--- title: "Sensitivity Analysis Framework for Bayesian Economic Disaggregation" author: "José Mauricio Gómez Julián" date: "`r Sys.Date()`" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Sensitivity Analysis Framework} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- > **How to read this manual.** > Sections 1–3 develop the **theory** (with equations); Sections 4–6 give **diagnostics** and **metrics**; Sections 7–8 provide **reproducible code**: a fast synthetic demo (evaluates on knit) and a full **real-data pipeline** (disabled by default for speed, enable by setting `eval=TRUE`). All code is consistent with the functions exported by the `BayesianDisaggregation` package. ```{r setup, include=TRUE} # Global chunk defaults knitr::opts_chunk$set( echo = TRUE, message = FALSE, warning = FALSE, fig.width = 9, fig.height = 6 ) # Libraries suppressPackageStartupMessages({ library(BayesianDisaggregation) library(dplyr) library(tidyr) library(ggplot2) library(readr) library(openxlsx) }) # Logging verbosity from the package log_enable("INFO") set.seed(2024) ``` # 1. Problem Setup We observe an **aggregate index** (e.g., CPI) by period $t=1,\dots,T$, and we want a **sectoral disaggregation** into $K$ components whose period-wise shares lie on the **unit simplex**: $$ W_t = (w_{t1},\dots,w_{tK}),\qquad w_{tk}\ge 0,\quad \sum_{k=1}^K w_{tk}=1. $$ We start with a **prior weight matrix** $P\in\mathbb{R}^{T\times K}$ (rows on the simplex), and construct a **likelihood of sectors** $L\in\Delta^{K-1}$ (a non-negative vector summing to one). A **temporal profile** then spreads $L$ to $LT\in\mathbb{R}^{T\times K}$. Finally, a **deterministic update rule** combines $P$ and $LT$ to obtain the posterior $W$. # 2. Constructing the Sectoral Likelihood $L$ ## 2.1 PCA/SVD of the centered prior matrix Let $P$ be validated (finite, non-negative, rows $\approx 1$; small deviations renormalized). We **center** columns over time: $$ X = P - \mathbf{1}\,\bar p^\top,\quad \bar p = \frac{1}{T}\sum_{t=1}^T P_{t\cdot}. $$ Compute the SVD $X = U\Sigma V^\top$. Let $v$ denote the **first right singular vector** (PC1 loadings). We map to non-negative salience via absolute values and normalize: $$ \ell_k = |v_k|,\qquad L_k = \frac{\ell_k}{\sum_j \ell_j}. $$ If PC1 is **degenerate** (near-zero variance or identical columns), we fall back to **column means** of $P$ (renormalized). This is implemented in: ```{r L_from_P, include=TRUE} # Example call (internals are in the package): # L <- compute_L_from_P(P) ``` **Diagnostics attached to `L`:** attributes `"pc1_loadings"`, `"explained_var"`, and `"fallback"`. ## 2.2 Temporal spreading of $L$ We create a time-varying matrix $LT$ by applying a non-negative weight profile $w_t$ and row-renormalizing: $$ LT_{t,k} \propto w_t L_k,\qquad \sum_k LT_{t,k}=1. $$ Built-in **patterns**: * `constant`: $w_t=1$ * `recent`: linearly increasing in $t$ (more weight to recent periods) * `linear`: affine ramp between endpoints * `bell`: symmetric Gaussian-like bump around $T/2$ ```{r spread-L, include=TRUE} # Example call: # LT <- spread_likelihood(L, T_periods = nrow(P), pattern = "recent") ``` # 3. Posterior Updating Rules (Deterministic, MCMC-free) Given $P$ and $LT$ (both row-wise on the simplex), we define four deterministic updates: * **Weighted average** (mixing parameter $\lambda\in[0,1]$): $$ W = \mathsf{norm}_1\{\lambda P + (1-\lambda)LT\}. $$ * **Multiplicative** (elementwise product with re-normalization): $$ W = \mathsf{norm}_1\{P\odot LT\}. $$ * **Dirichlet mean** (analytical conjugacy, $\gamma>0$, smaller $\gamma\Rightarrow$ sharper): $$ \alpha_{\text{post}} = \frac{P}{\gamma} + \frac{LT}{\gamma},\qquad W = \frac{\alpha_{\text{post}}}{\mathbf{1}^\top\alpha_{\text{post}}}. $$ * **Adaptive** (sector-wise mixing by prior volatility): $$ \phi_k=\min\!