Applied Statistical Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Applied Statistical Mathematics layer within the Kireth Research Semantic Library. Applied statistical mathematics studies the mathematical methods used to collect, model, estimate, test, predict and reason from uncertain or variable data. This parent ontology connects probability, sampling, estimation, hypothesis testing, regression, Bayesian inference, multivariate analysis, time series, stochastic processes, experimental design, survival analysis, spatial statistics, statistical learning, uncertainty quantification and computational statistics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Applied Statistical Mathematics"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Applied Statistical Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/applied-statistical-mathematics.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = AppliedStatisticalMathematics
 ENTITY_CLASS = AppliedStatisticalInferenceFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Applied statistical mathematics is the branch of applied mathematics concerned with modelling data, variability and uncertainty using probability, estimation, inference, prediction and statistical computation."
 SYMBOLIC_FORMS = [
  "theta_hat=argmax_theta L(theta|x)",
  "CI=estimate +/- critical_value*SE",
  "Y=X*beta+epsilon",
  "P(theta|D) proportional_to P(D|theta)P(theta)"
 ]
 SEMANTIC_ROLE = "Provides the parent semantic framework for applied probability, statistical modelling, estimation, hypothesis testing, regression, Bayesian inference, time series, multivariate analysis, experimental design, stochastic modelling and statistical computation."
}

SCOPE {
 INCLUDE = [
  AppliedStatisticalMathematics,
  AppliedStatistics,
  StatisticalInference,
  DescriptiveStatistics,
  InferentialStatistics,
  ProbabilityModel,
  RandomVariable,
  ProbabilityDistribution,
  Expectation,
  Variance,
  Covariance,
  Correlation,
  Sampling,
  Population,
  Sample,
  SamplingDistribution,
  Estimator,
  Estimate,
  Bias,
  Consistency,
  Efficiency,
  Sufficiency,
  MaximumLikelihood,
  MethodOfMoments,
  ConfidenceInterval,
  HypothesisTest,
  NullHypothesis,
  AlternativeHypothesis,
  TestStatistic,
  PValue,
  SignificanceLevel,
  TypeIError,
  TypeIIError,
  StatisticalPower,
  EffectSize,
  Regression,
  LinearRegression,
  MultipleRegression,
  LogisticRegression,
  GeneralisedLinearModel,
  NonlinearRegression,
  RegularisedRegression,
  Residual,
  ModelFit,
  ModelSelection,
  CrossValidation,
  BayesianStatistics,
  PriorDistribution,
  Likelihood,
  PosteriorDistribution,
  PosteriorPredictiveDistribution,
  CredibleInterval,
  BayesianDecisionTheory,
  MultivariateStatistics,
  MultivariateDistribution,
  CovarianceMatrix,
  PrincipalComponentAnalysis,
  FactorAnalysis,
  DiscriminantAnalysis,
  ClusterAnalysis,
  TimeSeries,
  Stationarity,
  Autocorrelation,
  AutoregressiveModel,
  MovingAverageModel,
  ARIMA,
  StateSpaceModel,
  Forecasting,
  SpectralAnalysis,
  StochasticProcess,
  MarkovProcess,
  GaussianProcess,
  ExperimentalDesign,
  Randomisation,
  Blocking,
  FactorialDesign,
  AnalysisOfVariance,
  ANOVA,
  SurvivalAnalysis,
  HazardFunction,
  SurvivalFunction,
  Censoring,
  KaplanMeierEstimator,
  CoxProportionalHazardsModel,
  SpatialStatistics,
  SpatialProcess,
  Kriging,
  Geostatistics,
  PointProcess,
  Resampling,
  Bootstrap,
  Jackknife,
  PermutationTest,
  MonteCarloMethod,
  MarkovChainMonteCarlo,
  MCMC,
  StatisticalLearning,
  Classification,
  Prediction,
  UncertaintyQuantification,
  MissingData,
  Imputation,
  RobustStatistics,
  NonparametricStatistics,
  KernelDensityEstimation,
  ComputationalStatistics,
  StatisticalSimulation,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  PureProbabilityTheoryWithoutAppliedStatisticalInference,
  PureDataVisualisationWithoutStatisticalModel,
  MachineLearningWithoutStatisticalInterpretation,
  DeterministicNumericalAnalysisWithoutUncertaintyOrData
 ]
 DISAMBIGUATION_RULE = "When data, uncertainty, sampling variation, parameter estimation, statistical testing, prediction or stochastic modelling are central, interpret the context as applied statistical mathematics."
}

