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Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
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
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K = 50
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0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
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Data Mining Lectures Lecture 3: EDA and Visualization 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
Q3-Q1
box contains middle 50% of data
Q2 (median)
healthy diabetic
plots all dataoutside
whiskers
up to1.5 x Q3-Q1
(or shorter,if no datathat far
above Q3)
Data Mining Lectures 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
steady growth trendNew Year bumps
summer peaks
summer bifurcations in air travel (favor early/late)
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Time-Series Example 3
Scotland experiment:“ milk in kid diet better health” ?
20,000 kids: 5k raw, 5k pasteurize,
10k control (no supplement)
mean weight vs mean agefor 10k control group
Would expect smooth weight growth plot.
Visually reveals unexpected pattern (steps),
not apparent from raw data table.
Possible explanations:
Grow less early in year than later?
No steps in height plots; so whyheight uniformly, weight spurts?
Kids weighed in clothes: summer garb lighter than winter?
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
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 = 11Mean of X = 9.0Mean of Y = 7.5Intercept = 3Slope = 0.5Residual standard deviation = 1.237Correlation = 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
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
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Summary Statistics
Summary Statistics of Data Set 2
N = 11Mean of X = 9.0Mean of Y = 7.5Intercept = 3Slope = 0.5Residual standard deviation = 1.237Correlation = 0.816
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Summary Statistics
Summary Statistics of Data Set 2
N = 11Mean of X = 9.0Mean of Y = 7.5Intercept = 3Slope = 0.5Residual standard deviation = 1.237Correlation = 0.816
Summary Statistics of Data Set 3
N = 11Mean of X = 9.0Mean of Y = 7.5Intercept = 3Slope = 0.5Residual standard deviation = 1.237Correlation = 0.816
Summary Statistics of Data Set 4
N = 11Mean of X = 9.0Mean of Y = 7.5Intercept = 3Slope = 0.5Residual standard deviation = 1.237Correlation = 0.816
Data Mining Lectures Lecture 3: EDA and Visualization 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. Are variables X and Y related? 2. Are variables X and Y linearly related? 3. Are variables X and Y non-linearly related? 4. 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
Data Mining Lectures 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
Data Mining Lectures 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
• useful to answer:– x,y related?
• no• linearly• nonlinearly
– variance(y) depend on x?– outliers present?
• MATLAB:– plot(X(1,:),X(2,:),’.’);
credit card repayment: low-low, high-high
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
0 2 4 6 8 10 12 14
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MEDIAN PERCAPITA INCOME
MEDIANHOUSEHOLD INCOME
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Problems with Scatter Plots of Large Data
96,000 bank loan applicants
appears: later apps older; reality: downward slope (more apps, more variance)
scatter plot degrades into black smudge ...
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Contour Plots Can Help
(same 96,000 bank loan apps as before)
recall:
unimodal
skewed
shows variance(y) with x is indeed due to horizontalskew in density
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
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Trellis Plot
Younger
Older
Male Female
Data Mining Lectures Lecture 3: EDA and Visualization Padhraic Smyth, UC Irvine
Star Plots: Using Icons to Encode Information
• 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
1 Price 2 Mileage (MPG) 3 1978 Repair Record (1 = Worst, 5 =
Best) 4 1977 Repair Record (1 = Worst, 5 =
Best)
5 Headroom 6 Rear Seat Room 7 Trunk Space 8 Weight
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
interactive“brushing” is useful
for seeing such distinctions
dimensions(possibly all d of them!)
often (re)orderedto better distinguishamong interesting
subsets of n total cases
(epileptic seizure data again)
1 (of n) cases
(this case isa “brushed”one, with a darker line,to standout from the n-1other cases)
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