Fig. 4

spVC’s estimation and inference on simulated data. A spVC’s power in detecting spatially varying covariate effects. B True log-transformed expected expression, \(\mu (\varvec{x}_i, \varvec{s}_i)\ (i=1,\dots ,5000)\), of the four example genes. C Relative expression levels of the four example genes based on the simulated data. The simulated counts were normalized by library size, log-transformed, and then scaled by the min-max normalization to obtain the relative expression levels. D True spatial effects, \(\gamma _{0}(\varvec{s}_i)\ (i=1,\dots ,5000)\), of the four example genes. Data were scaled to the range of \([-1,1]\) for visualization. E spVC’s Estimated spatial effects, \(\hat{\gamma }_{0}(\varvec{s}_i)\ (i=1,\dots ,5000)\), of the four example genes. Data were scaled to the range of \([-1,1]\) for visualization. F (Top) True and estimated spatially varying effects of cell type 2’s proportion, \(\gamma _{2}(\varvec{s}_i)\ \text {and}\ \hat{\gamma }_{2}(\varvec{s}_i)\ (i=1,\dots ,5000)\), of Gene 4. (Bottom) True and estimated spatially varying effects of cell type 4’s proportion, \(\gamma _{4}(\varvec{s}_i)\ \text {and}\ \hat{\gamma }_{4}(\varvec{s}_i)\ (i=1,\dots ,5000)\), of Gene 4. Data were scaled to the range of \([-1,1]\) for visualization