An application of the GAM-PCA-VAR model to respiratory disease and air pollution data

Transcription

1 An application of the GAM-PCA-VAR model to respiratory disease and air pollution data Márton Ispány 1 Faculty of Informatics, University of Debrecen Hungary Joint work with Juliana Bottoni de Souza, Valdério A. Reisen, Glaura C. Franco, Pascal Bondon, Jane Meri Santos International work-conference on Time Series Granada, Spain September 18th-20th, Márton Ispány was supported by the EFOP project.the project is co-financed by the European Union and the European Social Fund. ITISE2017 Márton Ispány Respiratory diseases and air pollution data 1 / 28

2 Outline 1 Respiratory diseases and air pollution in Vitória (Brazil) 2 Dependence in the data: inter and temporal correlation 3 Generalized additive models (GAM) and relative risk (RR) 4 Simulation study of GAM under dependent covariates 5 GAM-PCA model 6 GAM-PCA-VAR model 7 Goodness-of-fit and RR 8 Conclusions ITISE2017 Márton Ispány Respiratory diseases and air pollution data 2 / 28

3 1. Data Time period: January 1, 2005 and December 31, 2010, sample size n=2191. Discrete response Y : the number of hospital admissions for respiratory diseases (RD) was obtained from the main children s emergency department in the Metropolitan Area (called Hospital Infantil Nossa Senhora da Gloria). Continuous covariates X: atmospheric pollutants as particulate material (PM 10 ), sulphur dioxide (SO 2 ), nitrogen dioxide (NO 2 ), ozone (O 3 ) and carbon monoxide (CO). Confounding variables: temperature, relative humidity The concentrations of the pollutants considered exceeded neither the primary air quality standard recommended by the Brazilian National Council for the Environment (CONAMA), nor the guidelines suggested by the World Health Organization (WHO). ITISE2017 Márton Ispány Respiratory diseases and air pollution data 3 / 28

4 1. Descriptive statistics Pollutants are measured as the 24-hour average concentration for PM 10 and SO 2, 8-hour moving average concentrations for CO and O 3, and the 24-hour maximum concentration for NO 2 by daily averages among the stations. Percentile Mean Std. dev. Min Max PM 10 (µg/m 3 ) SO 2 (µg/m 3 ) O 3 (µg/m 3 ) NO 2 (µg/m 3 ) CO (µg/m 3 ) Min temp ( C) Ave temp ( C) Max temp ( C) Air relative humidity (%) Number of treatments for RD ITISE2017 Márton Ispány Respiratory diseases and air pollution data 4 / 28

5 1. Concentration of CO; NO 2 ; SO 2 ; PM 10 ; O 3 and the Number of treatments for RD CO NO Concentration Concentration Jan 06 Jan 08 Jan 10 Jan 12 Time Jan 06 Jan 08 Jan 10 Jan 12 Time SO 2 PM Concentration Concentration Jan 06 Jan 08 Jan 10 Jan 12 Time Jan 06 Jan 08 Jan 10 Jan 12 Time O Treatments for respiratory Concentration Admissions Jan 06 Jan 08 Jan 10 Jan 12 Time Jan 06 Jan 08 Jan 10 Jan 12 Time ITISE2017 Márton Ispány Respiratory diseases and air pollution data 5 / 28

6 2. Correlation between pollutants, meteorological variables and number of treatments Although some sample correlations appear not to be numerically significant, the non-parametric Pearson correlation test indicated that correlation among the atmospheric pollutants is significant for all pairs of variables at level 5% and for most pairs at level 0.1%. PM 10 SO 2 NO 2 CO O 3 T(max) T(min) RH Number of treatments PM SO NO CO O T(max) T(min) RH Number of treatments T= Temperature ( C); RH= Air relative humidity (%) All correlations were significant at a 5% level ITISE2017 Márton Ispány Respiratory diseases and air pollution data 6 / 28

7 2. Sample of the pollutants 1.00 CO 1.00 O Lag Lag SO 2 NO Lag Lag PM Lag ITISE2017 Márton Ispány Respiratory diseases and air pollution data 7 / 28

