# Linear Re-representation with regress()

When you already have a known set of basis functions—such as Fourier components, wavelets, splines, or principal components from a reference dataset—the goal isn’t to discover a latent space, but rather to express new data in terms of that existing basis. The `regress()` function facilitates this by wrapping several multi-output linear models within the `bi_projector` API.

This integration means you automatically inherit standard `bi_projector` methods for tasks like:

- `project()`: Map new data from the original space into the basis coefficients.
- `inverse_projection()`: Map basis coefficients back to the original data space.
- `reconstruct()` / `reconstruct_new()`: Reconstruct data using all or a subset of basis components.
- `coef()`: Retrieve the basis coefficients.
- `truncate()`: Keep only a subset of basis components.
- Plus caching, variable tracking helpers, etc.

This vignette demonstrates a typical workflow using an orthonormal Fourier basis, but the `regress()` function works equally well with arbitrary, potentially non-orthogonal, basis dictionaries.

# 1. Build a design matrix of basis functions

First, let’s define our basis. We’ll use sines and cosines.

    set.seed(42)
    n  <- 128                       # Number of observations (e.g., signals)
    p  <- 32                        # Original variables per observation (e.g., time points)
    k  <- 20                        # Number of basis functions (<= p often, <= n for lm)

    ## Create toy data: smooth signals + noise
    t   <- seq(0, 1, length.out = p)
    Y   <- replicate(n,  3*sin(2*pi*3*t) + 2*cos(2*pi*5*t) ) +
           matrix(rnorm(n*p, sd = 0.3), p, n) # Note: Y is p x n here

    ## Orthonormal Fourier basis (columns = basis functions)
    # We create k/2 sine and k/2 cosine terms, plus an intercept
    freqs <- 1:(k %/% 2) # Integer division for number of frequencies
    B <- cbind(rep(1, p), # Intercept column
               do.call(cbind, lapply(freqs, function(f) sin(2*pi*f*t))),
               do.call(cbind, lapply(freqs, function(f) cos(2*pi*f*t))))
    colnames(B) <- c("Intercept", paste0("sin", freqs), paste0("cos", freqs))

    # Make columns orthonormal (length 1, orthogonal to each other)
    B <- scale(B, center = FALSE, scale = sqrt(colSums(B^2)))

    cat(paste("Dimensions: Y is", nrow(Y), "x", ncol(Y),
              ", Basis B is", nrow(B), "x", ncol(B), "\n"))
    #> Dimensions: Y is 32 x 128 , Basis B is 32 x 21
    # We want coefficients C (k x n) such that Y ≈ B %*% C.

# 2. Fit multi-output regression

Now, we use `regress()` to find the coefficients \(C\) that best represent each signal in \(Y\) using the basis \(B\).

    library(multivarious)

    # Fit using standard linear models (lm)
    # Y is p x n (32 x 128): 32 time points x 128 signals
    # B is p x k (32 x 21): 32 time points x 21 basis functions
    # regress will fit 128 separate regressions, each with 32 observations and 21 predictors
    fit <- regress(X = B,          # Predictors = basis functions (p x k)
                   Y = Y,          # Response = signals (p x n)
                   method    = "lm",
                   intercept = FALSE) # Basis B already includes an intercept column

    # The result is a bi_projector object
    print(fit)
    #> Regression bi_projector object:
    #>   Method: lm
    #>   Input dimension: 128
    #>   Output dimension: 21
    #>   Coefficients dimension:  128 x 21

    ## Conceptual mapping to bi_projector slots:
    # fit$v : Coefficients (n x k) - Basis coefficients for each signal.
    # fit$s : Design Matrix (p x k) - The basis matrix B.
    #         Stored for reconstruction.

The `bi_projector` structure provides a consistent way to access the core components: the basis (`fit$s`) and the coefficients (`fit$v`).

# 3. Go back and forth between spaces

With the fitted `bi_projector`, we can move freely between the original data space and the basis coefficient space. This section demonstrates the key operations.

