Package 'bayesqm'

Description

assess_recovery() compares a point estimate and optional posterior draws to a known loading truth, returning RMSE, per-factor RMSE, bias, Tucker's congruence, credible-interval coverage, width, and Gneiting-Raftery interval score. assess_classification() compares a logical flag matrix to the true factor assignments using optimal permutation matching.

Usage

assess_recovery(Lambda_hat, Lambda_true, Lambda_draws = NULL, prob = 0.95)

assess_classification(flags, Lambda_true)

Arguments

Value

assess_recovery() returns a list of metrics; assess_classification() returns a list with accuracy and per_factor.

Description

Every plot in the package reads its palette through bayesqm_colors(). Call bayesqm_set_colors() to switch the active scheme for every subsequent plot. The available built-in schemes are "blue" (default), "teal", "red", "purple", and "grey". For full control, pass a named list with slots light, mid, dark, accent, grey, gridgrey, and fill.

Usage

bayesqm_colors()

bayesqm_set_colors(scheme)

Arguments

Value

bayesqm_colors() returns the active palette as a named list. bayesqm_set_colors() returns the previous scheme name, invisibly.

Examples

bayesqm_colors()
fit <- demo_fit(N = 6, J = 10, K = 2, Td = 50, seed = 1)
bayesqm_set_colors("teal")
plot(fit)
bayesqm_set_colors("blue")  # restore default

Description

S3 methods that make a bayesqm_fit behave like a standard R modelling object: coef() returns the posterior-mean loadings, fitted() the posterior-mean fitted Y on the original Q-sort scale, residuals() is Y - fitted(fit), sigma() is the posterior-mean residual scale, nobs() is the number of participants, and family() returns a small bayesqm_family list with ⁠$family⁠, ⁠$link⁠, and ⁠$nu⁠. as.matrix(), as.array(), and as.data.frame() return the posterior draws in Stan-style parameter naming (Lambda[i,k], F[j,k], nu, sigma, tau), which the posterior, bayesplot, and tidybayes packages consume natively.

update() re-fits the model with modified arguments; the original call and stored data are reused.

Usage

## S3 method for class 'bayesqm_fit'
coef(object, ...)

## S3 method for class 'bayesqm_fit'
fitted(object, ...)

## S3 method for class 'bayesqm_fit'
residuals(object, ...)

## S3 method for class 'bayesqm_fit'
nobs(object, ...)

## S3 method for class 'bayesqm_fit'
sigma(object, ...)

## S3 method for class 'bayesqm_fit'
family(object, ...)

## S3 method for class 'bayesqm_family'
print(x, ...)

## S3 method for class 'bayesqm_fit'
as.matrix(x, ...)

## S3 method for class 'bayesqm_fit'
as.array(x, ...)

## S3 method for class 'bayesqm_fit'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

## S3 method for class 'bayesqm_fit'
update(object, ..., evaluate = TRUE)

Arguments

Value

Depends on the method; see Description.

Description

print() shows a compact, brms-style header with convergence and the first few loadings. summary() expands with factor characteristics, the PSIS-LOO estimate, the divergence summary, and the MatchAlign Tucker-phi diagnostic. Both methods exist for bayesqm_fit (returned by fit_bayesian()) and bayesqm_run (returned by run_bayes()).

Usage

## S3 method for class 'bayesqm_fit'
print(x, digits = 2, length = 10, ...)

## S3 method for class 'bayesqm_fit'
summary(object, digits = 3, ...)

## S3 method for class 'bayesqm_run'
print(x, digits = 2, ...)

## S3 method for class 'bayesqm_run'
summary(object, ...)

Arguments

Value

The input, invisibly.

Description

Posterior summaries of factor membership and statement interpretation, each computed entirely from posterior draws:

• compute_threshold_prob() returns the ⁠N x K⁠ posterior probability that ⁠|lambda_ik| > threshold⁠, i.e. the Bayesian version of the Brown (1980) flagging rule.

• compute_dominant_prob() returns the ⁠N x K⁠ posterior probability that factor k is the dominant factor for participant i.

• compute_dominant_sign() returns the length-N posterior probability that the dominant loading is positive; participants with probability below 0.5 are negative exemplars.

• compute_divergence() returns the posterior of the viewpoint divergence D_j for every statement, together with the distinguishing and consensus probabilities it implies.

