Thursday 15h45 vicent_ribas ESANN

Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
A Quotient Basis Kernel for the prediction of
mortality in severe sepsis patients
Vicent J. Ribas
LSI - SOCO
Technical University of Catalonia (UPC)
Barcelona
April 19, 2013
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

Contents
1 Introduction
2 Database Description
3 Risk of Death Assessment from Observed Data
4 Conclusions

Introduction
Sepsis is a clinical syndrome deﬁned by the presence of
infection and Systemic Inﬂammatory Response Syndrome
(SIRS).

Introduction
(SIRS).
This can lead to severe sepsis or to septic shock (severe sepsis
with hypotension refractory to ﬂuid administration) and
multi-organ failure.

Introduction
(SIRS).
In western countries, septic patients account for as much as
25% of ICU bed utilization and the pathology occurs in 1% -
2% of all hospitalizations.

Introduction
(SIRS).
In western countries, septic patients account for as much as
25% of ICU bed utilization and the pathology occurs in 1% -
2% of all hospitalizations.
The mortality rates of sepsis range from 12.8% for sepsis and
20.7% for severe sepsis, and up to 45.7% for septic shock.

The medical management of sepsis and the study of its
prognosis and outcome is a relevant medical research
challenge.

The medical management of sepsis and the study of its
prognosis and outcome is a relevant medical research
challenge.
Provided that such methods are to be used in a clinical
environment (ICU), it requires prediction methods that are
robust, accurate and readily interpretable.

Dataset
A prospective observational cohort study of adult patients with
severe sepsis was conducted at the Critical Care Department
of the Vall d’ Hebron University Hospital (Barcelona, Spain).

Dataset
Data from 354 patients with severe sepsis was collected in
this ICU between June, 2007 and December, 2010.

Dataset
55% of cases correspond to ‘medical’ sepsis.

Dataset
The mean age of the patients in the database was 57.08 (with
standard deviation ±16.65) years.

Dataset
The mean age of the patients in the database was 57.08 (with
standard deviation ±16.65) years.
40% of patients were female.

Available Attributes
The collected data show the worst values for all variables
during the ﬁrst 24 hours of evolution of Severe Sepsis.

The collected data show the worst values for all variables
during the ﬁrst 24 hours of evolution of Severe Sepsis.
Organ dysfunction was evaluated by means of the SOFA score
system, which objectively measures organ dysfunction for 6
organs/systems.
Cardiovascular (CV) 2.86 (1.62) Haematologic (HAEMATO) 0.78 (1.14)
Respiratory (RESP) 2.31 (1.15) Global SOFA score 7.94 (3.86)
Central Nerv. Sys. (CNS) 0.48 (1.00) Dysf. Organs 1.68 (1.09)
Hepatic (HEPA) 0.48 (0.92) Failure Organs 1.51 (1.02)
Renal (REN) 1.06 (1.20) Total Dysf. Organs 3.18 (1.32)

Severity was evaluated by means of the APACHE II score,
which was 23.03 ± 9.62 for the population under study.

Mechanical ventilation was also assessed (66.71%).

Compliance with the SSC resuscitation bundles was 31.41%.

Compliance with the SSC resuscitation bundles was 31.41%.
The mortality rate intra-ICU for our study population was
26.32%.

Why generative kernels?
Selection of kernel function for a given problem is not trivial.

One normally must have good insight about the problem at
hand.

hand.
Mapping over higher dimensions simpliﬁes the problem but
computational cost grows with ∼ d3.

hand.
Solution: exploit the statistical structure of the data to build
the kernel.

hand.
Solution: exploit the statistical structure of the data to build
the kernel.
Requirement: pdf must be a regular exponential family. This
requirement is fulﬁlled by the dataset analysed in this work.

Quotient Basis Kernel
Let A be a set of n unique points A = {a1, . . . , an} and τ a term
ordering. A Gr¨obner basis of A, G = g1, . . . , gt, is a Gr¨obner basis
of I(A). Therefore, the points in A can be presented as the set of
solutions of 


g1(a) = 0
g2(a) = 0
· · ·
gt(a) = 0

Let A, be a set of n × s unique support points A = {a1, . . . , an}
and τ a term ordering. A monomial basis of the set of polynomial
functions over A is
ESTτ = {xα
: xα
/∈ LT(g) : g ∈ I(A) }
This means that ESTτ comprises the elements xα that are not
divisible by any of the leading terms of the elements of the Gr¨obner
basis of I(A).

Let τ be a term ordering and let us consider an ordering over the
support points A = {a1, . . . , an}. We call design matrix (i.e. ESTτ
evaluated in A) the following n × c matrix
Z = [ESTτ ] A
where c is the cardinality of ESTτ and n is the number of support
points. The covariance of the design matrix of ESTτ , which is a
kernel, is the QBK.

