A Spatio-Temporal Point Process Model for Ambulance Demand

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1 A Spatio-Temporal Point Process Model for Ambulance Demand David S. Matteson Department of Statistical Science Department of Social Statistics Cornell University Joint work with: Zhengyi Zhou (Applied Math, Cornell); Dawn B. Woodard & Shane G. Henderson (ORIE, Cornell); Athanasios C. Micheas (Statistics, UM-Columbia) Sponsorship: National Science Foundation 2014 September 18 avid S. Matteson S-T Ambulance Demand 2014 September 18 1 / 56

2 Introduction AHA American Heart Association: avid S. Matteson S-T Ambulance Demand 2014 September 18 2 / 56

Introduction Toronto Spatio-Temporal Ambulance Demand in Toronto February 1-28, 2007; 15, 393 observations Mean number of call arrivals per hour 10 15 20 25 30 35 Sunday Monday Tuesday Wednesday

3 Introduction Toronto Spatio-Temporal Ambulance Demand in Toronto February 1-28, 2007; 15, 393 observations Mean number of call arrivals per hour Sunday Monday Tuesday Wednesday Thursday Friday Saturday Mean 04:00:00 08:00:00 12:00:00 16:00:00 20:00:00 24:00:00 Hour of day Spatial demand locations (bases); mean demand per hour, by day David S. Matteson S-T Ambulance Demand 2014 September 18 3 / 56

4 Introduction Outline A Spatio-Temporal Model for Ambulance Demand Notation and Assumptions A Mixture Model Time invariant component distributions Mixture weight constraints Estimation Data augmentation Autoregressive priors Birth-and-death MCMC Imposing a boundary Comparison Methods Predictive Accuracy avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September 18 4 / 56

5 Introduction Notation Notation A continuous spatial domain S S = R 2 S = S R 2 (incorporate given spatial boundary) A discretized temporal domain T = {1, 2,..., T } T consecutive, equally spaced, time periods (1 or 2-hour blocks) s t,i spatial location of ith point at time t, for i = 1,..., n t S t = {s t,i : i = 1,..., n t } avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September 18 5 / 56

6 Introduction Assumptions Assumptions S t independently follows a non-homogeneous Poisson point process over S Positive, integrable intensity function λ t Decompose intensity λ t (s) = δ t f t (s), for s S avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September 18 6 / 56

7 Introduction Assumptions Assumptions S t independently follows a non-homogeneous Poisson point process over S Positive, integrable intensity function λ t Decompose intensity λ t (s) = δ t f t (s), for s S δ t = S λ t(s) ds aggregate event intensity for period t f t (s) spatial density of events for period t f t (s) > 0, for s S & S f t(s) ds = 1 n t λ t Poisson(δ t ) s t,i λ t, n t iid ft (s) for i = 1,..., n t avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September 18 6 / 56

8 Aggregate Demand Hourly Toronto EMS Data Su 8am 4pm Mo 8am 4pm Tu 8am 4pm We 8am 4pm Th 8am 4pm Fr 8am 4pm Sa 8am 4pm Su 8am 4pm Mo 8am 4pm Tu 8am 4pm We 8am 4pm Th 8am 4pm Fr 8am 4pm Sa 8am 4pm Hourly call arrivals Day/hour index Hourly call arrivals Hourly call arrivals Sunday Friday 04:00:00 08:00:00 12:00:00 16:00:00 20:00:00 24:00:00 hour of day 04:00:00 08:00:00 12:00:00 16:00:00 20:00:00 24:00:00 hour of day avid S. Matteson S-T Ambulance Demand 2014 September 18 7 / 56

9 Aggregate Demand Constrained and Smooth Factor Model: k f kl k (a) Factor levels (log linear scale) f 1 f 2 f 3 f 4 04:00:00 08:00:00 12:00:00 16:00:00 20:00:00 24:00:00 Hour of day (b) (c) Daily factor loadings (log linear scale) L L 2 L 3 L 4 Daily factor loadings (log linear scale) L L 2 L 3 L 4 F F F F First 28 days of 2008 Day Index for 2008 (Tic marks at Fridays) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September 18 8 / 56