8 2. Result of the explorative data analysis Evidences for: Interdependence between the continuous covariates (pollutants) Serial (temporal) dependence in the multivariate time series of covariates (pollutants) The aim of the analysis: Quantify the association between respiratory diseases and air pollution concentrations, especially, PM 10, SO 2, NO 2, CO and O 3. ITISE2017 Márton Ispány Respiratory diseases and air pollution data 8 / 28

9 3. Generalized additive models (GAM) Hastie and Tibshirani (1990) {Y t } {Y t } t Z is a count time series with conditional Poisson distribution of mean µ t : P (Y t = y t F t 1 ) = e µ t µ y t t y t!, y t = 0, 1,... For a sample Y 1,..., Y n of {Y t }, the conditional log-likelihood function is given by n l(µ) (Y t ln µ t µ t ), t=1 where µ t depends on the pollutants covariates and confounding variables thorugh the canonical link ln(µ t ) = q p β j X jt + f j (X jt ) with q < p j=0 j=q+1 ITISE2017 Márton Ispány Respiratory diseases and air pollution data 9 / 28

10 3. Relative risk (RR) Frequently used in epidemiological studies to measure the impact of atmospheric pollutant concentrations on the health of the exposed population, Baxter et al. (1997) The relative change in the expected count of respiratory disease events per ξ unit change in the covariate while keeping the other covariates fixed: RR Xj (ξ) := E(Y X j = ξ, X i = x i, i j) E(Y X j = 0, X i = x i, i j) For Poisson regression: RR Xj (ξ) = exp ( β j ξ ) RR and its CI are estimated as follows: ( ) ( ) RR Xj (ξ) = exp ˆβ j ξ, CI(RR Xj (ξ)) = exp ˆβ j ξ z α/2 se( ˆβ j )ξ Hypothesis testing: H 0 : RR Xj (1) = 1 against H 1 : RR Xj (1) > 1 ITISE2017 Márton Ispány Respiratory diseases and air pollution data 10 / 28

11 4. Simulation study of GAM Scenarios: independent data (S1); the dependent variable is a time series and the covariates are independent random vectors in time (S2); both the dependent and independent variables are time series (S3) Independent means N(0, 1). Dependent means AR(1) process. For the 3 scenarios, the data were generated from a conditional Poisson model, Y t X t Po(µ t ), with canonical link function. The sample size n = 100 and the number of Monte Carlo simulations was equal to Similar result was obtained in Dionisio et al. (2016). ITISE2017 Márton Ispány Respiratory diseases and air pollution data 11 / 28

12 4. Simulation results for a single covariate Model Parameter Mean Bias MSE S1: Independent β 0 = β 1 = S2: ϕ=0.1 β 0 = β 1 = S2: ϕ=0.5 β 0 = β 1 = S2: ϕ=0.9 β 0 = β 1 = S3: ϕ=0.1 β 0 = β 1 = S3: ϕ=0.5 β 0 = β 1 = S3: ϕ=0.9 β 0 = β 1 = ITISE2017 Márton Ispány Respiratory diseases and air pollution data 12 / 28

13 4. Simulation results for two covariates, X 1 and X 2 Model Parameter Mean Bias MSE S1: Independent β 0 = β 1 = β 2 = S2 β 0 = β 1 = β 2 = S3: φ 11 = 0.7, φ 12 = 0 β 0 = φ 21 = 0, φ 22 = 0.5 β 1 = β 2 = S3: φ 11 = 0.7, φ 12 = 0.4 β 0 = φ 21 = 0, φ 22 = 0.5 β 1 = β 2 = ITISE2017 Márton Ispány Respiratory diseases and air pollution data 13 / 28

