Local weighted regression across analog neighborhoods

Fits a weighted local regression of y on covariates within each focal location's analog neighborhood. Analog neighborhoods are defined by climatic similarity, geographic proximity, or both, while covariates capture additional predictors that influence outcomes within each neighborhood but are not captured by to the search dimensions. This generalizes the weighted mean — which averages over all within-neighborhood variation — by resolving variation driven by these auxiliary predictors. Supports ordinary and ridge-penalized weighted least squares. With purely geographic neighborhoods, this is equivalent to geographically weighted regression (GWR); with climate-based neighborhoods, it extends the analog impact model (AIM) framework to incorporate local covariate effects. This function is a wrapper that calls analog_search() with "regression" included in stat.

Usage

analog_regression(
  x,
  pool,
  y,
  weight = NULL,
  covariates,
  x_covariates = NULL,
  max_geog = NULL,
  max_clim = NULL,
  select = "all",
  k = NULL,
  kernel = c("gaussian_clim", "inverse_clim", "gaussian_geog", "inverse_geog",
    "gaussian_joint", "inverse_joint", "uniform"),
  theta = NULL,
  lambda = 0,
  stat = c("count", "ess", "regression"),
  se = c("none", "ess", "design"),
  normalize = "auto",
  x_cov = NULL,
  coord_type = "auto",
  index_res = "auto",
  cell_area_weight = "auto",
  n_threads = NULL,
  progress = FALSE,
  ...
)

Arguments

x

Focal locations for which regressions will be fit. Should be a matrix/data.frame with columns x, y, and climate variables, or a SpatRaster with climate variable layers.

pool

The reference dataset to search for analogs. Either:

Matrix/data.frame with columns x, y, and climate variables, or SpatRaster with climate variable layers, OR
An analog_index object created by build_analog_index() (for repeated queries).

y

Response variable(s) to model via local regression. Can be a numeric vector, matrix, data.frame, or SpatRaster. Must have exactly the same number of rows/cells as pool. A separate regression is fit for each variable.

weight

Optional pool site weights for use in aggregation. Numeric vector, single-column matrix/data.frame, or single-layer SpatRaster, with one value per row/cell of pool. For aggregation stats like "weighted_mean", "regression", etc., weights multiply through the weighted aggregation alongside any kernel weighting and cell-area weighting; they do not influence which analogs are selected by knn_* modes (selection remains distance-only). They are reported in pair mode as a user_weight column. Values must be non-negative; NA is allowed and treated as 0 (the point is excluded from aggregation). Default NULL means no user-supplied weights.

If you want to exclude a static subset of pool sites entirely, masking pool (and any associated y / covariates) upfront is more efficient than passing weight = 0 for those sites, since the lattice index will not have to scan or distance-compute against them. Use weight = 0 for cases where the mask varies per query against a shared index, or where some sites have a continuous weight and others should be excluded.

covariates

Predictor variables for local regression, supplied at pool locations. Can be a numeric vector (single covariate), matrix, data.frame, or SpatRaster. Must have exactly the same number of rows/cells as pool. Column/layer names carry through to output. These variables are NOT used for the analog search itself – only for regression within each neighborhood.

x_covariates

Optional predictor values at focal (x) locations, at which to evaluate the local regressions. When supplied, the output includes one additional column (pred) or one column per response variable (pred_{yname} for multi-y), giving the fitted value at the focal. Must have exactly the same number of rows/cells as x, and the same column/layer names as covariates. Default NULL returns only coefficients.

max_geog

Maximum geographic distance constraint (default: NULL = no geographic constraint). When specified, only reference locations within this distance are considered. Radius units should be specified in kilometers if coord_type = "lonlat", or in projected coordinate units if coord_type = "projected".

max_clim

Maximum climate distance constraint (default: NULL = no climate constraint). Can be either:

A scalar: Euclidean radius in climate space (e.g., 0.5)
A vector: Per-variable absolute differences (length must equal number of climate variables)

Only reference locations within this climate distance are considered. When x_cov is provided, scalar thresholds are interpreted in Mahalanobis distance units.

select

Character string specifying the analog selection strategy. One of:

"all" (default): Select all analogs that satisfy the max_clim and max_geog constraints.
"knn_clim": For each focal, select up to k analogs with smallest climate distance, subject to filters.
"knn_geog": For each focal, select up to k analogs with smallest geographic distance, subject to filters.