10 Aggregate Demand avid S. Matteson S-T Ambulance Demand 2014 September 18 9 / 56

11 Spatio-Temporal Demand Spatio-Temporal Ambulance Demand in Toronto One Month avid S. Matteson S-T Ambulance Demand 2014 September / 56

12 Spatio-Temporal Demand Spatio-Temporal Ambulance Demand in Toronto One Month One Hour David S. Matteson S-T Ambulance Demand 2014 September / 56

13 Spatio-Temporal Demand MEDIC Current Industry Practice: MEDIC avid S. Matteson S-T Ambulance Demand 2014 September / 56

14 Spatio-Temporal Demand MEDIC Current Industry Practice: MEDIC avid S. Matteson S-T Ambulance Demand 2014 September / 56

15 Spatio-Temporal Demand MEDIC Current Industry Practice: MEDIC avid S. Matteson S-T Ambulance Demand 2014 September / 56

16 Spatio-Temporal Demand MEDIC Current Industry Practice: MEDIC avid S. Matteson S-T Ambulance Demand 2014 September / 56

Spatio-Temporal Demand Dependence Spatio-Temporal Ambulance Demand in Toronto Figure: (a) training data, downtown subregion outlined by a rectangle; (b) time series (top) and autocorrelation function

17 Spatio-Temporal Demand Dependence Spatio-Temporal Ambulance Demand in Toronto Figure: (a) training data, downtown subregion outlined by a rectangle; (b) time series (top) and autocorrelation function (ACF, bottom) of the proportions of observations arising from downtown. Each period is a two-hour interval. Weekly (84 time periods) and daily (12 time periods) seasonality, as well as low-order autocorrelation. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

Spatio-Temporal Demand Dependence Spatio-Temporal Ambulance Demand in Toronto (a) (b) Figure: (a) training data, York subregion outlined by a rectangle; (b) time series (top) and

18 Spatio-Temporal Demand Dependence Spatio-Temporal Ambulance Demand in Toronto (a) (b) Figure: (a) training data, York subregion outlined by a rectangle; (b) time series (top) and autocorrelation function (ACF, bottom) of the proportions of observations arising from downtown. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

Spatio-Temporal Demand Dependence Spatio-Temporal Ambulance Demand in Toronto Figure: (a) Northwest region; (b) time series (top) and ACF (bottom) of the

19 Spatio-Temporal Demand Dependence Spatio-Temporal Ambulance Demand in Toronto Figure: (a) Northwest region; (b) time series (top) and ACF (bottom) of the proportions of observations arising from the region. Null ACF in this region. avid S. Matteson S-T Ambulance Demand 2014 September / 56

20 Spatio-Temporal Model Gaussian Mixture A Spatial Gaussian Mixture Model We assume that {δ t } is known or obtained from temporal aggregate demand analysis: Channouf et al. (2007) or Matteson et al. (2011). Consider modeling spatial densities {f t (s)} via Gaussian mixture K f t (s; {p t,j }, {µ t,j }, {Σ t,j }) = p t,j φ(s; µ t,j, Σ t,j ), s S j=1 avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

21 Spatio-Temporal Model Gaussian Mixture A Spatial Gaussian Mixture Model We assume that {δ t } is known or obtained from temporal aggregate demand analysis: Channouf et al. (2007) or Matteson et al. (2011). Consider modeling spatial densities {f t (s)} via Gaussian mixture K f t (s; {p t,j }, {µ t,j }, {Σ t,j }) = p t,j φ(s; µ t,j, Σ t,j ), s S j=1 φ is the bivariate Gaussian density, mean µ and covariance Σ Mixture weights {p t,j }: p t,j > 0 and K j=1 p t,j = 1 ( j, t) Number of mixture components K, fixed or variable avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

