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Partial projection: working with incomplete feature sets

1. Why partial projection?

Assume you trained a dimensionality-reduction model (PCA, PLS …) on p variables but, at prediction time,

one sensor is broken,
a block of variables is too expensive to measure,
you need a quick first pass while the “heavy” data arrive later.

You still want the latent scores in the same component space, so downstream models, dashboards, alarms, … keep running.

That’s exactly what

partial_project(model, new_data_subset, colind = which.columns)

does:

new_data_subset (n × q) ─► project into latent space (n × k)

with q ≤ p. If the loading vectors are orthonormal this is a simple dot product; otherwise a ridge-regularised least-squares solve is used.

2. Walk-through with a toy PCA

set.seed(1)
n  <- 100
p  <- 8
X  <- matrix(rnorm(n * p), n, p)

# Fit a centred 3-component PCA (via SVD)
# Manually center the data and create fitted preprocessor
Xc <- scale(X, center = TRUE, scale = FALSE)
svd_res <- svd(Xc, nu = 0, nv = 3)

# Create a fitted centering preprocessor
preproc_fitted <- fit(center(), X)

pca <- bi_projector(
  v     = svd_res$v,
  s     = Xc %*% svd_res$v,
  sdev  = svd_res$d[1:3] / sqrt(n-1), # Correct scaling for sdev
  preproc = preproc_fitted
)

2.1 Normal projection (all variables)

scores_full <- project(pca, X)        # n × 3
head(round(scores_full, 2))
#>       [,1]  [,2]  [,3]
#> [1,] -0.02  1.20  0.49
#> [2,] -2.77 -0.18  1.25
#> [3,]  0.57  0.99 -0.06
#> [4,] -0.36 -0.81  0.58
#> [5,]  0.91 -0.60 -0.28
#> [6,]  1.81  2.50  0.35

2.2 Missing two variables ➜ partial projection

Suppose columns 7 and 8 are unavailable for a new batch.

X_miss      <- X[, 1:6]               # keep only first 6 columns
col_subset  <- 1:6                    # their positions in the **original** X

scores_part <- partial_project(pca, X_miss, colind = col_subset)

# How close are the results?
plot_df <- tibble(
  full = scores_full[,1],
  part = scores_part[,1]
)

ggplot(plot_df, aes(full, part)) +
  geom_point() +
  geom_abline(col = "red") +
  coord_equal() +
  labs(title = "Component 1: full vs. partial projection") +
  theme_minimal()

Even with two variables missing, the ridge LS step recovers latent scores that lie almost on the 1:1 line.

3. Caching the operation with a partial projector

If you expect many rows with the same subset of features, create a specialised projector once and reuse it:

# Assuming partial_projector is available
pca_1to6 <- partial_projector(pca, 1:6)   # keeps a reference + cache

# project 1000 new observations that only have the first 6 vars
new_batch <- matrix(rnorm(1000 * 6), 1000, 6)
scores_fast <- project(pca_1to6, new_batch)
dim(scores_fast)   # 1000 × 3
#> [1] 1000    3

Internally, partial_projector() stores the mapping v[1:6, ] and a pre-computed inverse, so calls to project() are as cheap as a matrix multiplication.

4. Block-wise convenience

For multiblock fits (created with multiblock_projector()), project_block() provides a convenient wrapper around partial_project():

# Create a multiblock projector from our PCA
# Suppose columns 1-4 are "Block A" (block 1) and columns 5-8 are "Block B" (block 2)
block_indices <- list(1:4, 5:8)

mb <- multiblock_projector(
  v = pca$v,
  preproc = pca$preproc,
  block_indices = block_indices
)

# Now we can project using only Block 2's data (columns 5-8)
X_block2 <- X[, 5:8]
scores_block2 <- project_block(mb, X_block2, block = 2)

# Compare to full projection
head(round(cbind(full = scores_full[,1], block2 = scores_block2[,1]), 2))
#>       full block2
#> [1,] -0.02  -0.30
#> [2,] -2.77  -3.28
#> [3,]  0.57  -0.09
#> [4,] -0.36   0.03
#> [5,]  0.91   1.24
#> [6,]  1.81   1.85

This is equivalent to calling partial_project(mb, X_block2, colind = 5:8) but reads more naturally when working with block structures.

5. Not only “missing data”: regions-of-interest & nested designs

Partial projection is handy even when all measurements exist:

Region of interest (ROI). In neuro-imaging you might have 50,000 voxels but care only about the motor cortex. Projecting just those columns shows how a participant scores within that anatomical region without refitting the whole PCA/PLS.
Nested / multi-subject studies. For multi-block PCA (e.g. “participant × sensor”), you can ask “where would subject i lie if I looked at block B only?” Simply supply that block to project_block().
Feature probes or “what-if” analysis. Engineers often ask “What is the latent position if I vary only temperature and hold everything else blank?” Pass a matrix that contains the chosen variables and zeros elsewhere.