k

Number of nearest analogs to return per focal location for kNN selection modes. Required when select is "knn_geog" or "knn_clim"; must be NULL for select = "all".

kernel

Kernel decay function for weighting matches, used only when stat includes a weighted aggregation ("sum_weights", "mean_weights", "weighted_sum", "weighted_mean", "ess", "regression", or "tabulate"). One of:

"uniform": All matches weighted equally (kernel weight = 1.0).
"inverse_clim": Inverse climate distance, kernel weight = 1 / (climate_distance + eps), with epsilon given by theta.
"inverse_geog": Inverse geographic distance, kernel weight = 1 / (geographic_distance + eps), with epsilon given by theta.
"gaussian_clim": Gaussian kernel on climate distance, kernel weight = exp(-climate_distance^2 / (2 sigma^2)), with sigma given by theta.
"gaussian_geog": Gaussian kernel on geographic distance, kernel weight = exp(-geographic_distance^2 / (2 sigma^2)), with sigma given by theta.
"gaussian_joint": Gaussian kernel on combined distance, kernel weight = exp(-(clim_dist^2 / (2 sigma_clim^2) + geog_dist^2 / (2 sigma_geog^2))), with sigmas given by theta.
"inverse_joint": Inverse joint distance, kernel weight = 1 / (sqrt(clim_dist^2 + geog_dist^2) + eps), with epsilon given by theta.

theta

Optional numeric parameter controlling the shape of the weighting kernel, used whenever kernel is active (i.e. whenever stat includes a weighted aggregation) and kernel is not "uniform". Interpretation depends on kernel:

For "inverse_clim" or "inverse_geog": epsilon value added to distances (scalar; default: 1e-12 for climate, 1e-6 for geography).
For "gaussian_clim" or "gaussian_geog": sigma bandwidth parameter (scalar; larger values = slower decay with distance).
For "gaussian_joint" or "inverse_joint": 2-element vector c(theta_clim, theta_geog) (defaults: 1 for climate, 1 for geography).

See kernel_params() for help choosing theta and max_clim / max_geog values that work well together.

lambda

Ridge penalty parameter (default: 0, giving ordinary weighted least squares). Higher values shrink high-variance coefficients toward zero, causing the intercept to approach the weighted mean as lambda -> Inf. Useful when some neighborhoods have few analogs relative to the number of covariates, or when covariates are strongly inter-correlated.

stat

Statistic(s) to compute. "regression" is always included. Additional stats like "count", "ess", and "weighted_mean" can be requested alongside regression coefficients. Default includes "count" and "ess" for diagnostics.

se

Standard-error framing to apply to SE-supporting stats ("weighted_mean" and "regression"). One of:

"none" (default): no SE columns are returned.
"ess": effective-sample-size framing. For weighted_mean, SE = sqrt(var_w(y) / n_eff), where n_eff = (Σw)² / Σw² is Kish's effective sample size and var_w(y) = Σwy²/Σw - ȳ_w². For regression, Var(β̂) = σ²_ess · (X'WX + λI)⁻¹, with residual variance corrected using n_eff - p degrees of freedom.
"design": design-based framing (no assumption that weights are precisions). For weighted_mean, SE = sqrt(Σ w²(y - ȳ_w)²) / Σw.

normalize

One of TRUE, FALSE, or "auto" (default). Only used if stat includes "sum_weights" or "tabulate" and pool is a raster. When active, results for these stats are divided by a global scalar so that they represent a fraction of a theoretically "perfect" scenario where the full search area within max_geog is occupied wall-to-wall by cells whose climate exactly matches x. See details under analog_search() for more info.

x_cov

Optional focal-specific covariance matrices for Mahalanobis distance calculations. Should be a matrix or data.frame with one row per focal location and one column per unique covariance component, or a SpatRaster with a layer for each component. For n climate variables, there are n*(n+1)/2 unique components, ordered as: variances first (diagonals), then covariances (upper triangle by row).

coord_type

Coordinate system type:

"auto" (default): Automatically detect from coordinate ranges.
"lonlat": Unprojected lon/lat coordinates (uses great-circle distance; assumes max_geog is in km).
"projected": Projected XY coordinates (uses planar distance; assumes max_geog is in projection units).

index_res

Tuning parameter giving the number of bins per dimension of the internally-used lattice search index. Either:

A positive integer.
"auto" (the default): Automatically tune the index resolution by optimizing compute time on a subsample of focal points. If focal has relatively few rows, auto-tuning is skipped and a default resolution of 16 is used. Auto-tuning is not supported when downsample < 1, because the speed-optimal resolution can sometimes result in higher uncertainty of stat results under downsampling. In that case set index_res explicitly; finer values (e.g. 32) generally give better accuracy at the possible cost of query speed.