22 Spatio-Temporal Model Gaussian Mixture Means and Covariances (ellipses at 90% level) for K = 15 David S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

23 Spatio-Temporal Model Model 1 Time Invariant Component Distributions K f t (s; {p t,j }, {µ j }, {Σ j }) = p t,j φ(s; µ j, Σ j ), s S (1) j=1 Constant component mixture means µ t,j = µ j ( j, t) Constant component mixture covariances Σ t,j = Σ j ( j, t) Mixture weights {p t,j }: p t,j > 0 and K j=1 p t,j = 1 ( j, t) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

24 Spatio-Temporal Model Model 1 Mixture Weights avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

25 Model 1 Estimation Bayesian Estimation: Priors and Data Augmentation Following Richardson and Green (1997) and Stephens (2000) For j = 1,..., K and t = 1,..., T, with p t = (p t,1,..., p t,k ), let p t Dirichlet(γ,..., γ), µ j Normal(ξ, κ 1 ), Σ 1 j β Wishart(2α, (2β) 1 ), β Wishart(2g, (2h) 1 ), avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

26 Model 1 Estimation Bayesian Estimation: Priors and Data Augmentation Following Richardson and Green (1997) and Stephens (2000) For j = 1,..., K and t = 1,..., T, with p t = (p t,1,..., p t,k ), let p t Dirichlet(γ,..., γ), µ j Normal(ξ, κ 1 ), Σ 1 j β Wishart(2α, (2β) 1 ), β Wishart(2g, (2h) 1 ), Augment data: latent component labels z t,i denotes membership for s t,i Full conditionals: P(z t,i = j ) p t,j φ(s t,i ; µ j, Σ j ), [β ] Wishart [p t ] Dirichlet [µ j ] Normal [Σ 1 j ] Wishart avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

27 Mixture Weight Constraints Model 2 Modeling Seasonality: Mixture Weight Constraints Let W N (W << T ) denote length of a given cycle W = 84: 2-hour time periods within a week Each t T is matched to some b {1,..., W } such that b mod W = t mod W We assume s S f t (s; {p t,j }, {µ j }, {Σ j }) = f b (s; {p b,j }, {µ j }, {Σ j }) K = p b,j φ(s; µ j, Σ j ), (2) All time periods with same position within the cycle have the same density j=1 David S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

28 Mixture Weight Constraints Model 2 Mixture Weights avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

29 Model 2 CAR Prior (Conditional) Autoregressive Prior avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

30 Model 2 CAR Prior Summation Constraint avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

31 Model 2 CAR Prior A Reparameterization avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

32 Model 2 Reparameterization A Reparameterization Multinomial logit transformation of the mixture weights [ ] p b,r π b,r = log 1 K 1 j=1 p, r = 1,..., K 1, b,j avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

33 Model 2 Reparameterization A Reparameterization Multinomial logit transformation of the mixture weights [ ] p b,r π b,r = log 1 K 1 j=1 p, r = 1,..., K 1, b,j with inverse transformation p b,j = p b,k = exp(π b,j ) 1 + K 1 r=1 exp(π, for j = 1,..., K 1, b,r ) K 1 r=1 exp(π b,r ). avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

34 Model 2 Reparameterization A Reparameterization Multinomial logit transformation of the mixture weights [ ] p b,r π b,r = log 1 K 1 j=1 p, r = 1,..., K 1, b,j with inverse transformation p b,j = p b,k = exp(π b,j ) 1 + K 1 r=1 exp(π, for j = 1,..., K 1, b,r ) K 1 r=1 exp(π b,r ). Autoregressive priors to model dependence in transformed weights {π b,j } (i) first-order autocorrelation (ii) first-order autocorrelation with daily seasonality avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

