1 Introduction

Due to advancements in science and technology, the amount of available, often open-source, data is growing exponentially. New opportunities for statistical research arise by formulating relevant and interesting questions which can be answered with these data. This work is motivated by questions in urban analytics and three temporal datasets, namely COPD hospital admissions, NO2 measurements and visits to gyms in Leeds from 2013 to 2018. The NO2 data are measured at ten locations and extrapolated to postcode level. The visits to gyms and the COPD admissions are aggregated to postcode level. The health and pollution data are sampled densely and regularly in time (we call this dense functional data) and the physical exercise data are obtained at irregular time intervals with few measurements (we call this sparse longitudinal data). The questions we aim to answer with these data are: Do life style and pollution affect COPD hospital admissions at postcode level? Does a rise in pollution have a direct effect on COPD hospital admissions or is there a delay? A first step to answer these questions is to model health outcomes over time as function of pollution and physical exercise over time.

Relationships between temporal data are often not synchronous and involve a delay in the effects. Exposure of a person to high pollution might show to affect this person’s health with some delay and the effect might fade away after some time. Other examples are, a treatment with a medicine might take some time to start affecting the patient and after the end of the treatment it might still be having an effect for limited amount of time. Historical exposure to high temperatures might not have an effect on the growth of trees anymore after a certain period and it may also take some time before high temperatures result in lower growth rates.

Motivated by the problem of dense temporal and sparse longitudinal predictors, we consider estimation and prediction for a functional regression model [7]. We estimate the intervals through the corresponding lags of the effect of predictors on response. Specifically for our motivating example, we estimate the influence of dense functional pollutants (nitrogen dioxide (NO2) concentrations) and sparse longitudinal predictor (physical activities) on dense COPD hospital admissions. Moreover, we want to estimate the effect lags of pollutants on COPD hospital admission, i.e. from when the predictors have an influence on the response and until when this influence disappears.

The classical function-on-function linear model, which was first formulated by Ramsay and Dalzell [17], reads as follows:

$$\begin{aligned} Y(t)=\beta _0(t)+\int _0^T\beta _1(s,t)X(s)ds+\epsilon (t),\ t\in [0,T] \end{aligned}$$

where Y(t) is the response trajectory, X(s) is the predictor trajectory, \(\epsilon (t)\) is the error process, \(\beta _0(t)\) is the intercept process, \(\beta _1(s, t)\) is the two-dimensional regression coefficient function which shows the influence of X on Y. A drawback of this model is that the entire predictor trajectory X(s) including the future values, i.e. when \(s>t\), is assumed to influence the current value of response trajectory Y at time t. Clearly this is not appropriate in many applications. Malfait and Ramsay [5, 8, 12] proposed historical functional linear models, where only the past of the predictor trajectory influences the response at the current time:

$$\begin{aligned} Y(t)=\beta _0(t)+\int _{t-\delta _2}^{t-\delta _1}\beta _1(s,t)X(s)ds+\epsilon (s),\ t\in [0,T] \end{aligned}$$

where \(\delta _1\) and \(\delta _2\) (\(0<\delta _1<\delta _2<T\)) are the lags for the influence of predictor trajectory on response trajectory. For one dense functional predictor, Malfait and Ramsay [12] consider the triangular basis expansion of the coefficient function which is estimated at each observation point. A penalized approach which allows varying lags for the historical functional linear model has been developed by [5]. Kim et al. [8] considers the situation that both predictor process and response process are sparsely and irregularly observed. Pomann et al. [16] has extended the historical functional linear model to multiple homogeneous predictors, and the response is influenced by the predictors from a fixed starting effect time to current time. In this paper we extend this work to a model with both dense functional and sparse longitudinal predictors.

Our contribution of this paper is threefold: firstly, a model with multiple heterogeneous (sparse longitudinal and dense functional) predictors subject to time lags (both starting and ending points) that are fixed but unknown is proposed. We propose estimators for the coefficient regression functions. The effect lags (i.e. the fixed starting and ending effect times) are determined by minimizing the prediction error. Secondly, the asymptotic behavior of the estimated coefficient functions, and the predicted response curve is investigated. Thirdly, the three temporal datasets are integrated and the relationships between COPD hospital admissions and lifestyle and pollution are modelled.

The paper is organized as follows: In Section 2, the historical function-on-function linear model for multiple heterogeneous predictors is introduced. In Section 3, we consider the estimation of the coefficient functions and the uniform consistency of the estimators is established. In Section 4, the prediction of the response trajectories is proposed and the asymptotic property of the predicted trajectories is established. The determination of the lags is proposed in Section 5. Extensive numerical simulations are considered in Section 6 to show the asymptotic properties of our proposed estimators. In Section 7, the COPD and NO2 datasets are analysed and the lags of the influence of NO2 on COPD are determined. We finish by drawing some conclusions and further discussion.

2 Model

Suppose the observations are \(\{Y_{ij}, t_{ij}: i =1,...,n, \ j=1,...,m_{Yi} \}\), \(\{W_{1ij}, s_{1ij}: i =1,...,n, \ j=1,...,m_{1i} \}\) and \(\{W_{2ij}, s_{2ij}: i=1,...,n, \ j=1,...,m_{2i} \}\), where \(Y_{ij}\) is the j-th observation of response for the i-th subject (at time \(t_{ij}\in [0,1]\)) and \(m_{Yi}\) is the number of observations of response for the i-th subject, similarly, \(W_{1ij}, \ W_{2ij}\) are the j-th observation of subject i for two predictors respectively and \(m_{1i}\) and \(m_{2i}\) are the numbers of observations of the two predictors (at time \(s_{1ij}, s_{2ij}\in [0,1]\)) respectively. For example, the response \(Y_{ij}\) corresponds to the standardised COPD hospital admissions in the i-th postcode district at time \(t_{ij}\). The predictor \(W_{1ij}\) corresponds to the NO2 concentrations in the i-th postcode district at time \(s_{1ij}\). The predictor \(W_{2ij}\) corresponds to the standardised number of physical activities in the i-th postcode district at time \(s_{2ij}\).

Let \(W_{1ij}=X_{1i}(s_{1ij})+\epsilon _{1ij}\), \(W_{2ij}=X_{2i}(s_{2ij})+\epsilon _{2ij}\) and \(X_{1i}(t)\), \(X_{2i}(t)\) be independent copies of underlying square-integrable random functions \(X_1(t)\) and \(X_2(t)\) over [0, 1] respectively. Without loss of generality, we assume \(\mu _{X_1}(t)=E[X_{1}(t)]=0\) and \(\mu _{X_2}(t)=E[X_{2}(t)]=0\). See Remark 2 for guidance on how to estimate the means in real data applications. We denote by \(C_{X_1}(s, t)=cov(X_1(s), X_1(t))\) the covariance of \(X_1(t)\) and \(C_{X_2}(s, t)=cov(X_2(s), X_2(t))\) the covariance of \(X_2(t)\). We assume that the first predictor curves \(X_{1i}(t)\) are observed on a dense and regular grid of points, i.e. \(m_{11}=...=m_{1n}:=m_1\), while the second predictor curves \(X_{2i}(t)\) are observed on a sparse and irregular grid of points \(s_{2ij}\). The observations \(W_{1ij}\) and \(W_{2ij}\) are the discrete versions of \(X_{1i}(t)\) and \(X_{2i}(t)\) with iid mean-zero and variance-finite noise \(\epsilon _{1ij}\) and \(\epsilon _{2ij}\) respectively. The noise \(\epsilon _{1ij}\) and \(\epsilon _{2ij}\) are independent of \(X_{1i}(t)\) and \(X_{2i}(t)\) respectively. For the responses \(Y_{ij}\), they are observed either on a sparse and irregular grid or dense grid of \(t_{ij}\).