## 3.1 Inspecting the fitted coefficients

The coefficient matrix `fit$v` has dimensions \(n \times k\) (signals \(\times\) basis functions). Each row contains the weights that express one signal as a linear combination of basis functions.

    coef_matrix_first3 <- fit$v[1:3, ]
    cat("Coefficient matrix shape (first 3 signals):",
        nrow(coef_matrix_first3), "x", ncol(coef_matrix_first3), "\n\n")
    #> Coefficient matrix shape (first 3 signals): 3 x 21

    cat("Coefficients for signal 1:\n")
    #> Coefficients for signal 1:
    print(coef_matrix_first3[1, ])
    #>    Intercept         sin1         sin2         sin3         sin4         sin5
    #>  0.135054901  0.587137215 -0.183816033 12.150464524 -0.007984793 -0.162281276
    #>         sin6         sin7         sin8         sin9        sin10         cos1
    #> -0.593723657  0.581637564 -0.370023776 -0.108499186  0.173654284  0.394261726
    #>         cos2         cos3         cos4         cos5         cos6         cos7
    #> -0.590122835  0.288835048  0.448031764  8.196393043  0.080690010 -0.612238653
    #>         cos8         cos9        cos10
    #>  0.678547189 -0.030446715  0.088104811

Notice that `sin3` and `cos5` have the largest coefficients—this matches our generating function \(3\sin(2\pi \cdot 3t) + 2\cos(2\pi \cdot 5t)\).

## 3.2 Reconstructing the fitted data

Reconstruction recovers the original data from the basis representation. The `bi_projector` stores both the design matrix \(B\) (in `fit$s`) and the coefficients (in `fit$v`). Reconstruction is simply their product:

\[\hat{Y} = B \cdot C^T = \texttt{fit\$s} \times \texttt{t(fit\$v)}\]

    Y_hat <- fit$s %*% t(fit$v)
    max_reconstruction_error <- max(abs(Y_hat - Y))

    cat("Reconstruction shape:", nrow(Y_hat), "x", ncol(Y_hat), "\n")
    #> Reconstruction shape: 32 x 128
    cat("Maximum reconstruction error:", format(max_reconstruction_error, digits=3), "\n")
    #> Maximum reconstruction error: 0.7

The error is non-zero because our signals contain random noise that cannot be captured by the smooth Fourier basis. This is expected and acceptable.

## 3.3 Projecting new data onto the basis

A key use case is expressing new observations in terms of the same basis. For an orthonormal basis \(B\), projection is straightforward:

\[\text{coefficients} = B^T \cdot Y_{\text{new}}\]

Let’s create a new signal with the same underlying pattern but different noise:

    Y_new_signal <- 3*sin(2*pi*3*t) + 2*cos(2*pi*5*t) + rnorm(p, sd=0.3)
    Y_new_matrix <- matrix(Y_new_signal, nrow = p, ncol = 1)

    coef_new <- t(fit$s) %*% Y_new_matrix

    cat("Basis coefficients for new signal:\n")
    #> Basis coefficients for new signal:
    print(coef_new[, 1])
    #>   Intercept        sin1        sin2        sin3        sin4        sin5
    #> -0.16459096  0.35921181 -0.09452109 11.90760497  0.37429968 -0.13899361
    #>        sin6        sin7        sin8        sin9       sin10        cos1
    #>  0.44535459 -0.16769033  0.02891241  0.54613517 -0.41145969  0.53832581
    #>        cos2        cos3        cos4        cos5        cos6        cos7
    #>  0.45371511  0.24009459  0.81766812  8.25333978  0.75410708 -0.42128770
    #>        cos8        cos9       cos10
    #>  0.49377610  0.45618692  0.79379189

Again, the `sin3` and `cos5` components dominate, as expected.

## 3.4 Reconstructing new data

Finally, we can reconstruct the new signal from its basis coefficients to see how well the basis captures the underlying structure:

    Y_new_recon <- fit$s %*% coef_new
    reconstruction_error <- sqrt(mean((Y_new_matrix - Y_new_recon)^2))

    cat("Reconstruction RMSE for new signal:", format(reconstruction_error, digits=3), "\n")
    #> Reconstruction RMSE for new signal: 0.362

The reconstruction error reflects the noise that the basis cannot represent. Because \(B\) is orthonormal, projection and reconstruction are exact inverses for the signal components that lie within the basis span.

# 4. Regularisation & PLS

If the basis is ill-conditioned or you need feature selection/shrinkage, simply change the `method` argument. `regress()` wraps common regularized models.

    # Ridge regression (requires glmnet)
    fit_ridge <- regress(X = B, Y = Y, method = "mridge", lambda = 0.01, intercept = FALSE)

    # Elastic Net (requires glmnet)
    fit_enet  <- regress(X = B, Y = Y, method = "enet", alpha = 0.5, lambda = 0.02, intercept = FALSE)

    # Partial Least Squares (requires pls package) - useful if k > p or multicollinearity
    fit_pls   <- regress(X = B, Y = Y, method = "pls", ncomp = 15, intercept = FALSE)

    # All these return bi_projector objects, so downstream code using
    # project(), reconstruct(), coef() etc. remains the same.

