Statistical downscaling with spatial misalignment: Application to wildland fire PM_2.5 concentration forecasting

Abstract

Fine particulate matter, PM_2.5, has been documented to have adverse health effects and wildland fires are a major contributor to PM_2.5 air pollution in the US. Forecasters use numerical models to predict PM_2.5 concentrations to warn the public of impending health risk. Statistical methods are needed to calibrate the numerical model forecast using monitor data to reduce bias and quantify uncertainty. Typical model calibration techniques do not allow for errors due to misalignment of geographic locations. We propose a spatiotemporal downscaling methodology that uses image registration techniques to identify the spatial misalignment and accounts for and corrects the bias produced by such warping. Our model is fitted in a Bayesian framework to provide uncertainty quantification of the misalignment and other sources of error. We apply this method to different simulated data sets and show enhanced performance of the method in presence of spatial misalignment. Finally, we apply the method to a large fire in Washington state and show that the proposed method provides more realistic uncertainty quantification than standard methods.

1. Introduction

Air pollution associated with wildland fire smoke is an increasingly pressing health concern (Dennekamp and Abramson, 2011; Rappold et al., 2011; Johnston et al., 2012; Dennekamp et al., 2015; Haikerwal et al., 2015, 2016; Wettstein et al., 2018). Reliable short-term forecasts of fire-associated health risk using numerical models facilitate informed decision making for local populations. Numerical models produce forecasts on a coarse grid and are prone to bias. Assimilating point-level monitor data with numerical-model output can reduce bias and provide more realistic uncertainty quantification (e.g., Berrocal et al., 2010a,b; Kloog et al., 2011; Zhou et al., 2011, 2012; Berrocal et al., 2012; Reich et al., 2014; Chang et al., 2014). However, most downscaling methods only correct for additive and scaling biases and fail to guard against spatial misalignment errors where a forecasted event occurs in a different spatial location than forecasted. Spatial misalignment error in this context implies errors corresponding to predicting the location of a feature, such as fire plume, wrong. Not accounting for it is problematic for wildland fire smoke forecasting because a common source of error is in predicting the direction of the fire plume which cannot be accounted for by additive and scaling correction to the forecast. This motivates us to develop a statistical downscaling method that accounts for spatial misalignment errors.

Spatial misalignment correction can be achieved using standard image registration (or warping) techniques, ranging from simple affine and polynomial transformations to more sophisticated methods such as Fourier based transforms (Kuglin, 1975; De Castro and Morandi, 1987), nonparametric approaches like elastic deformation (Burr, 1981; Tang and Suen, 1993; Barron et al., 1994) and thin-plate splines (Bookstein, 1989; Mardia and Little, 1994; Mardia et al., 1996). Such applications of warping in image processing, especially medical imaging, has allowed us to use information from multiple sources simultaneously to improve our understanding. Beside image processing and medical imaging, warping is also popular in speech processing (Sakoe and Chiba, 1978), handwriting analysis (Burr, 1983), determination of alignment of boundaries of ice floes (McConnell et al., 1991) where they are used to improve pattern recognition capabilities. More recently, warping allowed improvement of weather forecast analysis and verification (Hoffman et al., 1995; Alexander et al., 1999; Sampson and Guttorp, 1999; Reilly et al., 2004; Gilleland et al., 2010).

Sampson and Guttorp (1992) used warping of spatial coordinates to model non-stationary and non-isotropic spatial covariance structures. Anderes and Stein (2008) and Anderes and Chatterjee (2009) developed methods for estimating deformation of isotropic Gaussian random fields. The first attempt at using warping for forecast verification in statistics, to our knowledge, was proposed by Aberg et al. (2005). Image warping in wind field modelling was proposed by Ailliot et al. (2006) and Fuentes et al. (2008) used warping to assimilate two different sources of rainfall data in a single model. Kleiber et al. (2014) used warping in the context of model emulation and calibration framework. They assume the observations lie on a grid and the spatial features are completely observed so that standard image registration techniques such as landmark registration can be used for estimating the warping function. However, this approach does not apply to our downscaling problem because the monitoring stations are spatially sparse and the shape and direction of the fire plume are not observed. The estimation of the warping function is challenging and further complicated by the dynamic environment, such as changes in the wind pattern.

We propose a new statistical downscaling method that optimizes the information from available forecasts and real-time monitoring data. We achieve this through (1) introducing a warping function to allow for flexible model discrepancy beyond the additive and multiplicative biases and (2) multi-resolution modeling to allow the data to determine the appropriate spatial resolution to inform prediction. We estimate the spatial misalignment between the forecast and the observed data using a penalized B-spline approach. We also use spectral smoothing (Reich et al., 2014) to capture important patterns more vividly and reduce noise simultaneously. By coalescing these two methods in a single model, we propose a novel downscaling model that accounts for spatial misalignment as well as the usual additive and scaling biases while smoothing out the forecast to improve prediction.

The remainder of the paper proceeds as follows. Section 2 introduces the motivating dataset and Section 3 describes the proposed method. The performance of the model and its component models are studied extensively using a simulation study in Section 4. The method is applied to forecasting air pollution during a major fire in Washington State in Section 5, where we show that accounting for spatial misalignment provides better assessment of uncertainty. We finish with some concluding remarks in Section 6.