• classify_membership() turns dominant probabilities into a per-participant descriptive tier (Strong / Moderate / Weak).

Usage

compute_threshold_prob(Lambda_draws, threshold)

compute_dominant_prob(Lambda_draws)

compute_dominant_sign(Lambda_draws)

compute_divergence(F_draws, delta = NULL, delta_grid = NULL)

classify_membership(Lambda_draws, strong = 0.8, moderate = 0.6)

Arguments

Details

For statement j with standardized viewpoint scores ⁠f_{j1}, ..., f_{jK}⁠, the divergence estimand is the mean absolute pairwise difference ⁠D_j = 2 / (K (K - 1)) * sum_{k < l} |f_jk - f_jl|⁠. The mean is used rather than the maximum, which is an order statistic over the K (K - 1) / 2 contrasts and is inflated when the posterior is diffuse. compute_divergence() reports the posterior median and central 95% credible interval of D_j, pi_D = P(D_j > delta | Y) (distinguishing) and pi_C = P(D_j < delta | Y) = 1 - pi_D (consensus), and the per-viewpoint departure ⁠g_jk = f_jk - mean_{l != k} f_jl⁠ with its dominant viewpoint, sign, and ⁠P(|g_jk| > delta | Y)⁠. The probabilities are the reported quantities; no fixed probability cutoff defines a distinguishing or consensus statement.

Value

compute_threshold_prob() and compute_dominant_prob() return ⁠N x K⁠ matrices. compute_dominant_sign() returns a length-N named vector. compute_divergence() returns a list (see Details). classify_membership() returns a data frame.

Description

Returns a human-readable caption string summarising the model configuration (K, N, J, family), the sampler (backend, chains, post-warmup draws), the interval probability, and convergence diagnostics (max Rhat, divergent transitions).

Usage

caption_bayesqm(fit, include_ref = TRUE, include_diag = TRUE)

Arguments

Value

A length-1 character string.

Examples

fit <- demo_fit(N = 6, J = 10, K = 2, Td = 50, seed = 1)
cat(caption_bayesqm(fit))

Description

Posterior-mean factor z-scores ranked onto the study's forced distribution. The result is a tidy data frame with one row per statement and per-factor integer grid scores. For the continuous z-scores with credible intervals, use compute_zscores().

Usage

compute_factor_array(F_draws, Y)

Arguments

Value

A data frame with columns statement and ⁠f1_grid, f2_grid, ..., fK_grid⁠.

Description

Posterior mean and central credible-interval bounds for each participant-factor loading, returned as a tidy data frame with one row per participant and three columns per factor.

Usage

compute_loadings(Lambda_draws, prob = 0.95)

Arguments

Value

A data frame with columns participant, and three numeric columns per factor: fk_loa (posterior mean), fk_lower, and fk_upper, for ⁠k = 1..K⁠.

Description

Returns a tidy data frame with one row per scalar parameter and columns mean, median, sd, lower, upper.

Usage

compute_posterior_scalars(scalar_draws, prob = 0.95)

Arguments

Value

A data frame.

Description

Posterior mean and central credible-interval bounds for each statement-factor z-score, returned as a tidy data frame with one row per statement and three columns per factor.

Usage

compute_zscores(F_draws, prob = 0.95)

Arguments

Value

A data frame with columns statement, and three numeric columns per factor: fk_zsc (posterior mean), fk_lower, and fk_upper, for ⁠k = 1..K⁠.

Description

The default separation for the distinguishing and consensus probabilities: the reliability-adjusted critical difference used to flag distinguishing statements in classical Q analysis (Brown 1980; Zabala & Pascual 2016), here generalized by computing it from the posterior dominant-factor counts. With p_f the number of participants whose posterior dominant factor is f, ⁠r_f = p_f r0 / (1 + (p_f - 1) r0)⁠ and SE_f = sqrt(1 - r_f), ⁠delta = z * mean_{k < l} sqrt(SE_k^2 + SE_l^2)⁠. The reliability is the stable population reliability of the design, not the posterior estimation spread.

Usage

critical_delta(Lambda_draws, level = 0.05, r0 = 0.8)

Arguments

Value

A single numeric value, the critical-difference delta.