CORE_CONCEPTS {
 AppliedStatisticalMathematics = "Applied mathematical study of data, variability, uncertainty and inference."
 StatisticalModel = "Mathematical representation of how observed data are generated."
 Estimator = "Rule mapping observed data to an estimate of an unknown parameter."
 SamplingDistribution = "Distribution of a statistic across repeated samples under a sampling model."
 ConfidenceInterval = "Frequentist interval procedure with stated long-run coverage."
 HypothesisTest = "Procedure evaluating evidence against a specified null hypothesis."
 Regression = "Statistical modelling of response-predictor relationships."
 BayesianInference = "Inference based on posterior distributions formed from likelihood and prior information."
 TimeSeries = "Ordered stochastic observations indexed by time."
 UncertaintyQuantification = "Mathematical characterisation of uncertainty in parameters, predictions or models."
}

RELATIONS {
 AppliedStatisticalMathematics --is_branch_of--> AppliedMathematics
 AppliedStatisticalMathematics --uses--> Probability
 StatisticalInference --uses--> SamplingDistribution
 Estimator --estimates--> Parameter
 MaximumLikelihood --uses--> Likelihood
 ConfidenceInterval --quantifies--> SamplingUncertainty
 HypothesisTest --uses--> TestStatistic
 StatisticalPower --depends_on--> EffectSize
 Regression --models--> ConditionalRelationship
 Residual --measures--> ModelDiscrepancy
 ModelSelection --compares--> StatisticalModels
 BayesianStatistics --uses--> PriorDistribution
 PosteriorDistribution --combines--> PriorAndLikelihood
 MultivariateStatistics --uses--> CovarianceMatrix
 PrincipalComponentAnalysis --uses--> EigenDecomposition
 TimeSeries --uses--> Autocorrelation
 ARIMA --models--> TemporalDependence
 GaussianProcess --models--> CorrelatedRandomFunction
 ExperimentalDesign --controls--> DataCollectionStructure
 Randomisation --reduces--> Confounding
 SurvivalAnalysis --models--> TimeToEvent
 SpatialStatistics --models--> SpatialDependence
 Bootstrap --approximates--> SamplingDistribution
 MCMC --samples_from--> PosteriorDistribution
 RobustStatistics --reduces--> OutlierSensitivity
}

OPERATORS {
 SAMPLE_MEAN(x) := x_bar=(1/n)*sum_i x_i
 SAMPLE_VARIANCE(x) := s2=(1/(n-1))*sum_i(x_i-x_bar)^2
 STANDARD_ERROR_MEAN(s,n) := SE=s/sqrt(n)
 Z_STATISTIC(est,null,SE) := z=(est-null)/SE
 CONFIDENCE_INTERVAL(est,SE,z) := [est-z*SE,est+z*SE]
 LIKELIHOOD(theta,x) := L(theta|x)=product_i f(x_i|theta)
 MAXIMUM_LIKELIHOOD(theta,x) := theta_hat=argmax_theta L(theta|x)
 BAYES_RULE(theta,D) := P(theta|D) proportional_to P(D|theta)*P(theta)
 LINEAR_MODEL(X,beta,epsilon) := Y=X*beta+epsilon
 OLS_ESTIMATOR(X,Y) := beta_hat=(X^T X)^(-1)X^T Y when_invertible
 LOGISTIC_MODEL(X,beta) := p=1/(1+exp(-X*beta))
 AUTOCOVARIANCE(X,tau) := gamma(tau)=Cov(X_t,X_(t+tau))
 BOOTSTRAP_SAMPLE(x) := sample_with_replacement(x,n)
}

AXIOMS {
 A1 = "Applied statistical inference requires an explicit data-generating or sampling model."
 A2 = "Population, sample and parameter semantics must be distinguished."
 A3 = "Statistical estimates require uncertainty quantification."
 A4 = "Association does not by itself establish causation."
 A5 = "Model assumptions affect the validity of inference and prediction."
 A6 = "Dependence between observations must be modelled when present."
 A7 = "Statistical significance and practical importance are distinct concepts."
 A8 = "Applied statistical conclusions require model checking, uncertainty analysis and reproducibility."
}

