Effective sparse imputation of patient conditions in electronic medical records for emergency risk predictions

Abstract

Electronic medical records (EMRs) are being increasingly used for “risk” prediction. By “risks,” we denote outcomes such as emergency presentation, readmission, and the length of hospitalizations. However, EMR data analysis is complicated by missing entries. There are two reasons—the “primary reason for admission” is included in EMR, but the comorbidities (other chronic diseases) are left uncoded, and many zero values in the data are accurate, reflecting that a patient has not accessed medical facilities. A key challenge is to deal with the peculiarities of this data—unlike many other datasets, EMR is sparse, reflecting the fact that patients have some but not all diseases. We propose a novel model to fill-in these missing values and use the new representation for prediction of key hospital events. To “fill-in” missing values, we represent the feature-patient matrix as a product of two low-rank factors, preserving the sparsity property in the product. Intuitively, the product regularization allows sparse imputation of patient conditions reflecting common comorbidities across patients. We develop a scalable optimization algorithm based on Block coordinate descent method to find an optimal solution. We evaluate the proposed framework on two real-world EMR cohorts: Cancer (7000 admissions) and Acute Myocardial Infarction (2652 admissions). Our result shows that the AUC for 3-month emergency presentation prediction is improved significantly from (0.729 to 0.741) for Cancer data and (0.699 to 0.723) for AMI data. Similarly, AUC for 3-month emergency admission prediction from (0.730 to 0.752) for Cancer data and (0.682 to 0.724) for AMI data. We also extend the proposed method to a supervised model for predicting multiple related risk outcomes (e.g., emergency presentations and admissions in hospital over 3, 6, and 12 months period) in an integrated framework. The supervised model consistently outperforms state-of-the-art baseline methods.

Appendix

The optimization steps of supervised framework of modeling multiple risk outcomes: the formulation of the framework can be expressed as

$$\begin{aligned}&\min _{\varvec{\mathbf {U}},\varvec{\mathbf {V}},\varvec{\mathbf {E}},\varvec{\mathbf {Z}}\ge \mathbf {0}}||\varvec{\mathbf {E}}||_{F}^{2}+\frac{\lambda _{y}}{2}||\varvec{\mathbf {Y}}-\mathbf {H}^{T}\mathbf {G}||_{F}^{2}++\lambda _{uv}||\varvec{\mathbf {Z}}||_{1}+\frac{\lambda }{2}\bigg [||\varvec{\mathbf {U}}\Vert _{F}^{2}+||\varvec{\mathbf {V}}\Vert _{F}^{2}\bigg ]+\lambda _{w}||\mathbf {W}||_{*}\nonumber \\&\quad \hbox {s.t. }\varvec{\mathbf {X}}-\varvec{\mathbf {U}}\varvec{\mathbf {V}}=\varvec{\mathbf {E}},\varvec{\mathbf {U}}\varvec{\mathbf {V}}=\varvec{\mathbf {Z}},\mathbf {H}=\varvec{\mathbf {U}}\varvec{\mathbf {V}},\mathbf {G}=\mathbf {W}\end{aligned}$$

(27)

The Augmented Lagrangian of the above formulation can be expressed as

$$\begin{aligned} \mathcal {F}&=||\varvec{\mathbf {E}}||_{F}^{2}+\frac{\lambda _{y}}{2}||\varvec{\mathbf {Y}}-\mathbf {H}^{T}\mathbf {G}||_{F}^{2}+\lambda _{uv}||\varvec{\mathbf {Z}}||_{1}+\frac{\lambda }{2}\bigg [||\varvec{\mathbf {U}}||_{F}^{2}+||\varvec{\mathbf {V}}||_{F}^{2}\bigg ]++\text {tr}(\mathbf {L}^{T}(\varvec{\mathbf {U}}\varvec{\mathbf {V}}+\varvec{\mathbf {E}}\nonumber \\&\quad -\varvec{\mathbf {X}}))+\frac{\beta }{2}||\varvec{\mathbf {U}}\varvec{\mathbf {V}}+\varvec{\mathbf {E}}-\varvec{\mathbf {X}}||_{F}^{2}+\text {tr}(\mathbf {Q}^{T}(\varvec{\mathbf {U}}\varvec{\mathbf {V}}-\varvec{\mathbf {Z}}))+\frac{\beta }{2}||\varvec{\mathbf {U}}\varvec{\mathbf {V}}-\varvec{\mathbf {Z}}||_{F}^{2}\nonumber \\&\quad +\frac{\beta }{2}||\varvec{\mathbf {U}}\varvec{\mathbf {V}}-\mathbf {H}||_{F}^{2}+\text {tr}(\varvec{{{\varLambda }}}_{1}^{T}(\varvec{\mathbf {U}}\varvec{\mathbf {V}}-\mathbf {H}))+\frac{\beta }{2}||\mathbf {G}-\mathbf {W}||_{F}^{2}+\text {tr}(\varvec{{{\varLambda }}}_{2}^{T}(\mathbf {G}-\mathbf {W}))\nonumber \\&\quad +\lambda _{w}||\mathbf {W}||_{*} \end{aligned}$$