We define the lag historical functional linear model with two heterogeneous predictors \(X_1(t)\) and \(X_2(t)\) and the response Y(t) as

$$\begin{aligned} Y_{ij}\,= \beta _0(t_{ij})+&\int _{\delta _{11}}^{\delta _{12}}\beta _1(s, t_{ij})X_{1i}(t_{ij}-s)ds\nonumber \\ +&\int _{\delta _{21}}^{\delta _{22}}\beta _2(s, t_{ij})X_{2i}(t_{ij}-s)ds+e_{ij} \text{ for } t_{ij} \ge \max (\delta _{12},\delta _{22}) \end{aligned}$$
(1)

where \(i\in \{1,...,n\}, \ j\in \{1,...,m_{Yi}\}, \ \beta _0: [0, 1]\rightarrow {\mathbb {R}}\), \(\Delta _1=[\delta _{11}, \delta _{12}]\), \(\Delta _2=[\delta _{21}, \delta _{22}]\), \(\beta _1:\Delta _1\times [0, 1]\rightarrow {\mathbb {R}}\) and \(\beta _2:\Delta _2\times [0, 1]\rightarrow {\mathbb {R}}\) are continuous two-dimensional coefficient functions, and \(e_{ij}\) are independent measurement errors with mean zero and finite variance \(\sigma _e^2\). Errors \(e_{ij}\) are assumed to be independent of \(X_{1i}(t)\) and \(X_{2i}(t)\).

In Equation (1), we assume that the \(\delta\)’s are fixed, similar to [8]. Given the \(\delta\)’s, the observations \(Y_{ij}\) are modelled at times \(t_{ij}\ge \max \{\delta _{12}, \delta _{22}\}\). Considering a subset only of the observations based on the lags is not unusual and is similar to the approach for AR(p) models, where model selection (determine p) comes first and then parameter estimation where observations at time \(t<p\) are not available.

Notice that (1) is equivalent to

$$\begin{aligned} Y_{ij}= \beta _0(t_{ij})&+\int _{t_{ij}-\delta _{12}}^{t_{ij}-\delta _{11}}\beta _1(t_{ij}-s, t_{ij})X_{1i}(s)ds\nonumber \\&+\int _{t_{ij}-\delta _{22}}^{t_{ij}-\delta _{21}}\beta _2(t_{ij}-s, t_{ij})X_{2i}(s)ds +e_{ij} \end{aligned}$$
(2)

then the model (1) means that given the entire predictor curves \(X_{1i}\) and \(X_{2i}\), the response for the i-th subject at time \(t_{ij}\ge \max \{\delta _{12}, \delta _{22}\}\) is only affected by the values of \(X_{1i}\) over time-window \([t_{ij}-\delta _{12}, t_{ij}-\delta _{11}]\) and by the values of \(X_{2i}\) over time-window \([t_{ij}-\delta _{22}, t_{ij}-\delta _{21}]\).

Whether \(X_{1i}(s)\) for s in the time-window \([t_{ij}-\delta _{12}, t_{ij}-\delta _{11}]\) and \(X_{2i}(s)\) for s the time-window \([t_{ij}-\delta _{22}, t_{ij}-\delta _{21}]\) have an effect on the response \(Y_{ij}\) also depends on the coefficient functions \(\beta _1(t_{ij}-s,t_{ij})\) and \(\beta _2(t_{ij}-s,t_{ij})\) respectively. These functions represent the effect of the predictors on the outcome variable and can be equal to zero.

3 Estimation of the model parameters

Let \(\{B_{11}(s),..., B_{1K_1}(s)\}\) and \(\{B_{21}(s),..., B_{2K_2}(s)\}\) be two pre-specified basis functions on \(\Delta _1\) and \(\Delta _2\). Then the two-dimensional coefficient functions \(\beta _1(s, t)\) and \(\beta _2(s, t)\) are assumed to be represented as

$$\begin{aligned} \beta _1(s, t)=\sum _{k=1}^{K_1}B_{1k}(s)b_{1k}(t),\ s\in \Delta _1,\ t\in [0, 1] \end{aligned}$$
(3)

and

$$\begin{aligned} \beta _2(s, t)=\sum _{k=1}^{K_2}B_{2k}(s)b_{2k}(t),\ s\in \Delta _2,\ t\in [0, 1] \end{aligned}$$
(4)

respectively, where \(K_1\) and \(K_2\) capture the resolution of the fit and should be chosen accordingly and \(b_{1k}(t)\) and \(b_{2k}(t)\) are the unknown time-varying coefficient functions defined on [0, 1]. Various basis functions such as Fourier, B-spline, wavelet basis can be used depending on the specific features of the coefficient functions.

Substituting (3) and (4) into equation (1), we have

$$\begin{aligned} Y_{ij} =&\beta _0(t_{ij}) +\sum _{k=1}^{K_1}b_{1k}(t_{ij})\int _{\delta _{11}}^{\delta _{12}}B_{1k}(s)X_{1i}(t_{ij}-s)ds\\ \nonumber&\qquad\;\;\,+\sum _{k=1}^{K_2}b_{2k}(t_{ij})\int _{\delta _{21}}^{\delta _{22}}B_{2k}(s)X_{2i}(t_{ij}-s)ds +e_{ij}\\ \nonumber =:&\beta _0(t_{ij})+\sum _{k=1}^{K_1}b_{1k}(t_{ij})\tilde{X}_{1ik}(t_{ij}) +\sum _{k=1}^{K_2}b_{2k}(t_{ij})\tilde{X}_{2ik}(t_{ij})+e_{ij}\\ \nonumber =&\beta _0(t_{ij})+{\textbf{b}}_1^T(t_{ij})\tilde{{\textbf{X}}}_{1i}(t_{ij}) +{\textbf{b}}_2^T(t_{ij})\tilde{{\textbf{X}}}_{2i}(t_{ij})+e_{ij}, \nonumber \end{aligned}$$
(5)

where \(\tilde{X}_{1ik}(t_{ij})=\int _{\delta _{11}}^{\delta _{12}}B_{1k}(s)X_{1i}(t_{ij}-s)ds\), \(\tilde{X}_{2ik}(t_{ij})=\int _{\delta _{21}}^{\delta _{22}}B_{2k}(s)X_{2i}(t_{ij}-s)ds\), \({\textbf{b}}_1(t_{ij})=(b_{11}(t_{ij}),..., b_{1K_1}(t_{ij}))^T\), \({\textbf{b}}_2(t_{ij})=(b_{21}(t_{ij}),..., b_{2K_2}(t_{ij}))^T\), \(\tilde{{\textbf{X}}}_{1i}(t_{ij})=({\tilde{X}}_{1i1}(t_{ij}),..., {\tilde{X}}_{1iK_1}(t_{ij}))^T\), and \(\tilde{{\textbf{X}}}_{2i}(t_{ij})=({\tilde{X}}_{2i1}(t_{ij}),..., {\tilde{X}}_{2iK_2}(t_{ij}))^T\). Note that the observed times \(t_{ij}\) depend on subject i. Model (1) reduces to a varying coefficient model with \(K_1\) induced predictors \(\tilde{X}_{1ik}(t_{ij})\) and \(K_2\) induced predictors \(\tilde{X}_{2ik}(t_{ij})\) which can be regarded as realizations of \(\tilde{X}_{1k}(t)\) and \(\tilde{X}_{1k}(t)\) at \(t_{ij}\) respectively.