2. PM_2.5 Data for Washington State

We have two sources of PM_2.5 data: numerical model forecasts on a grid and ground monitoring station scattered around the state. Both data sets give hourly PM_2.5 measurements for the state of Washington from August 13, 2015 to September 16, 2015, a period with severe wildland fires.

The numerical forecasts were generated by the BlueSky modeling system on a 4km × 4km grid resulting in a 200×95 grid covering Washington. The model is run daily at midnight and provides an hourly forecast for the next 84 hours of which we use only the forecasts for the first 24 hours for our analysis. The model only forecasts PM_2.5 levels created by wildland fires and does not contain any information about PM_2.5 generated from other sources such as traffic or industry. The forecasts also do not assimilate observed PM_2.5 data, instead they are mostly driven by location and intensity of the fire.

The second source of data is from the ground monitoring stations which measure the total PM_2.5 level at the corresponding locations. We have 55 monitor stations throughout the state of Washington. These monitors include both permanent monitors that are likely to be preferentially located near high population areas and temporary monitors that are placed near the areas impacted by the fire. Approximately 7% of the observations from these monitors are missing.

Figure 1 shows the concentration of log(1+PM_2.5) (in log μg/m³, called logPM_2.5 henceforth) on 22 August, 2015 at 04:00 GMT. The circles indicate the locations of the stations and their colors correspond to the observed logPM_2.5 values; missing observations are colored in gray. The background map shows the numerical model forecast. There is obvious difference in the spatial resolution of the two sources of data as well as the type of the data. The numerical model forecast only give information about wildland fire PM_2.5 emissions, whereas the monitor station data includes both wildland fire PM_2.5 emission and PM_2.5 emission from other sources. This adds an additional level of difficulty to model and infer about the same phenomenon from the two sources of data. Naturally excluding the other sources of PM_2.5 from the forecast model probably results in the system ignoring some complex feedback between fire related PM_2.5 and PM_2.5 from other sources. However, in a period with large fire, this effect is likely to be minimal and therefore can be overlooked.

Figure 1:

logPM_2.5 concentration in log μg/m³ in Washington at 04:00 GMT on August 22, 2015. The background map shows the forecast from the numerical model, while the circle shows the location of the monitoring station. The color of the circle indicates the concentration level of the observed logPM_2.5 with missing value colored in light gray.

3. Statistical model

Let Y_t(s) denote the measured logPM_2.5 from the monitor at spatial location s = (s₁, s₂)^T on day t, and X_t(s) be corresponding numerical forecast in log scale. Instead of directly relating these variables, we associate Y_t(s) to a smoothed and warped forecast to account for model discrepancies. Let $w:{\mathcal{R}}^2\to {\mathcal{R}}^2$ be a warping function that maps s to a new location w(s) = (w₁(s), w₂(s))^T to account for spatial misalignment (as discussed in Section 1) and ${\tilde{X}}_t(s)$ to be smoothed forecast. The model is then

$$Y_t(\mathbf{s})={\beta }_0(\mathbf{s})+\beta {\tilde{X}}_t(w(\mathbf{s}))+{ϵ}_t(\mathbf{s}),$$ (3.1)

where ${ϵ}_t(\mathbf{s})\stackrel{iid}{~}\mathrm{Normal}\left(0,{\sigma }^2\right)$ is error. Since the smoothed and warped forecast is a product of a atmospheric dispersion model that already takes into account the spatiotemporal variability as well as the effects of meteorological components and other factors, we assume that the errors ϵ_t(s) are independent over space and time. The slope parameter β is included to calibrate the difference in scale of the forecast and monitor data, perhaps due to the areal nature of the forecast and the point nature of the monitor.

3.1. Model for the Spatially Varying Intercept

A spatially varying intercept is employed to correct for possible additive bias. In our motivating example, additive bias in the monitor station observations come from other sources of PM_2.5, such as traffic and industry that are not included in the numerical forecast. We model the spatially varying intercept using finite basis function expansion

$${\beta }_0(\mathbf{s})=b_0+\sum _{j=1}^J\sum _{k=1}^K{A}_j^0\left(s_1\right)B_k^0\left(s_2\right)b_{jk}.$$ (3.2)

We use known basis functions for the two coordinates, $A_j^0\left(s_1\right)$ and $B_k^0\left(s_2\right)$, and estimate the coefficients b_jk and b₀. Although other choices of basis functions are possible, we use an outer product of B-spline basis functions, that is, $A_j^0\left(s_1\right)$ and $B_k^0\left(s_2\right)$ are univariate B-spline basis functions with J and K knots, respectively. Cubic B-splines basis functions are a sensible choice as they can approximate any smooth function in a bounded domain.