Examples

critical_delta(array(rnorm(200 * 8 * 3), c(200, 8, 3)))

Description

Returns a bayesqm_fit with realistic Q-methodology structure: every participant has a dominant factor, roughly 40 percent of the statements polarise the factor pair, 10 percent are consensus, and the remainder are weakly partial. Use it for documentation, teaching materials, and the package vignette; it is not a substitute for fit_bayesian() on real data.

Usage

demo_fit(N = 20, J = 22, K = 2, Td = 400, seed = 1L)

Arguments

Value

A bayesqm_fit.

Examples

fit <- demo_fit(N = 12, J = 15, K = 2)
plot(fit)
summary(fit)

Description

Returns a bayesqm_run object carrying a plausible ELPD trajectory across K = 1..K_max, with user-chosen peak K, Sivula K, and case label. Use it to demonstrate run_bayes() output and plot_elpd() without a Stan backend; it is not a substitute for run_bayes() on real data.

Usage

demo_run(
  K_max = 4L,
  k_peak = 3L,
  k_sivula = 2L,
  case = c("gap", "agree", "reversed"),
  seed = 1L
)

Arguments

Value

A bayesqm_run.

Examples

run <- demo_run()
run
plot_elpd(run)

Description

Fits the low-rank Bayesian factor model to Q-sort data. Samples the posterior with Stan (via cmdstanr or rstan), resolves rotational ambiguity with MatchAlign, and returns a classed bayesqm_fit object carrying posterior-mean loadings and factor scores, credible intervals, raw draws, LOO, PPC, and diagnostics.

Usage

fit_bayesian(
  Y,
  K,
  stan_dir = NULL,
  robust = TRUE,
  nu = "estimate",
  chains = 4,
  iter = 2000,
  warmup = 1000,
  seed = NULL,
  adapt_delta = 0.9,
  max_draws = 2000,
  prior_loading_scale = 1,
  prior_sigma_scale = 1,
  prior_nu_alpha = 2,
  prior_nu_beta = 0.1,
  use_half_cauchy = FALSE,
  prob = 0.95,
  delta = NULL
)

Arguments

Value

A bayesqm_fit object. See bayesqm-fit-methods for print() and summary(), and coef.bayesqm_fit() for the standard R accessors.

References

Poworoznek et al. (2025). Efficiently Resolving Rotational Ambiguity in Bayesian Matrix Sampling with Matching. Bayesian Analysis.

Examples

# Needs a working Stan backend; skipped when cmdstanr/CmdStan is absent.
has_stan <- requireNamespace("cmdstanr", quietly = TRUE) &&
  !inherits(try(cmdstanr::cmdstan_path(), silent = TRUE), "try-error")
if (has_stan) {
  sim <- generate_data(N = 8, J = 12, K = 2, seed = 1)
  fit <- fit_bayesian(sim$Y, K = 2, chains = 1, iter = 600, warmup = 300)
  summary(fit)
}

Description

generate_data() is the top-level data-generating function used by the package's simulation studies and tests. It builds a loading matrix (generate_loadings()), factor scores, noise of the chosen type (generate_noise()), and discretises the continuous signal onto a forced Q-sort grid (discretize_to_grid()). See also get_distribution() for the standard forced-distribution lookup.

Usage

generate_data(
  N,
  J,
  K,
  noise_sd = 1,
  error_type = "normal",
  nu = 5,
  contam_prop = 0.1,
  contam_scale = 4,
  loading_type = "simple",
  primary_range = c(0.55, 0.85),
  cross_range = c(-0.15, 0.15),
  seed = NULL
)

generate_loadings(
  N,
  K,
  primary_range = c(0.55, 0.85),
  cross_range = c(-0.15, 0.15),
  type = "simple"
)

generate_noise(
  J,
  N,
  type = "normal",
  sd = 1,
  nu = 5,
  contam_prop = 0.1,
  contam_scale = 4
)

discretize_to_grid(Y_cont, distr)

get_distribution(J)

Arguments

Value

generate_data() returns a list with Y, Lambda_true, F_true, distribution, N, J, K. The component helpers return their respective raw objects.

Description

Thin aliases that forward to read_pqmethod(), read_qsort() (HTMLQ auto-detection), read_kenq(), and read_easyhtml_firebase(). These exist only so scripts written against the qmethod package continue to work; new code should call the ⁠read_*⁠ functions directly.

Usage

import.pqmethod(file, ...)

import.htmlq(file, ...)

import.kenq(file, ...)

import.easyhtmlq(file, ...)