5.1 Mini-demo: projecting an ROI

Assume columns 1–5 (instead of 50 for brevity) of X form our ROI.

roi_cols   <- 1:5                 # pretend these are the ROI voxels
X_roi      <- X[, roi_cols]       # same matrix from Section 2

roi_scores <- partial_project(pca, X_roi, colind = roi_cols)

# Compare component 1 from full vs ROI
df_roi <- tibble(
  full = scores_full[,1],
  roi  = roi_scores[,1]
)

ggplot(df_roi, aes(full, roi)) +
  geom_point(alpha = .6) +
  geom_abline(col = "red") +
  coord_equal() +
  labs(title = "Component 1 scores: full data vs ROI") +
  theme_minimal()

Interpretation: If the two sets of scores align tightly, the ROI variables are driving this component. A strong deviation would reveal that other variables dominate the global pattern.

5.2 Single-subject positioning in a multiblock design

Using the multiblock projector from Section 4, we can see how individual observations score when viewed through just one block:

# Get scores for observation 1 using only Block 1 variables (columns 1-4)
subject1_block1 <- project_block(mb, X[1, 1:4, drop = FALSE], block = 1)

# Get scores for the same observation using only Block 2 variables (columns 5-8)
subject1_block2 <- project_block(mb, X[1, 5:8, drop = FALSE], block = 2)

# Compare: do both blocks tell the same story about this observation?
cat("Subject 1 scores from Block 1:", round(subject1_block1, 2), "\n")
#> Subject 1 scores from Block 1: 3.43 0.72 -1.06
cat("Subject 1 scores from Block 2:", round(subject1_block2, 2), "\n")
#> Subject 1 scores from Block 2: -0.3 1.36 0.9
cat("Subject 1 scores from full data:", round(scores_full[1,], 2), "\n")
#> Subject 1 scores from full data: -0.02 1.2 0.49

This lets you assess whether an observation’s position in the latent space is consistent across blocks, or whether one block tells a different story.

6. Cheat-sheet: why you might call `partial_project()`

Scenario	What you pass	Typical call
Sensor outage / missing features	matrix with observed cols only	`partial_project(mod, X_obs, colind = idx)`
Region of interest (ROI)	ROI columns of the data	`partial_project(mod, X[, ROI], ROI)`
Block-specific latent scores	full block matrix	`project_block(mb, blkData, block = b)`
“What-if”: vary a single variable set	varied cols + zeros elsewhere	`partial_project()` with matching `colind`

The component space stays identical throughout, so downstream analytics, classifiers, or control charts continue to work with no re-training.

Session info

sessionInfo()
#> R version 4.5.1 (2025-06-13)
#> Platform: aarch64-apple-darwin20
#> Running under: macOS Sonoma 14.3
#> 
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
#> 
#> locale:
#> [1] C/en_CA.UTF-8/en_CA.UTF-8/C/en_CA.UTF-8/en_CA.UTF-8
#> 
#> time zone: America/Toronto
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] glmnet_4.1-10      Matrix_1.7-3       knitr_1.51         tibble_3.3.1      
#> [5] dplyr_1.1.4        ggplot2_4.0.1      multivarious_0.3.1
#> 
#> loaded via a namespace (and not attached):
#>  [1] GPArotation_2025.3-1 utf8_1.2.6           sass_0.4.10         
#>  [4] future_1.68.0        generics_0.1.4       shape_1.4.6.1       
#>  [7] lattice_0.22-7       listenv_0.10.0       digest_0.6.39       
#> [10] magrittr_2.0.4       evaluate_1.0.5       grid_4.5.1          
#> [13] RColorBrewer_1.1-3   iterators_1.0.14     fastmap_1.2.0       
#> [16] foreach_1.5.2        jsonlite_2.0.0       ggrepel_0.9.6       
#> [19] RSpectra_0.16-2      survival_3.8-3       mgcv_1.9-3          
#> [22] scales_1.4.0         pls_2.8-5            codetools_0.2-20    
#> [25] jquerylib_0.1.4      cli_3.6.5            crayon_1.5.3        
#> [28] rlang_1.1.7          chk_0.10.0           parallelly_1.45.1   
#> [31] future.apply_1.20.0  splines_4.5.1        withr_3.0.2         
#> [34] cachem_1.1.0         yaml_2.3.12          otel_0.2.0          
#> [37] tools_4.5.1          parallel_4.5.1       corpcor_1.6.10      
#> [40] globals_0.18.0       rsvd_1.0.5           assertthat_0.2.1    
#> [43] vctrs_0.7.0          R6_2.6.1             matrixStats_1.5.0   
#> [46] proxy_0.4-27         lifecycle_1.0.5      MASS_7.3-65         
#> [49] irlba_2.3.5.1        pkgconfig_2.0.3      pillar_1.11.1       
#> [52] bslib_0.9.0          geigen_2.3           gtable_0.3.6        
#> [55] glue_1.8.0           Rcpp_1.1.1           xfun_0.55           
#> [58] tidyselect_1.2.1     svd_0.5.8            farver_2.1.2        
#> [61] nlme_3.1-168         htmltools_0.5.9      labeling_0.4.3      
#> [64] rmarkdown_2.30       compiler_4.5.1       S7_0.2.1

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