At first, notice that \(\mu _{X_1}(t)=\mu _{X_2}(t)=0\) implies \(\beta _0(t_{ij})=E[Y_{ij}]\), so \(\beta _0(t)\) can be estimated by smoothing \(Y_{ij}\) via local smoothing method based on the pooled data, see for example Yao et al. [19]. We denote \(Y_{ij}-{\hat{\beta }}_0(t_{ij})\) by \(Y_{ij}\), where \({\hat{\beta }}_0(t_{ij})\) is an estimator of \(\beta _0(t)\) evaluated at time \(t_{ij}\).

In order to derive the estimator of \(\{b_{11}(t),...,b_{1K_1}(t)\}\) and \(\{b_{21}(t),...,b_{2K_2}(t)\}\), we assume \(t_{ij}=t_j^0\) to simplify the notation in this paragraph, i.e. the observation times for different subjects are the same. We then estimate \(b_{1k}(t_j^0)\) and \(b_{2k}(t_j^0)\) by minimizing the penalized sum of squared errors (PSSE):

$$\begin{aligned} PSSE_{b_1, b_2} =&\sum _{i=1}^n e_{ij}^2 +\rho _1\Vert {\textbf{b}}_1(t_j^0)\Vert ^2 +\rho _2\Vert {\textbf{b}}_2(t_j^0)\Vert ^2\\\nonumber =&\Vert {\textbf{Y}}_{j}-\tilde{{\textbf{X}}}_{1}(t_j^0){\textbf{b}}_1(t_j^0) -\tilde{{\textbf{X}}}_{2}(t_j^0){\textbf{b}}_2(t_j^0)\Vert ^2\\\nonumber&+\rho _1\Vert {\textbf{b}}_1(t_j^0)\Vert ^2 +\rho _2\Vert {\textbf{b}}_2(t_j^0)\Vert ^2,\nonumber \end{aligned}$$
(6)

where \(\Vert \cdot \Vert\) is the Euclidean norm of a vector, \(\textbf{Y}_{j}=(Y_{1j},..., Y_{nj})^T\), \(\tilde{\textbf{X}}_{1}(t_j^0)=\left( \tilde{{\textbf{X}}}_{11}(t_j^0),..., \tilde{{\textbf{X}}}_{1n}(t_j^0)\right) ^T_{n\times K_1}\), \(\tilde{{\textbf{X}}}_{2}(t_j^0)=\left( \tilde{{\textbf{X}}}_{21}(t_j^0),..., \tilde{{\textbf{X}}}_{2n}(t_j^0)\right) ^T_{n\times K_2}\), \(\rho _1>0\) and \(\rho _2>0\) are the regularization parameters which are assumed to be time independent in order to reduce the high variability. The penalization does not only prevent over-fitting but also guarantees the invertibility of matrix while solving the minimization problem. Then the minimizer of (6) is

$$\begin{aligned} \begin{bmatrix} \hat{ {\textbf{b}}}_1(t_j^0) \\ \hat{ {\textbf{b}}}_2(t_j^0) \end{bmatrix} =\left( {\textbf{Z}}_{j}^T{\textbf{Z}}_{j} + \begin{bmatrix} \rho _1I_{K_1} &{} 0 \\ 0 &{} \rho _2I_{K_2} \end{bmatrix} \right) ^{-1} \left( {\textbf{Z}}_{j}^T{\textbf{Y}}_{j} \right) , \end{aligned}$$

where \(I_K\) is the \(K\times K\) identity matrix and

$$\begin{aligned} {\textbf{Z}}_{j} =\begin{bmatrix} {\tilde{X}}_{111}(t_j^0)&{}\cdots &{} {\tilde{X}}_{11K_1}(t_j^0) &{}{\tilde{X}}_{211}(t_j^0)&{}\cdots &{} {\tilde{X}}_{21K_2}(t_j^0) \\ \vdots &{} &{} \vdots &{}\vdots &{} &{}\vdots \\ {\tilde{X}}_{1n1}(t_j^0)&{}\cdots &{} {\tilde{X}}_{1nK_1}(t_j^0) &{}{\tilde{X}}_{2n1}(t_j^0)&{}\cdots &{} {\tilde{X}}_{2nK_2}(t_j^0) \end{bmatrix}. \end{aligned}$$

Notice that when \(t_{ij} \ne t_j^0\), \(\hat{ {\textbf{b}}}_1(t_j^0)\) needs to be replaced by \(\hat{{\textbf{b}}}_1(t_{ij})\) and \(\textbf{Z}_{j}\) can be adapted correspondingly by replacing \(t_j^0\) by \(t_{ij}\) and the formulas in this paragraph hold.

Therefore, for arbitrary \(t\in [0, 1]\), we have

$$\begin{aligned} \begin{bmatrix} \hat{ {\textbf{b}}}_1(t) \\ \hat{ {\textbf{b}}}_2(t) \end{bmatrix} =\left( \begin{bmatrix} \hat{{\textbf{C}}}_{11}(t) &{} \hat{{\textbf{C}}}_{12}(t) \\ \hat{{\textbf{C}}}_{21}(t) &{} \hat{{\textbf{C}}}_{22}(t) \end{bmatrix} + \begin{bmatrix} \frac{\rho _1}{n}I_{K_1} &{} 0 \\ 0 &{} \frac{\rho _2}{n}I_{K_2} \end{bmatrix} \right) ^{-1} \begin{bmatrix} \hat{{\textbf{C}}}_{1Y}(t) \\ \hat{{\textbf{C}}}_{2Y}(t) \end{bmatrix} \end{aligned}$$
(7)

