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Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing

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Abstract

Datasets which are identical over a number of statistical properties, yet produce dissimilar graphs, are frequently used to illustrate the importance of graphical representations when exploring data. This paper presents a novel method for generating such datasets, along with several examples. Our technique varies from previous approaches in that new datasets are iteratively generated from a seed dataset through random perturbations of individual data points, and can be directed towards a desired outcome through a simulated annealing optimization strategy. Our method has the benefit of being agnostic to the particular statistical properties that are to remain constant between the datasets, and allows for control over the graphical appearance of resulting output.
Same Stats, Different Graphs:
Generating Datasets with Varied Appearance and
Identical Statistics through Simulated Annealing
Justin Matejka and George Fitzmaurice
Autodesk Research, Toronto Ontario Canada
{first.last}@autodesk.com
Figure 1. A collection of data sets produced by our technique. While different in appearance, each has the same summary statistics
(mean, std. deviation, and Pearson’s corr.) to 2 decimal places. (x
͞ =54.02, y
͞ = 48.09, sdx = 14.52, sdy = 24.79, Pearson’s r = +0.32)
ABSTRACT
Datasets which are identical over a number of statistical
properties, yet produce dissimilar graphs, are frequently used
to illustrate the importance of graphical representations when
exploring data. is paper presents a novel method for
generating such datasets, along with several examples. Our
technique varies from previous approaches in that new
datasets are iteratively generated from a seed dataset through
random perturbations of individual data points, and can be
directed towards a desired outcome through a simulated
annealing optimization strategy. Our method has the benefit
of being agnostic to the particular statistical properties that
are to remain constant between the datasets, and allows for
control over the graphical appearance of resulting output.
INTRODUCTION
Anscome’s Quartet [1] is a set of four distinct datasets each
consisting of 11 (x,y) pairs where each dataset produces the
same summary statistics (mean, standard deviation, and
correlation) while producing vastly different plots (Figure
2A). is dataset is frequently used to illustrate the
importance of graphical representations when exploring
data. e effectiveness of Anscombe’s Quartet is not due to
simply having four different data sets which generate the
same statistical properties, it is that four clearly dierent and
identifiably distinct datasets are producing the same
statistical properties. Dataset I appears to follow a somewhat
noisy linear model, while Dataset II is following a parabolic
distribution. Dataset III appears to be strongly linear, except
for a single outlier, while Dataset IV forms a vertical line
with the regression thrown off by a single outlier. In contrast,
Figure 2B shows a series of datasets also sharing the same
summary statistics as Anscombe’s Quartet, however without
any obvious underlying structure to the individual datasets,
this quartet is not nearly as effective at demonstrating the
importance of graphical representations.
While very popular and effective for illustrating the
importance of visualizations, it is not known how Anscombe
came up with his datasets [5]. Our work presents a novel
method for creating datasets which are identical over a range
of statistical properties, yet produce dissimilar graphics. Our
method differs from previous by being agnostic to the
particular statistical properties that are to remain constant
between the datasets, while allowing for control over the
graphical appearance of resulting output.
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DOI: http://dx.doi.org/10.1145/3025453.3025912
Figure 2. (A) Anscombe’s Quartet, with each dataset having
the same mean, standard deviation, and correlation. (B)
Four unstructured datasets, each also having the same
statistical properties as those in Anscombe’s Quartet.
A
B
I II III IV
RELATED WORK
As alluded to above, producing multiple datasets with similar
statistics and dissimilar graphics was introduced by
Anscombe in 1973 [1]. “Graphs in Statistical Analysis” starts
by listing three notions prevalent about graphs at the time:
(1) Numerical calculations are exact, but graphs are
rough;
(2) For any particular kind of statistical data there
is just one set of calculations constituting a
correct statistical analysis;
(3) Performing intricate calculations is virtuous,
whereas actually looking at the data is cheating.
While one cannot argue that there is currently as much
resistance towards graphical methods as when Anscombe's
paper was originally published, the datasets described in the
work (Figure 1A) are still effective and frequently used for
introducing or reinforcing the importance of visual methods.