Arguments

Value

A qsort_data object.

Description

Draws a blue-gradient heatmap of ⁠P(dominant factor = k)⁠ per participant, with a right-hand "Assignment" strip (orange-red gradient) showing the verdict "Strong / Mod. / Weak F-k" for each participant.

Usage

make_dominant_panel(fit, title = NULL, anonymize = TRUE)

Arguments

Value

A ggplot object.

Description

Draws $\Delta$ELPD against K with +/- 1.96 SE whiskers, shades the Sivula rejection band ($|\Delta \mathrm{ELPD}| < 4$), and marks the Sivula-selected K (red triangle), the ELPD peak (blue square), and an optional adopted K (orange diamond).

Usage

make_elpd_diff(run, title = NULL, adopted = NULL)

Arguments

Value

A ggplot object.

Description

Draws a ridgeline density of the posterior predictive correlation-matrix RMSE ($RMSE_R$) at every K in run, with a median tick per ridge.

Usage

make_ppc_ridge(run, title = NULL)

Arguments

Value

A ggplot object.

Description

Resolves rotational, sign, and label-permutation ambiguity in posterior draws of a factor model by the three-step MatchAlign procedure of Poworoznek et al. (2025): varimax rotation per draw, median-condition pivot selection, greedy L2 signed-permutation matching, and Procrustes rotation.

Usage

matchalign(Lambda_draws, F_draws)

Arguments

Value

A list with aligned Lambda and Fmat arrays, a congruence matrix of per-draw Tucker-phi per factor, and the pivot index used.

References

Poworoznek, E., Anceschi, N., Ferrari, F., & Dunson, D. (2025). Efficiently Resolving Rotational Ambiguity in Bayesian Matrix Sampling with Matching. Bayesian Analysis.

Description

Horizontal dot-and-whisker plot of the posterior viewpoint divergence D_j for every statement: posterior median with a 95% credible-interval whisker, statements ordered by the distinguishing probability P(D_j > delta | Y). A dashed rule marks the substantive separation delta and each point is shaded by P(D_j > delta | Y). By default the divergence summary stored on the fit is used (computed at the fit's delta); pass delta to recompute at a different separation without refitting.

Usage

plot_dist_cons(fit, delta = NULL, ...)

Arguments

Value

The input, invisibly.

Description

ELPD against K with +/- 1.96 SE whiskers, a solid vertical rule at the ELPD peak, and a dashed rule at the Sivula (parsimony) K. Both rules are drawn in the primary blue, distinguished by line type and by text annotations at the top of the axis. The title reports the peak-plus-Sivula case (agree, gap, reversed).

Usage

plot_elpd(run, ...)

Arguments

Value

The input, invisibly.

Description

One panel per scalar hyperparameter (default nu, sigma, tau) showing a two-tone kernel density: the full posterior in the light shade, the central credible interval re-shaded in the mid shade, and the posterior median marked by a short tick at the baseline. Coverage defaults to fit$brief$prob. When a parameter's support spans more than one order of magnitude (max / min > 20, all positive), the x-axis is automatically rendered on a log scale so the density shape is not crushed against the lower bound.

Usage

plot_hyper(fit, pars = c("nu", "sigma", "tau"), log = NULL, ...)

Arguments

Value

The input, invisibly.

Description

Horizontal dotchart of every participant's loading, one panel per factor, ranked by posterior mean. Each loading is drawn as a median point with nested 50 percent (thick) and 95 percent (thin) credible-interval whiskers. A faint grey vertical rule marks Brown's descriptive cut-off ⁠+/- 1.96 / sqrt(J)⁠. When highlight_flagged = TRUE, participants in fit$flagged[, k] are drawn as filled points in the accent colour.

Usage

plot_loading_posterior(fit, factors = NULL, highlight_flagged = TRUE, ...)

Arguments

Value

The input, invisibly.

Description

Tiled heatmap of ⁠P(argmax_k |Lambda[i, k]| = k)⁠ with one row per participant, one column per factor, rendered on a sequential blue ramp. A right-side tier strip encodes Strong / Moderate / Weak membership (per classify_membership()). A horizontal colourbar under the plot gives the probability scale. Rows are sorted by dominant factor first, then tier, then probability – so the block structure a reader wants to see is preserved.

Usage

plot_membership(fit, sort = TRUE, ...)

Arguments

Value

The input, invisibly.

