Transcript ICS 278: Data Mining Lecture 1: Introduction to Data Mining

ICS 278: Data Mining
Lecture 3: Exploratory Data Analysis
and Visualization
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Lecture 3
• Finish up material from Lecture 2
• Homework due this Thursday
• Discuss projects in some detail
• Exploratory Data Analysis and Visualization
– Reading: Chapter 3 in the text
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Exploratory Data Analysis (EDA)
• get a general sense of the data
• interactive and visual
– (cleverly/creatively) exploit human visual power to see patterns
• 3 to 5 dimensions (e.g. spatial, color, time, sound)
– e.g. plot raw data/statistics, reduce dimensions as needed
• data-driven (model-free)
• especially useful in early stages of data mining
– detect outliers
(e.g. assess data quality)
– test assumptions (e.g. normal distributions?)
– identify useful raw data & transforms (e.g. log(x))
• http://www.itl.nist.gov/div898/handbook/eda/eda.htm
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Summary Statistics
• not visual
• sample statistics of data X
–
–
–
–
mean:  = i Xi / n
{  minimizes i (Xi - )2 }
mode: most common value in X
median: X=sort(X), median = Xn/2 (half below, half above)
quartiles of sorted X: Q1 value = X0.25n , Q3 value = X0.75 n
• interquartile range: value(Q3) - value(Q1)
• range:
max(X) - min(X) = Xn - X1
– variance: 2 = i (Xi - )2 / n
– skewness: i (Xi - )3 / [ (i (Xi - )2)3/2 ]
• zero if symmetric; right-skewed more common (e.g. us … Gates)
– number of distinct values for a variable (see unique.m in MATLAB)
– Note: all of these are estimates based on the sample at hand – they
may be different from the “true” values (e.g., median age in US).
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Exploratory Data Analysis
Tools for Displaying Single Variables
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Histogram
•
Most common form: split data range into equal-sized bins Then for each
bin, count the number of points from the data set that fall into the bin.
–
–
•
Vertical axis: Frequency (i.e., counts for each bin)
Horizontal axis: Response variable
The histogram graphically shows the following:
1. center (i.e., the location) of the data;
2. spread (i.e., the scale) of the data;
3. skewness of the data;
4. presence of outliers; and
5. presence of multiple modes in the data.
These features can provide useful information of both
- the proper distributional model for the data
-
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Issues with Histograms
• For small data sets, histograms can be misleading. Small changes in
the data or to the bucket boundaries can result in very different
histograms.
• For large data sets, histograms can be quite effective at illustrating
general properties of the distribution.
• example
• Can smooth histogram using a variety of techniques
– E.g., kernel density estimation (pages 59-61 in text)
• Histograms effectively only work with 1 variable at a time
– Difficult to extend to 2 dimensions, not possible for >2
– So histograms tell us nothing about the relationships among variables
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Histogram Example
classical bell-shaped, symmetric histogram with most of the frequency
counts bunched in the middle and with the counts dying off out in the
tails. From a physical science/engineering point of view, the
Normal/Gaussian distribution often occurs in nature (due in part to the
central limit theorem).
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
ZipCode Data: Population
900
8000
K = 50
7000
K = 500
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x 10
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K = 50
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Lecture 3: EDA and Visualization
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Padhraic Smyth, UC Irvine
ZipCode Data: Population
• MATLAB code:
X = zipcode_data(:,2)
% second column from zipcode array
histogram(X, 50)
% histogram of X with 50 bins
histogram(X, 500)
% 500 bins
index = X < 5000;
% identify X values lower than 5000
histogram(X(index),50) % now plot just these X values
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Histogram Detecting Outlier (Missing Data)
blood pressure = 0 ?
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Right Skewness Example: Credit Card Usage
similarly right-skewed are
Power law distributions
(Pi ~ 1/ia, where a >= 1)
e.g. for a = 1 we have “Zipf’s law”
For word frequencies in text
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Box (and Whisker) Plots: Pima Indians Data
plots
all data
outside
whiskers
Q3-Q1
box contains middle 50% of data
up to
1.5 x
Q3-Q1
Q2
(median)
healthy
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(or shorter,
if no data
that far
above Q3)
diabetic
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Time Series Example 1
annual fees introduced in UK
(many users cutback to 1 credit card)
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Time Series Example 2
summer bifurcations in air travel
(favor early/late)
summer
peaks
New Year bumps
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steady growth
trend
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Time-Series Example 3
mean weight vs mean age
for 10k control group
Scotland experiment:
“ milk in kid diet  better health” ?
20,000 kids:
5k raw, 5k pasteurize,
10k control (no supplement)
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Possible explanations:
Would expect smooth weight growth plot.
Visually reveals
unexpected pattern (steps),
not apparent from raw data table.
Lecture 3: EDA and Visualization
Grow less early in year than later?
No steps in height plots; so why
height  uniformly, weight  spurts?
Kids weighed in clothes: summer
garb lighter than winter?
Padhraic Smyth, UC Irvine
Exploratory Data Analysis
Tools for Displaying Pairs of Variables
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
A simple data set
Data
X 10.00 8.00 13.00 9.00 11.00 14.00 6.00 4.00 12.00 7.00 5.00
Y 8.04 6.95 7.58 8.81 8.33 9.96 7.24 4.26 10.84 4.82 5.68
Anscombe, Francis (1973), Graphs in Statistical Analysis,
The American Statistician, pp. 195-199.
