A Versatile Approach to Evaluating and Testing Automated Vehicles
based on Kernel Methods
Zhiyuan Huang1, Yaohui Guo2, Henry Lam3, and Ding Zhao2,4,∗
Abstract—Evaluation and validation of complicated control
systems are crucial to guarantee usability and safety. Usually,
failure happens in some very rarely encountered situations,
but once triggered, the consequence is disastrous. Accelerated
Evaluation is a methodology that efficiently tests those rarely-
occurring yet critical failures via smartly-sampled test cases. The
distribution used in sampling is pivotal to the performance of
the method, but building a suitable distribution requires case-
by-case analysis. This paper proposes a versatile approach for
constructing sampling distribution using kernel method. The
approach uses statistical learning tools to approximate the critical
event sets and constructs distributions based on the unique
properties of Gaussian distributions. We applied the method to
evaluate the automated vehicles. Numerical experiments show
proposed approach can robustly identify the rare failures and
significantly reduce the evaluation time.
I. INTRODUCTION
The auto companies have been competing to get their
automated vehicles (AVs) ready on road for years, yet there
is still none available in the market. Partly, this is due to
the challenging task of robustly testing and guaranteeing the
safety of an AV before its release. Companies have been
trying different methods such as road test [1], [2], computer
simulation test [3] and human-vehicle interaction test [4],
[5], yet providing safety certificate for an AV system is still
open for solving [1]. Assisting the endeavors of solving this
problem, the U.S Department of Transportation has released
a new AV policy: A Vision for Safety 2.0 [6]. This official
document standardizes the required safety features of an
autonomous vehicle, providing guidance and clearer pathways
for the various stakeholders aiming to certify the safety of their
AV systems. However, even with this newly published official
guideline, the testing standard remains unclear while the AV
target release is quickly approaching. Thus, an effective and
efficient testing method for an autonomous vehicle is an urgent
need under this background.
Traditional vehicle safety tests are based on crash databases
collected from crashes or dangerous scenarios, such as the
CSD and GIDAS crash databases [7]. However, the informa-
tion logged in these databases is limited so that it is difficult to
reconstruct and analyze the dangerous scenarios. As a result,
*This work was funded by the Mobility Transformation Center at the
University of Michigan with grant No. N021552.
1Department of Industrial and Operations Engineering, University of Michi-
gan, Ann Arbor, MI 48109, USA.
2Robotics Institute, University of Michigan, Ann Arbor, MI 48109, USA.
3Department of Industrial Engineering and Operations Research, Columbia
University, New York, NY 10027, USA.
4Department of Mechanical Engineering, University of Michigan, Ann
Arbor, MI 48109, USA.
∗Corresponding author. E-mail: zhaoding@umich.edu (Z. D.)
the Naturalistic-Field Operational Test (N-FOT) has been ac-
tively used instead in recent auto safety evaluation. An N-FOT
logs naturalistic driving data such as the drivers maneuver,
driving environment, and vehicle dynamics, providing detailed
data for reconstructing the desired traffic scenario. Among
these N-FOT records are the 100-Car Naturalistic Driving
Study [8], the Safety Pilot Model Deployment (SPMD) [9],
and Strategic Highway Research Program (SHRPII) [10]. To
evaluate the safety of an AV algorithm, an N-FOT based safety
evaluation tests the performance of driving policy under a
certain sampled environment drawn from the databases. The
recent research includes modeling driver failure into safety
evaluation [11], [12] and analyzing the contributing factors in
dangerous scenarios [13], [14], [15]. In these researches, the
safety is evaluated under certain conditions, from which it is
then generalized under various naturalistic driving conditions.
While sampling from the N-FOT data is a safe method to
imitate the reality, it turns out that such approach is deemed
inefficient. According to National Highway Safety Administra-
tion [16], the fatality rate is 1.08 per 100 million vehicle miles,
which means it requires approximately 100 million miles
data to generate a fatal crash in a simulation under natural
conditions. This enormous simulation effort is indeed costly
and time-consuming. Thus, an accelerated method is necessary
to attain an effective and efficient auto safety evaluation.