where \(\hat{{\textbf{C}}}_{11}(t)=\left[ {\hat{C}}_{\tilde{X}_{1k}, \tilde{X}_{1\,l}}(t)\right] _{kl}\) is a \(K_1\times K_1\) matrix with \({\hat{C}}_{\tilde{X}_{1k}, \tilde{X}_{1l}}(t)\) an estimator of \(C_{\tilde{X}_{1k}, \tilde{X}_{1\,l}}(t)=cov\left( \tilde{X}_{1k}(t), \tilde{X}_{1\,l}(t)\right)\), \(\hat{{\textbf{C}}}_{12}(t)=\left[ {\hat{C}}_{\tilde{X}_{1k}, \tilde{X}_{2\,l}}(t)\right] _{kl}\) is a \(K_1\times K_2\) matrix with \({\hat{C}}_{\tilde{X}_{1k}, \tilde{X}_{2l}}(t)\) an estimator of \(C_{\tilde{X}_{1k}, \tilde{X}_{2\,l}}(t)=cov\left( \tilde{X}_{1k}(t), \tilde{X}_{2\,l}(t)\right)\), \(\hat{{\textbf{C}}}_{21}(t)=\left[ {\hat{C}}_{\tilde{X}_{2k}, \tilde{X}_{1\,l}}(t)\right] _{kl}\) is a \(K_2\times K_1\) matrix with \({\hat{C}}_{\tilde{X}_{2k}, \tilde{X}_{1l}}(t)\) an estimator of \(C_{\tilde{X}_{2k}, \tilde{X}_{1\,l}}(t)=cov\left( \tilde{X}_{2k}(t), \tilde{X}_{1\,l}(t)\right)\), \(\hat{{\textbf{C}}}_{22}(t)=\left[ {\hat{C}}_{\tilde{X}_{2k}, \tilde{X}_{2\,l}}(t)\right] _{kl}\) is a \(K_2\times K_2\) matrix with \({\hat{C}}_{\tilde{X}_{2k}, \tilde{X}_{2l}}(t)\) an estimator of \(C_{\tilde{X}_{2k}, \tilde{X}_{2\,l}}(t)=cov\left( \tilde{X}_{2k}(t), \tilde{X}_{2\,l}(t)\right)\), \(\hat{{\textbf{C}}}_{1Y}(t)=\left[ {\hat{C}}_{{\tilde{X}}_{11}, Y}(t),..., {\hat{C}}_{{\tilde{X}}_{1K_1}, Y}(t)\right] ^T\) is a vector and \({\hat{C}}_{{\tilde{X}}_{1l}, Y}(t)\) is an estimator of \(C_{{\tilde{X}}_{1\,l}, Y}(t)=cov\left( \tilde{X}_{1\,l}(t), Y(t)\right)\), and \(\hat{{\textbf{C}}}_{2Y}(t)=\left[ C_{{\tilde{X}}_{21}, Y}(t),..., C_{{\tilde{X}}_{2K_2}, Y}(t)\right] ^T\) is a vector and \({\hat{C}}_{{\tilde{X}}_{2l}, Y}\) is an estimator of \(C_{{\tilde{X}}_{2\,l}, Y}(t)=cov\left( \tilde{X}_{2\,l}(t), Y(t)\right)\).

To obtain the necessary quantities in (7), we have to consider the embedding covariance functions and their corresponding estimations. We only give the details for \(C_{\tilde{X}_{1k}, \tilde{X}_{1l}}(t)\) here and for the other covariance functions see Appendix A. For \(C_{\tilde{X}_{1k}, \tilde{X}_{1\,l}}(t)\), we have

$$\begin{aligned} C_{\tilde{X}_{1k}, \tilde{X}_{1l}}(t)&=cov\left( \tilde{X}_{1k}(t), \tilde{X}_{1l}(t)\right) \\&=\int _{\delta _{11}}^{\delta _{12}}\int _{\delta _{11}}^{\delta _{12}} B_{1k}(s)B_{1l}(u)E[X_1(t-s)X_1(t-u)]duds\\&=\int _{\delta _{11}}^{\delta _{12}}\int _{\delta _{11}}^{\delta _{12}} B_{1k}(s)B_{1l}(u)C_{X_1}(t-s, t-u)duds, \end{aligned}$$

where \(C_{X_1}(s, u)\) is the covariance between \(X_1(s)\) and \(X_1(u)\). Since predictor \(X_1\) is densely observed, \(C_{X_1}(s, u)\) can be estimated by bivariate kernel smoothing, see [2]:

$$\begin{aligned} {\hat{C}}_{X_1}(s, u)= \frac{1}{(m_{1}b)^2}\sum _{j,k=1}^{m_{1}} K\left( \frac{s-s_{1j}}{b}, \frac{u-s_{1k}}{b}\right) \frac{1}{n}\sum _{i=1}^n W_{1ij}W_{1ik}, \end{aligned}$$

where b is a bandwidth, \(m_1\) is the number of observations of the dense and regular predictor \(X_1\) for each subject, and K is a bivariate kernel function.

Once \({\hat{\textbf{b}}}_1(t)\) and \({\hat{\textbf{b}}}_2(t)\) are obtained (for given lags \(\delta\)’s and regularization parameters \(\rho\)’s), we can estimate coefficient functions by

$$\begin{aligned} {\hat{\beta }}_1(s, t)=\sum _{k=1}^{K_1}B_{1k}(s){\hat{b}}_{1k}(t),\ s\in \Delta _1,\ t\in [0, 1] \end{aligned}$$

and

$$\begin{aligned} {\hat{\beta }}_2(s, t)=\sum _{k=1}^{K_2}B_{2k}(s){\hat{b}}_{2k}(t),\ s\in \Delta _2,\ t\in [0, 1]. \end{aligned}$$

Theorem 1

Under assumptions (A1–A5) and (B1–B3) specified in Appendix B, denote \(I_t=[\max \{\delta _{12}, \delta _{22}\}, 1]\),

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{s, t\in \Delta _1\times I_t}|{\hat{\beta }}_1(s, t)-\beta _1(s, t)| =0\quad {\text {in probability}}. \\ \lim _{n\rightarrow \infty }\sup _{s, t\in \Delta _2\times I_t}|{\hat{\beta }}_2(s, t)-\beta _2(s, t)| =0\quad {\text {in probability}}. \end{aligned}$$

Remark 1

Here the convergence rate depends on the tuning parameters, e.g. the bandwidth b, used in estimating the corresponding covariance structures (including auto-covariance structure of the predictor processes and cross-covariance structure between the predictor processes and response process). Note that the bandwidth b should be chosen in such a way that there are a sufficient number of observations in the interval to estimate the covariance. For details of the proof, see Appendix C.

Remark 2

If the mean \(\mu _{X_1}(t)\) and \(\mu _{X_2}(t)\) of the predictors \(X_1(t)\) and \(X_2(t)\) are not equal to 0, they can be estimated nonparametrically. Specifically, for the sparse and irregular longitudinal predictor, one can use the method proposed by Yao et al. [19, 20] or by [9, 14]. For the dense and regular functional predictor, methods such as kernel estimation proposed by [2] can be used.

