Supplementary Figure 1 Annual number of F0-F5 (grey) and F2-F5 (black) tornado observations over 30 years ( ) for Canada and United States.
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1 SUPPLEMENTARY FIGURES Supplementary Figure 1 Annual number of F0-F5 (grey) and F2-F5 (black) tornado observations over 30 years ( ) for Canada and United States.
2 Supplementary Figure 2 Differences in the conditional autoregressive term between the CAPE-HLCY F2-F5 and CAPE-HLCY-VWSH F2-F5 monthly models. a, December; b, January; c, February; d, March; e, April; f, May; g, June; h, July; i, August; j, September; k, October; l, November.
3 Supplementary Figure 3 Scatterplots of the monthly number of F0-F5 tornadoes against the population density per grid cell. Red line depicts the population threshold for the month where p i (β) a, December; b, January; c, February; d, March; e, April; f, May; g, June; h, July; i, August; j, September; k, October; l, November.
4 Supplementary Figure 4 Scatterplots of the monthly number of F2-F5 tornadoes against the population density per grid cell. Red line depicts the population threshold for the month where p i (β) a, December; b, January; c, February; d, March; e, April; f, May; g, June; h, July; i, August; j, September; k, October; l, November.
5 Supplementary Figure 5 Results of the CAPE-HLCY F0-F5 model. The observed tornado counts per grid cell, Tobs: a, Spring; b, Summer; c, Autumn; d, Winter; The predictions of the total number of tornado occurrence per grid cell, Tlatent: e, Spring; f, Summer; g, Autumn; h, Winter; Values for the conditional autoregressive term, φ: i, Spring; j, Summer; k, Autumn; l, Winter.
6 Supplementary Figure 6 Results of the CAPE-HLCY F2-F5 model. The observed tornado counts per grid cell, Tobs: a, Spring; b, Summer; c, Autumn; d, Winter; The predictions of the total number of tornado occurrence per grid cell, Tlatent: e, Spring; f, Summer; g, Autumn; h, Winter; Values for the conditional autoregressive term, φ: i, Spring; j, Summer; k, Autumn; l, Winter.
7 Supplementary Figure 7 Predictions of the CAPE-HLCY model - Peak months of tornado activity for US and Canada. a, May (F0-F5); b, July (F0-F5); c, May (F2-F5); d, July (F2-F5) tornadoes.
8 Supplementary Figure 8 Difference between the model estimate Tlatent and tornado observations, Tobs, of the CAPE-HLCY-VWSH model for F0-F5 tornadoes. a, Spring; b, Summer; c, Autumn; d, Winter.
9 Supplementary Figure 9 Difference between the model estimate Tlatent and tornado observations, Tobs, of the CAPE-HLCY-VWSH model for F2-F5 tornadoes. a, Spring; b, Summer; c, Autumn; d, Winter.
10 Supplementary Figure 10 CAPE and HLCY phase space for each month. Red contours indicate the CAPE- HLCY phase space for grid cells with tornado occurrence (F0-F5). a, December; b, January; c, February; d, March; e, April; f, May; g, June; h, July; i, August; j, September; k, October; l, November. CAPE values are expressed in logarithmic (base 10) scale (m 2 s -2 ) and HLCY values in original units (m 2 s -2 ).
11 Supplementary Figure 11 CAPE and SHEAR phase space for each month. Red contours indicate the CAPE-SHEAR phase space for grid cells with tornado occurrence (F0-F5). a, December; b, January; c, February; d, March; e, April; f, May; g, June; h, July; i, August; j, September; k, October; l, November. CAPE values are expressed in logarithmic (base 10) scale (m 2 s -2 ) and SHEAR values in original units (m s -1 ).
12 Supplementary Figure 12 CAPE and VWSH phase space for each month. Red contours indicate the CAPE-VWSH phase space for grid cells with tornado occurrence (F0-F5). a, December; b, January; c, February; d, March; e, April; f, May; g, June; h, July; i, August; j, September; k, October; l, November. CAPE values are expressed in logarithmic (base 10) scale (m 2 s -2 ) and VWSH in original units (s -1 ) multiplied by 10-3.