A natural problem in finite basis function expansion based modelling is the choice of number of knots and their position. We select J and K to be large enough to capture the variability in the data with enough detail and use a penalized B-spline approach to prevent overfitting. Penalization is achieved by employing a Gaussian prior distribution on the coefficients b = (b₁₁,b₁₂, … b_JK)^T with mean 0 and covariance ${\sigma }_0^2{\Sigma }_0.{\Sigma }_0$ has a conditional autoregressive (CAR) covariance structure, i.e, Σ₀ = (M₀ − ρ₀E₀)⁻¹, where E₀ is the adjacency matrix for the coefficients in b and M₀ is a diagonal matrix with the number of neighbors for each knot on the diagonal. The coefficients b_jk and b_j0_k0 are considered neighbors if |j – j′| + |k − k′| = 1.

3.2. Model for the Warping Function

We approximate the warping function using the finite basis function expansion

$$w_l(\mathbf{s})=s_l+\sum _{j=1}^{J_1}\sum _{k=1}^{J_2}{A}_j\left(s_1\right)B_k\left(s_2\right)a_{jkl},l=1,2.$$ (3.3)

The warping function is defined by basis functions for the two coordinates A_j(s₁) and B_k(s₂) and the corresponding coefficients a_jkl. We use an outer product of B-spline basis functions for our model here as well, that is, A_j(s₁) and B_k(s₂) are univariate cubic B-spline basis functions with J₁ and J₂ knots, respectively. However, B-spline would not be a good choice if the warped location is outside the bounded domain. This is tackled by forcing any point outside the grid to be remapped to its closest point on the grid boundary.

Other applications of warping in spatial statistics have used some restrictions on the form of the warping function. For example, Sampson and Guttorp (1992) restricted the class of warping functions to one-to-one functions and Snelson et al. (2004) restricted the warping functions to be monotone and have the entire real line as its range. Such restrictions are not necessary here since warping the space for covariates does not present problems of preserving measure-theoretic properties or positive definiteness of the covariance structure. Therefore, we can apply warping functions that map multiple locations to one point in the warped image. This may be unavoidable if the forecast is available only on a coarse spatial grid and multiple monitors reside in the same grid cell.

While insisting that the warping function is one-to-one is unnecessary and overly restrictive, we do impose a prior penalty to avoid overfitting. Our prior encourages the warping function to be smooth and centered around identity warp, w(s) = s. We consider identical priors for the coefficients ${\mathbf{a}}_l=\left(a_{11l},\dots ,a_{J_1{J}_{2}l}\right)$ for each l = 1, 2 and that a₁ and a₂ are independent. To ensure a smooth warping function, we use a spatial prior for a_l defined as a neighboring scheme based on the indices that involves the rook neighbors for each index when viewed to be placed on a two-dimensional integer grid. That is, a_jkl and a_j0_k0_l are neighbors if |j – j′| + |k − k′| = 1. A correlation structure for such a neighboring scheme is created by assigning a CAR covariance structure Σ_w = (M₁ − ρ_wE₁)⁻¹ to the normally distributed coefficients, with E₁ being the adjacency matrix and M₁ being the diagonal matrix with ith diagonal entry equal to the number of neighbors of the ith point. This means that a_l has a Gaussian distribution with mean 0 and covariance ${\sigma }_a^2{\Sigma }_w$. By setting E(a) = 0, we shrink the warping function towards the identity function.

3.3. Model for the Smoothing Function

Smoothing the forecast eliminates spurious small-scale variation and allows aligning large-scale features of the forecast such as smoke plumes with the monitor data. Since the forecast is on a regular grid, the smoothing can be achieved using the spectral downscalar proposed by Reich et al. (2014). The spectral representation of the forecast is

$$X_t(\mathbf{s})=\int \exp \left(-i{\omega }^{\top }\mathbf{s}\right)Z_t(\omega )d\omega ,$$ (3.4)

where $\omega \in {ℝ}^2$ is a frequency and

$$Z_t(\omega )=\int \exp \left(i{\omega }^{\top }\text{s}\right)X_t(\text{s})d\mathbf{s}$$ (3.5)

is the inverse Fourier transform of the forecast. This decomposes the forecast’s signals at different frequencies Z_t(ω). Processes that comprises of lower frequencies contain the information about the large-scale patterns, while processes corresponding to higher frequencies holds local information. We capture the forecast features at L different resolutions using Z

$${\tilde{X}}_{lt}(\mathbf{s})=\int V_l(\omega )\exp \left(-i{\omega }^{\top }\mathbf{s}\right)Z_t(\omega )d\omega ,$$ (3.6)

where V_l(ω) are known basis functions that serve as weights based on frequencies satisfying ^R V_l(ω)dω = 1, ∀l = 1, · · ·, L where L is the number of basis functions. A useful choice for the basis functions are Bernstein polynomials, as suggested by Reich et al. (2014) (see the Appendix-A for details). We then reconstruct the smoothed process by

$${\tilde{X}}_t(\mathbf{s})=\sum _{l=1}^L{\alpha }_l{\tilde{X}}_{lt}(\mathbf{s}).$$ (3.7)

Smoothing is achieved if α_l ≈ 0 for terms with large ‖ω‖ as this essentially filters out high resolution features. On the other hand, if α_l = 1 for all l, the smoothed forecast reduces to the original forecast, i.e, ${\tilde{X}}_t(\mathbf{s})=X_t(\mathbf{s})$.

