Table 2.

Optimization algorithm for single pollutant DLNM.

Step	Description
1	$\text{Y}$ is a daily count data that originates from quasi-Poisson distribution. Depending on the type of outcome, it can be assumed as one of the exponential families of distributions
2	Consider $2^k$ of DLNMs by best subset selection, where $k$ is the number of covariates (e.g. meteorological factors) except the terms to describe seasonality, trend, holiday effects, etc. In other words, comparing the model performance by considering everything from the non-covariate single-exposure model to the full-covariates single-exposure model
3	Tune hyper-parameters of each $2^k$ of DLNM based on QAIC (the AIC for quasi-Poisson) • Maximum lag days: [$m_1,m_2,\cdots$] • Degrees of freedom in predictor space(${\text{df}}_p$): [$v_1,v_2,\cdots$] • Degrees of freedom in additional lag dimension(${\text{df}}_l$): [$l_1,l_2,\cdots$] Knots are equally spaced, and a natural cubic spline is selected as a basis function. To tune and optimize each $2^k$ of DLNM, use one of the search methods (e.g. grid search, random search)
4	Among the $2^k$ of DLNMs optimized in step 3, The model with the smallest QAIC value is selected as the best model. However, if there is a model with a QAIC difference of less than 2 from the optimal model with the smallest QAIC as follows: ${\mathrm{\Delta }}_{\text{i}}={\text{QAIC}}_{\text{i}}-{\text{QAIC}}_{\text{min}}<2$ The simplest model is the best model by comparing the models, including the optimal one

DLNM distributed lag non-linear models.