13 Supplementary Figure 13 Parameter posteriors of the CAPE-HLCY-VWSH model based on the 30-yr ( ) and 15-yr ( ) record of F0-F5 tornadoes. The lower and upper edges of the bottom and top whiskers represent the 2.5 and 97.5 percentiles, respectively. The boxes encompass the parameter values that correspond to 68.27% of the probability mass (1 standard deviation around the mean) of the posterior distributions, while the lines inside represent the medians. a,cape; b, HLCY; c, VWSH; d, β.
14 Supplementary Figure 14 Calibration of the CAPE-HLCY-VWSH model ( ): Observed (a, March; e, April; i, May) and model tornado counts (b, March; f, April; j, May); Model predictive confirmation ( ): Observed tornado counts (c, March; g, April; k, May) and posterior predictions (d, March; h, April; i, May).
15 Supplementary Figure 15 Calibration of the CAPE-HLCY-VWSH model ( ): Observed (a, June; e, July; i, August) and model tornado counts (b, June; f, July; j, August); Model predictive confirmation ( ): Observed tornado counts (c, June; g, July; k, August) and posterior predictions (d, June; h, July; i, August).
16 Supplementary Figure 16 Calibration of the CAPE-HLCY-VWSH model ( ): Observed (a, September; e, October; i, November) and model tornado counts (b, September; f, October; j, November); Model predictive confirmation ( ): Observed tornado counts (c, September; g, October; k, November) and posterior predictions (d, September; h, October; i, November).
17 Supplementary Figure 17 Calibration of the CAPE-HLCY-VWSH model ( ): Observed (a, December; e, January; i, February) and model tornado counts (b, December; f, January; j, February); Model predictive confirmation ( ): Observed tornado counts (c, December; g, January; k, February) and posterior predictions (d, December; h, January; i, February).
18
19 Supplementary Figure 18 Posterior parameter quantile values of the binomial-poisson and the zero-inflated Poisson models that consider the variables CAPE, HLCY, and VWSH to predict F0-F5 tornadoes. The bottom and top line of each box plot represents the 2.5 and 97.5 percentile values. The bottom and top box range represents the 1 standard deviation away from the mean; The line in the box represents the median. a, CAPE F0-F5 tornadoes; b, CAPE F2-F5 tornadoes; c, HLCY F0-F5 tornadoes; d, HLCY F2-F5 tornadoes; e, VWSH F0-F5 tornadoes; f, VWSH F2-F5 tornadoes; g, β F0-F5 tornadoes; h, β F2-F5 tornadoes.
20 Supplementary Figure 19 Posterior parameter quantile values of CAPE of the binomial- Poisson and the zero-inflated Poisson models for all 7 combinations of the variables CAPE, HLCY, SHEAR and VWSH to predict F0-F5 and F2-F5 tornadoes. The bottom and top line of each box plot represents the 2.5 and 97.5 percentile values. The bottom and top box range represents the 1 standard deviation away from the mean. The line in the box represents the median. a, F0-F5 binomial-poisson models; b, F0-F5 zero-inflated Poisson models; c, F2-F5 binomial-poisson models; d, F2-F5 zero-inflated Poisson models.
21 Supplementary Figure 20 Posterior parameter quantile values of HLCY of the binomial- Poisson and the zero-inflated Poisson models for all 7 combinations of the variables CAPE, HLCY, SHEAR and VWSH to predict F0-F5 and F2-F5 tornadoes. The bottom and top line of each box plot represents the 2.5 and 97.5 percentile values. The bottom and top box range represents the 1 standard deviation away from the mean. The line in the box represents the median. a, F0-F5 binomial-poisson models; b, F0-F5 zero-inflated Poisson models; c, F2-F5 binomial-poisson models; d, F2-F5 zero-inflated Poisson models.
22 Supplementary Figure 21 Posterior parameter quantile values of SHEAR of the binomial- Poisson and the zero-inflated Poisson models for all 7 combinations of the variables CAPE, HLCY, SHEAR and VWSH to predict F0-F5 and F2-F5 tornadoes. The bottom and top line of each box plot represents the 2.5 and 97.5 percentile values. The bottom and top box range represents the 1 standard deviation away from the mean. The line in the box represents the median. a, F0-F5 binomial-poisson models; b, F0-F5 zero-inflated Poisson models; c, F2-F5 binomial-poisson models; d, F2-F5 zero-inflated Poisson models.