Unfortunately, Anscombe does not report how the datasets
were created, nor suggest any method to create new ones.
e first attempt at producing a generalized method for
creating such datasets was published in 2007 by Chatterjee
and Firat [5]. ey proposed a genetic algorithm based
approach where 1,000 random datasets were created with
identical summary statistics, then combined and mutated
with an objective function to maximize the “graphical
dissimilarity” between the initial and final scatter plots.
While the datasets produced were graphically dissimilar to
the input datasets, they did not have any discernable structure
in their composition. Our technique differs by providing a
mechanism to direct the solutions towards a specific shape,
as well as allowing for variety in the statistical measures
which are to remain constant between the solutions.
Govindaraju and Haslett developed a method for regressing
datasets towards their sample means while maintaining the
same linear regression formula [7]. In 2009, the same authors
extended their procedure to creating “cloned” datasets [8]. In
addition to maintaining the same linear regression as the seed
dataset, their cloned datasets also maintained the same means
(but not the same standard deviations). While Chatterjee and
Firat [5] wanted to create datasets as graphically dissimilar
as possible, Govindaraju and Haslett’s cloned datasets were
designed to be visually similar, with a proposed application
of confidentializing sensitive data for publication purposes.
While our technique is primarily aimed at creating visually
distinct datasets, by choosing appropriate statistical tests to
remain constant through the iterations (such as a
Kolmogorov-Smirnov test) our technique can produce
datasets with similar graphical characteristics as well.
In the area of generating synthetic datasets, GraphCuisine [2]
allows users to direct an evolutionary algorithm to create
network graphs matching user-specified parameters. While
this work looks at a similar problem, it differs in that it is
focused on network graphs, is an interactive system, and
allows for directly specifying characteristics of the output,
while our technique looks at 1D or 2D distributions of data,
is non-interactive, and perturbs the data such that the initial
statistical properties are maintained throughout the process.
Finally, on the topic of using scatter plots to encode graphics,
Residual Sur(Realism) [11] produces datasets with hidden
images which are only revealed when appropriate statistical
measures are performed. Conversely, our technique encodes
graphical appearance into the data directly.
METHOD
e key insight behind our approach is that while generating
a dataset from scratch to have particular statistical properties
is relatively dicult, it is relatively easy to take an existing
dataset, modify it slightly, and maintain (nearly) the same
statistical properties. With repetition, this process creates a
dataset with a different visual appearance from the original,
while maintaining the same statistical properties. Further, if
the modifications to the dataset are biased to move the points
towards a particular goal, the resulting graph can be directed
towards a particular visual appearance.
e pseudocode for the high-level algorithm is listed below:
INITIAL_DS is the seed dataset from which the statistical
values we wish to maintain are calculated. e PERTURB
function is called at each iteration of the algorithm to modify
the latest version of the dataset (CURRENT_DS) by moving
one or more points by a small amount, in a random direction.
e “small amount” is chosen from a normal distribution and
is calibrated such that >95% of movements result in the
statistical properties of the overall dataset remaining
unchanged (to two decimal places).
Once the individual points have been moved, the FIT
function is used to check if perturbing the points has
increased the overall fitness of the dataset. e fitness can be
calculated in a variety of ways, but for conditions where we
want to coerce the dataset to into a shape, fitness is calculated
as the average distance of all points to the nearest point on
the target shape.
e naïve approach of accepting only datasets with an
improved fitness value results in possibly getting stuck in
locally-optimal solutions where other, more globally-optimal
solutions are possible. To mitigate this possibility, we
employ a simulated annealing technique [9]. With the
possible solutions generated in each iteration, simulated
annealing works by always accepting solutions which
1: current_ds initial_ds
2: for x iterations, do:
3: test_ds PERTURB(current_ds, temp)
4: if ISERROROK(test_ds, initial_ds):
5: current_ds test_ds
6:
7: function PERTURB(ds, temp):
8: loop:
9: test MOVERANDOMPOINTS(ds)
10: if FIT(test) > FIT(ds) or temp > RANDOM():
11: return test
improve the fitness, but also, if the fitness is not improved,
the solution may be accepted based on the “temperature” of
the simulated annealing algorithm. If the current temperature
is less than a random number between 0 and 1, the solution
is accepted even if it the fitness is worsened. We found that
using a quadratically-smoothed monotonic cooling schedule
starting with a temperature of 0.4 and finishing with a
temperature of 0.01 worked well for the sample datasets.