35 Model 2 Reparameterization (Conditional) Autoregressive Prior avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

36 Model 2 Autoregressive Priors First-Order (Conditional) Autoregressive Prior Transformed weight π b,r depends on neighboring periods π b 1,r, π b+1,r We impose the following priors π b,r π b,r N ( c r + ρ r [(π b 1,r c r ) + (π b+1,r c r )], νr 2 ) r = 1,..., K 1; b = 1,..., W ; π b,r = (π 1,r,..., π b 1,r, π b+1,r,..., π W,r ) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

37 Model 2 Autoregressive Priors First-Order (Conditional) Autoregressive Prior Transformed weight π b,r depends on neighboring periods π b 1,r, π b+1,r We impose the following priors π b,r π b,r N ( c r + ρ r [(π b 1,r c r ) + (π b+1,r c r )], νr 2 ) r = 1,..., K 1; b = 1,..., W ; π b,r = (π 1,r,..., π b 1,r, π b+1,r,..., π W,r ) ρ r U(0, 0.5) c r N(0, 10 4 ) ν 2 r U(0, 10 4 ) Autoregressive coefficients ρ r control the persistence over time Intercepts c r control for average levels Variances ν 2 r control the variability avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

38 Model 2 Autoregressive Priors First-Order (Conditional) Autoregressive Prior Transformed weight π b,r depends on neighboring periods π b 1,r, π b+1,r We impose the following priors π b,r π b,r N ( c r + ρ r [(π b 1,r c r ) + (π b+1,r c r )], νr 2 ) r = 1,..., K 1; b = 1,..., W ; π b,r = (π 1,r,..., π b 1,r, π b+1,r,..., π W,r ) ρ r U(0, 0.5) c r N(0, 10 4 ) ν 2 r U(0, 10 4 ) Autoregressive coefficients ρ r control the persistence over time Intercepts c r control for average levels Variances ν 2 r control the variability For any ρ r ( 0.5, 0.5), the prior distribution of π r = (π 1,r,..., π B,r ) exists and is a proper multivariate normal distribution (Besag, 1974) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

39 Model 2 Autoregressive Priors Posterior Weights Figure: Posterior p t across time for component 1 in Model 2 (top); posterior p t across time for component 1 in Model 2 with CAR(1) prior (bottom) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

40 Model 2 Autoregressive Priors Estimation Algorithm 1. Initialize β, {µ j }, {Σ j }, {π b,r }, {c r }, {ρ r } and {ν 2 r } 2. Sample {µ j }, β, {Σ j } via full conditionals, similar to Model (1) 3. Update {π b,r } via Metropolis-Hastings 4. Update hyperparameters {c r }, {ρ r } and {ν 2 r } via Metropolis-Hastings David S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

41 Model 2 Seasonal Autoregressive Priors Seasonal (Conditional) Autoregressive Model avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

42 Model 2 Seasonal Autoregressive Priors Seasonal (Conditional) Autoregressive Model avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

43 Model 2 Seasonal Autoregressive Priors Seasonal (Conditional) Autoregressive Model avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

44 Model 2 Seasonal Autoregressive Priors Seasonal First-Order (Conditional) Autoregressive Model Transformed weight π b,r depends on neighboring periods π b 1,r, π b+1,r and same period on neighboring days π b d,r, π b+d,r d is number of time periods in a day (d = 12) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

45 Model 2 Seasonal Autoregressive Priors Seasonal First-Order (Conditional) Autoregressive Model Transformed weight π b,r depends on neighboring periods π b 1,r, π b+1,r and same period on neighboring days π b d,r, π b+d,r d is number of time periods in a day (d = 12) Priors [ π b,r π b,r N (c r + ρ r (π b 1,r c r ) + (π b+1,r c r ) ] + (π b d,r c r ) + (π b+d,r c r ), νr 2 ρ r U(0, 0.25) ), avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