Ignored if pool is an analog_index (uses index's resolution).

cell_area_weight

Controls cell-area weighting when pool is a raster. One of "auto" (default; on for raster pools, off otherwise), TRUE (force on; errors if pool is not a SpatRaster), or FALSE (force off). Cell-area weights correct aggregation statistics for non-uniform cell areas (e.g. lonlat grids near the poles, or projected grids on non-equal-area projections); they are computed via terra::cellSize() and normalized to mean 1. When pool is a pre-built analog_index, this argument must agree with the index's stored configuration: cell_area_weight = FALSE errors if the index was built with cell-area weighting on (rebuild the index instead).

n_threads

Optional integer number of threads to use for the computation. If NULL (default), the global RcppParallel setting is used (see RcppParallel::setThreadOptions).

progress

Logical; if TRUE, display a progress bar during computation. Progress tracking works by splitting the focal dataset into chunks and processing them sequentially. Useful for large datasets. Default is FALSE.

...

additional arguments passed to analog_search() (usually unneeded)

Value

A data.frame (or SpatRaster if x is a SpatRaster) with one row per focal location containing:

index, x, y: Focal location identifiers
Columns for any additional stats requested (e.g., count, ess)
coef_intercept: Regression intercept
coef_{name}: Regression slope for each covariate
When se != "none": se_intercept and se_{name} giving standard errors of each coefficient
With multiple y variables: columns are named coef_{coeff}_{varname} (and se_{coeff}_{varname} when SEs are returned)
When x_covariates is supplied: pred (single-y) or pred_{varname} (multi-y) giving the fitted value at each focal.

See analog_search() for column-naming conventions across stats and metadata() for attached metadata attributes.

Details

Method

For each focal location, the function:

Selects analog pool locations based on select, max_clim, max_geog, and k
Computes distance-based kernel weights for each analog (via kernel and theta)
Fits a weighted least squares regression of y on covariates using these weights, with optional ridge penalty lambda
Returns the regression coefficients (intercept + slopes), and optionally their standard errors (see se in analog_search()).

The math: beta = (X'WX + lambda * I_p)^{-1} X'Wy, where W is diagonal weights, X is the design matrix (intercept + covariates), and I_p penalizes covariate coefficients only (not the intercept).

Relationship to Weighted Mean

With lambda -> Inf, covariate coefficients shrink to zero and the intercept converges to the weighted mean. With lambda = 0 (the default), the full local regression is used. If covariates are centered (weighted mean zero within each neighborhood), the intercept equals the weighted mean at any lambda. The lambda parameter thus provides smooth interpolation between a simple weighted average and a full local regression.

Common Configurations

Geographically weighted regression (select = "all" or "knn_geog", with max_geog and geographic weights): Local spatial regression using geographic proximity to define and weight neighborhoods, equivalent to GWR.
Climate-nearest regression (select = "knn_clim", with max_geog and k): Fixed-size neighborhoods of the k most similar climates within geographic range.

Prediction

To evaluate fitted values at the focal locations, pass covariate values at those locations via x_covariates. The output will then include a pred column (single-y case) or pred_{yname} columns (multi-y case) alongside the usual coefficient columns.

If you only need coefficients, leave x_covariates = NULL.

Examples

if (FALSE) { # \dontrun{
# GWR-style spatial regression
gwr_result <- analog_regression(
  x = sites,
  pool = sites,
  y = sites$income,
  covariates = data.frame(education = sites$edu, access = sites$access),
  select = "knn_geog",
  k = 50,
  max_clim = NULL,
  kernel = "gaussian_geog",
  theta = 20,
  se = "ess"
)

# With focal-side covariates, to obtain fitted predictions
pred_result <- analog_regression(
  x = future_sites,
  pool = sites,
  y = sites$income,
  covariates   = data.frame(education = sites$edu),       # at pool
  x_covariates = data.frame(education = future_sites$edu) # at focals
)
head(pred_result[, c("coef_intercept", "coef_education", "pred")])
} # }