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
A simple data set
Data
X 10.00 8.00 13.00 9.00 11.00 14.00 6.00 4.00 12.00 7.00 5.00
Y 8.04 6.95 7.58 8.81 8.33 9.96 7.24 4.26 10.84 4.82 5.68
Summary Statistics
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
A simple data set
Data
X 10.00 8.00 13.00 9.00 11.00 14.00 6.00 4.00 12.00 7.00 5.00
Y 8.04 6.95 7.58 8.81 8.33 9.96 7.24 4.26 10.84 4.82 5.68
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
3 more data sets
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X2
Y2
X3
Y3
X4
Y4
10.00
9.14
10.00
7.46
8.00
6.58
8.00
8.14
8.00
6.77
8.00
5.76
13.00
8.74
13.00
12.74
8.00
7.71
9.00
8.77
9.00
7.11
8.00
8.84
11.00
9.26
11.00
7.81
8.00
8.47
14.00
8.10
14.00
8.84
8.00
7.04
6.00
6.13
6.00
6.08
8.00
5.25
4.00
3.10
4.00
5.39
19.00
12.50
12.00
9.13
12.00
8.15
8.00
5.56
7.00
7.26
7.00
6.42
8.00
7.91
5.00
4.74
5.00
5.73
8.00
6.89
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Summary Statistics
Summary Statistics of Data Set 2
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Summary Statistics
Summary Statistics of Data Set 2
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
Summary Statistics of Data Set 3
Summary Statistics of Data Set 4
N = 11
N = 11
Data Mining Lectures
Lecture 3: EDA and Visualization
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
Padhraic Smyth, UC Irvine
Graphs reveals the mystery!
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Displaying high-dimensional data
• multiple bivariate graphs
– scatter plot matrix
– trellis plot
• Icon plots
– star graph
– Chernoff’s faces
• Parallel coordinates
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
2D: Scatter Plots
• standard tool for displaying relationship between two variables
• A scatter plot is a plot of the values of Y versus the corresponding
values of X:
– Vertical axis: variable Y--usually the response variable
– Horizontal axis: variable X--variable we suspect may be related
• Scatter plots can provide answers to the following questions:
1.
2.
3.
4.
Are variables X and Y related?
Are variables X and Y linearly related?
Are variables X and Y non-linearly related?
Does the variation in Y change
depending on X?
5. Are there outliers?
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Scatter Plot: No relationship
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Scatter Plot: Linear relationship
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Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Scatter Plot: Quadratic relationship
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Scatter plot: Homoscedastic
Variation of Y Does Not Depend on X
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Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Scatter plot: Heteroscedastic
variation in Y differs depending on the value of X
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
2D Scatter Plots
•
standard tool to display relation
between 2 variables
– e.g. y-axis = response, x-axis =
suspected indicator
•
credit card repayment: low-low, high-high
useful to answer:
– x,y related?
• no
• linearly
• nonlinearly
– variance(y) depend on x?
– outliers present?
•
MATLAB:
– plot(X(1,:),X(2,:),’.’);
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
5
2.5
x 10
MEDIAN
HOUSEHOLD
INCOME
2
1.5
1
0.5
0
0
2
4
6
8
10
MEDIAN PERCAPITA INCOME
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Lecture 3: EDA and Visualization
12
14
4
x 10
Padhraic Smyth, UC Irvine
Problems with Scatter Plots of Large Data
appears: later apps older; reality: downward slope (more apps, more variance)
96,000 bank loan applicants
scatter plot degrades into black smudge ...
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Contour Plots Can Help
recall:
(same 96,000 bank loan apps as before)
shows variance(y)  with x 
is indeed due to horizontal
skew in density
unimodal
skewed

skewed 
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Problems with Scatter Plots of Large Data
# weeks credit card buys gas vs groceries
(10,000 customers)
actual correlation (0.48) higher than appears (overprinting)
also demands explanation
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Exploratory Data Analysis
Tools for Displaying Pairs of Variables
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Scatter Plot Matrix
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Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Trellis Plot
Older
Younger
Male
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Female
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Star Plots: Using Icons to Encode Information
•
•
1
2
3
4
Price
Mileage (MPG)
1978 Repair Record (1 = Worst, 5 = Best)
1977 Repair Record (1 = Worst, 5 = Best)
5
6
7
8
Headroom
Rear Seat Room
Trunk Space
Weight
Each star represents a single
observation. Star plots are used to
examine the relative values for a
single data point
The star plot consists of a
sequence of equi-angular spokes,
called radii, with each spoke
representing one of the variables.
•
Useful for small data sets with up
to 10 or so variables
•
Limitations?
–
–
Small data sets, small dimensions
Ordering of variables may affect
perception
9 Length
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Chernoff’s Faces
•
described by ten facial characteristic parameters: head eccentricity, eye
eccentricity, pupil size, eyebrow slant, nose size, mouth shape, eye spacing,
eye size, mouth length and degree of mouth opening
• Chernoff faces applet
• more icon plots
• Limitations:
– Similar to star plots
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Parallel Coordinates
(epileptic seizure data again)
1 (of n)
cases
dimensions
(possibly all d of them!)
(this case is
a “brushed”
one, with a
darker line,
to standout
from the n-1
other cases)
often (re)ordered
to better distinguish
among interesting
subsets of n total cases
interactive
“brushing” is useful
for seeing such
distinctions
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
“Grand Tour”
• scatter plot matrix only multi-bivariate
• can achieve richer multivariate visualization by:
– rotate direction of projection over all d (not just pick two)
– user control over spin
– random projection (“Grand Tour”)
• e.g. XGOBI visualization package (available on the Web)
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine
Summary
• EDA and Visualization
– Can be very useful for
• data checking
• getting a general sense of individual or pairs of variables
– But…
• do not necessarily reveal structure in high dimensions
• Reading: Chapter 3
Data Mining Lectures
Lecture 3: EDA and Visualization
Padhraic Smyth, UC Irvine