# Example on Simulated Data

    library(miniLNM)
    library(dplyr)
    set.seed(20240904)

This vignette illustrates use of the `lnm` function using a simulated dataset. First, we create some example data coming from a true LNM model.

    example_data <- lnm_data(N = 200, K = 20)
    xy <- bind_cols(example_data[c("X", "y")])

Next, we define the regression. We are using an extension of the formula interface that allows for multiple outcomes. This allows us to use it for a wider range of integration problems, like in our multimedia package for mediation analysis.

    fit <- lnm(starts_with("y") ~ starts_with("x"), xy, refresh = 0)
    #> Warning: Pareto k diagnostic value is 15.87. Resampling is disabled. Decreasing
    #> tol_rel_obj may help if variational algorithm has terminated prematurely.
    #> Otherwise consider using sampling instead.
    fit

    #> [LNM Model]
    #> Regression formula: y1 + y2 + y3 + y4 ... ~ x1 + x2 + x3 + x4 ...
    #> 5-dimensional input and 20-dimensional output
    #> First few entries of estimated regression coefficients:
    #> # A tibble: 5 × 19
    #>      y1    y2    y3    y4    y5    y6    y7    y8    y9   y10   y11   y12   y13
    #>
    #> 1  0.79  0.14  0.67  0.86  0.7   0.43  0.41  0.63  0.37  0.54  0.43  0.16  0.63
    #> 2  0.98  0.86  0.29  0.39  0.57  0.18  0.27  0.07  0.34  0.13  0.39  0.88  0.06
    #> 3  0.46  0.35  0.69  0.5   0.42  0.15  0.86  0.43  0.88  0.82  0.16  0.53  0
    #> 4  0.85  0.71  0.53  0.83  0.13  0.05  0.49  0.28  1     0.1   0.9   0.91  0.44
    #> 5  0.79  0.26  0.84  0.43  0.85  0.77  0.63  0.03  0.82  0.18  0.09  0.8   0.67
    #> # ℹ 6 more variables: y14 , y15 , y16 , y17 , y18 ,
    #> #   y19

Once we have estimated the model, we can use predict to get the fitted compositions on the training data. We can also draw new samples at different read depths and use `newdata` to sample at new design points.

    newx <- lnm_data()$X
    new_p_hat <- predict(fit, newx, depth = 300)

Let’s verify that the estimates are close to the truth. First we’ll get some fitted values.

    p_hat <- predict(fit, example_data$X)
    y_star <- sample(fit, newdata = example_data$X, depth = 1e4)

The block below compares some posterior predictive samples with the original data. The overlap between red and black points means that the simulated data are a close match to the original samples.

    true <- colMeans(example_data$y / rowSums(example_data$y))
    fitted <- colMeans(y_star / rowSums(y_star))
    plot(true, colMeans(p_hat), asp = 1)
    points(true, fitted, col = "red")
    abline(a = 0, b = 1)

We can also check the estimated coefficients. Everything is slightly shrunk towards zero, but this is what you would expect given the normal prior.

    plot(example_data$B, beta_mean(fit))
    abline(a = 0, b = 1)

    sessionInfo()

    #> R version 4.4.1 Patched (2024-08-21 r87049)
    #> Platform: aarch64-apple-darwin20
    #> Running under: macOS Sonoma 14.5
    #>
    #> Matrix products: default
    #> BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib
    #> LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0
    #>
    #> locale:
    #> [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    #>
    #> time zone: America/Chicago
    #> tzcode source: internal
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] dplyr_1.1.4   miniLNM_0.1.0
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] tensorA_0.36.2.1     sass_0.4.9           utf8_1.2.4
    #>  [4] generics_0.1.3       digest_0.6.37        magrittr_2.0.3
    #>  [7] evaluate_0.24.0      grid_4.4.1           fastmap_1.2.0
    #> [10] operator.tools_1.6.3 jsonlite_1.8.8       pkgbuild_1.4.4
    #> [13] backports_1.5.0      gridExtra_2.3        fansi_1.0.6
    #> [16] QuickJSR_1.3.1       scales_1.3.0         codetools_0.2-20
    #> [19] jquerylib_0.1.4      abind_1.4-5          cli_3.6.3
    #> [22] rlang_1.1.4          munsell_0.5.1        withr_3.0.1
    #> [25] cachem_1.1.0         yaml_2.3.10          StanHeaders_2.32.10
    #> [28] parallel_4.4.1       tools_4.4.1          rstan_2.32.6
    #> [31] inline_0.3.19        rstantools_2.4.0     checkmate_2.3.2
    #> [34] colorspace_2.1-1     ggplot2_3.5.1        curl_5.2.2
    #> [37] vctrs_0.6.5          posterior_1.6.0      R6_2.5.1
    #> [40] matrixStats_1.4.1    stats4_4.4.1         lifecycle_1.0.4
    #> [43] V8_5.0.0             pkgconfig_2.0.3      RcppParallel_5.1.9
    #> [46] pillar_1.9.0         bslib_0.8.0          gtable_0.3.5
    #> [49] loo_2.8.0            glue_1.7.0           Rcpp_1.0.13
    #> [52] highr_0.11           xfun_0.47            tibble_3.2.1
    #> [55] tidyselect_1.2.1     knitr_1.48           htmltools_0.5.8.1
    #> [58] rmarkdown_2.28       formula.tools_1.7.1  compiler_4.4.1
    #> [61] distributional_0.4.0