4 Prediction

Suppose we observe a new discrete response curve \({\textbf{Y}}^*_j=(Y^*(t_1^*),...Y^*(t_{m^*}^*))\), discrete dense predictor trajectory \({\textbf{W}}_1^*=(W_1^*(s_{11}),..., W_1^*(s_{1m_{1}}))^T\) and discrete sparse predictor trajectory \({\textbf{W}}_2^*=(W_2^*(s_{21}^*),...W_2^*(s_{2m_{2}}^*))^T\). From the original model (1), the predicted response curve is

$$\begin{aligned} E\left[ Y^*(t)|X_1^*, X_2^*\right] =\beta _0(t)&+\int _{\delta _{11}}^{\delta _{12}}\beta _1(s,t)X_1^*(t-s)ds\nonumber \\&+\int _{\delta _{21}}^{\delta _{22}}\beta _2(s,t)X_2^*(t-s)ds. \end{aligned}$$
(8)

However, the lags \(\delta _{11}, \delta _{12}, \delta _{21}, \delta _{22}\) and regularization parameters \(\rho _1\) and \(\rho _2\) have to be determined and the functional representation of the predictor trajectories \(X_1^*(s)\) and \(X_2^*(s)\) have to be recovered from data.

For \(X_1^*(s)\), it can be easily recovered by kernel smoothing, since the sampling is dense. However for \(X_2^*(s)\), since the sampling is sparse and irregular, we use functional principal component analysis (FPCA) and approximate the curve by a limited number of functional principal components. We assume \(X_2^*(s)\sim X_2(s)\in L^2[0, 1]\) and \(E[X_2(s)]=0\). Denote the covariance of \(X_2(s)\) by \(C_{X_2}(s, u)=cov(X_2(s), X_2(u))\), then the Mercer’s theorem gives the following spectral decomposition of the covariance

$$\begin{aligned} C_{X_2}(s, t)=\sum _{l=1}^\infty \lambda _l\phi _l(s)\phi _l(u) \end{aligned}$$

where \(\lambda _1\ge \lambda _2\ge ...\ge 0\) are eigenvalues and \(\phi _l\) are orthonormal eigenfunctions. By Karhunen-Loève (KL) expansion, \(X_2^*(s)\) can be represented as

$$\begin{aligned} X_2^*(s)=\sum _{l=1}^\infty \xi _l^*\phi _l(s) \end{aligned}$$

where \(\xi _l^*=\int _0^1 X_2^*(s)\phi _l(s)ds\) are the functional principal component scores and are uncorrelated random variables with mean 0 and variance \(\lambda _l\). The truncated formula for \(X_2^*(s)\) is:

$$\begin{aligned} X_2^{*L}(s)=\sum _{l=1}^L\xi _l^*\phi _l(s). \end{aligned}$$

The covariance \(C_{X_2}(s, t)\) can be estimated as we discussed in last section and the eigenfunctions \(\phi _l\) can be estimated following the spectral decomposition of the estimated covariance. However, the scores \(\xi _l^*\) cannot be approximated by numerical integration which is used for dense functional data. In fact, under the Gaussian assumption, denote \(\varvec{\phi }_l=(\phi _l(s_{21}^*),...,\phi _l(s_{2m_{X_2}^*}^*))^T\), the best linear predictor for \(\xi _l^*\) is (see [11], Yao et al. [19, 20] or see the application in [10])):

$$\begin{aligned} {\tilde{\xi }}_l^*=\lambda _l\varvec{\phi }_l^T\Sigma ^{-1}{\textbf{W}}_2^* \end{aligned}$$

where \(\Sigma =var({\textbf{W}}_2^*)\). Then the estimate of \(\xi _l^*\) can be defined as

$$\begin{aligned} {\hat{\xi }}_l^*={\hat{\lambda }}_l\hat{\varvec{\phi }_l}^T{\hat{\Sigma }}^{-1}{\textbf{W}}_2^*. \end{aligned}$$

The number of eigenfunctions L can be selected to be the number of eigenfunctions that explain 99% of the variation. Once obtaining the estimation of eigenfunctions \(\phi _l\), scores \(\xi _{il}\) and L, \(X_2^*(s)\) can be recovered as

$$\begin{aligned} {\hat{X}}_2^*(s)=\sum _{l=1}^L{\hat{\xi }}_l^*{\hat{\phi }}_l^*(s). \end{aligned}$$

After plugging the functional representation of the predictor curves \({\hat{X}}_1^*(s)\) and \({\hat{X}}_2^*(s)\) into (8), we have

$$\begin{aligned} {\hat{Y}}_L^*(t)&=\int _{\delta _{11}}^{\delta _{12}}{\hat{\beta }}_1(s,t){\hat{X}}_1^*(t-s)ds +\int _{\delta _{21}}^{\delta _{22}}{\hat{\beta }}_2(s,t){\hat{X}}_2^*(t-s)ds \nonumber \\&=\int _{\delta _{11}}^{\delta _{12}}{\hat{\beta }}_1(s,t){\hat{X}}_1^*(t-s)ds +\int _{\delta _{21}}^{\delta _{22}}{\hat{\beta }}_2(s,t) \sum _{l=1}^L{\hat{\xi }}_l^*{\hat{\phi }}_l(t-s)ds. \end{aligned}$$
(9)

Define

$$\begin{aligned} {\tilde{Y}}^*(t)&=\int _{\delta _{11}}^{\delta _{12}}\beta _1(s,t)X_1^*(t-s)ds +\int _{\delta _{21}}^{\delta _{22}}\beta _2(s,t) \sum _{l=1}^\infty {\tilde{\xi }}_l^*\phi _l(t-s)ds. \end{aligned}$$

and

$$\begin{aligned} {\tilde{Y}}_L^*(t)&=\int _{\delta _{11}}^{\delta _{12}}\beta _1(s,t)X_1^*(t-s)ds +\int _{\delta _{21}}^{\delta _{22}}\beta _2(s,t) \sum _{l=1}^L{\tilde{\xi }}_l^*\phi _l(t-s)ds. \end{aligned}$$

Theorem 2

Under assumptions (A1–A5) and (B1–B3) in Appendix B, denote \(I_t=[\max \{\delta _{12}, \delta _{22}\}, 1]\), for all \(t\in I_t\), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }{\hat{Y}}_L^*(t)={\tilde{Y}}^*(t)\quad \text {in probability}. \end{aligned}$$

Remark 3

Note that the number of eigenfunctions L used in the KL expansion of the sparse and irregular predictor process is a function of sample size n and goes to infinity as n goes to infinity. For details of the proof, see Appendix C.

5 Computation of the Lags

The final task is to estimate the time lag \(\delta\)’s. For selecting \(\delta\)’s and \(\rho\)’s, we consider the normalized prediction error (NPE) criterion and the K-fold cross validation criterion. Specifically, for pre-specified \(\delta\) and \(\rho\), NPE in this situation is defined as

$$\begin{aligned} NPE\{(\delta , \rho )\} =\frac{1}{N}\sum _{i=1}^n\sum _{j=1}^{m_{Yi}} \frac{\left| {\hat{Y}}_{ij}-Y_{ij}\right| }{|Y_{ij}|} \end{aligned}$$
(10)

where \({\hat{Y}}_{ij}\) is the predicted value for the jth measurement on the ith response trajectory Y(t) obtained based on the pre-specified \(\delta\) and \(\rho\), \(N=\sum _{i=1}^nm_{Y_i}\). Similarly, for pre-specified \(\delta\) and \(\rho\), we define K-fold cross validation criterion as follows. We divide the dataset into K equal parts. For each \(k=1,..., K\), we fit the model to the other \(K-1\) parts, which gives the estimation of coefficient functions, and further gives the prediction \({\hat{Y}}^{-k}_{ij}\) in the kth part. Then the K-fold cross validation score is defined as,

$$\begin{aligned} CV\{(\delta , \rho )\} =\frac{1}{K}\sum _{k=1}^{K}\sum _{i\in k\text {th part} }\frac{1}{m_{Y_i}}\sum _{j=1}^{m_{Y_i}} \left( {\hat{Y}}^{-k}_{ij}-Y_{ij}\right) ^2. \end{aligned}$$
(11)

Similar criteria are considered in [8] and Pomann et al. [16].