Accelerated Evaluation (AE) procedure is proposed in [17]
to serve this notion of accelerated framework, and is further
studied in [18], [19], [20]. This approach utilizes statistical
models to represent the traffic environment and estimates the
rate of safety-critical events in the test scenarios. Obtaining
an unbiased estimator for these rates is accelerated by the use
of Importance Sampling (IS), alleviating the needs for exten-
sive simulation replications. The efficiency of this approach
depends primarily on the choice of accelerated distribution,
which needs to be constructed based on specific characteristics
of the problem of interest.
This paper extends the study of AE by proposing a new
scheme for constructing accelerated distribution. The scheme
merges supervised learning methods with rare event simulation
and utilizes the properties of Gaussian distribution in IS theory.
The proposed approach provides an alternative method for
constructing accelerated distribution for AV test scenarios
under Gaussian Mixture Models (GMM).
The structure of the paper is as the following: Section
II reviews the AE procedure and emphasizes the concepts
relevant to the proposed approach. Section III presents the
base knowledge and shows the procedure of the proposed ap-
proach. Section IV showcases the performance of the proposed
approach based on two problem instances.
arXiv:1710.00283v1  [cs.LG]  1 Oct 2017
II. REVIEW OF ACCELERATED EVALUATION
In this section, the AE procedure is reviewed in detail in
the following order. Section II-A gives the high-level overview
of the AE procedure, followed by Section II-B in which the
general problem setting for the approach is described. To
provide adequate knowledge regarding the statistical methods
used, the IS theory, that supports the validity of AE procedure,
will be concisely presented in Section II-C.
A. Procedure For Accelerated Evaluation
In evaluating the performance of an AV system, the AE pro-
cedure suggests to examine the AV under various different test
scenarios. The safety level of a vehicle is assessed by the rate
of safety-critical events in the test scenarios. In this case, the
goal of the AE is modeled as a probability estimation problem
(described in section II-B). The AE procedure consists of four
major steps [19]:
• Model the behaviors of the traffic environment repre-
sented by f(x) (original distribution) as the major dis-
turbance to the AV using large-scale naturalistic driving
data.
• Skew the disturbance statistics from f(x) to modified
statistics f ∗(x) (accelerated distribution) to generate
more frequent and intense interactions between AVs and
the traffic environment.
• Conduct accelerated tests with f ∗(x).
• Use the IS technique to skew back the results to under-
stand real-world behavior and safety benefits.
Note that the key part in the AE procedure is the use of IS
technique, whose theory will be reviewed in Section II-C.
The proposed approach provides an alternative in construct-
ing the accelerated distribution in Step 2. In the following sec-
tions, we focus on the construction of accelerated distribution.
B. Evaluation Problem
The AE evaluates testing vehicles by individually estimating
the rate of safety-critical events under various different test
scenarios. Here, we show the setting of the problem and the
notations that we use in the following sections.
In a test scenario, we assume that the uncertain environment
traffic is represented by a vector X ∈X and is regarded
as a stochastic model. Note that X is the domain of X.
The distribution model f(X) for X is estimated from a
sufficiently large set of naturalistic driving data. The safety-
critical events occurs with certain values of X. We represent
the set as ε = {X|X leads to a safety-critical event} and use
an indicator function
Iε(x) =
(
1
x ∈ε
0
otherwise.
(1)
to denote the response of a certain vector x. Iε(x) is usually
complicated, but can be evaluated by on-track experiments or
computer simulations.
Generally, the safety-critical events are very rare (smaller
than 10−6). For this reason, using crude Monte Carlo method
to estimate the objective is time-consuming due to large
sample size needed to observe a critical event, and even much
larger samples to achieve statistical confidence. Therefore, we
need to improve the efficiency of the evaluation.
C. Importance Sampling
IS [21], [22] is a technique to reduce the variance in sim-
ulation, while providing unbiased estimation. This technique
is crucial to the efficiency and unbiasedness of the AE.