REASONING {
 R1_CONTEXT = identify(question,data,population,sample,variables,dependence,uncertainty)
 R2_MODEL = select(distribution,regression,time_series,multivariate,survival,spatial,stochastic_or_Bayesian_model)
 R3_ESTIMATION = estimate(parameters,standard_errors,posterior_or_sampling_distribution)
 R4_TESTING = formulate(null,alternative,test_statistic,error_rates)
 R5_DIAGNOSTICS = check(residuals,assumptions,outliers,dependence,calibration)
 R6_PREDICTION = compute(predictions,forecast_intervals,posterior_predictive_or_confidence_bands)
 R7_MODEL_SELECTION = compare(models,penalised_criteria,cross_validation)
 R8_ROBUSTNESS = analyse(sensitivity,missingness,outliers,distributional_assumptions)
 R9_COMPUTATION = apply(bootstrap,MCMC,MonteCarlo,numerical_optimisation_if_required)
 R10_VALIDATION = assess(generalisation,reproducibility,uncertainty,practical_relevance)
}

CONSTRAINTS {
 C1 = "A p-value is not the probability that the null hypothesis is true."
 C2 = "A confidence interval is not ordinarily a posterior probability interval."
 C3 = "Correlation does not establish causal effect."
 C4 = "Model misspecification can invalidate standard errors and predictions."
 C5 = "Independent-observation methods can fail under serial, spatial or clustered dependence."
 C6 = "Large samples can make trivial effects statistically significant."
 C7 = "Small samples can produce unstable estimates and low power."
 C8 = "Missing-data mechanisms can materially change valid inference."
 C9 = "Model-selection procedures can bias naive post-selection inference."
 C10 = "Out-of-sample validation is distinct from in-sample fit."
}

SEMANTIC_EQUIVALENCE {
 "AppliedStatisticalMathematics" <=> "AppliedStatisticalMethods"
 "MaximumLikelihood" <=> "MaximumLikelihoodEstimation"
 "GeneralisedLinearModel" <=> "GLM"
 "PrincipalComponentAnalysis" <=> "PCA"
 "MarkovChainMonteCarlo" <=> "MCMC"
 "AnalysisOfVariance" <=> "ANOVA"
}

VALIDATION {
 V1 = "Population and sample are explicit."
 V2 = "Variables and measurement scales are documented."
 V3 = "Statistical assumptions are checked."
 V4 = "Parameter estimates include uncertainty."
 V5 = "Dependence structure is modelled where relevant."
 V6 = "Model diagnostics are reported."
 V7 = "Missing-data handling is explicit."
 V8 = "Predictive models are validated out of sample where feasible."
 V9 = "Computational procedures report convergence where relevant."
 V10 = "Results distinguish statistical from practical significance."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 APPLIED_STATISTICAL_MATHEMATICS_DEFINITION = 0.99
 ESTIMATION_REASONING = 0.99
 TESTING_REASONING = 0.99
 REGRESSION_REASONING = 0.99
 BAYESIAN_REASONING = 0.98
 TIME_SERIES_REASONING = 0.98
 MULTIVARIATE_REASONING = 0.98
 COMPUTATIONAL_REASONING = 0.98
 POLICY = "Reduce confidence when sampling design, dependence, distribution, missingness, model assumptions, prior structure, validation data or computational convergence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AppliedStatisticalMathematicsReasoning,
  SamplingReasoning,
  EstimationReasoning,
  HypothesisTestingReasoning,
  RegressionReasoning,
  BayesianReasoning,
  MultivariateReasoning,
  TimeSeriesReasoning,
  ExperimentalDesignReasoning,
  SurvivalAnalysisReasoning,
  SpatialStatisticsReasoning,
  ResamplingReasoning,
  MonteCarloReasoning,
  MCMCReasoning,
  StatisticalLearningReasoning,
  UncertaintyQuantificationReasoning,
  RobustStatisticsReasoning,
  ComputationalStatisticsReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "applied statistical mathematics" -> AppliedStatisticalMathematics
  "applied statistics" -> AppliedStatisticalMathematics
  "maximum likelihood estimation" -> MaximumLikelihood
  "GLM" -> GeneralisedLinearModel
  "PCA" -> PrincipalComponentAnalysis
  "MCMC" -> MarkovChainMonteCarlo
  "ANOVA" -> AnalysisOfVariance
 }
}