Once the perturbed dataset has been accepted, either through
an improved fitness value or from the simulated annealing
process, the perturbed dataset is compared to the initial
dataset for statistical equivalence. For the examples in this
paper we consider properties to be “the same” if they are
equal to two decimal places. e ISERROROK function
compares the statistics between the datasets, and if they are
equal (to the specified number of decimal places), the result
from the current iteration becomes the new current state.
Example Generated Datasets
Example 1: Coercion Towards Target Shapes
In this first example (Figure 1), each dataset contains 182
points and are equal (to two decimal places) for the
“standard” summary statistics (x/y mean, x/y standard
deviation, and Pearson’s correlation). Each dataset was
seeded with the plot in the top left.e target shapes are
specified as a series of line segments, and the shapes used in
this example are shown in Figure 3.
Figure 3. e initial data set (top-left), and line segment
collections used for directing the output towards specific
shapes. e results are seen in Figure 1.
With this example dataset, the algorithm ran for 200,000
iterations to achieve the final results. On a laptop computer
this process took ~10 minutes. Figure 4 shows the
progression of one of the datasets towards the target shape.
Figure 4. Progression of the algorithm towards a target
shape over the course of the cooling schedule.
Example 2: Alternate Statistical Measures
One benefit of our approach over previous methods is that
the iterative process is agnostic to the particular statistical
properties which remain constant between the datasets. In
this example (Figure 5) the datasets are derived from the
same initial dataset as in Example 1, but rather than being
equal on the parametric properties, the datasets are equal in
the non-parametric measures of x/y median, x/y interquartile
range (IQR), and Spearman’s rank correlation coecient.
Figure 5. Example datasets are equal in the non-parametric
statistics of x/y median (53.73, 46.21), x/y IQR (19.17, 37.92),
and Spearman’s rank correlation coecient (+0.31).
Example 3: Specific Initial Dataset
e previous two examples used a rather “generic” dataset of
a slightly positively correlated point cloud as the starting
point of the optimization. Alternately, it is possible to begin
with a very specific dataset to seed the optimization.
Figure 6. Creating a collection of datasets based on the
“dinosaurus” dataset. Each dataset has the same summary
statistics to two decimal places: (x
͞ =54.26, y
͞ = 47.83, sdx =
16.76, sdy = 26.93, Pearson’s r = -0.06).
Alberto Cairo produced a dataset called the “Datasaurus” [4].
Like Anscombe’s Quartet, this serves as a reminder to the
importance of visualizing your data, since, although the
dataset produces “normal” summary statistics, the resulting
plot is a picture of a dinosaur. In this example we use the
“datasaurus” as the initial dataset, and create other datasets
with the same summary statistics (Figure 6).
Example 4: Simpson’s Paradox
Another instrument for demonstrating the importance of
visualizing your data is Simpson’s Paradox [3, 10]. is
paradox occurs with data sets where a trend appears when
looking at individual groups in the data, but disappears or
reverses when the groups are combined.
To create a dataset exhibiting Simpson’s Paradox, we start
with a strongly positively correlated dataset (Figure 7A), and
then perturb and direct that dataset towards a series of
Iteration: 1
Temperature: 0.4
Iteration: 50,000
Temperature: 0.35
Iteration: 100,00
0
Temperature: 0.2
Iteration: 200,000
Temperature: 0.01
Iteration: 1 Iteration: 20,000 Iteration: 80,00
0
Iteration: 200,00
0
negatively sloping lines (Figure 7B). e resulting dataset
(Figure 7C) has the same positive correlation as the initial
dataset when looked at as a whole, while the individual
groups each have a strong negative correlation.
Figure 7. Demonstration of Simpson's Paradox. Both
datasets (A and C) have the same overall Pearson's
correlation of +0.81, however after coercing the data
towards the pattern of sloping lines (B), each subset of data
in (C) has an individually negative correlation.