Consider a random vector X ∈X with distribution f(X)
and a rare event set ε ⊂Ωon sample space Ω. Our goal is to
estimate the probability of the rare event
P(X ∈ε) = Ef[Iε(X)] =
Z
xf(x)dx,
(2)
where the event indicator function is defined as (1).
Given samples X1, ..., XN independently and identically
generated from f(x), the crude Monte Carlo computes the
sample mean of Iε(x)
ˆP(X ∈ε) = 1
N
N
X
n=1
Iε(Xn).
(3)
The IS technique is derived from a change of sampling
distribution as the following [21]:
E[Iε(X)] =
Z
Iε(x)f(x)dx =
Z
Iε(x) f(x)
f ∗(x)f ∗(x)dx, (4)
where f ∗is a distribution density function that has the same
support with f.
Note that
E∗[Iε(x) f(x)
f ∗(x)] =
Z
Iε(x) f(x)
f ∗(x)f ∗(x)dx,
(5)
where E∗denotes the expectation with regard to f ∗. Therefore
by taking the sample mean of the expectation,
ˆP(X ∈ε) = 1
N
N
X
n=1
Iε(Xn) f(x)
f ∗(x)f(x)dx
(6)
gives an unbiased estimator of the above expectation [21],
where Xi’s are generated from f ∗.
By appropriately selecting f ∗, the evaluation procedure
obtains an estimation with smaller variance. We refer to f ∗as
the accelerated distribution in our procedure. The construction
of a “good” accelerated distribution f ∗is discussed in section
III.
III. CONSTRUCTING ACCELERATED DISTRIBUTION VIA
CLASSIFICATION METHODS
As discussed in [20], GMM is a powerful distribution model
for representing traffic environment in AE. In this section, we
propose a new approach to construct accelerated distributions
for test scenarios under GMM setting.
In Section III-A, we review the existing strategies for con-
structing efficient accelerated distribution for GMM. Section
III-B shows the key idea of our approach. Section III-C
presents the procedure of the proposed approach.
Fig. 1. The procedure of the proposed approach.
Fig. 2. Scheme of constructing accelerated distribution for Gaussian Mixture
Model with convex rare event set.
A. Known Strategy for Efficient Accelerated Distribution
When the random vector X follows Gaussian distribution
with mean µ and covariance matrix Σ and the rare event set
ε satisfies the convexity assumption, there is a simple scheme
that obtains an efficient accelerated distribution [23], [24].
For a convex rare event set ε, we define the dominating
point of ε on φ(x; µ, Σ) to be
a∗= arg max
a∈ε φ(a; µ, Σ),
(7)
where φ(x; µ, Σ) is the density function for Gaussian distri-
bution with mean µ and covariance matrix Σ. By shifting
the mean µ of the Gaussian distribution to a∗, we obtain an
accelerated distribution that provides an efficient estimator for
P(x ∈ε) [24].
Now we consider the GMM with the density
f(x) =
k
X
i=1
piφ(x; µi, Σi).
(8)
where k is the number of mixture components, pi is the
proportion of the ith component, µi and Σi are the mean and
covariance for the ith component. We use a∗
i to denote the
dominating point with regard to the ith component and critical
event set ε. An efficient accelerated distribution is then given
by:
f ∗(x) =
k
X
i=1
piφ(x; ai, Σi).
(9)
Fig. 2 illustrates this scheme.
Fig. 3. The procedure of constructing an accelerated distribution f∗.
In practice, the critical event sets for automated vehicle
evaluation are generally non-convex. Therefore, it is hard to
apply this approach directly.
B. Constructing Accelerated Distribution via Kernel Method
We propose a procedure to construct an accelerated dis-
tribution for GMM as illustrated in Fig. 3. The procedure
utilizes the scheme in section III-A and uses kernel method
to transform the data on a higher dimensional space. The
Classification is used to obtain the linear boundaries for the
critical event set.