14 5. PCA for the pollutants Principal component analysis (PCA) for a stationary vector time series {X t } with the covariance matrix Σ X with eigenvalues/eigenvectors pair (λ i, a i ), i = 1,..., q: Z it = a i X t, i = 1,..., q Properties: Cov(Z it, Z j(t+h) ) = a i Cov(X t, X t+h )a j = a i Γ X (h)a j PC1 PC2 PC3 PC4 PC5 Standard deviation Proportion of variance Cumulative proportion of variance CO * NO * O * PM * SO * Standard deviation is the square root of the eigenvalue ITISE2017 Márton Ispány Respiratory diseases and air pollution data 14 / 28

15 5. GAM-PCA - Generalized additive modelling and principal component analysis A probabilistic latent variable model defined by with link function µ t (υ 0, υ, A) = exp Y t F t 1 Po(µ t ) and X t = AZ t { r i=0 } } υ i Z it = exp {υ 0 + υ A r X t where the latent variables {Z t } WN q (Λ), Λ is a diagonal variance matrix of dimension q and A is an orthogonal matrix of dimension q q. Given a sample (X 1, Y 1 ),..., (X n, Y n ), the log-likelihood: l n (Y t ln µ t µ t ) 1 2 t=1 n (A X t ) Λ 1 (A X t ) n ln det Λ 2 t=1 ITISE2017 Márton Ispány Respiratory diseases and air pollution data 15 / 28

16 5. Fitting GAM-PCA model and RR GAM-PCA, as a two-stage model, can be fitted by a two-stage procedure: 1 The parameter matrices A and Λ are estimated by applying the PCA for the estimated covariance matrix Σ X. 2 The parameters υ 0 and υ are estimated by fitting the GAM model with link function using the first r-th PCs. The estimate of RR per ξ unit change in the pollutant concentration is given as follows: RR ( ) X j (ξ) = exp ˆβ j ξ where r ˆβ j := â ji ˆυ i The standard error of ˆβ j se 2 ( ˆβ j ) = i=1 can be derived as r âji 2 se2 (ˆυ i ). i=1 ITISE2017 Márton Ispány Respiratory diseases and air pollution data 16 / 28

17 5. CCF of the main PCs of the pollutants PC1 PC1 x PC Lag Lag PC1 x PC3 PC2 x PC Lag Lag A counterexample to page 299 in Jolliffe (2002), in which the author argues that when the main objective of PCA is only descriptive, complications such as non-independence (temporal) does not seriously affect this objective ITISE2017 Márton Ispány Respiratory diseases and air pollution data 17 / 28

18 5. Impact of PCA for temporally dependent (VAR(1)) covariates Sample autocorrelation function () and cross-correlation function (CCF) of the PCs for simulated data: PC ₁ PC ₁ x PC₂ Lag Lag PC ₂ x PC₁ PC ₂ Lag Lag ITISE2017 Márton Ispány Respiratory diseases and air pollution data 18 / 28

19 6. GAM-PCA-VAR - GAM-PCA and vector autoregressive modelling A probabilistic latent variable model defined by Y t F t 1 Po(µ t ) and X t = ΦX t 1 + AZ t with link function r ln(µ t ) = υ i Z it = υ 0 + υ A r X t υ A r ΦX t 1, i=0 where the latent variables {Z t } WN q (Λ), Λ is a diagonal variance matrix of dimension q, A is an orthogonal matrix of dimension q q and Φ is a matrix of dimension q q. The quintuplet (υ 0, υ, A, Λ, Φ) forms the parameters of the GAM-PCA-VAR model. By the link function, Y t depends on both X t and X t 1 demonstrating the presence of serial dependence in the GAM-PCA-VAR model. ITISE2017 Márton Ispány Respiratory diseases and air pollution data 19 / 28

20 6. Fitting GAM-PCA-VAR model Given a sample (X 1, Y 1 ),..., (X n, Y n ), the log-likelihood: l n (Y t ln µ t µ t ) 1 2 t=2 n t=2 ε t AΛ 1 A ε t n 1 2 where ε t = X t ΦX t 1 is the error term. GAM-PCA-VAR is fitted by a three-stage procedure: ln det Λ, 1 (S)VAR(1) model is fitted to the original covariates by applying standard time series techniques. 2 Using PCA for the residuals defined by ˆε t = X t ˆΦX t 1, t = 2,..., n, where ˆΦ denotes the estimated autoregressive coefficient matrix in the fitted VAR(1) model, the first r-th PCs are computed. 3 GAM model is fitted using these PCs by maximizing the Poisson part of the log-likelihood. ITISE2017 Márton Ispány Respiratory diseases and air pollution data 20 / 28