Regular Exponential Family
Let P = (P|η ∈ N) be a regular exponential family with canonical
suﬃcient statistic T. Then the log likelihood function takes the
form
l(η|T) = n(ηt
T − G(η))

Kernels based on the Jensen Shannon Metric

Let P = (P|η ∈ N) be a regular exponential family with canonical
suﬃcient statistic T. Then the log likelihood function takes the
form
l(η|T) = n(ηt
T − G(η))
This function accepts a convex - conjugate (Legendre Dual) of the
form
l(γ|T) = n(γt
T − F(γ))
In our case, the dual F is the negative log-entropy function.

JS(γ1, γ2) =
F(γ1) + F(γ2)
2
− F
γ1 + γ2
2
.
Centred kernel:
φ(x, y) = JS(x, x0) + JS(y, x0) − JS(x, y) − JS(x0, x0).

JS(γ1, γ2) =
F(γ1) + F(γ2)
2
− F
γ1 + γ2
2
.
Centred kernel:
Exponentiated kernel: φ(x, y) = exp(−tJS(x, y)) ∀t > 0.

JS(γ1, γ2) =
F(γ1) + F(γ2)
2
− F
γ1 + γ2
2
.
Centred kernel:
Exponentiated kernel: φ(x, y) = exp(−tJS(x, y)) ∀t > 0.
Inverse kernel: φ(x, y) = 1
t+JS(x,y) ∀t > 0.

RoD with Generative Kernels - QBK -
x1 is the Number of Dysfunctional Organs as measured by the
SOFA Score.
x2 corresponds to Mechanical Ventilation (yes/no).
x3 corresponds to Severity as Measured by the APACHE II
Score.
x4 corresponds to the SSC Resuscitation Bundles (i.e.
administration of antibiotics, performance of haemocultures
and so on). This is also a binary variable.

RoD with Generative Kernels - QBK -
ESTτ =



1, x4, x3, x3x4,
x2, x2x4, x2x3, x2x3x4,
x2
2 , x2
2 x4, x2
2 x3, x2
2 x3x4,
x3
2 , x3
2 x4, x3
2 x3, x3
2 x3x4,
x4
2 , x4
2 x4, x4
2 x3, x4
2 x3x4,
x5
2 , x5
2 x4, x6
2 , x1,
x1x4, x1x3, x1x3x4, x1x2,
x1x2x4, x1x2x3, x1x2x3x4, x1x2
2 ,
x1x2
2 x4, x1x2
2 x3, x1x2
2 x3x4, x1x3
2 ,
x1x3
2 x4, x1x3
2 x3, x1x3
2 x3x4, x1x4
2 ,
x1x4
2 x4, x1x4
2 x3, x1x5
2




RoD with Generative Kernels
We have used Matlab’s Support Vector Machine QP solver
implemented in the BioInformatics and Optimization Toolboxes.
We have also used 10-fold cross validation to evaluate the
classiﬁcation performance for the diﬀerent kernels. A grid search
yielded the appropriate values for C parameters for each Kernel.
More particularly,
Quotient Basis and Fisher C = 1.
Generative Kernels C = 10. Also the parameter t for the
Exponential and Inverse Kernels was set to 2.
Gaussian, Linear and Polynomial Kernels C = 10.

The Risk-of-Death (ROD) formula based on the APACHE II score
is a standard method in use in the critical care ﬁeld.
ln
ROD
1 − ROD
= −3.517 + 0.146 · A + , (1)
where A is the APACHE II score and is a correction factor that
depends on clinical traits at admission in the ICU. For instance, if
the patient has undergone post-emergency surgery, is set to
0.613.

Results
Kernel AUC Error Rate Sens. Spec. CPU time [s]
Quotient 0.89 0.18 0.70 0.86 1.45
Exponential 0.75 0.21 0.70 0.82 1.64
Inverse 0.62 0.22 0.70 0.82 1.68
Centred 0.75 0.21 0.70 0.82 1.99
Gaussian 0.83 0.24 0.65 0.81 1.56
Poly (order 2) 0.69 0.28 0.71 0.76 1.59
Linear 0.62 0.26 0.62 0.78 1.35
Apache II 0.70 0.28 0.55 0.82 n/a

Conclusions
The investigated kernels provided accurate and medically
actionable results, whilst keeping an acceptable balance
between the different parameters of interest (accuracy rate,
sensitivity and specificity).
The QBK is defined through the Gröbner basis of an algebraic
ideal.
It outperforms all kernels presented in this work as well as the
clinical standard method based on the APACHE II score.
All kernels presented outperform the standard APACHE II
ROD formula in terms of accuracy.

THANK YOU!!

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