The following intuition motivates our approach: Suppose
we have a feature function (see Appendix B) φ : X →Rm,
where X ⊂Rn is the domain for X and n < m. We assume
that there is a subspace on the feature space that is a linear
transformation of the original space. (For instance, the feature
function for polynomial kernel φ(x) = (x2,
√
2x, 1) satisfies
the assumption.) We further assume that on the feature space,
the critical event set has a linear boundary β′x + b = 0
and ˜f is a Gaussian mixture distribution that well fit the the
model on the feature space. Thus, the scheme in section III-A
is directly applicable. Let ˜f ∗be the accelerated distribution
constructed on the feature space - we note that it is still a
GMM. If the feature space contains the a linear transformation
of the original space, we can easily obtain an accelerated
distribution f ∗on the original space by linearly transforming
the marginal distribution of
˜f ∗. Since the marginal of a
Gaussian distribution is still a Gaussian distribution [25], f ∗
is still a Gaussian mixture distribution.
To carry out the procedure described above, we need to
find a feature function φ(x), the boundary β′x + b = 0 and
the model ˜f. The selection of φ(x) depends on the shape of
the critical event set, which is problem dependent. To obtain
β′x+b = 0, we use classification methods, e.g. support vector
machine (SVM), to learn for the linear boundaries. The learned
critical event set is represented by ε = {x|β′x + b ≥0}.
If f is a Gaussian distribution, the true distribution on the
feature space is generally not Gaussian. In the proposed pro-
cedure, we use a GMM ˜f to approximate the true distribution
on the feature space.
C. A Sequential Procedure for the Accelerated Evaluation
To summarize the procedure of the proposed approach in
section III-B, we present the step-by-step description and elab-
orated discussion how to apply the approach in a sequential
procedure.
Suppose we have a GMM f(x) for random vector X and
we are interested in E[Iε(X)]. The procedure is as follows:
1) Construct a training set with n samples, {(Xi, Yi)}n
i=1,
where Yi = Iε(Xi). We suggest to select Xi’s using
space filling designs.
2) Select a feature function φ(x) and transform the training
set as {(φ(Xi), Yi)}n
i=1. We suggest to use the feature
function for polynomial kernel and use a small param-
eter d (d = 2 or d = 3).
3) Use a classification method with linear boundary on
the training set {(φ(Xi), Yi)}n
i=1 and obtain the linear
boundary β′x + b = 0. We use SVM (see Appendix A)
in the experiments.
4) Generate samples {Xj}m
j=1 from the model f and trans-
form the samples as {φ(Xj)}m
j=1.
5) Select K as the number of mixture components. Use
GMM with K mixtures to fit the samples {φ(Xj)}m
j=1
and obtain the approximation model ˜f. The discussion
about the selection of K refers to section IV-C
6) For each component in ˜f, find the dominating point
using (7) and construct ˜f ∗using (9).
7) Adjust the marginal distribution of ˜f ∗and obtain f ∗.
8) Generate sample Xi’s from the distribution f ∗and use
(6) to estimate the objective.
9) (Additional step) As new returns Iε(Xi) of Xi’s are
obtained in step 8, update {(Xi, Yi)}n
i=1 and go back to
step 1.
Note that if we use step 9, the procedure sequentially
update the training set and the information of the critical event
set. When the experiment Iε(x) is very time-consuming and
expensive, we suggest to start with small sample size and add
step 9 in the procedure. In the experiments in section IV, we
do not use step 9.
IV. NUMERICAL EXPERIMENTS
In this section, we present simulation results to illustrate
the procedure of the proposed approach and to show the
applicability of the approach on AV evaluation. In section
IV-A, we apply the proposed method on a simple problem
to provide an intuition of the mechanism. Section IV-B shows
the performance of the method in an AV test case.
A. Simple Example Problem
We use a simple example to discuss the implementation
of the proposed method. The example is set to be a low
dimensional problem for illustration purposes. Despite being
Fig. 4. 1,000 samples from the distribution model of X with critical events
labeled.