21 6. PCA for the filtered pollutants The time structure of the pollutants did not alter the cumulative proportion of the variance but the clustering of the pollutants by factor loadings resulted in a different interpretation, which is more coherent with the behaviour of the variables considered. PC1 PC2 PC3 PC4 PC5 Standard deviation Proportion of variance Cumulative proportion of variance CO NO O PM SO Standard deviation is the square root of the eigenvalue ITISE2017 Márton Ispány Respiratory diseases and air pollution data 21 / 28

22 6. CCF of the main PCs of the filtered pollutants Figure shows that the fitting of the seasonal VAR(1) model practically eliminated the autocorrelation of PC1 and the cross-correlation, as expected from the aforementioned discussion. PC1 PC1 x PC PC1 x PC PC2 x PC ITISE2017 Márton Ispány Respiratory diseases and air pollution data 22 / 28

23 7. Goodness-of-fit statistics The AIC and BIC information criteria indicate that the GAM-PCA-VAR model is the best to fit the data. Model MSE AIC BIC GAM GAM-PCA GAM-PCA-VAR ITISE2017 Márton Ispány Respiratory diseases and air pollution data 23 / 28

24 7. Fitted GAM-PCA-VAR model to the number of treatments for RD The graph shows that the model provided a good fit to the number of daily treatments for children under 6 years old in the Vitória metropolitan area. Number of hospital treatments Original series Adjusted series Time ITISE2017 Márton Ispány Respiratory diseases and air pollution data 24 / 28

25 7. Relative risk for air pollutants Relative risk (RR) and 95% confidence intervals for treatments for respiratory diseases in children under 6 years old for an interquartile variation in the pollutants PM 10, SO 2, NO 2, O 3 and CO in the Metropolitan Area from Jan 2005 to Dec RR RR RR PM (1.010,1.039) 1.029(1.001,1.090) 1.075(1.001,1.092) SO (1.010,1.080) 0.982(0.972,1.001) 1.027(1.010,1.040) CO 1.020(1.010,1.030) 1.048(1.002,1.071) 1.077(1.020,1.100) NO (0.990,1.020) 1.028(1.010,1.040) 1.012(1.010,1.030) O (0.972,1.001) 1.081(1.003,1.093) 0.992(0.992,1.020) RR: GAM, RR : GAM-PCA and RR : GAM-PCA-VAR The RR estimates for the pollutant PM 10 increased from 2% ( RR) to 3% ( RR ˆ ) and 7% ( RR ). Substantial increases in the RR estimates were also observed for the pollutant CO. In this case, RR = 1.020, RR = and RR = ITISE2017 Márton Ispány Respiratory diseases and air pollution data 25 / 28

26 8. Conclusions Clear evidences were found for inter- and temporal correlation between the pollutant covariates. Simulation study was performed to study the impact of the presence of inter- and temporal correlation between the covariates for estimating the parameters of GAM. A novel hybrid GAM-PCA-VAR model was proposed. GAM-PCA-VAR model was fitted to the data. By this new model more significant (and more realistic) RR estimates were obtained. ITISE2017 Márton Ispány Respiratory diseases and air pollution data 26 / 28

27 References Juliana Bottoni de Souza, Valdério A. Reisen, Glaura C. Franco, Márton Ispány, Pascal Bondon, Jane Meri Santos (2018) Generalized additive model with principal component analysis: An application to time series of respiratory disease and air pollution data Journal of the Royal Statistical Society: Series C (Applied Statistics), DOI: /rssc ITISE2017 Márton Ispány Respiratory diseases and air pollution data 27 / 28

28 Thank you for your attention! ITISE2017 Márton Ispány Respiratory diseases and air pollution data 28 / 28