(28)

$$\begin{aligned} \hbox {s.t.}&\quad \varvec{\mathbf {Z}}\ge \mathbf {0} \end{aligned}$$

(29)

where $\mathbf {L}$,$\mathbf {Q}$,$\varvec{{{\varLambda }}}_{1}$ and $\varvec{{{\varLambda }}}_{2}$ are the Lagrange multipliers and $\beta$ is a parameter to improve the numerical stability of the algorithm. After some manipulation, the Eq. (29) becomes

$$\begin{aligned} \mathcal {F}&=||\varvec{\mathbf {E}}||_{F}^{2}+\frac{\lambda _{y}}{2}||\varvec{\mathbf {Y}}-\mathbf {H}^{T}\mathbf {G}||_{F}^{2}+\lambda _{uv}||\varvec{\mathbf {Z}}||_{1}+\frac{\lambda }{2}\bigg [||\varvec{\mathbf {U}}||_{F}^{2}+||\varvec{\mathbf {V}}||_{F}^{2}\bigg ]\\&\quad +\frac{\beta }{2}\bigg [||\varvec{\mathbf {U}}\varvec{\mathbf {V}}+\varvec{\mathbf {E}}-\varvec{\mathbf {X}}-\frac{\mathbf {L}}{\beta }||_{F}^{2}\\&\quad +||\varvec{\mathbf {U}}\varvec{\mathbf {V}}-\varvec{\mathbf {Z}}-\frac{\mathbf {Q}}{\beta }||_{F}^{2}+||\varvec{\mathbf {U}}\varvec{\mathbf {V}}-\mathbf {H}-\frac{\varvec{{{\varLambda }}}_{1}}{\beta }||_{F}^{2}+||\mathbf {G}-\mathbf {W}-\frac{\varvec{{{\varLambda }}}_{2}}{\beta }||_{F}^{2}\bigg ] \end{aligned}$$

Assume $\varvec{\mathbf {C}}_{1}^{t}=\varvec{\mathbf {X}}+\frac{\mathbf {L}^{t}}{\beta }-\varvec{\mathbf {S}}^{t}$ and $\varvec{\mathbf {C}}_{2}^{t}=\varvec{\mathbf {Z}}^{k}+\frac{\mathbf {Q}^{t}}{\beta }$ and $\varvec{\mathbf {C}}_{3}^{t}=\mathbf {G}^{t}+\frac{\varvec{\mathbf {E}}_{1}^{t}}{\beta }$, the updating steps of the variables are as following