46 Model 2 Seasonal Autoregressive Priors Seasonal First-Order (Conditional) Autoregressive Model Transformed weight π b,r depends on neighboring periods π b 1,r, π b+1,r and same period on neighboring days π b d,r, π b+d,r d is number of time periods in a day (d = 12) Priors [ π b,r π b,r N (c r + ρ r (π b 1,r c r ) + (π b+1,r c r ) ] + (π b d,r c r ) + (π b+d,r c r ), νr 2 ρ r U(0, 0.25) ), For ρ r ( 0.25, 0.25), the joint prior distribution of π r is proper avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

47 Model 2 Seasonal Autoregressive Priors Posterior Means and Covariances Figure: Posterior means and covariances (ellipses at 90% level) of all components from fitting proposed mixture model using 15 components. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

48 Model 2 Seasonal Autoregressive Priors Posterior Means and Covariances Figure: Each ellipse (except the 15th) is shaded with the posterior mean of ρ r for that component. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

49 Model 2 Seasonal Autoregressive Priors Posterior log Spatial Densities, K = 15 Figure: Mixture model using 15 components: (a) posterior log spatial density for Wednesday 2-4pm (demand concentrated at downtown at midday); (b) posterior log spatial density for Wednesday 2-4am (demand more spread out at midnight). avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

50 Model 2 BDMCMC Estimating the Number of Components Varying number of components (usually) improves mixing & predictions Adapt the birth-and-death MCMC (BDMCMC) methods of Stephens (2000) Alternative to Reversible Jump MCMC (RJMCMC), Green (1995) and Richardson and Green (1997) BDMCMC widely used in pattern recognition applications (cf. Micheas et al., 2012; Elguebaly and Bouguila, 2012). Fix birth rate of new components at λ b Assume a truncated Poisson prior on the number of components, i.e., P(l) τ l /l!, l {1,..., l max }, for some l max David S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

51 Model 2 BDMCMC BDMCMC Algorithm Initialize l components, y = {({p b,1 } W b=1, µ 1, Σ 1 ),..., ({p b,l } W b=1, µ l, Σ l )} 1. For j {1,..., l}, calculate death rate for the j th component, d j (y) L ( y\({p b,j } W b=1 d j (y) = λ, µ j, Σ j ) ) P(l 1) P ( {p b,1,..., p b,j 1, p b,j+1,..., p b,l } ) W b=1 b L(y) l P(l) P ( ) {p b,1,..., p b,l } W b=1 L( ) is joint likelihood function, and P( ) represents the priors 2. Total death rate, d(y) = l j=1 d j(y) 3. Time to next jump: exponential with mean λ b + d(y) 4. Birth or Death? P(birth) = λ b /(λ b + d(y)) and P(death) = d(y)/(λ b + d(y)) 5. If birth, independently sample component s weight from Beta(1, l) 6. Normalize weights to sum 1, in all periods 7. Repeat 1 6 for a fixed time T 0 8. Fix the number of components, and update all other parameters via full-conditionals or Metropolis-Hastings avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

Model 2 BDMCMC Posterior log Spatial Densities, variable number of components Figure: Mixture model using variable number of components: (a) posterior log spatial density for

52 Model 2 BDMCMC Posterior log Spatial Densities, variable number of components Figure: Mixture model using variable number of components: (a) posterior log spatial density for Wednesday 2-4am using an average of 19 components; (b) that using an average of 24 components. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

53 Model 2 Boundary Estimation with a Boundary Point processes commonly occur on fixed bounded regions We restrict the spatial domain, S from R 2 to a bounded region S Toronto: Lake Ontario to the south and surrounding suburban areas Normalize the spatial densities f b with respect to the boundary f b (s; θ) = f b(s; θ)1 {s S} S f b(s; θ) ds, in which θ is a vectorization of all the parameters Numerically approximate the denominator avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

Model 2 Boundary Estimation with a Boundary Figure: Estimated first-order autoregressive-prior model using 15 components: (a) posterior means and covariances (ellipses at 90% level) of all 15