23 Supplementary Figure 22 Posterior parameter quantile values of VWSH of the binomial- Poisson and the zero-inflated Poisson models for all 7 combinations of the variables CAPE, HLCY, SHEAR and VWSH to predict F0-F5 and F2-F5 tornadoes. The bottom and top line of each box plot represents the 2.5 and 97.5 percentile values. The bottom and top box range represents the 1 standard deviation away from the mean. The line in the box represents the median. a, F0-F5 binomial-poisson models; b, F0-F5 zero-inflated Poisson models; c, F2-F5 binomial-poisson models; d, F2-F5 zero-inflated Poisson models.
24 SUPPLEMENTARY TABLES Supplementary Table 1 The seven combinations of atmospheric/climatological variables examined to predict the 30-yr tornado observations (F0-F5 and F2-F5) for each calendar month and season independently. Combinations of atmospheric/climatological variables CAPE-HLCY CAPE-SHEAR CAPE-VWSH CAPE-HLCY-SHEAR CAPE-HLCY-VWSH CAPE-SHEAR-VWSH CAPE-HLCY-SHEAR-VWSH
25 Supplementary Table 2 Comparison between the posterior mean and standard deviation values of the deviance (or -2 log[model likelihood]) among the different combinations of variables used to predict F0-F5 tornadoes. F0-F5 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Model mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd CAPE-HLCY CAPE-SHEAR CAPE-VWSH CAPE-HLCY-SHEAR CAPE-HLCY-VWSH CAPE-SHEAR-VWSH CAPE-HLCY-SHEAR-VWSH
26 Supplementary Table 3 Comparison between the posterior mean and standard deviation values of the deviance (or -2 log[model likelihood]) among the different combinations of variables used to predict F2-F5 tornadoes. F2-F5 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Model mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd mean sd CAPE-HLCY CAPE-SHEAR CAPE-VWSH CAPE-HLCY-SHEAR CAPE-HLCY-VWSH CAPE-SHEAR-VWSH CAPE-HLCY-SHEAR-VWSH
27 Supplementary Table 4 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE and HLCY to predict F0-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - HLCY β σ Model mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Supplementary Table 5 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE and HLCY to predict F2-F5 tornadoes. F2-F5 α0 α1 - CAPE α2 - HLCY β σ Model mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
28 Supplementary Table 6 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE and SHEAR to predict F0-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - SHEAR β σ Model mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Supplementary Table 7 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE and SHEAR to predict F2-F5 tornadoes. F2-F5 α0 α1 - CAPE α2 - SHEAR β σ Model mean sd mean sd mean sd mean sd mean sd Jan ,759 2, Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
29 Supplementary Table 8 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE and VWSH to predict F0-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - VWSH β σ Model mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Supplementary Table 9 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE and VWSH to predict F2-F5 tornadoes. F2-F5 α0 α1 - CAPE α2 - VWSH β σ Model mean sd mean sd mean sd mean sd mean sd Jan ,932 2, Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
30 Supplementary Table 10 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE, HLCY and SHEAR to predict F0-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - HLCY α3 - SHEAR β σ Model mean sd mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Supplementary Table 11 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE, HLCY and SHEAR to predict F2-F5 tornadoes. F2-F5 α0 α1 - CAPE α2 - HLCY α3 - SHEAR β σ Model mean sd mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
31 Supplementary Table 12 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE, SHEAR and VWSH to predict F0-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - SHEAR α3 - VWSH β σ Model mean sd mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Supplementary Table 13 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE, SHEAR and VWSH to predict F2-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - SHEAR α3 - VWSH β σ Model mean sd mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