Then \(\delta\)’s and \(\rho\)’s are chosen in a hierarchical manner. Let \(D_1\) and \(D_2\) be the sets of potential lags for the first and second predictor, i.e. \(\{(\delta _{11}, \delta _{12})\}\) and \(\{(\delta _{21}, \delta _{22})\}\), respectively. Let \(D_\rho\) be the set of potential regularization parameters \(\{(\rho _1, \rho _2)\}\). Firstly, for a fixed point of lags \(\delta ^0=\left( (\delta _{11}^0, \delta _{12}^0), (\delta _{21}^0, \delta _{22}^0)\right) \in D_1\times D_2\), we calculate the NPE values for all \(\rho \in D_\rho =\{(\rho _1, \rho _2)\}\), i.e. \(NPE\{(\delta ^0, \rho )\}_{\rho \in D_\rho }\). Then the \(\rho\) that achieves the smallest NPE value, i.e. \(\rho _{opt}(\delta ^0)=\min _{\rho \in D_\rho } NPE\{(\delta ^0, \rho )\}\), is chosen as the optimal \(\rho\) for the given fixed point of lags \(\delta ^0\). Secondly, we calculate the cross validation score \(CV\{(\delta ^0, \rho _{opt}(\delta ^0))\}\) for the given \(\delta ^0\) based on the \(\rho _{opt}(\delta ^0)\). At last, we repeat the above steps for all \(\delta \in D_1\times D_2\) and we obtain the cross validation score for all \(\delta \in D_1\times D_2\), \(CV\{(\delta , \rho )\}_{\delta \in D_1\times D_2, \rho _\delta \in D_\rho }\). Then, the optimal \(\delta\) is chosen to be the one with the smallest cross validation score.

Remark 4

Since for our simulation study and real data analysis, \(Y_{ij}\), for j such that \(t_j>\max \{\delta _{12},\delta _{22}\}\), is larger than 0.5, we define NPE as in formula (9). If \(\min \{|Y_{ij}|\}\) is very close to zero, one may define NPE as

$$\begin{aligned} NPE\{(\delta , \rho )\} =\frac{1}{N}\sum _{i=1}^n \frac{\sum _{j=1}^{m_{Yi}}\left| {\hat{Y}}_{ij}-Y_{ij}\right| }{\sum _{j=1}^{m_{Yi}}|Y_{ij}|}. \end{aligned}$$

In our simulation and real data analysis, the two NPE criteria give the same results.

Remark 5

Here, for pre-specified \(\delta\), we first chose \(\rho\) from a set of \(\rho\) values using the NPE criterion. Then we chose \(\delta\) based on the CV criterion from a set of \(\delta\) values. The advantage is that the computationally faster NPE criterion for choosing \(\rho\) and the more refined CV criterion for selecting the \(\delta\) are used.

6 Simulations

We study efficiency of the NPE criterion for selecting the time lags \(\delta\)’s and regularization parameters \(\rho\)’s.

For \(n=50,\ 100,\ 150,\ 200\) subjects, we first generate two predictor curves \(X_1(t)\) and \(X_2(t)\) on dense and equally spaced time points over [0, 1], i.e. \(\{j/99, j=0,..., 99\}\) and then accordingly generate the response curve Y(t) at time points \(\{j/99, j=0,..., 99\}\). The number of measurements made on the ith response \(m_{Yi}\) is randomly selected from 20 to 50, the number of measurements made on the ith predictor \(m_{1i}\) is 100 and the number of measurements made on the ith predictor \(m_{2i}\) is randomly selected from 30 to 50.

Define \(X_{1i}(t)=\xi _{i1}\sin (2\pi t)+\xi _{i2}t^2\) with \(\xi _{i1}\overset{iid}{\sim }N(0, 1)\) and \(\xi _{i2}\overset{iid}{\sim }N(0, 1)\), \(X_{2i}(t)=\zeta _i\cos (2\pi t)\) with \(\zeta _i\overset{iid}{\sim }N(0, 1)\). We take the same time lags for both \(X_1\) and \(X_2\), i.e. \(\delta _{11}=\delta _{21}=0.1\), \(\delta _{12}=\delta _{22}=0.4\). For coefficient functions, we choose \(\beta _0(t)=t+t^{1/5}\), \(\beta _1(s, t)=\sin (2\pi s)\cos (\pi t),\ t\in [0, 1],\ s\in [0.1, 0.4]\), \(\beta _2(s, t)=\sin (4\pi s)\cos (2\pi t),\ t\in [0, 1],\ s\in [0.1, 0.4]\). The measurement errors are taken to be independent normal with signal to noise ratio 20 for the predictors and response. Figure 1 shows the simulated data for the first replicate with \(n=100\).

Fig. 1
figure 1

Simulated data of first replicate: The left plot shows the discrete noisy observations of the first predictor which are observed for a dense and regular grid. The middle plot shows the discrete noisy observation of the second predictor which are observed for a sparse and irregular grid. The right plot shows the discrete noisy observations of the response which are sparsely and irregularly observed

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Since we could not assume any prior on the coefficients and B-spline basis are computationally fast and have good properties, we use B-spline functions for the estimation and prediction. With respect to \(K_1\) and \(K_2\), we use B-spline functions of degree 4 with 10 equally spaced interior knots over \(\Delta _1\) and \(\Delta _2\) (number of bases is 14). For details on B-spline basis, see for example [4]. The number of functional principal components is chosen such that 99% of variation is kept. The penalized parameters \(\rho _1\) and \(\rho _2\) are chosen on the dense grid of \(\rho _1,\ \rho _2 \in [10^{-5}, 10^{-2}]\) with 20 equally-spaced points in both directions. We use NPE criterion and 10-fold cross validation criterion to determine the regularization parameters and the lags. Notice that in order to check the estimation performance, the estimation procedure is done under the correct lags, i.e. \(\delta _{11}=\delta _{21}=0.1\), \(\delta _{12}=\delta _{22}=0.4\). Figure 2 shows the result of one simulation, where \(\rho _1\) is chosen as \(4.28\times 10^{-4}\), \(\rho _2\) is chosen as \(8.86\times 10^{-4}\) and the corresponding NPE is \(1.95\times 10^{-2}\). From Fig. 2, we conclude that our model successfully reveals the structure of coefficient functions.