54 Model 2 Boundary Estimation with a Boundary Figure: Estimated first-order autoregressive-prior model using 15 components: (a) posterior means and covariances (ellipses at 90% level) of all 15 components at the 50,000th iteration; (b) posterior log spatial density on Wednesday 2:00pm-4:00pm, averaged across the last 50,000 Monte Carlo samples. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

55 Evaluation Comparison Methods Comparison Methods (MEDIC) Histogram Estimator Partition spatial domain into C equally-sized rectangular spatial cells, each with area A C For each time period t, define f t (s) = C c=1 1 A C n c,t + a n t + ac 1 {s c}, Modified proportion (nc,t + a)/(n t + ac) vs. true proportion (n c,t )/(n t ) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

56 Evaluation Comparison Methods Comparison Methods (MEDIC) Histogram Estimator Partition spatial domain into C equally-sized rectangular spatial cells, each with area A C For each time period t, define f t (s) = C c=1 1 A C n c,t + a n t + ac 1 {s c}, Modified proportion (nc,t + a)/(n t + ac) vs. true proportion (n c,t )/(n t ) Kernel Density Estimator Bivariate normal kernel function Bandwidths: plug-in selector (PI, see Wand and Jones, 1994), or smoothed cross-validation selector (SCV, see Duong and Hazelton, 2005) avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

57 Evaluation Comparison Methods MEDIC and MEDIC-KDE avid S. Matteson S-T Ambulance Demand 2014 September / 56

58 Evaluation Comparison Methods MEDIC and MEDIC-KDE avid S. Matteson S-T Ambulance Demand 2014 September / 56

Evaluation Comparison Methods Log Predictive Densities, MEDIC and MEDIC-KDE Figure: Log predictive densities using two current industry estimation methods for 2-4am (midnight) on February 6, 2008

59 Evaluation Comparison Methods Log Predictive Densities, MEDIC and MEDIC-KDE Figure: Log predictive densities using two current industry estimation methods for 2-4am (midnight) on February 6, 2008 (Wednesday). Compared to mixture models, estimates from the MEDIC method is overly noisy while those from MEDIC with KDE are not very refined. avid S. Matteson S-T Ambulance Demand 2014 September / 56

60 Evaluation Predictive Accuracy Predictive Accuracy Average logarithmic score (Good, 1952; Gneiting and Raftery, 2007) PA({s t,i }) = 1 T t=1 n t ˆf t ( ) density estimate from training data s t,i test data T n t log ˆf t (s t,i ) t=1 i=1 avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

61 Evaluation Predictive Accuracy Predictive Accuracy Average logarithmic score (Good, 1952; Gneiting and Raftery, 2007) PA({s t,i }) = 1 T t=1 n t ˆf t ( ) density estimate from training data s t,i test data Monte Carlo estimate PA mix ({s t,i }) = 1 M [ M 1 T t=1 n t m=1 T n t log ˆf t (s t,i ) t=1 i=1 ] T n t log ˆf t (s t,i θ (m) ) t=1 i=1 θ (m) : m th -iteration posterior parameter estimates from the training data avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

62 Evaluation Predictive Accuracy Predictive Accuracy Gaussian Mixture Competing Methods Estimation method PA for Mar 07 PA for Feb components ± ± Variable number of comp: average 19 comp ± ± average 24 comp ± ± MEDIC MEDIC with KDE Table: Predictive accuracies of proposed Gaussian mixture models and competing methods on test data of March 2007 and February The predictive accuracies for mixture models are presented with their 95% batch means confidence intervals. avid S. Matteson S-T Ambulance Demand 2014 September / 56

63 Evaluation Predictive Accuracy Operational Predictive Accuracy: 44 bases avid S. Matteson S-T Ambulance Demand 2014 September / 56

64 Evaluation Predictive Accuracy Operational Predictive Accuracy: 44 bases avid S. Matteson S-T Ambulance Demand 2014 September / 56

65 Evaluation Predictive Accuracy Operational Predictive Accuracy: 44 bases avid S. Matteson S-T Ambulance Demand 2014 September / 56