Constructing ${\tilde{X}}_{lt}(s)$ requires computing the stochastic integrals in (3.5) and (3.6). For fast computing, these integrals are approximated using two dimensional discrete Fourier transform and inverse discrete Fourier transform as

$$Z_{pt}\approx \frac{1}{P}\sum _{q=1}^P\exp \left(i{\omega }_p^{\top }{\mathbf{s}}_q\right)X_t\left({\mathbf{s}}_q\right)\text{and}{\tilde{X}}_{lt}(\mathbf{s})\approx \sum _{p=1}^P{V}_l\left({\omega }_p\right)\exp \left(-i{\omega }_p^{\top }\mathbf{s}\right)Z_{pt},$$ (3.8)

where the forecast is on a grid of P₁ × P₂ and P = P₁P₂.

In (3.1), the scale of β and α₁, α₂, … , α_L are not identified, so we reparametrize to β = β(α₁, α₂, … , α_L)^T = (β₁, β₂, … , β_L)^T and place a prior on β. To prevent overfitting, we use the same penalized splines approach as before. We put another CAR covariance structure on β with the neighboring scheme based on their indices, as before,

$$\beta ~N\left(\mathbf{0},{\sigma }^2{\tau }^2{\mathbf{D}}_x\right),$$

where 0 is the zero vector of length L and D_x = (M₂ − ρ_xE₂)⁻¹ is the corresponding CAR covariance structure, E₂ being the adjacency matrix with terms l and k considered neighbors if |l−k| = 1 and M₂ being the corresponding diagonal matrix created similarly as before.

3.4. Model Details

Since the forecast, and the spectral covariates, can only be computed on a grid and the monitoring sites are non-gridded points in ${ℝ}^2$, we use the nearest grid neighbor as a proxy for forecast at the station. That is, we use the model

$$Y_t(\mathbf{s})={\beta }_0(\mathbf{s})+\sum _{l=1}^L{\beta }_l{\tilde{X}}_{lt}(\tilde{w}(\mathbf{s}))+{ϵ}_t(\mathbf{s}),$$

where $\tilde{w}(\mathbf{s})$ is the location of the closest forecast grid cell to w(s). Any point that goes outside the grid as a result of the warping is set at the nearest grid point, as mentioned earlier.

To complete the Bayesian model for jointly modeling the smoothing and warping, we specify the priors for the hyperparameters: ${\sigma }_0^2~IG(0.01,0.01)$ and σ² ~ IG(0.01, 0.01). We assume ${\sigma }_a^2~\text{Half-Normal}(0.15)$. This sets the 99th percentile for the prior to be 1. This choice of prior strongly suggest the warping to be adequately smooth. We put a Beta(10,1) prior on the hyperparameters ρ₀, ρ_a and ρ_x, suggesting a minimal level of spatial correlation being present. Instead of choosing a hyperprior for τ², we set τ² = 10. This helps avoid numerical instability in the computational process and yet provides enough prior uncertainty for the β parameter. We recommend choosing J,K, J₁ and J₂ to be large, e.g., so that the number of basis functions is roughly the same as the number of monitor stations, as the penalization should set the unnecessary coefficients to zero, thus reducing it to a simpler model. We use (J,K) = (J₁, J₂) = (6, 4),(10, 5) or (12, 8) for our simulation study scenarios with the corresponding number of monitor stations being 25,50 or 100. For the data example, we use (J, K) = (J₁, J₂) = (11, 5). We should chose L to be large as well since we added a penalization for that too. However, in this case we do not need to choose L to be as large as the number of monitor stations since we expect the smoothing operator to be a smooth function. For instance, we use L = 15 throughout our studies and data example as we believe decomposing the information in X_t(s) into 15 partitions would allow us to filter out enough needless small scale variations to achieve smoothing.

4. Simulation Study

In this section, we conduct a simulation study to explore the performance of the proposed method in different scenarios. We consider four data generation processes and three sets of monitor station locations for each of these processes and create 30 datasets for each combination of these factors. We use the forecast from the dataset described in Section 2 for August 18, 2015 to August 22, 2015 as X_t(s). The grid size for the simulation study was therefore the same as the forecast grid of the data, 200 × 95.

We consider five data generation processes. In the first case, data is generated by a simple linear regression (SLR) model with the forecast as the predictor, i.e.,

$$Y_t(\mathbf{s})={\beta }_0+{\beta }_1{X}_t(\mathbf{s})+{ϵ}_t(\mathbf{s}),$$

with ${ϵ}_t(\mathbf{s})\stackrel{iid}{~}N(0,1)$, β₀ = 1.5 and β₁ = 0.25. Second, we use the smoothed forecast predictor

$$Y_t(\mathbf{s})={\beta }_0+\sum _{l=1}^L{\beta }_l{\tilde{X}}_{lt}(\mathbf{s})+{ϵ}_t(\mathbf{s}),$$

where $L=10,{ϵ}_t(\mathbf{s})\stackrel{iid}{~}N(0,1)$, β₀ = 1.5 and β_l were decreasingly ordered realizations of a N(0.25, 0.0625) random variate for l = 1, 2, …, 10. The descending order of the coefficients ensures that the low frequency terms have higher weights than high frequency terms. The next two cases have a warped and smoothed forecast as predictor

$$Y_t(\mathbf{s})={\beta }_0+\sum _{l=1}^L{\beta }_l{\tilde{X}}_{lt}(\tilde{w}(\mathbf{s}))+{ϵ}_t(\mathbf{s})$$

with $\tilde{w}(\cdot )$ being a warping function and the remaining components of the model being the same as in the previous scenario. These two cases are distinguished by their warping function. The first warping function is the translation warp

$$w(\mathbf{s})=\mathbf{s}+\left(\begin{array}{c}0.16\\ 0.16\end{array}\right).$$