Fig. 5. 1,000 samples from uniform distribution on the domain of X with
critical events labeled.
simple, the problem setting maintains some similarity with AV
test scenarios.
Assume we have a 2-dimensional random vector X ∈X,
where X ∈R. Here we have X = {(x, y)|0 ≤x ≤5, 0 ≤y ≤
5}. Suppose the distribution model f(X) of X is known and
we have f(X) to be a multivariate Gaussian distribution with
mean (1, 1) and identity covariance matrix I2. The critical
event set is defined as ε = {(x, y)|x2 + y2 ≤0.22 or (x −
5)2 + (y −5)2 ≤1.52 or (x −3)2 + (y −5)2 ≤0.72 or (x −
5)2 +(y −3)2 ≤0.52}. The indicator function Iε(x) = 1(x ∈
ε) shows whether a vector x is in the critical event set. Our
objective is to estimate the probability of the critical event,
i.e. P(X ∈ε) = E[Iε(x)].
Fig. 4 shows 1,000 samples generated from the distribution
model of X. We could observe that distribution concentrates
on the area with relatively small value on both axes (consider
that the upper bound for both coordinates is 5).
Now, we start to explore the domain of the X and to
learn the critical event set. We use 1,000 samples randomly
generated using a uniform distribution on the domain and label
these samples with regard to the critical events. Fig. 5 shows
the samples and the return of the indicator function Iε(x).
Let us denote these samples as Xn and the returns as Yn.
It is obvious that the two types of events cannot be linearly
classified and the critical event set is obviously not convex.
Therefore if we want to construct an accelerated distribution
for estimating the objective probability, we cannot directly use
the scheme mentioned in section III-A.
Fig. 6.
The classification boundary obtained from linear support vector
machine on the original space.
Fig. 7. The estimation of the probability P(X ∈ε).
We follow the procedure described in section III-C to
construct an accelerated distribution for the problem. Here we
use a polynomial kernel which maps vectors X = (x, y) in the
original space to the feature space through the transformation
φ(X) = φ(x, y) = (x2, y2, xy, x, y). We then use linear SVM
to classify the transformed samples φ(Xn) with regard to the
returns Yn. A linear boundary on the feature space is obtained
as β′x + b = 0. In Fig. 6, we plot this boundary on the
original space. We should note that the classification captures
the property of the critical set.
Next, we want to approximate the distribution model on
the feature space. We generate extra 20,000 samples from the
original distribution f(X), and transform the new generated
samples with φ(X). To investigate the effect of the accuracy
of the approximation model, we use two cases with different
number of mixture for comparison, i.e. K1 = 20, K2 = 3. We
fit Gaussian mixture distribution with K1 and K2 components
on the transformed samples respectively. For each component
in the mixture distribution, we find the dominant point and
construct a sampling distribution on the feature space. Finally,
we use the marginal distribution on the original space as the
accelerated distribution.
The simulation performance is shown in Fig. 7 and 8. In
the figures, we use “AE Ki” to denote the proposed approach
with Ki components in the approximation mixture distribution
and “Crude MC” to denote the crude Monte Carlo approach.
In Fig. 7, we could observe that the proposed approach
attains stability within 2,000 samples. The crude Monte Carlo
Fig. 8.
The relative half width with 95% confidence of the estimation for
P(X ∈ε).
is obviously oscillating. This shows that the proposed approach
provides better estimation using smaller sample size. This
argument is further confirmed by Fig. 8.
In Fig. 8, we plot the 95% relative half-width, which is
given by
w = α0.95
ˆσ
√
N ˆP
,
(10)
where α0.95 denotes the 0.95 quantile of the normal distribu-
tion, N denotes the number of samples used in the estimation,
ˆσ denotes the standard deviation of the sampled objective, and
ˆP denotes the estimation of the objective.
The 95% relative half-width for the crude Monte Carlo is
much larger than the proposed approaches. To reach the same
level of 95% relative half-width as the proposed approaches,
the crude Monte Carlo requires roughly 100 times more
samples.