Evaluation Predictive Accuracy Operational Predictive Accuracy: Error(M, r) = 1 T T t=1 P M,t(r) P test,t (r) Figure: (a) all 44 ambulance bases in Toronto; (b) and (c) average absolute error in

66 Evaluation Predictive Accuracy Operational Predictive Accuracy: Error(M, r) = 1 T T t=1 P M,t(r) P test,t (r) Figure: (a) all 44 ambulance bases in Toronto; (b) and (c) average absolute error in measuring operational performance made by our mixture model (15 components), MEDIC, and MEDIC with KDE, using test data from March 2007 and February 2008, respectively (with 95% confidence intervals in gray). avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

67 Conclusions Thank you! Conclusions Flexible, parsimonious framework to model spatio-temporal point processes A Constrained Factor Model A Finite Mixture Model Time invariant component distributions Mixture weight constraints Autoregressive priors Birth-and-death MCMC Spatial boundary Large Toronto EMS dataset, spatio-temporal ambulance demand Average logarithmic score Future work: forecasting and covariate models avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

68 Conclusions Bibliography Bibliography Besag, J. E. (1974), Spatial interaction and the statistical analysis of lattice systems, Journal of the Royal Statistical Society, Series B, 36, Channouf, N., L Ecuyer, P., Ingolfsson, A., and Avramidis, A. (2007), The application of forecasting techniques to modeling emergency medical system calls in Calgary, Alberta, Health Care Management Science, 10, Duong, T., and Hazelton, M. L. (2005), Cross-validation bandwidth matrices for multivariate kernel density estimation, Scandinavian Journal of Statistics, 32, Elguebaly, T., and Bouguila, N. (2012), Medical image classification using birth-and-death MCMC,, in Circuits and Systems (ISCAS), 2012 IEEE International Symposium on, Seoul, South Korea, pp Gneiting, T., and Raftery, A. (2007), Strictly proper scoring rules, prediction, and estimation, Journal of the American Statistical Association, 102, Good, I. J. (1952), Rational decisions, Journal of the Royal Statistical Society: Series B, 14, Green, P. (1995), Reversible jump MCMC computation and Bayesian model determination, Biometrika, 82, Matteson, D. S., and James, N. A. (2014), A nonparametric approach for multiple change point analysis of multivariate data, Journal of the American Statistical Association, 109(505), Matteson, D. S., McLean, M. W., Woodard, D. B., and Henderson, S. G. (2011), Forecasting emergency medical service call arrival rates, Annals of Applied Statistics, 5, Micheas, A. C., Wikle, C. K., and Larsen, D. R. (2012), Random set modeling of three-dimensional objects in a hierarchical Bayesian context, Journal of Statistical Computation and Simulation, 0, Richardson, S., and Green, P. (1997), On Bayesian analysis of mixtures with an unknown number of components (with discussion), Journal of the Royal Statistical Society: Series B, 59, Stephens, M. (2000), Bayesian analysis of mixture models with an unknown number of components - an alternative to reversible jump methods, Annals of Statistics, 28, Wand, M. P., and Jones, M. C. (1994), Multivariate plug-in bandwidth selection, Computational Statistics, 9, Zhou, Z., Matteson, D. S., Woodard, D. B., Henderson, S. G., and Micheas, A. C. (2014), A Spatio-Temporal Point Process Model for Ambulance Demand, in press, Journal of the American Statistical Association; arxiv preprint arxiv: ,. avid S. Matteson (matteson@cornell.edu) S-T Ambulance Demand 2014 September / 56

A Spatio-Temporal Point Process Model for Ambulance Demand

A Spatio-Temporal Point Process Model for Ambulance Demand Zhengyi Zhou, David S. Matteson, Dawn B. Woodard, Shane G. Henderson and Athanasios C. Micheas arxiv:1401.5547v2 [stat.ap] 27 May 2014 May 29,