32 Supplementary Table 14 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE, HLCY, SHEAR and VWSH to predict F0-F5 tornadoes. F0-F5 α0 α1 - CAPE α2 - HLCY α3 - SHEAR α4 - VWSH β σ Model mean sd mean sd mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Supplementary Table 15 Posterior parameter mean and standard deviation values of the model that considers the variables CAPE, HLCY, SHEAR and VWSH to predict F2-F5 tornadoes. F2-F5 α0 α1 - CAPE α2 - HLCY α3 - SHEAR α4 - VWSH β σ Model mean sd mean sd mean sd mean sd mean sd mean sd mean sd Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
33 Supplementary Table 16 Comparison between observed monthly numbers of tornadoes in Canada and United States and those predicted by the CAPE-HLCY model. F0-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F2-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
34 Supplementary Table 17 Comparison between observed monthly numbers of tornadoes in Canada and United States and those predicted by the CAPE-SHEAR model. F0-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F2-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
35 Supplementary Table 18 Comparison between observed monthly numbers of tornadoes in Canada and United States and those predicted by the CAPE-VWSH model. F0-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F2-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
36 Supplementary Table 19 Comparison between observed monthly numbers of tornadoes in Canada and United States and those predicted by the CAPE-HLCY-SHEAR model. F0-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F2-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
37 Supplementary Table 20 Comparison between observed monthly numbers of tornadoes in Canada and United States and those predicted by the CAPE-SHEAR-VWSH model. F0-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F2-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
38 Supplementary Table 21 Comparison between observed monthly numbers of tornadoes in Canada and United States and those predicted by the CAPE-HLCY-SHEAR-VWSH model. F0-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F2-F5 ƩTobs ƩTlatent Difference Model Can US Total Can US Total Can US Total Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
39 Supplementary Table 22 Climatological/atmospheric variables from the North American Regional Reanalysis. Input Parameters Category Abbreviation Accumulated convective precipitation Precipitation acpcp Air temperature at 2 m Buoyancy air Accumulated total precipitation Precipitation apcp Convective available potential energy (surface-based) Buoyancy cape Mean of convective cloud cover Cloud cdcon Convective inhibition (surface-based) Buoyancy cin Dew point temperature at 2 m Moisture dpt Accumulated total evaporation Moisture evap High cloud area fraction Cloud hcdc Geopotential height (at tropopause) Pressure hgt Storm relative helicity (0-3 km) Wind shear hlcy Low cloud area fraction Cloud lcdc Best (4-layer) lifted index (180-0mb above ground) Buoyancy lftx4 Latent heat flux Buoyancy lhtfl Medium cloud area fraction Cloud mcdc Horizontal moisture divergence Moisture mconv Mean sea level pressure Pressure mslet Moisture availability Moisture mstav Accumulated potential evaporation Moisture pevap Precipitable water Moisture pr_wtr Precipitation rate Precipitation prate Pressure at mean sea level Pressure prmsl Relative humidity at 2 m Moisture rhum Vector magnitude difference (0-6 km) Wind shear shear Specific humidity at 2 m Moisture shum u-component of storm motion (0-6 km) Kinematic ustm u-wind at 10m Kinematic uwnd v-component of storm motion (0-6 km) Kinematic vstm Pressure vertical velocity Kinematic vvel v-wind at 10 m Kinematic vwnd Vertical wind shear (at tropopause) Wind shear vwsh Water vapor convergence accumulation Moisture wvconv
40 Supplementary Table 23 Performance of the CAPE-HLCY-VWSH model based on the Pearson correlation coefficient values between the observed number of tornadoes and the posterior mean predictions per grid cell. The predictive confirmation was based on the separation of the 30-yr dataset into two subsets; the calibration ( ) and predictive confirmation ( ) datasets. The former one was used to obtain parameter estimates through Bayesian updating and the derived model predictive posteriors were then tested independently against the latter dataset. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