Description

Histogram of the replicated correlation-matrix RMSE stored on fit$ppc$rmse.r: per draw, the RMSE between cor(Y_rep) and cor(Y_obs). Rendered in the bayesplot-idiom (filled bars, no border, suppressed y-axis) with the median and central credible interval marked.

Usage

plot_ppc(fit, breaks = 30, ...)

Arguments

Value

The input, invisibly.

Description

Boxplot of the per-draw Tucker's phi between each aligned loading column and the MatchAlign pivot, stored on fit$align_info$congruence. A semi-transparent strip of the individual draws is overlaid so bimodality – the visible signature of residual label-switching – is not hidden by the box. A dashed rule at 0.95 marks the conventional near-identity threshold.

Usage

plot_tucker(fit, ...)

Arguments

Value

The input, invisibly.

Description

For a single statement, draws the posterior median z-score per factor with nested 50 percent (thick) and 95 percent (thin) credible-interval whiskers, stacked vertically. The x-axis is symmetric around zero so the zero reference is centred.

Usage

plot_zscore_posterior(fit, statement, ...)

Arguments

Value

The input, invisibly.

Description

One panel per factor: horizontal dotchart of every statement's posterior median z-score with nested 50 percent (thick) and prob-coverage (thin) credible-interval whiskers. Statements are sorted once – by default, by the first factor's posterior mean – and that ordering is reused across panels so the reader can scan horizontally to compare factors. For a loadings-only view, see plot_loading_posterior(); for a single statement across factors, see plot_zscore_posterior().

Usage

## S3 method for class 'bayesqm_fit'
plot(x, sort_by = 1L, prob = NULL, cex.lab = 0.7, ...)

Arguments

Value

The input, invisibly.

Description

Posterior credible intervals for any subset of parameters in a bayesqm_fit. Method for rstantools::posterior_interval().

Usage

## S3 method for class 'bayesqm_fit'
posterior_interval(object, prob = 0.95, pars = NULL, regex_pars = NULL, ...)

Arguments

Value

A matrix with one row per parameter and two columns for the lower and upper interval bounds (as percent strings).

Description

Returns the priors actually used when the model was fit, as a printable bayesqm_prior object. Method for rstantools::prior_summary().

Usage

## S3 method for class 'bayesqm_fit'
prior_summary(object, ...)

## S3 method for class 'bayesqm_prior'
print(x, ...)

Arguments

Value

A bayesqm_prior data frame with columns parameter and prior.

Description

qsort_data() is the canonical constructor for a Q-sort dataset. validate_qsort(), check_distribution(), and infer_distribution() are the validation helpers used internally by the constructor and by the file readers. parse_distribution() accepts a numeric vector, a comma-/semicolon-/space-separated string, or a text file containing one of those.

Usage

qsort_data(
  Y,
  statements = NULL,
  participants = NULL,
  distribution = NULL,
  metadata = list(),
  source = "manual",
  validate = TRUE
)

validate_qsort(qdata, distribution = NULL)

check_distribution(Y, distribution)

infer_distribution(Y)

parse_distribution(x)

Arguments

Value

qsort_data() returns a qsort_data S3 list with fields Y, statements, participants, distribution, metadata, and source. validate_qsort() returns a list with valid, issues, warnings, and summary. check_distribution() returns a list with ok, non_conforming, and grid_values. infer_distribution() and parse_distribution() return integer vectors.

Description

Print, summary, and matrix conversion for qsort_data

Usage

## S3 method for class 'qsort_data'
print(x, ...)

## S3 method for class 'qsort_data'
summary(object, ...)

## S3 method for class 'qsort_data'
as.matrix(x, ...)

Arguments

Value

print() and summary() return the input invisibly; as.matrix() returns the ⁠J x N⁠ Q-sort matrix.

Description

read_qsort() auto-detects the file format from extension and content and dispatches to a specialised reader. The specialised readers are also exported for explicit use:

• read_qsort_csv(), read_qsort_excel() for generic CSV / Excel (with HTMLQ / FlashQ / Ken-Q auto-detection baked in)

• read_pqmethod() for PQMethod .DAT files

• read_kenq() for Ken-Q JSON or CSV

• read_kenq_excel() for multi-sheet Ken-Q Excel (Type 1 and Type 2, both old and Ver2 sub-formats)

• read_kade_zip() for KADE ZIP archives

• read_easyhtml_firebase() for Easy-HTMLQ Firebase JSON

• read_statements() for a standalone statement-text file

All readers return a qsort_data object in ⁠J x N⁠ orientation (statements as rows, participants as columns).

