PART III APPLICATIONS

Transcription

1 S. Vieira PART III APPLICATIONS Fuzz IEEE 2013, Hyderabad India 1 Applications Finance Value at Risk estimation based on a PFS model for density forecast of a continuous response variable conditional on a high-dimensional set of covariates. Healthcare Mortality Risk estimation for intensive care septic shock patients Fuzz IEEE 2013, Hyderabad India 2 Tutorial on Probabilistic Fuzzy Systems 1

2 Value at Risk VaR indicates the maximum loss that a portfolio of assets will suffer over a horizon of h (days) with a confidence of c. Fuzz IEEE 2013, Hyderabad India 3 Value at Risk Motivation Value-at-risk (VaR) is a way to quantify the market risk. Large banks must nowadays base their market risk capital requirements on the VaR estimate. Time series data exhibits complex behavior, e.g. path-dependency (conditional variance) non-linearity skewness and multimodality Value at Risk (VaR) has been successfully estimated using single covariate probabilistic fuzzy systems (PFS) Fuzz IEEE 2013, Hyderabad India 4 Tutorial on Probabilistic Fuzzy Systems 2

3 Value at Risk Goal: VaR estimation based on a PFS model for density forecast of a continuous response variable conditional on a high-dimensional set of covariates. Dataset: VaR estimation of the S&P 500 index 3718 daily returns from February 18, 1997 to November 23, The response variable is the percentage return. Fuzz IEEE 2013, Hyderabad India 5 S&P 500 index example Model estimation with 3208 days and out-of-sample evaluation on days. Fuzz IEEE 2013, Hyderabad India 6 Tutorial on Probabilistic Fuzzy Systems 3

4 S&P 500 index example Multi-covariate probabilistic fuzzy model The PFS we consider approximate the distribution of returns at time t + 1 conditional on multi-covariates at time t. In this work we use a nine variable model using LastDay, LastWeek, LastMonth, CloseAbs95, CloseAbs80, CloseSqr95, CloseSqr80, MaxMin95 and MaxMin80. Fuzz IEEE 2013, Hyderabad India 7 S&P 500 index example Multi-covariate probabilistic fuzzy model Determine PFS parameters Number of rules Antecedent membership functions (number, type, location) Consequent membership functions (number, type, location) Probability parameters Fuzz IEEE 2013, Hyderabad India 8 Tutorial on Probabilistic Fuzzy Systems 4

5 S&P 500 index example Fuzz IEEE 2013, Hyderabad India 9 S&P 500 index example Fuzz IEEE 2013, Hyderabad India 10 Tutorial on Probabilistic Fuzzy Systems 5

6 S&P 500 index - Results Conditional density estimation results Table: Estimated quantiles for the in-sample and out-of-sample. of Quantile Estimation 0.94% 2.37% 4.58% 9.57% 19.73% Out-of-sample 1.37% 3.53% 6.47% 10.39% 19.02% Fuzz IEEE 2013, Hyderabad India 11 S&P 500 index - Results Fuzz IEEE 2013, Hyderabad India 12 Tutorial on Probabilistic Fuzzy Systems 6

7 S&P 500 index - Results Fuzzy histogram model Antecedent membership functions Fuzz IEEE 2013, Hyderabad India 13 S&P 500 index - Results Fuzzy histogram model Fuzzy histogram rule output 0.6 R 1 R R 3 R 4 R Fuzz IEEE 2013, Hyderabad India 14 Tutorial on Probabilistic Fuzzy Systems 7

8 S&P 500 index - Results Fuzzy system Optimized probability parameters μ Aq r t (a) Original μ Aq r t Fuzz IEEE 2013, Hyderabad India 15 (b) After optimization Conclusions and Future Work This model can estimate a probability density function of a non-linear system while keeping a linguistic link between variables. Besides the information provided by the linguistic interpretation of the rules, the probabilistic fuzzy model allows to gain more information and process understanding given by the different reasoning mechanisms. These advantages are illustrated in a financial application of conditional density estimation. Fuzz IEEE 2013, Hyderabad India 16 Tutorial on Probabilistic Fuzzy Systems 8