Fig. 2
figure 2

The estimated functions for the first replicate: The left upper corner plot is the true \(\beta _1\): abscissa is s with the domain [0.1, 0.4], ordinate is the values of \(\beta _1\), there are 60 curves and they are \(\beta _1(s, t_j)\) for \(t_j=j/99,\ j=40,...,99\). The second left upper plot is the estimation of \(\beta _1\). The third left upper plot is the contour line of the true \(\beta _1\). The fourth left upper plot is the contour line of the estimated \({\hat{\beta }}_1\). The bottom panel shows the true and estimated \(\beta _2\)

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Table 1 shows the asymptotic properties of our estimation. For different number of observations \(n=50,\ 100,\ 150,\ 200\), the NPE values are shown and also the estimation is based on the correct lags. As we can see, the NPE decreases as n increases which is expected based on Theorem 1.

Table 1 NPEs based on correct lags
Full size table

For evaluating the performance of our model on selecting the effect lags, the \(\rho\)’s are determined based on the NPE criterion and the \(\delta\)’s are determined based on 10-fold cross-validation score. Since the true \(\delta _{11}=\delta _{21}=0.1\) and \(\delta _{12}=\delta _{22}=0.4\), in order to save computational time, we fix the ending point i.e. \(\delta _{11}=\delta _{21}=0.1\), use the same starting point, i.e. \(\delta _{12}=\delta _{22}\) and search over \(\{0.3, 0.4, 0.5\}\). That is we have three combinations but there is only one correct combination. Our model has 65 correct choices out of 100 simulations.

7 Data Analysis

Chronic Obstructive Pulmonary Disease (COPD) continues to be one of the leading causes of morbidity and mortality in the world and a burden on many national health systems [13]. Many of the risk factors associated with the disease are controlled by each patient’s personal decisions, an example of such a risk factor is how much a patient smokes (see [1]). Other risk factors are more complicated and are determined by socioeconomic factors such as where a patient lives (for example the presence of sport facilities and the amount of air pollution in the neighbourhood) and their access to healthcare. These factors are more likely to be effected by government and policy. In 2003, the US National Heart, Lung, and Blood Institute estimated that the total costs (direct and indirect) of COPD was approximately $32.1 billion for the year (see [13]). Modelling COPD in a predictive way could be a useful asset in allocating health resources to better deal with predicted spikes in COPD exacerbations and can be used to identify areas in which preventative measures can be taken to better COPD health outcomes. Both of these approaches could potentially make significant headway in reducing the morbidity related to COPD and the economic costs of the disease.

Air quality is a known factor that affects a person’s health and quality of life. Previous research suggests a link between air quality and COPD hospitalisation (see [21]). More specifically, a study looking at the effect of pollutants on 94 COPD sufferers living in London found that a rise in NO2 concentration accounted for a 6% increase in the odds of a symptomatic fall in peak flow rate (see [15]). We aim to link heterogeneous data sets regarding NO2 concentration and exercise intensity to model COPD hospital admissions for the city of Leeds.

Healthcare, air quality and lifestyle datasets are typically a mixture of static, temporal, dense and sparse data. The pollution data is usually fairly regular over time (the data we present is collected every 15 min), but there is only a limited number of monitoring sites in a city meaning that the data are spatially sparse. Physical activity data can come from a variety of sources stretching from a small group of volunteers who allow their daily movements to be tracked, to a large sample based on information from logged visits to local gym facilities.

7.1 Pollution Data

Pollution data was taken from 10 automatic monitoring points from across Leeds taking measurements of NO2 concentration every 15 min. The dataset from 2013 to 2018 can be found Ratified air quality - nitrogen dioxide. The location of the 10 monitors is shown in Fig. 6 and a snapshot of the data is shown in Table 7 in Appendix D. A mean value at each collection point was calculated daily for all years. Some collection points reported N/As for long periods of time making the data sparse at some points in time.

The 10 collection points were mapped onto a 861 point grid of Leeds. The location of the grid is shown in Fig. 8 in Appendix D. Using the Krige function in R, an interpolation over this grid was carried out using the inverse distance weighted interpolation method with a power of 2 (default). N/As were excluded. The points of the grid were then allocated to postcode district by matching them to the closest postcode (Euclidean distance). This was done using the website http://www.geodojo.net/uk/converter/. The data was then aggregated down to district level by taking the mean of all data points within a district. From this the daily NO2 for each postcode district in Leeds is calculated along with the standard deviation between all the points that lie within the district boundaries, see Table 9 in Appendix D.

The daily mean NO2 concentrations from 2013 to 2018 for the 18 postcode districts in Leeds and their kernel smoothing with data-adaptive local plug-in bandwidth selection curves are shown in the left and right panel of Fig. 3 respectively. As expected a yearly seasonal effect for daily mean NO2 is observed with peaks in the winter.

Fig. 3
figure 3

Raw and smoothed daily mean NO2 concentration of 18 postcode districts

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7.2 Physical Activity Data

Raw physical activity data was obtained from a local authority concerning use of their gym and sports facilities for the period 2013 to 2018. These data were given per postcode sector level and we further aggregated these to postcode district level (e.g. LS1).

To standardise the physical activity data, the population size per district was used, i.e. the counts per postcode district were divided by the population size of the district. For the population size, we used mid year population estimates per lower layer super output area (LSOA) which were obtained from the Office for National Statistics (ONS). These were converted to postcode district by using a postcode lookup which maps postcodes onto a LSOA. When the LSOA covers multiple postcode districts, the LSOA population was split equally between them.

The physical activity curves for the 18 postcode districts in Leeds are given in Fig. 4. These curves show little variation in time.

Fig. 4
figure 4

Raw physical activity data for 18 postcode districts around Leeds. Left panel shows the standardised number of individuals using gym facilities for 18 postcode districts across Leeds from 2013 to 2018. Right panel shows the number of visits to gym facilities per person per week for 18 postcode districts across Leeds from 2013 to 2018

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7.3 COPD Data

Temporal data on hospital admissions due to COPD in Leeds was provided by the Leeds Teaching Hospitals NHS Trust. Raw data include admission date, age, disease, unique ID, sex, and area by postcode information. We removed records of subjects with ages that were not numbers. We restricted the dataset to ages above 20 years and to the time window of 2013 to 2018. The obtained dataset comprised 7944 COPD hospital admissions.

The data was then grouped by day. To standardise, the expected count of hospital admissions \(E_i\) was calculated. We divided the total number of counts over the whole 6 year period by the number of days in this period. Then to take the size of the population into account, again, the mid year population estimates obtained from the ONS at LSOA level were used (see above). Thus, for each district i, \(i\in \{1, 2,..., 18\}\), with a population size \(P_i\), and an expected admission count \(E_i\), the standardised count is defined as

$$\begin{aligned} Standardised \ Count_i= \frac{Count_i}{P_iE_i}, \end{aligned}$$

where \(Count_i\) is the weekly admission count in the ith district. The weekly admissions after standardisation based on district population (per 10,000) are shown in the left panel of Fig. 5 and the corresponding kernel smoothing with data-adaptive local plug-in bandwidth selection curves [3, 6] are given in the right figure. It appears that there is a yearly seasonality phenomenon with peaks every late winter and early spring.

