods,
tools
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Reading
• Reading this week, will span week 7 lecture
(Data Analysis II)
• No reading discussion in Week 5 or 6
• Note reading for week 7 – data sources for
project definitions
– There is a lot of material to review
– Might be worth reviewing it before week 7
• Why – week 7 defines the group projects,
come familiar with the data out there!
Working with someone else's data
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What you tripped over
• Security – theft versus integrity
• Interoperability – using .xls or .accdb
• Logical collections (please notice plural) – did
pretty well!
• Specific versus generic (need details) mostly
well done
• Not enough searching on data formats,
metadata, standards, etc.
• Naming the assignment, seriously 
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Data Collection Minimums
• 100 data points or more
• Think15 % more
• Is too many too much?
– Subset your data
– Remember provenance
• Don’t forget references!
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Practical details for week 5
• The preparation for collection is Assignment 1
which is the theoretical exercise
• Week 5 will be to see how much of this
translates into practice
• Ground rules – must attend the start of class
– do ONE of your data collections
– No one off collections, i.e. must be something
you could repeat
– This is an individual exercise, you will see what
others have done in week 6 class
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Practical details for week 5 (ctd)
• A write up is required, details in Assignment 2
• No “analysis” is required but you will need to
present your data (week 6) so interpretation
may be required
• Sources??
– Images
– Sound
– Existing devices, sensors
– Others?
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Hosting data
• Access to a computer to place data?
– http
– ftp
– Dropbox ;-)
– USB drive?
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