The second warping function we used was diffeomorphism warp (Guan et al., 2019) that preserves the boundaries of the image. For 0 ≤ s₁, s₂ ≤ 1,

$$w_1(\text{s})=s_1-2{\theta }_1{s}_2\sin s_1\cos s_2\left(\cos \pi s_1+1\right)\left(\cos \pi s_2+1\right)$$

$$w_2(\mathbf{s})=s_2-2{\theta }_2{s}_1\sin s_1\sin s_2\cos \frac{\pi s_1}{2}\cos \frac{3\pi s_2}{2},$$

where θ₁ and θ₂ are tuning parameters jointly deciding the location, direction and extent of the warp set equal to θ₁ = 0.1 and θ₂ = 0.5. The fifth scenario we considered had a spatially varying intercept β₀(s) = 0.5 + 1.25s₁ − 0.5s₂ and we also set β = 1.2 for this scenario (which increases the magnitude of the prediction errors for all methods). We use a spectral smoothing and translation warp as before to generate the data. To investigate the effect of number of monitor stations, we select 25, 50 or 100 monitor station observations randomly on the grid for each of the data generation processes.

A visualization of the data generation process for the fourth scenario can be seen in Figure 2. The top left panel shows the original forecast which was smoothed using spectral smoothing. The smoothed forecast is shown in the top right panel. A diffeomorphism warp, shown in the bottom left panel was then applied to this smoothed output. The resulting output in the bottom right panel shows a shrunken plume around the top middle part as a result.

Figure 2:

Original forecast X_t(s) in log-scale on August 22, 2015 at 4 : 00 AM (top left); The smoothed forecast ${\tilde{X}}_t(s)$ in log-scale (top right); the diffeomorphism warp w(s) (bottom left); the warped and smoothed forecast ${\tilde{X}}_t(w(\mathbf{s}))$ in log-scale, created by using the diffeomorphism warp w(s) on the ${\tilde{X}}_t(s)$, used to generate the data. The log-concentrations are measured in log μg/m³.

For each of these scenarios, we fit a simple linear model to the data as well as three versions of the model proposed in Section 3 with or without the warping and smoothing components. For each method, a Markov Chain Monte Carlo (MCMC) chain was run for 20,000 iterations, of which the first 10,000 iterations were discarded as burn-in samples.

To compare models we use mean squared error (MSE) and mean absolute deviation (MAD) computed using the posterior mean as point forecast and pointwise coverage of 95% intervals and continuous ranked probability score (CRPS). The posterior predictive densities for the warped outputs can be skewed, heavy-tailed or even multi-modal and so metrics based on point predictions (MSE or MAD) may not capture the uncertainty properly. To evaluate the entire predictive distribution, CRPS (Gneiting and Raftery, 2007) is therefore a more meaningful choice since it is a measure of integrated squared difference between the cumulative distribution (CDF) function of the forecast and the corresponding CDF of the observations.

We compute the 3-day ahead forecast and compute the MSE, MAD, coverage and CRPS for the forecast of n monitor stations. For each of the 15 scenarios and for the corresponding 30 datasets in each scenario, MSE, MAD, coverage and CRPS for each of the four models are computed and averaged over space and time for all datasets. The MSE (MAD is similar) and CRPS for these cases are reported in Table 1. For all methods and cases, coverage is always between 94% and 100% and so it is not reported. Figure 3 presents a comparison of the true and estimated (posterior mean) of warping function w(s) for data sets with n = 25 and n = 100 from scenarios 3 and 4.

Figure 3:

True (red) and estimated (green) warps for the translation warp (top row) and diffeomorphism warp (bottom row) for simulated data with n = 25 (left) and n = 100 (right).

From Table 1, all methods perform similarly when data are generated from the SLR model. Therefore the added complexities of the full model do not result in overfitting in this case. The full model has smaller MSE and CRPS than the SLR model in the second case. Although the model with only smoothing component is somewhat better as that matches the true data generation model. In the later three cases, the full model provides the best results. The performance of all methods improve with increasing values of n. This is reflected in Figure 3 which compares the true and estimated warp for both the warps used in this study. In both cases, the estimates are closer to the true value for n = 100 than for n = 25. The estimation is more accurate for the translation warp, compared to the more complicated diffeomorphism warp.