Example 5: Cloned Dataset with Similar Appearance
As discussed by Govindaraju and Haslett [8] another use for
datasets with the same statistical properties is the creation of
“cloned” datasets to anonymize sensitive data [6]. In this
case, it is important that individual data points are changed
while the overall structure of the data remains similar. is
can be accomplished by performing a Kolmogorov-Smirnov
test within the ISERROROK function for both x and y. By only
accepting solutions where both the x and y K-S statistic is
<0.05 we ensure that the result will have a similar shape to
the original (Figure 8). is approach has the benefit of
maintaining the x/y means and correlation as accomplished
in previous work [8], and additionally the x/y standard
deviations as well. is could also be useful for “graphical
inference” [12] to create a collection of variant plots
following the same null hypothesis.
Figure 8. Example of creating a “mirror” dataset as in [8].
Example 6: 1D Boxplots
To demonstrate the applicability of our approach to non 2D-
scatterplot data, this example uses a 1D distribution of data
as represented by a boxplot. e most common variety of
boxplot, the “Tukey Boxplot”, presents the 1st quartile,
median, and 3rd quartile values on the “box”, with the
“whiskers” showing the location of the furthest datapoints
within 1.5 interquartile ranges (IQR) from the 1st and 3
rd
quartiles. Starting with the data in a normal distribution
(Figure 9A) and perturbing the data to the left (B), right (C),
edges (D, E), and arbitrary points along the range (F) while
ensuring that the boxplot statistics remain constant produces
the results shown in Figure 9.
Figure 9. Six data distributions, each with the same 1st
quartile, median, and 3rd quartile values, as well as equal
locations for points 1.5 IQR from the 1st and 3rd quartiles.
Each dataset produces an identical boxplot.
LIMITATIONS AND FUTURE WORK
When the source dataset and the target shape are vastly
different, the produced output might not be desirable. An
example is show Figure 10, where the data set from Figure
7A is coerced into a star (Figure 10). is problem can be
mitigated by coercing the data towards “simpler” patterns
with more coverage of the coordinate space such as lines
spanning the grid, or pre-scaling and positioning the target
shape to better align with the initial dataset.
Figure 10. Undesirable outcome (C) when coercing a
strongly positively correlated dataset (A) into a star (B).
e currently implemented fitness function looks only at the
position of individual points in relation to the target shape,
which can result in “clumping” of data points and sparse
areas on the target shape. A future improvement could
consider an additional goal to “separate” the points to
encourage better coverage of the target shape in the output.
e parameters chosen for the algorithm (95% success rate,
quadratic cooling scheme, start/end temperatures, etc.) were
found to work well, but should not be considered “optimal”.
Such optimization is left as future work.
e code and datasets presented in this work are available at
www.autodeskresearch.com/publications/samestats.
CONCLUSION
We presented a technique for creating visually dissimilar
datasets which are equal over a range of statistical properties.
e outputs from our method can be used to demonstrate the
importance of visualizing your data, and may serve as a
starting point for new data anonymization techniques.
AC
B
“Cloned” DataOriginal Data Comparison
−10 −5 0 5 10
A
B
C
D
E
F
ABC
REFERENCES
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3. Blyth, C.R. (1972). On Simpson’s Paradox and the
Sure-ing Principle. Journal of the American
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4. Cairo, A. Download the Datasaurus: Never trust
summary statistics alone; always visualize your data.
http://www.thefunctionalart.com/2016/08/download-
datasaurus-never-trust-summary.html.
5. Chatterjee, S. and Firat, A. (2007). Generating Data
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10. Simpson, E.H. (1951). e Interpretation of Interaction
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This paper introduces an interactive system called GraphCuisine that lets users steer an Evolutionary Algorithm (EA) to create random graphs that match user-specified measures. Generating random graphs with particular characteristics is crucial for evaluating graph algorithms, layouts and visualization techniques. Current random graph generators provide limited control of the final characteristics of the graphs they generate. The situation is even harder when one wants to generate random graphs similar to a given one, all-in-all leading to a long iterative process that involves several steps of random graph generation, parameter changes, and visual inspection. Our system follows an approach based on interactive evolutionary computation. Fitting generator parameters to create graphs with pre-defined measures is an optimization problem, while assessing the quality of the resulting graphs often involves human subjective judgment. In this paper we describe the graph generation process from a user’s perspective, provide details about our evolutionary algorithm, and demonstrate how GraphCuisine is employed to generate graphs that mimic a given real-world network. An interactive demo of GraphCuisine can be found on our website http://www.aviz.fr/Research/Graphcuisine .
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