From the performance of this simple example, we shall
note that the proposed approach provides good accelerated
distributions. By comparing the performance with K1 and K2,
we conclude that a more accurate approximation model on the
feature space leads to a more efficient accelerated distribution.
B. Lane Change Scenario
The lane change model for AE is proposed by [17]. The
setting is illustrated in Fig. 9, which shows that a leading
human driving vehicle is conducting a lane change in front
of an automated vehicle. Our objective is to evaluate the rate
of safety-critical events in this scenario, where safety-critical
events includes conflict, collision, injury, etc. In this example,
we are interested in the rate of collision during the lane change
procedure. We assume that the initial status of the two vehicles
is captured by a vector X = (v, ˙R, R−1) ∈X, which consists
of three important parameters: v, the velocity of the leading
vehicle, ˙R, the relative range of the two vehicles, and R−1, the
inverse of the range between the two vehicles. We assume that
X is stochastic and the distribution of X, f(X), is a known
GMM. Given X, the lane change procedure is supposed to be
deterministic with regard to the testing vehicle. We use ε ⊂X
to denote the critical event set and our objective is represented
by P(X ∈ε).
In this example, we use the vehicle control model shown in
Fig. 9 as the testing vehicle. The model consists of Adaptive
Fig. 9. The setting of the lane change scenario and the related parameters.
Fig. 10. The estimation of the probability P(X ∈ε).
Cruise Control (ACC) and Autonomous Emergency Braking
(AEB) [26]. (In the figure, TTC is defined as TTC = R/ ˙R.
TTCAEB is a threshold that triggers the AEB system.)
We apply the proposed approach to obtain accelerated
distributions. We construct a training set with 20, 000 samples
by using a grid on the domain (v and ˙R are bounded; R−1 is
unbounded, we sample between the minimum and maximum
data in a naturalistic driving data). In this case, we use a poly-
nomial kernel with feature function φ(X) = φ(v, ˙R, R−1) =
(v2, ˙R2, (R−1)2, v ˙R, vR−1, ˙RR−1, v, ˙R, R−1)
to
map
the
model onto a higher dimensional space. Again, we use differ-
ent value, K1 = 8 and K2 = 20, for the number of mixtures
of the approximation model in the feature space.
Fig. 10 presents the estimation results of the objective,
P(X ∈ε). We observe that the estimation converges as we
increase the number of samples. In this case, using different
value for the number of mixtures for the approximation
model does not result in significant difference in the resulting
estimation.
To illustrate the efficiency of the proposed method, we
will compare the performance of the proposed method with
crude Monte Carlo. Since the probability we are estimating is
very small, running crude Monte Carlo would require a huge
computing time. Here we use the probability we estimated
to approximate the number of samples required for the crude
Monte Carlo. Using the formula for the standard deviation of
the crude Monte Carlo estimation ˆP(x ∈ε) by
std( ˆP(x ∈ε)) =
s
ˆP(x ∈ε)(1 −ˆP(x ∈ε))
n
,
(11)
TABLE I
COMPARISON BETWEEN THE PROPOSED APPROACH AND CRUDE MONTE
CARLO.
Standard Deviation
Required Samples
Proposed AE
2 × 10−7
2 × 105
Crude MC
5 × 10−5
1.2 × 1010
we found that the crude Monte Carlo method requires roughly
6 × 104 times more samples to reach the sample confidence
level as the proposed approach.
Table I summarizes the comparison of the proposed AE
with crude Monte Carlo. Column 2 shows the approximated
standard deviation of the estimators. Column 3 presents the
approximated number of the required samples to obtain an
estimation with a 95% confidence relative half-width less than
0.4. Note that the proposed AE uses an extra of 2×104 samples
to learn the critical event set and construct the accelerated
distribution. These samples are much smaller than the scale
of the required samples for estimation.
C. Discussion on Implementation
In the implementation of the proposed approach, there
are some open choices in the procedure. We know from
the experiment that the efficiency of the approach is largely
influenced by these choices. Here, we discuss how they are
related to the efficiency of the method.