5. Application to PM_2.5 Forecasting in Washington State

In our simulation study, we fix the warping function to be constant over time. However, for the wildland fire application, the warp likely varies over time, following changes in the location of the fires and wind field. Therefore, we analyze the data separately by day with the first 18 hours of data as training and forecast on the next 6 hours for each of the 35 days. This strikes a balance between flexibility to capture dynamics of the warping function while still providing sufficient training data to estimate the warping function. The priors, models, computational details and metric of comparison are the same as the simulation study.

For each day, we compute predictive MSE, MAD, coverage and CRPS averaged over space and time. These metrics, averaged over days, are presented in Table 2. Day-by-day comparisons for MSE, MAD, coverage and CRPS are available in Appendix-A.

The models with smoothing, warping or both perform significantly better than SLR. Smoothing leads to the largest reduction in MSE and MAD while the full model leads to the largest reduction in CRPS. Therefore, smoothing appears to be sufficient if only a point estimate is required, but including the warping function provides a better fit to the full predictive distribution.

To further illustrate how the warping models provide richer uncertainty quantification, we compute the posterior mean, standard deviation, skewness and kurtosis for each test set observation and present the distribution of these summary statistics as boxplots in Figure 4. The mean values are similar for all models, but the warp based models exhibit higher skewness and kurtosis.

Figure 4:

Boxplots of mean (in log μg/m³), standard deviation (in log μg/m³), skewness and kurtosis of the posterior predictive distributions for each model. The color schemes for each model is black (SLR), blue (Smoothing), green (Warp) and Red (Full) for each of the subfigures.

Skewness and kurtosis often result from uncertainty in the warping function. For example, we compare the posterior predictive densities (PPD) of the four methods for a particular station located at the edge of the wildfire on August 22, 2015 at 7:00 PM in the left panel of Figure 5. One would expect high uncertainty in estimation for such a location. The PPD from SLR method misses the true value (magenta) by quite some margin, while the three methods capture the true value within their respective PPDs. The PPD for the full model estimator has a heavier right tail and smaller peak, indicating high variance and kurtosis capturing the uncertainty of estimation in such a location. On the other hand, comparing the densities for a location that is in the middle of the wildfire for the same day and time, shows that the PPDs behave similar to each other and have low skewness and thin tails, as can be seen in the right panel of Figure 5. Figure 5 also shows the estimated warping function for the day. The red arrows imply a significant warp at the location at its base, while the green ones are non-significant (where warp at location s is significant if the 95% credible set of either component of w(s) − s excludes zero). The trace plots for the estimate of displacement due to the warping function (w(s) − s) for the two locations marked in Figure 5 are presented in Appendix-A showing adequate convergence for the MCMC procedure.

Figure 5:

The numerical forecast of log-concentration of PM_2.5(μg/m³) on August 22, 2015 at 7:00 PM (top left) with two stations highlighted, one near the edge of fire (triangle) and one in the middle of fire (rectangle). The estimated warp for the day (top right) showing significant (red) and non-significant (green) warps. Comparisons of PPDs are made for the four competing models for the location marked as triangle (bottom left) and the location marked as rectangle (bottom right).

6. Concluding Remarks

Motivated by an wildland fire application, we develop a new downscaling method that incorporates spectral smoothing and image warping techniques into a single downscaling method which is shown to improve forecast distributions for simulated and real data.

The proposed method could be extended in several ways. A simple extension would be to use forecasts that include background PM_2.5 information as well. This will likely diminish the problem of having different quantities being measured by the two sources. Changing the slope in Eq. (3.1) to a spatially varying one could also be contemplated. Another extension could see the error distribution to have spatiotemporal dependence, especially in applications where the numerical model is unable to capture the important spatiotemporal trends observed in the data. While this would be straightforward to implement, it would add to the computational burden making it more challenging to apply in real time. Another possible extension is to allow for warping in both space and time. Another extension would be use spatially varying the slope parameter β in Eq. 3.1. Warping the forecast in time would adjust for timing errors, such as model misspecification of the rate of wildland fire expansion or the speed of a storm traveling through the spatial domain. We have not done this because we refit the model in fairly small spatiotemporal windows, but in other applications warping in time could be just as important as warping in space.

Table 1:

MSE estimates (in μg²/m⁶) and CRPS for the proposed model with both smoothing and warping components, only smoothing component and only warping component along with a SLR model for different scenarios. Each horizontal block represents a true data generation scheme dictated by the first three columns and the last two sets of 4 columns each represent the performance of the 4 competing models in terms of MSE and CRPS. The lowest MSE and CRPS values in each case are in bold.