9 Healthcare Motivation: For clinical decision-making, PFS is particularly appealing as it allows estimating the probability of observing a certain class label which is useful for both providing statistical support to the rules as well as risk stratification against mortality. PFS rules increase the transparency of the learned system for humans and provide an additional means to validate the classifier by experts knowledge regarding the system. Fuzz IEEE 2013, Hyderabad India 17 Mortality Risk Estimation Goal: estimate the hourly mortality risk of a septic shock patient using probabilistic fuzzy models. Compare the estimates to the daily and hourly computed evolution of SOFA score. Data set: Retrospective cohort study involving 618 septic shock patients from four high-resolution temporal ICU databases acquired from Germany, the Netherlands, Portugal and US. Fuzz IEEE 2013, Hyderabad India 18 Tutorial on Probabilistic Fuzzy Systems 9

10 Databases MEDAN database - ICU abdominal septic shock patients admitted to 70 different hospitals in Germany collected from 1998 to patients and 103 different variables CATHARINA Hospital database ICU patients admitted to a large university affiliated general hospital in Eindhoven, The Netherlands 23,779 patients Fuzz IEEE 2013, Hyderabad India 19 Databases Hospital da LUZ database ICU patients admitted to a private Hospital in Lisbon, Portugal collected from 2010 to ,271 patients MIMIC II database - ICU patients admitted to the Beth Israel Deaconess Medical Center in Boston, US collected from 2001 to , patients and more than 200 variables Fuzz IEEE 2013, Hyderabad India 20 Tutorial on Probabilistic Fuzzy Systems 10

11 Mortality Risk Outcome Definition The primary outcome variable is the patient condition (alive or deceased) within a particular time-window from a given time point. This variable was defined as having a binary format, taking value 1 if the patient died within that period of time, and 0 if not. This time-window, also named as lead-time, was varied between 24h, 48h and 72h. Fuzz IEEE 2013, Hyderabad India 21 Mortality Risk Outcome Definition Ideal mortality risk estimation for a patient surviving. Ideal mortality risk estimation for a patient dying. Fuzz IEEE 2013, Hyderabad India 22 Tutorial on Probabilistic Fuzzy Systems 11

12 Probabilistic Fuzzy Classifier Rules R j : If x is A j then y j = w 1 with probability Pr(w 1 A j ) and y j = w 2 with probability Pr(w 2 A j ) and y j = w c with probability Pr(w c A j ) Fuzz IEEE 2013, Hyderabad India 23 Probabilistic Fuzzy Classifier Estimation of antecedents A j and the probability parameters Pr(w c A j ) : maximum-likelihood (ML) method Fuzz IEEE 2013, Hyderabad India 24 Tutorial on Probabilistic Fuzzy Systems 12

13 Features Considered Variables (units) Heart Rate (beats/min) Platelets (103/μL) Systolic arterial blood pressure (mmhg) PTT (sec) Diastolic arterial blood pressure (mmhg) AT3 (%) Temperature (C) Sodium (mmol/l) CVP (mmhg) Potassium (mmol/l) Arterial ph Calcium (mmol/l) Arterial po2 (mmhg) Creatinine (mg/dl) Arterial pco2 (mmhg) Urea (mg/dl) Arterial Base Excess GOT(ASAT) (U/L) Bicarbonate (mmol) GPT(ALAT) (U/L) SpO2 (%) Bilirubin (mg/dl) White blood cells (103/μL) CRP (mg/l) Hematocrit (%) Glucose (mg/dl) FiO2 (%) Fuzz IEEE 2013, Hyderabad India 25 Model Assessment Each of the 4 datasets was initially randomly divided into three subsets: 40% for feature selection (FS) 40% for model assessment (MA) and 20% for model validation (MV) Sequential forward selection for FS using area under the receiver-operating curve (AUC). Validity and robustness - 10-fold cross-validation was performed in the MA data Estimate of mortality risks in an individual patient basis for the MV subset. Fuzz IEEE 2013, Hyderabad India 26 Tutorial on Probabilistic Fuzzy Systems 13

