This document describes research on using a quotient basis kernel (QBK) machine learning method to predict mortality in patients with severe sepsis using clinical data. It introduces sepsis and its impact, describes the dataset of 354 sepsis patients from an ICU in Spain, and explains the QBK approach which exploits the statistical structure of the data to build an interpretable kernel function. The goal is to develop a robust and accurate method for predicting patient outcomes to support clinical decision making.

1. 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

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

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

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

5. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Introduction
Sepsis is a clinical syndrome defined by the presence of
infection and Systemic Inflammatory Response Syndrome
(SIRS).
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

6. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Introduction
Sepsis is a clinical syndrome defined by the presence of
infection and Systemic Inflammatory Response Syndrome
(SIRS).
This can lead to severe sepsis or to septic shock (severe sepsis
with hypotension refractory to fluid administration) and
multi-organ failure.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

7. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Introduction
Sepsis is a clinical syndrome defined by the presence of
infection and Systemic Inflammatory Response Syndrome
(SIRS).
This can lead to severe sepsis or to septic shock (severe sepsis
with hypotension refractory to fluid administration) and
multi-organ failure.
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.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

8. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Introduction
Sepsis is a clinical syndrome defined by the presence of
infection and Systemic Inflammatory Response Syndrome
(SIRS).
This can lead to severe sepsis or to septic shock (severe sepsis
with hypotension refractory to fluid administration) and
multi-organ failure.
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.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

9. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
The medical management of sepsis and the study of its
prognosis and outcome is a relevant medical research
challenge.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

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

12. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Contents
1 Introduction
2 Database Description
3 Risk of Death Assessment from Observed Data
4 Conclusions
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

14. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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).
Data from 354 patients with severe sepsis was collected in
this ICU between June, 2007 and December, 2010.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

15. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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).
Data from 354 patients with severe sepsis was collected in
this ICU between June, 2007 and December, 2010.
55% of cases correspond to ‘medical’ sepsis.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

16. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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).
Data from 354 patients with severe sepsis was collected in
this ICU between June, 2007 and December, 2010.
55% of cases correspond to ‘medical’ sepsis.
The mean age of the patients in the database was 57.08 (with
standard deviation ±16.65) years.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

17. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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).
Data from 354 patients with severe sepsis was collected in
this ICU between June, 2007 and December, 2010.
55% of cases correspond to ‘medical’ sepsis.
The mean age of the patients in the database was 57.08 (with
standard deviation ±16.65) years.
40% of patients were female.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

18. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Available Attributes
The collected data show the worst values for all variables
during the first 24 hours of evolution of Severe Sepsis.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

20. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Available Attributes
Severity was evaluated by means of the APACHE II score,
which was 23.03 ± 9.62 for the population under study.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

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

23. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Available Attributes
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%.
The mortality rate intra-ICU for our study population was
26.32%.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

24. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Why generative kernels?
Selection of kernel function for a given problem is not trivial.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

26. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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.
Mapping over higher dimensions simplifies the problem but
computational cost grows with ∼ d3.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

27. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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.
Mapping over higher dimensions simplifies the problem but
computational cost grows with ∼ d3.
Solution: exploit the statistical structure of the data to build
the kernel.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

28. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
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.
Mapping over higher dimensions simplifies the problem but
computational cost grows with ∼ d3.
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.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

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

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

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

33. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Kernels based on the Jensen Shannon Metric
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

35. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Kernels based on the Jensen Shannon Metric
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).
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

36. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Kernels based on the Jensen Shannon Metric
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).
Exponentiated kernel: φ(x, y) = exp(−tJS(x, y)) ∀t > 0.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

37. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Kernels based on the Jensen Shannon Metric
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).
Exponentiated kernel: φ(x, y) = exp(−tJS(x, y)) ∀t > 0.
Inverse kernel: φ(x, y) = 1
t+JS(x,y) ∀t > 0.
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

38. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Contents
1 Introduction
2 Database Description
3 Risk of Death Assessment from Observed Data
4 Conclusions
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

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

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

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

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

44. Introduction Database Description Risk of Death Assessment from Observed Data Conclusions
Contents
1 Introduction
2 Database Description
3 Risk of Death Assessment from Observed Data
4 Conclusions
Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona
A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

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

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