Warp	Smoothing	Spatially Varying Intercept	n	MSE				CRPS
				SLR	Proposed Model			SLR	Proposed Model
					Smooth	Warp	Both		Smooth	Warp	Both
		No	25	1.01	1.02	1.02	1.03	0.78	0.78	0.78	0.77
None	None	No	50	1.00	1.00	1.00	1.01	0.78	0.77	0.77	0.77
		No	100	1.00	1.00	1.00	1.00	0.79	0.79	0.79	0.78
		No	25	1.27	1.02	1.35	1.03	0.87	0.78	0.85	0.78
None	Spectral	No	50	1.30	1.00	1.33	1.01	0.89	0.77	0.85	0.77
		No	100	1.31	1.00	1.35	1.00	0.90	0.79	0.86	0.79
		No	25	1.69	1.34	1.60	1.19	1.00	0.85	0.88	0.78
Translation	Spectral	No	50	1.62	1.25	1.45	1.07	0.99	0.82	0.84	0.77
		No	100	1.64	1.25	1.49	1.09	1.00	0.83	0.90	0.78
		No	25	1.53	1.32	1.46	1.17	0.96	0.84	0.85	0.76
Diffeomorphism	Spectral	No	50	1.50	1.23	1.39	1.10	0.95	0.82	0.83	0.75
		No	100	1.56	1.26	1.49	1.12	0.97	0.83	0.90	0.77
		Yes	25	32.56	8.76	14.50	5.37	4.56	1.85	2.59	1.35
Translation	Spectral	Yes	50	27.38	7.08	12.25	2.80	4.23	1.73	2.38	1.00
		Yes	100	28.69	7.04	25.53	1.84	4.37	1.74	3.76	0.84

Table 2:

Values of MSE (in (log μg/m³)²), MAD (in log μg/m³), coverage and CRPS for the data analysis for the different methods, averaged over days.

	SLR	Smoothing	Warp	Full
MSE	0.4624	0.3357	0.3517	0.3675
MAD	0.4983	0.4081	0.4198	0.4219
Coverage	0.9237	0.9294	0.9126	0.9019
CRPS	0.4842	0.3688	0.3570	0.3500

A Technical Details for the Model

A.1. Construction of Basis Functions for Spectral Smoothing

As mentioned in Section 3, we smooth our forecast using a spectral smoothing approach proposed by Reich et al. (2014). This process, using fast Fourier transform and inverse fast Fourier transform, breaks the original forecasts X_t(s) into several layers X_lt(s) by weighting them with basis functions V_l(ω). Each of the X_lt(s₎s contains information about phenomenons of different scales. We mentioned some restrictions on the basis functions to be used in Section 3.

A common choice for choosing this basis function is the Bernstein polynomial basis function, as suggested by Reich et al. (2014). This approach assumes that the dependence of frequency ω in constructing the basis functions is solely on the magnitude of the frequency ‖ω‖. With this assumption, the basis functions can be written as

$$V_l(\omega )=V_l(\Vert \omega \Vert )=\left(\begin{array}{c}L-1\\ l-1\end{array}\right){\left(\frac{\Vert \omega \Vert }{2\pi }\right)}^{l-1}{\left(1-\frac{\Vert \omega \Vert }{2\pi }\right)}^{L-l},$$ (A.1)

for l = 1,2, … ,L. This set up ensures that $\int V_l(\omega )d\omega =1$, ∀l.

However, such representation of ${\tilde{X}}_{lt}(\mathbf{s})$ may be subject to identifiability issues because of how the basis functions are defined in Equation (A.1). To avoid such issues, we follow Reich et al. (2014) and define

$$\delta =\{\begin{array}{l}\omega \text{if}\Vert \omega \Vert \le \Vert \overline{\omega }\Vert \hfill \\ \overline{\omega }\text{if}\Vert \omega \Vert >\Vert \overline{\omega }\Vert \hfill \end{array}\in [0,2\pi ),$$ (A.2)

where $\overline{\omega }={\left[\mathbb{I}\left({\omega }_1>0\right)\left(2\pi -{\omega }_1\right),\mathbb{I}\left({\omega }_2>0\right)\left(2\pi -{\omega }_2\right)\right]}^{\top }$. After this, we define our basis functions as V_l(ω) = V_l(‖δ‖). This ensures we avoid aliasing issues while retaining the other properties.

A.2 Computing

The warping function is not completely identifiable, that is to say that for two different warping functions w₁(s) ≠ w₂(s), we may have the same warped output X_t(w₁(s)) = X_t(w₂(s)) for some s and at some timepoint t. If the two warping function differ only on how they map points with zero values to other points with zero values, then it is not possible to distinguish them. Assuming that the forecast would be non-constant over any region is unrealistic as it is bound to have regions with zero values, in general. This is not necessary for us to have the warping function identifiable, but it does create problems with convergence as parameters can fluctuate between two sets of values both of which give the same warped output.

Another concern for convergence is the large number of parameters in the model. The smoothing coefficients needed to be marginalized to achieve convergence in the full model. Convergence of component models (smoothing-only or warping-only) is much quickly achieved compared to the full model scenario and usually require no tricks such as marginalization, although we used marginalization for them as well. We used the simple Metropolis within Gibbs algorithm to run our MCMC chains throughout. Metropolisadjusted Langevin algorithm (MALA) or Hamiltonian Monte Carlo (HMC) methods may provide quicker convergence but would add much complexity to each iteration.

Codes for generic purpose use for these methods are available in the author’s GitHub repository.

B Supplemental Tables and Figures

B.1 Additional Tables for MAD and Coverage Estimates from the Simulation Study

We present here additional tables obtained from the simulation study detailed in Section 4. These tables show the performance of the four models, the OLS model, the full model (see Section 3) and the two sub-models, smoothing only and warping only model (see Section 4), in four different data generation scenarios with three different values of n for each of the four cases. The results obtained here are similar to those in Section 4.