14 Results Most predictive variables: Database Variables AUC MEDAN (7) Diastolic arterial blood pressure; Temperature; 0.81 ± 0.02 Platelets; Prothrombin time; BUN; FiO2; Bilirubin MIMIC II CATHARINA LUZ (7) Arterial ph; Platelets; Prothrombin time; Sodium; Diastolic arterial blood pressure; Glucose; FiO2 (10) Arterial ph; Platelets; Glasgow Coma Score; Prothrombin time; Bilirubin; Creatinine; Diastolic arterial blood pressure; BUN; Carbon dioxide (7) Arterial ph; Hematocrit; Prothrombin time; Bilirubin; FiO2; Glucose; Platelets; Diastolic arterial blood pressure 0.74 ± ± ± 0.04 Fuzz IEEE 2013, Hyderabad India 27 Results AUC for the best-fitted model in the MV subset for the 5 most predictive variables: Variables Method AUC Hourly SFS PFS 0.65 ± 0.04 LR 0.69 ± 0.08 Fuzz IEEE 2013, Hyderabad India 28 Tutorial on Probabilistic Fuzzy Systems 14

15 Results Average correlation (p-value) between the predicted mortality risk signal and the step signal, with a 72h lead time: Variables Method MEDAN MIMIC II CATHARINA LUZ Daily SOFA Hourly SOFA Daily SFS Hourly SFS PFS 0.07 (0.24) 0.05 (0.15) 0.01 (0.11) (0.10) LR 0.08 (0.22) 0.04 (0.17) 0.02 (0.09) (0.13) PFS 0.07 (0.19) 0.05 (0.21) 0.03 (0.12) (0.09) LR 0.07 (0.17) 0.06 (0.20) 0.02 (0.14) (0.11) PFS 0.35 (0.04) 0.18 (0.08) 0.08 (0.06) 0.07 (0.14) LR 0.37 (0.05) 0.16 (0.09) 0.11 (0.05) (0.10) PFS 0.43 (0.04) 0.26 (0.04) 0.13 (0.05) 0.10 (0.12) LR 0.40 (0.03) 0.24 (0.05) 0.14 (0.04) 0.13 (0.08) Fuzz IEEE 2013, Hyderabad India 29 Surviving Patient Fuzz IEEE 2013, Hyderabad India 30 Tutorial on Probabilistic Fuzzy Systems 15

16 Patient Dying 72h lead-time Fuzz IEEE 2013, Hyderabad India 31 Conclusions The present work attempted to present a measure that goes beyond the daily and tries to capture the hourly yprogress. Results showed that the best discrimination and correlation are obtained when a lead-time of 72h is used. This suggests that the deterioration of a septic shock patient s condition is in general better captured along a 72h window picture. Fuzz IEEE 2013, Hyderabad India 32 Tutorial on Probabilistic Fuzzy Systems 16

17 Conclusions From the set of selected variables, only non-invasive diastolic blood pressure is actually feasible to be determined in an hourly basis in the ICU setting. Remaining variables are more likely to be acquired in average every 8 to 16 hours. However, despite this lower frequency of collection, the prolongation of the last available information in an hourly basis still points to an improvement of performance, in opposition to a single snapshot considering the worst daily values. Fuzz IEEE 2013, Hyderabad India 33 Future Work Personal hourly estimates may assist clinicians in making decisions in several distinct ways: stratifying patients according to their mortality risk, by providing greater certainty of the expected effects of treatment, by improving understanding of specific prognostic elements and their relative influence on outcomes, by reducing reliance on commonly used clinical rules that may be biased, and by providing an explicit opportunity to review and compare explicit probability thresholds for important clinical decisions Fuzz IEEE 2013, Hyderabad India 34 Tutorial on Probabilistic Fuzzy Systems 17

18 Probabilistic Fuzzy Systems Tutorial Fuzz IEEE 2013, Hyderabad India Fuzz IEEE 2013, Hyderabad India 36 Tutorial on Probabilistic Fuzzy Systems 18