\Big(\frac{\sigma_k}{\bar\sigma},\,0.8\Big),\quad W_{t\cdot}=\mathsf{norm}_1\{(1-\phi)\odot P_{t\cdot} + \phi\odot LT_{t\cdot}\}. $$ All are exposed in the package: ```{r posteriors, include=TRUE} # posterior_weighted(P, LT, lambda = 0.7) # posterior_multiplicative(P, LT) # posterior_dirichlet(P, LT, gamma = 0.1) # posterior_adaptive(P, LT) ``` # 4. Coherence, Stability, and Interpretability ## 4.1 Coherence with respect to $L$ Define prior/posterior **temporal means**: $$ \bar p = \frac{1}{T}\sum_t P_{t\cdot},\qquad \bar w = \frac{1}{T}\sum_t W_{t\cdot}. $$ Let $\rho(\cdot,\cdot)$ be a **robust correlation** (max of |Pearson| and |Spearman|). The **coherence** scales the **increment** $\Delta\rho=\max(0,\rho(\bar w,L)-\rho(\bar p,L))$: $$ \text{coherence}=\min\{1,\ \text{const} + \text{mult}\cdot\Delta\rho\}. $$ ## 4.2 Numerical and temporal stability * **Numerical stability (exponential penalty)** on row-sum deviation and negatives: $$ S_{\text{num}}=\exp\{-a\cdot\overline{|\sum_k W_{tk}-1|} - b\cdot \#(W<0)\}. $$ * **Temporal stability** via average $|\Delta|$ (lower variation $\Rightarrow$ higher score): $$ S_{\text{tmp}} = \frac{1}{1+\kappa\cdot \overline{|\Delta W|}},\quad \overline{|\Delta W|}=\frac{1}{K}\sum_k \frac{1}{T-1}\sum_{t}|W_{t+1,k}-W_{t,k}|. $$ * **Composite stability** (default weights 60% numeric, 40% temporal): $$ S_{\text{comp}} = 0.6\,S_{\text{num}} + 0.4\,S_{\text{tmp}}. $$ The package functions: ```{r metrics-fns, include=TRUE} # coherence_score(P, W, L, mult = 3.0, const = 0.5) # numerical_stability_exp(W, a = 1000, b = 10) # temporal_stability(W, kappa = 50) # stability_composite(W, a = 1000, b = 10, kappa = 50) ``` ## 4.3 Interpretability Two principles: 1. **Preservation** of the sectoral structure (correlation between $\bar p$ and $\bar w$); 2. **Plausibility** of average sector changes (penalize extreme relative shifts). Implementation: $$ \text{pres}=\max\{0,\rho(\bar p,\bar w)\},\qquad r_k=\frac{|\,\bar w_k-\bar p_k\,|}{\bar p_k+\epsilon},\quad \text{plaus}= \frac{1}{1+2\cdot Q_{0.9}(r_k)}. $$ Then $\text{interp}=0.6\,\text{pres}+0.4\,\text{plaus}$. ```{r interp-fn, include=TRUE} # interpretability_score(P, W, use_q90 = TRUE) ``` # 5. End-to-End API (`bayesian_disaggregate`) The convenience pipeline: 1. `read_cpi()` and `read_weights_matrix()` (Excel) 2. `compute_L_from_P(P)` and `spread_likelihood(L, T, pattern)` 3. posterior rule (`weighted` / `multiplicative` / `dirichlet` / `adaptive`) 4. metrics: coherence, stability (composite), interpretability, efficiency (heuristic), composite score 5. export helpers: `save_results()` and a one-file workbook for “best” config ```{r api, include=TRUE} # Example signature (see Section 8 for real data): # bayesian_disaggregate(path_cpi, path_weights, # method = c("weighted","multiplicative","dirichlet","adaptive"), # lambda = 0.7, gamma = 0.1, # coh_mult = 3.0, coh_const = 0.5, # stab_a = 1000, stab_b = 10, stab_kappa = 50, # likelihood_pattern = "recent") ``` # 6. Interpreting Key Visualizations * **Heatmap of posterior $W$**: each **cell** is a sector share in a year; **rows** are years, **columns** are sectors. *Read it as*: darker tiles = larger sector share; **horizontal smoothness** indicates temporal stability; **vertical patterns** (bands) show persistent sectoral importance. * **Top-sectors lines**: for the most relevant sectors by average share, **lines** track the sector’s share over time. *Read it as*: consistent levels = stability; trend changes coincide with macro structure shifts. * **Sectoral CPI sheet**: $\hat Y_{t,k} = \text{CPI}_t \times W_{t,k}$. *Read it as*: dollarized (or index-scaled) decomposition of the aggregate. # 7. Reproducible Synthetic Demo (evaluates on knit) This chunk synthesizes a small example you can knit safely. ```{r demo, include=TRUE} # Synthetic prior matrix (rows on simplex) T <- 10; K <- 6 set.seed(123) P <- matrix(rexp(T*K), nrow = T) P <- P / rowSums(P) # Likelihood vector from P (PCA/SVD; robust with fallback) L <- compute_L_from_P(P) # Spread over time with "recent" pattern LT <- spread_likelihood(L, T_periods = T, pattern = "recent") # Try a couple of posteriors W_weighted <- posterior_weighted(P, LT, lambda = 0.7) W_adaptive <- posterior_adaptive(P, LT) # Metrics for adaptive coh <- coherence_score(P, W_adaptive, L) stab <- stability_composite(W_adaptive, a = 1000, b = 10, kappa = 50) intr <- interpretability_score(P, W_adaptive) eff <- 0.65 comp <- 0.30*coh + 0.25*stab + 0.25*intr + 0.20*eff data.frame(coherence = coh, stability = stab, interpretability = intr, efficiency = eff, composite = comp) %>% round(4) ``` # 8. Full Real-Data Pipeline (disable/enable evaluation) > **Switch to `eval=TRUE` after setting your paths**. By default we keep this chunk off to render quickly on any machine. ```{r real-pipeline, eval=FALSE} # === Create synthetic data for CRAN-compliant demo === demo_dir <- tempdir() # Create synthetic CPI data (mimicking your structure) set.seed(123) cpi_demo <- data.frame( Year = 2000:2010, CPI = cumsum(c(100, rnorm(10, 0.5, 2))) # Random walk starting at 100 ) cpi_file <- file.path(demo_dir, "synthetic_cpi.xlsx") openxlsx::write.xlsx(cpi_demo, cpi_file) # Create synthetic weights matrix (mimicking VAB weights structure) set.seed(456) years <- 2000:2010 sectors <- c("Agriculture", "Manufacturing", "Services", "Construction", "Mining") weights_demo <- data.frame(Year = years) for(sector in sectors) { weights_demo[[sector]] <- runif(length(years), 0.05, 0.35) } # Normalize rows to sum to 1 (simplex constraint) weights_demo[, -1] <- weights_demo[, -1] / rowSums(weights_demo[, -1]) weights_file <- file.path(demo_dir, "synthetic_weights.xlsx") openxlsx::write.xlsx(weights_demo, weights_file) # Use synthetic data paths path_cpi <- cpi_file path_w <- weights_file out_dir <- demo_dir cat("Using synthetic data for CRAN demo:\n") cat("CPI file:", path_cpi, "\n") cat("Weights file:", path_w, "\n") cat("Output directory:", out_dir, "\n") # --- Base run (robust defaults) --- base_res <- bayesian_disaggregate( path_cpi = path_cpi, path_weights = path_w, method = "adaptive", lambda = 0.7, # recorded in metrics; not used by "adaptive" gamma = 0.1, coh_mult = 3.0, coh_const = 0.5, stab_a = 1000, stab_b = 10, stab_kappa = 60, likelihood_pattern = "recent" ) xlsx_base <- save_results(base_res, out_dir = file.path(out_dir, "base")) print(base_res$metrics) # --- Minimal grid search for demo (reduced size) --- n_cores <- 1 # Use single core for CRAN compliance grid_df <- expand.grid( method = c("weighted", "adaptive"), # Reduced methods lambda = c(0.5, 0.7), # Reduced options gamma = 0.1, # Single option coh_mult = 3.0, # Single option coh_const = 0.5, # Single option stab_a = 1000, stab_b = 10, stab_kappa = 60, # Single option likelihood_pattern = "recent", # Single option KEEP.OUT.ATTRS = FALSE, stringsAsFactors = FALSE ) grid_res <- run_grid_search( path_cpi = path_cpi, path_weights = path_w, grid_df = grid_df, n_cores = n_cores ) write.csv(grid_res, file.path(out_dir, "grid_results.csv"), row.names = FALSE) best_row <- grid_res %>% arrange(desc(composite)) %>% slice(1) print("Best configuration from grid search:") print(best_row) # --- Re-run the best configuration for clean export --- best_res <- bayesian_disaggregate( path_cpi = path_cpi, path_weights = path_w, method = best_row$method, lambda = if (!is.na(best_row$lambda)) best_row$lambda else 0.7, gamma = if (!is.na(best_row$gamma)) best_row$gamma else 0.1, coh_mult = best_row$coh_mult, coh_const = best_row$coh_const, stab_a = best_row$stab_a, stab_b = best_row$stab_b, stab_kappa = best_row$stab_kappa, likelihood_pattern = best_row$likelihood_pattern ) xlsx_best <- save_results(best_res, out_dir = file.path(out_dir, "best")) # --- One Excel with everything (including hyperparameters) --- sector_summary <- tibble( Sector = colnames(best_res$posterior)[-1], prior_mean = colMeans(as.matrix(best_res$prior[, -1])), posterior_mean = colMeans(as.matrix(best_res$posterior[, -1])) ) wb <- createWorkbook() addWorksheet(wb, "Hyperparameters"); writeData(wb, "Hyperparameters", best_row) addWorksheet(wb, "Metrics"); writeData(wb, "Metrics", best_res$metrics) addWorksheet(wb, "Prior_P"); writeData(wb, "Prior_P", best_res$prior) addWorksheet(wb, "Posterior_W"); writeData(wb, "Posterior_W", best_res$posterior) addWorksheet(wb, "Likelihood_t"); writeData(wb, "Likelihood_t", best_res$likelihood_t) addWorksheet(wb, "Likelihood_L"); writeData(wb, "Likelihood_L", best_res$likelihood) addWorksheet(wb, "Sector_Summary"); writeData(wb, "Sector_Summary", sector_summary) for (sh in c("Hyperparameters","Metrics","Prior_P","Posterior_W", "Likelihood_t","Likelihood_L","Sector_Summary")) { freezePane(wb, sh, firstRow = TRUE) addFilter(wb, sh, rows = 1, cols = 1:ncol(readWorkbook(wb, sh))) setColWidths(wb, sh, cols = 1:200, widths = "auto") } # --- Add sectoral CPI (aggregate times posterior weights) --- W_post <- best_res$posterior # Year + sectors cpi_df <- read_cpi(path_cpi) # Year, CPI sector_cpi <- dplyr::left_join(W_post, cpi_df, by = "Year") %>% dplyr::mutate(dplyr::across(-c(Year, CPI), ~ .x * CPI)) # Quality check: sector sums vs CPI check_sum <- sector_cpi %>% dplyr::mutate(row_sum = rowSums(dplyr::across(-c(Year, CPI))), diff = CPI - row_sum) cat("Quality check (first 5 rows):\n") print(head(check_sum, 5)) addWorksheet(wb, "Sector_CPI") writeData(wb, "Sector_CPI", sector_cpi) freezePane(wb, "Sector_CPI", firstRow = TRUE) addFilter(wb, "Sector_CPI", rows = 1, cols = 