41 Supplementary Note 1 WinBUGS code for the Binomial-Poisson model approach associated with the CAPE-HLCY-VWSH binomial Poisson models. 1) Binomial- Poisson model approach model { for (i in 1 : N) { Tobs[i] ~ dbin(p[i],tlatent[i]) Tlatent[i]~dpois(lamda[i]) log(lamda[i]) <-a0 + a1*cape[i] + a2*hlcy[i] + a3*vwsh[i] + b[i] p[i]<-exp(-beta/exp(popd[i]))} b[1:n] ~ car.normal(adj[], weights[], num[], tau) for(k in 1:sumNumNeigh) {weights[k] <- 1} a0 ~ dnorm(0, 0.001) a1 ~ dnorm(0, 0.001) a2 ~ dnorm(0, 0.001) a3 ~ dnorm(0, 0.001) beta<-exp(betaaux) betaaux~dnorm(1, 0.001) tau ~ dgamma(0.01,0.01) sigma <- sqrt(1 / tau) } 2) Inference Data List(N=7373, cape=c(paste supplementary data1.xlsx worksheet1 row-(month-standardized cape) as comma separate values here), hlcy=c(supplementary data1.xlsx worksheet1 row-(month-standardized hlcy) as comma separate values here), vwsh=c(paste supplementary data1.xlsx worksheet1 row-(month-standardized vwsh) as comma separate values here), Tobs=c(paste supplementary data1.xlsx worksheet1 row-(month-tobs F0-F5) as comma separate values here), Popd=(paste supplementary data1.xlsx worksheet1 row-(all-population density) as comma separate values here), List( num =c(paste upplementary data1.xlsx worksheet2 as comma separate values here), adj=c(paste supplementary data1.xlsx worksheet3 as comma separate values here), sumnumneigh = 56996) 3) Initial values 1 (Binomial- Poisson model) list(tau = 1, a0 = 2, a1 = 2, a2 = 2, a3 = 1, betaaux = 2, b=c(paste supplementary data1.xlsx worksheet1 row-(all-b) as comma separate values here), Tlatent=c(paste supplementary data1.xlsx worksheet1 row-(all-tlatent) as comma separate values here)) 3) Initial values 2 (Binomial- Poisson model)
42 list(tau = 2, a0 = 1, a1 = 1, a2 = 1, a3 = 1, betaaux = 1, b=c(paste supplementary data1.xlsx worksheet1 row-(all-b) as comma separate values here), Tlatent=c(paste supplementary data1.xlsx worksheet1 row-(all-tlatent) as comma separate values here))
43 Supplementary Note 2 WinBUGS code for the Zero-inflated Poisson model approach associated with the CAPE-HLCY-VWSH binomial Poisson models. 1) Zero-inflated Poisson model approach model { for (i in 1 : N) { Tobs[i] ~ dpois(mu[i]) mu[i]<-u[i]*lamda[i] u[i]~dbern(p[i]) log(lamda[i]) <-a0 + a1*cape[i] + a2*hlcy[i] + a3*vwsh[i] + b[i] p[i]<-exp(-beta/exp(popd[i]))} b[1:n] ~ car.normal(adj[], weights[], num[], tau) for(k in 1:sumNumNeigh) {weights[k] <- 1} a0 ~ dnorm(0, 0.001) a1 ~ dnorm(0, 0.001) a2 ~ dnorm(0, 0.001) a3 ~ dnorm(0, 0.001) beta<-exp(betaaux) betaaux~dnorm(1, 0.001) tau ~ dgamma(0.01,0.01) sigma <- sqrt(1 / tau)} 2) Inference Data List(N=7373, cape=c(paste supplementary data1.xlsx worksheet1 row-(month-standardized cape) as comma separate values here), hlcy=c(supplementary data1.xlsx worksheet1 row-(month-standardized hlcy) as comma separate values here), vwsh=c(paste supplementary data1.xlsx worksheet1 row-(month-standardized vwsh) as comma separate values here), Tobs=c(paste supplementary data1.xlsx worksheet1 row-(month-tobs F0-F5) as comma separate values here), Popd=(paste supplementary data1.xlsx worksheet1 row-(all-population density) as comma separate values here), List( num =c(paste upplementary data1.xlsx worksheet2 as comma separate values here), adj=c(paste supplementary data1.xlsx worksheet3 as comma separate values here), sumnumneigh = 56996) 3) Initial values 1 (Zero-Inflated-Poisson model) list(tau = 2, a0 = 1, a1 = 1, a2 = 1, a3 = 1, betaaux = 1, b=c(paste supplementary data1.xlsx worksheet1 row-(all-b) as comma separate values here), u=c(paste supplementary data1.xlsx worksheet1 row-(all-u) as comma separate values here)) 3) Initial values 2 (Zero-Inflated-Poisson model) list(tau = 1, a0 = 2, a1 = 2, a2 = 2, a3 = 1, betaaux = 2,
44 b=c(paste supplementary data1.xlsx worksheet1 row-(all-b) as comma separate values here), u=c(paste supplementary data1.xlsx worksheet1 row-(all-u) as comma separate values here))
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