mathmodels: A Powerful R Toolkit for Mathematical Modeling

An R package providing a versatile toolkit for mathematical modeling, developed as a companion to the book Mathematical Modeling: Algorithms and Programming Implementation (China Machine Press). It focuses on implementing rigorous algorithms in a user-friendly manner.

Current Version (0.0.14) introduces a data-cleaning toolkit — missing-value diagnostics with Little’s MCAR test (na_mcar_test()), missingness visuals (plot_na_heatmap(), plot_na_bar()), simple and model-based imputation (impute(), impute_model() with kNN / linear regression / decision trees), multiple imputation by FCS chained equations (impute_multiple()) with Rubin’s-rules pooling (rubin_pool(), pool_fit(); Barnard-Rubin small-sample df), and outlier handling (outlier()) — plus the Excel-style pivot_table(). Together with the previous release’s tidy statistical inference toolkit (stat_*() tests, bootstrap and permutation frameworks, export_table()), the multivariate statistics toolkit (mv_*()), the factor-analysis weighting weight_fa(), and the renamed weight_* / mf_* families. All this builds on the regression prediction toolkit, time-series workflow, epidemic modeling, differential equation models, grey prediction, Markov chain models, interpolation and curve fitting, and a rich suite of evaluation methods (AHP, Entropy, CRITIC, PCA, TOPSIS, Fuzzy, RSR, DEA).

Key Features

• Data Cleaning (new) — missing-value diagnostics (na_summary(), na_mcar_test() for Little’s MCAR test) with heatmap / bar visuals; imputation: simple fills (group mean / median / mode, linear & spline interpolation, constants), a unified model-based framework (impute_model() with kNN, linear regression, decision trees), and multiple imputation (impute_multiple(): FCS chained equations, pmm/norm for numeric columns, logreg/polyreg for categorical ones) with Rubin’s-rules pooling (rubin_pool() for scalar estimates, pool_fit() for lm/glm fits, incl. the Barnard-Rubin small-sample df adjustment); every changed cell is recorded for audit. Outlier detection and handling (outlier(): IQR / z-score / Hampel-MAD bounds, set-NA or cap). Excel-style pivot_table().
• Multivariate Statistics (new) — PCA (mv_pca()), EFA (mv_efa()), hierarchical and k-means clustering with validity metrics (mv_hclust(), mv_kmeans(), mv_cluster_metrics()), discriminant analysis (mv_discrim()), correspondence analysis (mv_corresp()), MDS (mv_mds()), canonical and partial correlation (mv_cancor(), mv_pcor()), all with tidy components and plot_mv_*() visuals.
• Statistical Inference — tidy hypothesis tests: t-test / Wilcoxon, ANOVA / Kruskal-Wallis, ANCOVA, correlation, chi-square, normality and variance-homogeneity tests (stat_*()), each returning one tidy row per test with effect sizes; generic bootstrap and permutation frameworks; export_table() renders results as Word/LaTeX three-line tables.
• Evaluation Models — AHP, Entropy weighting, CRITIC, CV, PCA and factor-analysis weighting, TOPSIS, Grey Relational Analysis (GRA), Rank Sum Ratio (RSR), Fuzzy Comprehensive Evaluation (FCE) with the mf_* family of membership functions, Data Envelopment Analysis (CCR/BCC/SBM, Malmquist), plus inequality measures (Gini, Theil Index), coupling coordination degree, and obstacle degree models.
• Prediction Models —
  • Grey prediction (GM(1,1), GM(1,N), Verhulst), Markov chain prediction (markov_chain(),GM11_markov()),
  • regression prediction toolkit (reg_lm(), reg_logistic(), reg_poisson(), reg_negbin() with stepwise selection; reg_predict(), reg_diagnostics(), plot_reg_predict(), plot_reg_residuals()),
  • lightweight time series toolkit built around ts_df: as_ts_df() for conversion, complete_ts_df() / impute_ts_df() for gap handling, ts_transform() / ts_back_transform() for Box-Cox / log / differencing workflows, ts_test() for stationarity tests, ts_stl() for STL decomposition, ts_ets(), ts_sarima(), and ts_arimax() for modeling, ts_forecast() for unified forecasting, plus plot_ts(), plot_ts_forecast(), plot_ts_acf(), plot_ts_pacf(), plot_ts_stl(), and plot_ts_residuals().
• interpolation and curve fitting — piecewise linear, cubic spline, polynomial, and Hermite interpolation (interp_linear(), interp_spline(), interp_poly(), interp_hermite()), plus curve fitting (poly_fit(), curve_fit(), growth_fit()) with automatic starting values for logistic, Gompertz, and Michaelis-Menten nonlinear models.
• Differential Equation Models — String-formula ode_solver() for arbitrary ODE systems; ready-to-use population models (Malthus, Logistic), epidemic compartment models (SI, SIS, SIR, SEIR), and Lotka–Volterra predator–prey model, all with a unified init + params interface. Epidemic visualization (plot_compartments(), plot_incidence(), plot_phase_si(), plot_Rt_estimate()) and metrics (epi_metrics() for R0, peak, attack rate).
• Tidyverse Integration — Seamlessly works with |> and dplyr/ggplot2 for smooth data manipulation and batch processing.