B.2 Additional Figures from Data Analysis

We present additional images from data analysis here. The Figures 6 and 7 shows the performance of the four models, as in Section 5, for every run (each run being based on each day) based on the metrics MAD and coverage. The inference is similar to that in Section 5. On most days with large fires, and thereby large plumes, the full model works better than the smoothing only model. All other models work better than the SLR model on almost any day.

Table 3:

MAD (standard error) estimates (in μg/m³) for the proposed model with both smoothing and warping components, only smoothing component and only warping component along with a SLR model for different scenarios. The lowest MAD value in each case is in bold.

Warp	Smoothing	Spatially Varying Intercept	n	SLR	Proposed Model
					Smooth	Warp	Both
		No	25	0.80(0.02)	0.81(0.02)	0.81(0.02)	0.81(0.02)
None	None	No	50	0.80(0.01)	0.80(0.01)	0.80(0.01)	0.80(0.01)
		No	100	0.80(0.01)	0.80(0.01)	0.80(0.01)	0.80(0.01)
		No	25	0.90(0.02)	0.81(0.02)	0.93(0.02)	0.81(0.02)
None	Spectral	No	50	0.91(0.01)	0.80(0.01)	0.92(0.01)	0.80(0.01)
		No	100	0.91(0.01)	0.80(0.01)	0.93(0.01)	0.80(0.01)
		No	25	1.03(0.02)	0.92(0.02)	1.00(0.03)	0.87(0.04)
Translation	Spectral	No	50	1.01(0.01)	0.89(0.01)	0.95(0.02)	0.82(0.03)
		No	100	1.02(0.01)	0.89(0.01)	0.97(0.01)	0.83(0.04)
		No	25	0.98(0.03)	0.91(0.02)	0.96(0.03)	0.86(0.03)
Diffeomorphism	Spectral	No	50	0.97(0.01)	0.88(0.01)	0.93(0.01)	0.83(0.02)
		No	100	0.99(0.01)	0.89(0.01)	0.96(0.01)	0.84(0.01)
		Yes	25	4.69(0.04)	2.11(0.04)	2.98(0.13)	1.67(0.04)
Translation	Spectral	Yes	50	4.31(0.03)	1.90(0.02)	2.70(0.07)	1.17(0.32)
		Yes	100	4.43(0.02)	1.91(0.01)	4.06(0.06)	0.98(0.23)

Table 4:

Coverage (standard error) estimates for the proposed model with only smoothing component, only warping component and the full model along with an SLR model for different scenarios.

Warp	Smoothing	Spatially Varying Intercept	n	SLR	Proposed Model
					Smooth	Warp	Both
		No	25	0.95(0.01)	0.95(0.01)	0.95(0.01)	0.95(0.01)
None	None	No	50	0.95(0.00)	0.95(0.00)	0.95(0.00)	0.95(0.00)
		No	100	0.95(0.00)	0.95(0.00)	0.95(0.00)	0.95(0.00)
		No	25	0.95(0.01)	0.95(0.01)	0.95(0.01)	0.95(0.01)
None	Spectral	No	50	0.95(0.01)	0.95(0.00)	0.94(0.01)	0.95(0.00)
		No	100	0.95(0.00)	0.95(0.00)	0.94(0.00)	0.95(0.00)
		No	25	0.95(0.01)	0.95(0.01)	0.94(0.01)	0.95(0.01)
Translation	Spectral	No	50	0.95(0.00)	0.95(0.00)	0.95(0.01)	0.95(0.01)
		No	100	0.95(0.00)	0.95(0.00)	0.95(0.00)	0.95(0.00)
		No	25	0.95(0.01)	0.94(0.01)	0.94(0.01)	0.95(0.01)
Diffeomorphism	Spectral	No	50	0.95(0.00)	0.94(0.00)	0.95(0.00)	0.95(0.00)
		No	100	0.95(0.00)	0.94(0.00)	0.95(0.01)	0.95(0.00)
		Yes	25	0.96(0.00)	0.97(0.00)	0.96(0.01)	0.97(0.00)
Translation	Spectral	Yes	50	0.95(0.00)	1.00(0.00)	1.00(0.00)	1.00(0.00)
		Yes	100	0.96(0.00)	0.93(0.00)	0.96(0.00)	0.98(0.01)

Figure 8 shows the trace plots for for the estimated displacement due to warp for the two locations in Figure 5. The location in the middle of the fire (right panel) has values around zero, meaning a non-significant warp at the location. The estimate for the location at the edge the fire is has a jagged trace with values away from zero, indicating a significant warp. In both cases, the MCMC algorithm mixes well.

Figure 6:

Daily prediction MSE (left panel) in μg²/m⁶ and CRPS (right panel) for the four models

Figure 7:

Daily prediction MAD (left panel) in μg/m³ and coverage (right panel) for the four models

Figure 8:

Trace plots for x (top) and y (bottom) coordinates of the estimated displacement due to warp (w(s) − s) for the two locations flagging in Figure 5. Left panel is for the location at the edge of fire and right panel for the location in the middle of it.