Usage

read_qsort(file, format = "auto", ...)

read_qsort_csv(
  file,
  orientation = c("auto", "statements_rows", "participants_rows"),
  id_col = c("auto", "first", "none"),
  statements = NULL,
  distribution = NULL,
  ...
)

read_qsort_excel(
  file,
  sheet = 1,
  orientation = c("auto", "statements_rows", "participants_rows"),
  id_col = c("auto", "first", "none"),
  statements = NULL,
  distribution = NULL,
  ...
)

read_pqmethod(file, statements_file = NULL)

read_kenq(file, format = c("auto", "json", "csv"))

read_kenq_excel(file)

read_kade_zip(file)

read_easyhtml_firebase(file)

read_statements(file, column = 1, id_column = NULL)

Arguments

Value

A qsort_data object, except read_statements() which returns a named character vector.

Description

Replaces the default f1..fK factor labels everywhere they appear on the fit: posterior-mean and credible-interval loading matrices, the factor-score matrices, the factor-characteristics table, the correlation matrix, the flagged matrix, the distinguishing / consensus table column names, and the factor dimension of the raw Lambda_draws / F_draws arrays.

Usage

rename_factors(fit, new_names)

Arguments

Value

The input fit with every factor label replaced.

Examples

fit <- demo_fit(N = 6, J = 10, K = 3, Td = 50, seed = 1)
fit <- rename_factors(fit, c("tradition", "innovation", "caution"))
colnames(fit$loa)

Description

Fits fit_bayesian() for K = 1..K_max and reports the ELPD peak (automated adoption), the Sivula (2025) parsimony diagnostic, and a case label (agree, gap, reversed) summarising their relationship. When the two agree the data supports that K; when they disagree the gap is itself a reportable finding.

Usage

run_bayes(
  Y,
  K_max = 5,
  stan_dir = NULL,
  elpd_diff_threshold = 4,
  se_ratio_threshold = 2,
  ...
)

select_k_peak(elpds, K_candidates)

select_k_sivula(
  elpds,
  loo_list,
  K_candidates,
  elpd_diff_threshold = 4,
  se_ratio_threshold = 2
)

Arguments

Value

run_bayes() returns a bayesqm_run object carrying the list of fits, the ELPD comparison table, and the peak / Sivula / case verdict. select_k_peak() and select_k_sivula() return an integer K.

Description

Opens a graphics device chosen from the file extension (.pdf, .svg, .png, .tiff, .jpeg), evaluates expr so whatever it draws lands on that device, and closes the device. expr is lazily evaluated, so a call like save_bayesqm_plot("fig.pdf", plot(fit)) does not draw to the current screen device first. If expr returns a ggplot object (for example, from ggplot2::autoplot()), it is print()ed onto the device.

Usage

save_bayesqm_plot(file, expr, width = 7.2, height = 5, dpi = 300)

Arguments

Value

The file path, invisibly.

Examples

fit <- demo_fit(N = 6, J = 10, K = 2, Td = 50, seed = 1)
f <- file.path(tempdir(), "fig_loadings.pdf")
save_bayesqm_plot(f, plot_loading_posterior(fit))
unlink(f)

Description

Returns the standardized-scale width of one forced-distribution category, ⁠1 / sd(forced positions)⁠. This is an alternative separation to the package default, the reliability-adjusted critical difference of critical_delta().

Usage

suggest_delta(distribution)

Arguments

Value

A single numeric value: the standardized width of one forced-distribution category.

Examples

suggest_delta(c(2, 3, 4, 6, 8, 6, 4, 3, 2))

Description

Linear-algebra utilities shared by MatchAlign and the recovery assessments. tucker_congruence(x, y) computes sum(x*y) / sqrt(sum(x^2) * sum(y^2)). procrustes_rotation(X, Target) finds the orthogonal R minimising ⁠||X R - Target||_F⁠ and returns X %*% R with R attached as attribute "rotation".

Usage

tucker_congruence(x, y)

procrustes_rotation(X, Target)

Arguments

Value

Tucker returns a scalar; Procrustes returns the rotated matrix with a "rotation" attribute.