Fig. 5
figure 5

Raw and smoothed weekly standardised COPD hospital admissions in 10,000. The left panel shows the raw weekly standarised COPD hospital admissions in 18 postcode districts in Leeds from 1st January 2013 to 31st December 2018. The right panel shows the corresponding smoothing curves that are estimated through kernel smoothing with data-adaptive local plug-in bandwidth selection

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7.4 Data Analysis Results

As a preliminary analysis, we first fitted a standard linear regression model with average standardised COPD hospital admissions Y as response variable and average standardised weekly number of gym visits and average daily NO2 value as covariates. Both covariates appeared to have a statistically significant effect on the outcome. The estimate of \(\beta _1\) is \(-2.86\) with p value of 0.003 and the estimates of \(\beta _2\) of 0.007 with p value of 0.003.

Next we are interested in the time aspect. Figure 3 and Fig. 5 show temporal fluctuations and a potential effect lag (delay) of the influence of NO2 concentrations on COPD hospital admissions. On the other hand, Fig. 4 shows that physical activity curves for 18 postcode districts across Leeds have little variation in time. Therefore, we did not consider physical activity as a covariate in a functional regression model.

In order to obtain a rough idea of the effect lag and duration of influence, we compared the position of local maximum of NO2 concentrations and hospital admissions. Specifically, based on the background, we restrict the starting point and ending point within one period (one year). And then we designed quite dense searching grid (with respect to the smoothness of the curves) for ending point as \(\{7, 14, 21, 24.5, 28, 31.5, 35\}\) (days) and that for starting point as \(\{84, 98, 112, 126, 140, 154, 168, 182, 196,199.5, 203, 206.5, 210, 213.5, 217, 220.5, 224, 238, 252, 266, 280\}\) (days). The mean functions of both predictor and outcome are estimated using kernel smoothing method as mentioned in Remark 2 on page 9. We used the same basis system and number of basis for estimation and prediction as in the simulation section, i.e. we use B-spline functions of degree 4 with 10 equally spaced interior knots (number of basis is 14). Here, we did 7*21 searches and the computation time is 4.6 h (on a standard laptop using Rstudio) which is affordable.

The optimal combination of ending point and starting point appears to be 24.5 and 203 days, i.e. 3.5 and 29 weeks based on leave-one-curve-out cross validation squared prediction error criterion. This means that \(t-203\) is the starting effective time and \(t-24.5\) is the ending effective time for NO2 concentration to have effect on COPD hospital admissions at time t. In other words, it takes around 3.5 weeks for NO2 concentration to have effect on COPD hospital admissions and this effect lasts around 25.5 weeks.

We did another 7*21 searches on a slightly different grid, i.e. ending points are \(\{7, 14, 21, 22.75, 26.25, 29.75, 33.25\}\) (days) and starting points are \(\{84, 98, 112, 126, 140, 154, 168, 182, 196, 197.75, 201.25, 204.75, 208.25, 211.75, 215.25, 218.75, 224, 238, 252, 266, 280\}\) (days). The results are only slightly different: the optimal combination of ending point and starting point is 26.25 and 210 days, i.e. 3.75 and 28.25 weeks based on leave-one-curve-out cross validation squared prediction error criterion.

8 Discussion

In this paper, we have developed a functional regression model that combines heterogeneous predictors, in particular sparse and dense functional predictors, with their effects on the response being restricted by time lags that in effect create an interval where the predictors affect the response. We prove the consistency of the coefficient functions of both the dense and sparse predictors. For prediction, we recovered new dense predictors using kernel smoothing, whereas we used FPCA for the sparse ones and proved the asymptotic property of the predictions. Finally, we minimized the normalized prediction error in order to find the optimal lags. Simulation studies are conducted to evaluate the model, the coefficient functions are estimated and the lags are determined with high accuracy.

The lags are estimated by using a grid search. For the simulation study, a grid search is adopted over several points that are close to the true effect lags. For real applications, we do not know the true effect lags and a larger grid is necessary or preliminary information can be used. One can also consider to run a preliminary search for each predictor separately in order to get a rough estimate of the lags and then design the grid accordingly. In our data application, we did 7*21 searches, and the computation time is 4.6 h (on a standard laptop using Rstudio). We did another search and obtained almost the same estimates. For these searches, the assumption is made that the true lag is included in the grid or close to one if the elements in the grid. Preliminary runs can provide some evidence that the true lags are included in the run. Moreover, if the final estimated lags are near the end points of the intervals/grid it is recommended that the grid is extended and a new fit is implemented.

Our model assumes that the lags do not change over time. Since the coefficient functions changes over time, the effective length of the historical interval of covariate values having an effect on the outcome can still be smaller for some periods. It might be interesting to investigate whether the length of the historical interval changes over time or is periodic. However such a model will be computationally intensive. Another extension is to use multivariate dense functional and sparse and irregular longitudinal predictors. As the estimator is derived from formula (6), if more than two predictors exist, one can estimate their covariance structure based on the type of the predictors (dense, sparse and irregular) then the corresponding \({\hat{b}}(t)\) can be obtained from formula (6). Note that this does not result in an over-parameterisation.

Our model is used to analyse the influence of daily mean NO2 concentrations on the COPD hospital admissions in 18 postcode districts in Leeds. It appeared that NO2 concentrations of 25.5 weeks ago still have an effect on COPD hospital admissions and that the NO2 concentrations of the last 3.5 weeks have not yet an effect. While it makes sense that historical NO2 concentrations have an effect and that recent values have not yet an effect on COPD hospital admissions, the estimates of the lags themselves need careful interpretation. First we do not provide standard errors. One way to obtain standard errors might be a bootstrap method, however such a method is very intensive computationally. Also the lags have to be interpreted together with the \(\beta\) functions which can vary over a time. Thus for a part of the time interval between lags, the effect of the covariate can be zero if the coefficient function is zero in this interval. Further, the models are fitted on aggregated data.

COPD hospital admissions are essentially Poisson distributed in each day, however in order to validate our methodology we aggregate weekly and standardised based on district populations and then consider these weekly aggregated data as continuous. Therefore, future research on general response is of great importance both in theory development and applications. Another extension is to model the spatial correlation of the COPD admissions. More efficient parameter estimates can be obtained by using this information.

Our model can be used to manage resources of hospitals, since it predicts the delay of hospital admissions after NO2 pollution as well as the duration. Another potential application is decision making around policies for improving the health in a city. For example should a city invest in more sport facilities or in a greener transport system? To answer such questions, the model should probably be extended with more factors. Moreover one might want to address possible missingness processes, for example the observed use of sport facilities is an underestimation since people might do sports using other facilities or in other districts. Recently, we developed methods for appropriate estimation of the mean function for temporal data subject to a detection limit [9].

To conclude we presented methods for functional analysis of temporal data where the effect of the temporal covariate might be delayed. We used this method for an interesting problem from urban analytics. We identified a delayed effect of 3.5 weeks for NO2 air pollution on COPD hospital admissions.