1:ncol(sector_cpi)) setColWidths(wb, "Sector_CPI", cols = 1:200, widths = "auto") excel_onefile <- file.path(out_dir, "best", "Best_Full_Output_withSectorCPI.xlsx") saveWorkbook(wb, excel_onefile, overwrite = TRUE) cat("Full results saved to:", excel_onefile, "\n") # --- Quick plots (saved as PNGs) --- dir_plots <- file.path(out_dir, "best", "plots") if (!dir.exists(dir_plots)) dir.create(dir_plots, recursive = TRUE) W_long <- best_res$posterior %>% pivot_longer(-Year, names_to = "Sector", values_to = "Weight") p_heat <- ggplot(W_long, aes(Year, Sector, fill = Weight)) + geom_tile() + scale_fill_viridis_c() + labs(title = "Posterior weights (W): heatmap", x = "Year", y = "Sector", fill = "Share") + theme_minimal(base_size = 11) + theme(axis.text.y = element_text(size = 6)) ggsave(file.path(dir_plots, "posterior_heatmap.png"), p_heat, width = 12, height = 9, dpi = 220) top_sectors <- best_res$posterior %>% summarise(across(-Year, mean)) %>% pivot_longer(everything(), names_to = "Sector", values_to = "MeanShare") %>% arrange(desc(MeanShare)) %>% slice(1:3) %>% pull(Sector) # Reduced to top 3 for demo p_lines <- best_res$posterior %>% select(Year, all_of(top_sectors)) %>% pivot_longer(-Year, names_to = "Sector", values_to = "Weight") %>% ggplot(aes(Year, Weight, color = Sector)) + geom_line(linewidth = 0.9) + labs(title = "Top 3 sectors by average share (posterior W)", y = "Share", x = "Year") + theme_minimal(base_size = 11) ggsave(file.path(dir_plots, "posterior_topSectors.png"), p_lines, width = 11, height = 6, dpi = 220) cat("Demo completed successfully. All files written to temporary directory.\n") ``` # 9. Practical Guidance and Defaults * Prefer `method="adaptive"` when prior sector volatilities are heterogeneous; otherwise `weighted` with $\lambda\in[0.7,0.9]$ is strong and often tops the grid. * The default **coherence** parameters `(mult=3.0, const=0.5)` produce a bounded, interpretable 0–1 score that emphasizes **improvement** over the prior. * The **exponential** numerical penalty is intentionally sharp: it keeps row-sum deviations and negatives at bay in automated runs and grid searches. * For reports, export **Sector\_CPI** to illustrate the economic decomposition $\hat Y_{t,k}$. # Appendix A. Invariants and Quick Checks ```{r invariants, include=TRUE} # Example: invariants on a fresh synthetic run T <- 6; K <- 5 set.seed(7) P <- matrix(rexp(T*K), nrow = T); P <- P / rowSums(P) L <- compute_L_from_P(P) LT <- spread_likelihood(L, T, "recent") W <- posterior_multiplicative(P, LT) # Invariants stopifnot(all(abs(rowSums(P) - 1) < 1e-12)) stopifnot(all(abs(rowSums(LT) - 1) < 1e-12)) stopifnot(all(abs(rowSums(W) - 1) < 1e-12)) c( coherence = coherence_score(P, W, L), stability = stability_composite(W), interpret = interpretability_score(P, W) ) %>% round(4) ``` # Appendix B. Session Info ```{r session, include=TRUE} sessionInfo() ``` **Notes** - The *real-data* chunk is set to `eval=FALSE` so the manual renders anywhere. Flip it to `TRUE` on your machine to run fully against your Excel files. - The “best one-file” export includes **Hyperparameters, Metrics, Prior_P, Posterior_W, Likelihood_t, Likelihood_L, Sector_Summary, Sector_CPI**, with frozen headers and filters for quick analysis. - Plots are written to `.../best/plots/` and match the interpretation guidance in Section 6.