$$\begin{aligned} \varvec{\mathbf {V}}^{t+1}&=\min _{\varvec{\mathbf {V}}}\frac{1}{2}||\varvec{\mathbf {U}}^{t}\varvec{\mathbf {V}}-\varvec{\mathbf {C}}_{1}^{t}||_{F}^{2}+\frac{1}{2}||\varvec{\mathbf {U}}^{t}\varvec{\mathbf {V}}-\varvec{\mathbf {C}}_{2}^{t}||_{F}^{2}+\frac{1}{2}||\varvec{\mathbf {U}}^{t}\varvec{\mathbf {V}}-\varvec{\mathbf {C}}_{3}^{t}||_{F}^{2}+\frac{\lambda }{2\beta }||\varvec{\mathbf {V}}||_{F}^{2}.\\ \varvec{\mathbf {U}}^{t+1}&=\min _{\varvec{\mathbf {U}}}\frac{1}{2}||\varvec{\mathbf {U}}\varvec{\mathbf {V}}^{t+1}-\varvec{\mathbf {C}}_{1}^{t}||_{F}^{2}+\frac{1}{2}||\varvec{\mathbf {U}}\varvec{\mathbf {V}}^{t+1}-\varvec{\mathbf {C}}_{2}^{t}||_{F}^{2}+\frac{1}{2}||\varvec{\mathbf {U}}\varvec{\mathbf {V}}^{t+1}-\varvec{\mathbf {C}}_{3}^{t}||_{F}^{2}+\frac{\lambda }{2\beta }||\varvec{\mathbf {U}}||_{F}^{2}\\ \varvec{\mathbf {Z}}^{t+1}&=\min _{\varvec{\mathbf {Z}}\ge \mathbf {0}}\lambda _{uv}||\varvec{\mathbf {Z}}||_{1}+\frac{\beta }{2}||\varvec{\mathbf {Z}}-(\varvec{\mathbf {U}}^{t+1}\varvec{\mathbf {V}}^{t+1}-\frac{\mathbf {Q}^{t}}{\beta })||_{F}^{2}\\ \varvec{\mathbf {E}}^{t+1}&=\min _{\varvec{\mathbf {S}}}||\varvec{\mathbf {S}}||_{F}^{2}+\frac{\beta }{2}||\varvec{\mathbf {S}}-(\varvec{\mathbf {X}}+\frac{\mathbf {L}^{t}}{\beta }-\varvec{\mathbf {U}}^{t+1}\varvec{\mathbf {V}}^{t+1})||_{F}^{2}\\ \mathbf {H}^{t+1}&=\min _{\mathbf {H}}\bigg [\frac{\lambda _{y}}{2}||\varvec{\mathbf {Y}}-\mathbf {H}^{T}\mathbf {G}^{t}||_{F}^{2}+||\varvec{\mathbf {U}}^{t+1}\varvec{\mathbf {V}}^{t+1}-\mathbf {H}-\frac{\varvec{{{\varLambda }}}_{1}^{t}}{\beta }||_{F}^{2}\bigg ]\\ \mathbf {G}^{t+1}&=\min _{\mathbf {G}}\bigg [\frac{\lambda _{y}}{2}||\varvec{\mathbf {Y}}-(\mathbf {H}^{t+1})^{T}\mathbf {G}||_{F}^{2}+||\mathbf {G}-\mathbf {W}^{t}-\frac{\varvec{{{\varLambda }}}_{2}^{t}}{\beta }||_{F}^{2}\bigg ]\\ \mathbf {W}^{t+1}&=\min _{\mathbf {W}}\lambda _{w}||\mathbf {W}||_{*}+||\mathbf {G}^{t+1}-\frac{\varvec{{{\varLambda }}}_{2}^{t}}{\beta }-\mathbf {W}||_{F}^{2}\\ \mathbf {L}^{t+1}&=\mathbf {L}^{t}+\beta (\varvec{\mathbf {X}}-\varvec{\mathbf {U}}^{t+1}\varvec{\mathbf {V}}^{t+1}-\varvec{\mathbf {S}}^{t+1})\\ \mathbf {Q}^{t+1}&=\mathbf {Q}^{t}+\beta (\varvec{\mathbf {Z}}^{t+1}-\varvec{\mathbf {U}}^{t+1}\varvec{\mathbf {V}}^{t+1})\\ \varvec{{{\varLambda }}}_{1}^{t+1}&=\varvec{{{\varLambda }}}_{1}^{t}+\beta (\varvec{\mathbf {U}}^{t+1}\varvec{\mathbf {V}}^{t+1}-\mathbf {H}^{t+1})\\ \varvec{{{\varLambda }}}_{2}^{t+1}&=\varvec{{{\varLambda }}}_{2}^{t}+\beta (\mathbf {G}^{t+1}-\mathbf {W}^{t+1}) \end{aligned}$$

As updating variables $\varvec{\mathbf {U}}$, $\varvec{\mathbf {V}}$, $\varvec{\mathbf {Z}}$, $\varvec{\mathbf {E}}$, $\mathbf {H}$, $\mathbf {G}$ and the Lagrange multipliers are similar to optimization methods of unsupervised framework (Sect. 3.3), for $\mathbf {W}$ with nuclear norm, the solution is given by

$$\begin{aligned}&\mathcal {P}(\mathbf {A},\lambda _{w})=\min _{\mathbf {W}}\bigg [||\mathbf {A}-\mathbf {W}||_{F}^{2}+\lambda _{w}||\mathbf {W}||_{*}\bigg ] \end{aligned}$$

where $\mathbf {A}=\mathbf {G}^{t+1}-\frac{\varvec{{{\varLambda }}}_{2}^{t}}{\beta }$. If $\varvec{\mathbf {U}}_{\ell }$,$\varvec{\mathbf {V}}_{r}$ are the left and right singular vectors of $\mathbf {A}$ and $\varvec{{{\varSigma }}}_{\mathbf {A}}$ is diagonal matrix of singular values, then $\mathcal {P}(\mathbf {A},\lambda _{w})=\varvec{\mathbf {U}}_{\ell }\varvec{{{\varSigma }}}_{(\mathbf {A},\lambda _{w})}\varvec{\mathbf {V}}_{\ell }^{T}$ where $diag(\varvec{{{\varSigma }}}_{(\mathbf {A},\lambda _{w})})=\max (0,diag(\varvec{{{\varSigma }}}_{\mathbf {A}}-\lambda _{w})).$