Installation

Once released on CRAN, install mathmodels with:

install.packages("mathmodels")

You can install the latest development version directly from GitHub:

remotes::install_github("zhjx19/mathmodels")

Getting Started

library(mathmodels)

# --- AHP (Analytic Hierarchy Process) ---
A = matrix(c(1,   1/2, 4, 3,   3,
             2,   1,   7, 5,   5,
             1/4, 1/7, 1, 1/2, 1/3,
             1/3, 1/5, 2, 1,   1,
             1/3, 1/5, 3, 1,   1), byrow = TRUE, nrow = 5)
AHP(A)

# --- Epidemic compartment modeling ---
result = ode_sir(
  init   = c(S = 990, I = 10, R = 0),
  params = c(beta = 0.15, gamma = 0.1),  # R0 = 1.5
  times  = seq(0, 100, by = 0.5))

# Visualize
plot_compartments(result, compartments = c("S", "I", "R"))

# Compute epidemic metrics
metrics = epi_metrics(result, beta = 0.15, gamma = 0.1, N = 1000)
metrics$R0         # basic reproduction number

# --- Time series ---
x = as_ts_df(log(AirPassengers))
fit = ts_sarima(x)
fc  = ts_forecast(fit, h = 12)
plot_ts_forecast(x, fc)           # forecast with CIs

# --- Statistical inference (tidy in, tidy out) ---
res = stat_t_test(sleep, .cols = extra, group = group)
res                               # one tidy row: statistic, p, CI, Cohen's d
export_table(res, output = "latex", caption = "Two-sample t-test")

# --- Multivariate statistics ---
r = mv_pca(mtcars, cyl:carb)
plot_mv_scree(r, type = "cum")    # cumulative variance scree

Learning More

For detailed documentation, tutorials, and in-depth examples on using the mathmodels package, please refer to the comprehensive online manual:

📘 mathmodels Package Manual - Simplifying Mathematical Modeling (Online Book, In Chinese)

This online book is the definitive guide to the package’s functionalities. Currently implemented modules include:

• Differential equation models: Malthus, Logistic, SI, SIS, SIR, SEIR, Lotka–Volterra with ode_solver() and ode_*() functions; epidemic visualization (plot_compartments(), plot_incidence(), plot_phase_si(), plot_Rt_estimate()) and metrics (epi_metrics())
• Statistical inference (0.0.13): tidy tests — stat_describe(), stat_t_test(), stat_wilcox_test(), stat_anova(), stat_kruskal(), stat_ancova(), stat_cor(), stat_chisq(), stat_normality(), stat_hov_test() — all with .by slice grouping and effect sizes; generic frameworks stat_bootstrap(), stat_perm_test(); export_table() for Word/LaTeX three-line tables
• Multivariate statistics (0.0.12): mv_pca(), mv_efa(), mv_hclust(), mv_kmeans(), mv_cluster_metrics(), mv_discrim(), mv_corresp(), mv_mds(), mv_cancor(), mv_pcor() with plot_mv_*() visuals
• Data cleaning (0.0.14): na_summary(), na_mcar_test() (Little’s MCAR test), plot_na_heatmap() / plot_na_bar(); imputation via impute() (group mean/median/mode, linear/spline, constant), the unified model-based framework impute_model() (kNN, linear regression, decision trees), and multiple imputation impute_multiple() (FCS chained equations) with Rubin’s-rules pooling rubin_pool() / pool_fit(); outlier handling with outlier() (detect / set NA / cap)
• Pivot table (0.0.14): pivot_table() for Excel-style aggregation and spreading
• Time series: ts_df(), as_ts_df(), validate_ts_df(), complete_ts_df(), impute_ts_df(), drop_na_ts_df(), ts_transform(), ts_back_transform(), ts_ets(), ts_sarima(), ts_arimax(), ts_stl(), ts_test(), ts_forecast(); visualization: plot_ts(), plot_ts_acf(), plot_ts_pacf(), plot_ts_forecast(), plot_ts_stl(), plot_ts_residuals()
• Regression prediction: reg_lm(), reg_logistic(), reg_poisson(), reg_negbin() with stepwise selection; reg_predict(), reg_diagnostics(), plot_reg_predict(), plot_reg_residuals()
• Interpolation & curve fitting: interp_linear(), interp_spline(), interp_poly(), interp_hermite() for interpolation; poly_fit(), curve_fit(), growth_fit() for curve fitting
• Markov chain prediction: markov_chain() and GM11_markov()
• Indicator data preprocessing
• AHP, Entropy weighting, CRITIC, CV, PCA and factor analysis weighting (weight_*() family, incl. weight_fa())
• Weight combination techniques (combine_weights())
• TOPSIS, Grey Relational Analysis (GRA)
• Rank Sum Ratio (RSR), Fuzzy Comprehensive Evaluation (FCE) with the mf_* membership-function family
• Data Envelopment Analysis (CCR/BCC/SBM, Malmquist)
• Inequality Measures (Gini, Theil Index)
• Regional Economics (LQ/HHI/EG Index)
• Coupling coordination degree and obstacle degree