# 5. Partial / custom mappings

The `bi_projector` interface allows for flexible manipulation:

    # Truncate: Keep only the first 5 basis functions (Intercept + 2 sine + 2 cosine)
    fit5   <- truncate(fit, ncomp = 5)
    cat("Dimensions after truncating to 5 components:",
        "Basis (s):", paste(dim(fit5$s), collapse="x"),
        ", Coefs (v):", paste(dim(fit5$v), collapse="x"), "\n")
    #> Dimensions after truncating to 5 components: Basis (s): 32x5 , Coefs (v): 128x5
    # Reconstruction using only first 5 basis functions (manual)
    # Equivalent to: scores(fit5) %*% t(coef(fit5)) for the selected components
    Y_hat5 <- fit5$s %*% t(fit5$v)

    # Partial inverse projection: Map only a subset of coefficients back
    # e.g., reconstruct using only components 2 through 6 (skip intercept)
    # Note: partial_inverse_projection is not a standard bi_projector method,
    # this might require manual slicing of the basis matrix B (fit$s) or coefs (fit$v).
    # Manual reconstruction example for components 2:6
    coef_subset <- fit$v[2:6, , drop=FALSE] # k_sub x n
    basis_subset <- fit$s[, 2:6, drop=FALSE] # p x k_sub
    Y_lowHat <- basis_subset %*% coef_subset # p x n reconstruction

    # Variable usage helpers (Conceptual - actual functions might differ)
    # `variables_used(fit)` could show which basis functions have non-zero coefficients (esp. for 'enet').
    # `vars_for_component(fit, k)` isn't directly applicable here as components are the basis functions themselves.

# 6. Under-the-hood: Matrix View

The core idea is to represent the \(p \times n\) data matrix \(Y\) as a product of the \(p \times k\) basis matrix (stored in `s`) and the \(k \times n\) coefficient matrix (stored in `v`): \[ \underbrace{Y}_{p\times n} \approx \underbrace{s}_{p\times k} \underbrace{v}_{k\times n} \]

`regress()` estimates \(v\) (coefficients) based on the chosen method:

`method` | Solver Used (Conceptual) | Regularisation | Key Reference
--- | --- | --- | ---
“lm” | QR decomposition (`lm.fit`) | None | Classical OLS
“mridge” | `glmnet` (alpha=0) | Ridge (\(\lambda ||\beta||_2^2\)) | Hoerl & Kennard 1970
“enet” | `glmnet` | Elastic Net (\(\alpha\)-mix) | Zou & Hastie 2005
“pls” | `pls::plsr` | Latent PLS factors | Wold 1984

The resulting object stores: * `v`: The estimated coefficient matrix (\(k \times n\)). * `s`: The basis/design matrix \(B\) (\(p \times k\)), possibly centered or scaled by the underlying solver. * `sdev`, `center`: Potentially stores scaling/centering info related to \(s\).

All other `bi_projector` methods (`project`, `reconstruct`, `inverse_projection`) are derived from these core matrices.

# 7. Internal Checks (Developer Focus)

This section contains internal consistency checks, primarily for package development and CI testing.

The code chunk below only runs if the environment variable `_MULTIVARIOUS_DEV_COVERAGE` is set.

    # This chunk only runs if _MULTIVARIOUS_DEV_COVERAGE is non-empty
    message("Running internal consistency checks for regress()...")
    #> Running internal consistency checks for regress()...
    tryCatch({
      stopifnot(
        # Check reconstruction fidelity for lm
        max(abs(reconstruct(fit) - Y)) < 1e-10,
        # Check dimensions of inverse projection matrix (n x p)
        # inverse_projection maps coefficients (k x n) back to data (p x n)
        # The matrix itself maps k -> p implicitly. Let's check coef matrix dims.
        nrow(fit$v) == ncol(B), # k rows
        ncol(fit$v) == ncol(Y)  # n columns
        # Add checks for other methods if evaluated
      )
      message("Regress internal checks passed.")
    }, error = function(e) {
      warning("Regress internal checks failed: ", e$message)
    })
    #> Warning in value[[3L]](cond): Regress internal checks failed:
    #> max(abs(reconstruct(fit) - Y)) < 1e-10 is not TRUE

# 8. Take-aways

- `regress()` provides a convenient way to fit multiple linear models simultaneously when expressing data \(Y\) in a known basis \(B\).
- It returns a `bi_projector`, giving immediate access to projection, reconstruction, truncation, and coefficient extraction.
- Supports standard OLS (`lm`), Ridge (`mridge`), Elastic Net (`enet`), and PLS (`pls`) out-of-the-box.
- Works with any basis dictionary (orthonormal or not).
- Can be integrated into larger analysis pipelines using composition (`%>>%`) or cross-validation helpers.

Happy re-representing!