To obtain an efficient accelerated distribution, we want to
achieve two tasks: a) the classification must learn the property
of the critical event set, at least a rough bound needs to be
found; b) the approximation on the feature space needs to
roughly capture the shape of the transformed distribution. We
discuss these two tasks separately.
In the classification step, we need to construct a train set
and choose a kernel function. Since the collection of train data
requires test experiment, the size of the training set need to be
balanced. In practice, to select a reasonable number requires
prior knowledge of the test scenario. When such knowledge is
not available, we suggest to start with a small size of data that
includes some critical events and then follow the sequential
procedure described in section III-C. Given the number of data
in the training set, we can use a deterministic design (e.g. use
a grid) or a random design (e.g. uniformly sample), as long
as samples fill the domain well.
The kernel functions to select need to have an explicit fea-
ture function. We suggest to use polynomial kernel. A feature
space with higher dimension would achieve better accuracy
for the critical event learning, however, to approximate the
distribution model on the higher dimension is generally harder.
For this reason, we want the dimension of the feature space
to be as small as possible.
For approximating the distribution model on the feature
space, we need to choose the number of mixture components.
Although a larger number of components always provides
a better fitting, the efficiency of the constructed accelerated
distribution might not always improve. We suggest to use
smaller number of components for less computing efforts.
For high dimensional feature space, regularization should
be applied in fitting the approximation model to make the
fitting algorithm converge in fewer iterations. The selection of
regularization parameter is suggested to choose through cross
validation [27].
APPENDIX
A. Classification Methods for Critical Event Set Learning
Here we review SVM as an example of classification
methods that suits the proposed approach. Please refer to [27]
for more details.
SVM is a popular algorithm for classification and regres-
sion. Here we briefly introduce the SVM with hard margin for
binary classification. Denote the training dataset as (xi, ti), i =
1, 2, 3, ..., n, where xi is the feature vector and ti ∈{+1, −1}
is the label; suppose the data xi is linear separable. The SVM
returns the linear classifier of the form:
y(x) = β′x + b
where β′ is the coefficient of the hyper plane in the feature
space; b is the bias; y(xi) > 0 for ti = 1; y(xi) < 0 for
ti = −1. SVM wants to find the hyperplane to separate the
points while maximizing the minimum distance between the
hyperplane and the nearest points xi from either side. This
can be formulated with the constraints as
arg max
β,b
{ 1
∥β∥arg min
n
ti(β′xi + b)}
ti(β′xi + b) > 0,
i = 1, 2, 3, ..., n
A common solution to SVM is to use the Lagrange Multiplier,
which can be found in [27].
Note that other classification methods, e.g. logistic regres-
sion, also provide a linear boundary and are compatible with
kernel method. These methods also fits the proposed approach.
B. Kernel Method for Critical Event Set Learning
The kernel is a generalized inner product, computing the
inner product of two vectors in a higher dimensional space
without explicitly define the feature function. Here we intro-
duce the feature function, and more detail on kernel method
can be found in [28] .The feature function is defined as
φ : X →Rm, where X ⊂Rn is the domain for input xi,
i = 1, 2, 3, ..., n, and n < m. A benefit of introducing feature
space is non-separable data can be separable in some feature
spaces.
For instance, consider two sets on a 2-D plane: S1 =
{(x, y) | x2 + y2 < 3} ⊂R2 and S2 = {(x, y) | x2 + y2 >
3} ⊂R2. There is no line can separate S1 from S2 since
S1 is surrounded by S2. But if we map the point (x, y) on
R2 to R3 by a feature function φ(x, y) = (x, y, x2 + y2), the
mapped sets are linearly separable. Denoting the original sets
S1, S2 in the feature space as Φ(S1) = {φ(x) | x ∈S1} and
Φ(S2) = {φ(x) | x ∈S2}, then Φ(S1) and Φ(S2) are linearly
separable by the plane z = 3 in R3.
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