EXECUTION_RULES {
 E1 = "If data and uncertainty are analysed through statistical models, activate AppliedStatisticalMathematics."
 E2 = "If parameter estimation is central, identify estimator properties and uncertainty."
 E3 = "If evidence against a null hypothesis is requested, use hypothesis-testing reasoning."
 E4 = "If predictor-response relationships are central, route to regression or generalised linear models."
 E5 = "If prior information is explicit, consider Bayesian inference."
 E6 = "If temporal ordering is central, route to time-series or stochastic-process reasoning."
 E7 = "If spatial dependence is central, route to spatial statistics."
 E8 = "If analytical sampling distributions are unavailable, consider bootstrap or Monte Carlo methods."
 E9 = "Any statistical result should be validated for assumptions, uncertainty, robustness and reproducibility."
}

RECURSION {
 ENTRY = AppliedStatisticalMathematics
 EXPAND(AppliedStatisticalMathematics) -> [
  AppliedStatistics, StatisticalInference, DescriptiveStatistics,
  InferentialStatistics, ProbabilityModel, RandomVariable,
  ProbabilityDistribution, Expectation, Variance, Covariance, Correlation,
  Sampling, Population, Sample, SamplingDistribution, Estimator, Estimate,
  Bias, Consistency, Efficiency, Sufficiency, MaximumLikelihood,
  MethodOfMoments, ConfidenceInterval, HypothesisTest, NullHypothesis,
  AlternativeHypothesis, TestStatistic, PValue, SignificanceLevel,
  TypeIError, TypeIIError, StatisticalPower, EffectSize, Regression,
  LinearRegression, MultipleRegression, LogisticRegression,
  GeneralisedLinearModel, NonlinearRegression, RegularisedRegression,
  Residual, ModelFit, ModelSelection, CrossValidation, BayesianStatistics,
  PriorDistribution, Likelihood, PosteriorDistribution,
  PosteriorPredictiveDistribution, CredibleInterval, BayesianDecisionTheory,
  MultivariateStatistics, MultivariateDistribution, CovarianceMatrix,
  PrincipalComponentAnalysis, FactorAnalysis, DiscriminantAnalysis,
  ClusterAnalysis, TimeSeries, Stationarity, Autocorrelation,
  AutoregressiveModel, MovingAverageModel, ARIMA, StateSpaceModel,
  Forecasting, SpectralAnalysis, StochasticProcess, MarkovProcess,
  GaussianProcess, ExperimentalDesign, Randomisation, Blocking,
  FactorialDesign, AnalysisOfVariance, SurvivalAnalysis, HazardFunction,
  SurvivalFunction, Censoring, KaplanMeierEstimator,
  CoxProportionalHazardsModel, SpatialStatistics, SpatialProcess, Kriging,
  Geostatistics, PointProcess, Resampling, Bootstrap, Jackknife,
  PermutationTest, MonteCarloMethod, MarkovChainMonteCarlo,
  StatisticalLearning, Classification, Prediction, UncertaintyQuantification,
  MissingData, Imputation, RobustStatistics, NonparametricStatistics,
  KernelDensityEstimation, ComputationalStatistics, StatisticalSimulation,
  Reproducibility
 ]
 EXPAND(StatisticalFamilies) -> [
  StatisticalInference,
  Regression,
  BayesianStatistics,
  MultivariateStatistics,
  TimeSeries,
  ExperimentalDesign,
  SurvivalAnalysis,
  SpatialStatistics,
  ComputationalStatistics
 ]
 STOP_CONDITION = "Stop when data structure, sampling assumptions, model family, estimation, uncertainty, diagnostics, prediction and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"
 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Computational Mathematics Ontologies",
  "Optimisation Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Regression Ontologies",
  "Bayesian Statistics Ontologies",
  "Time Series Ontologies",
  "Stochastic Process Ontologies",
  "Multivariate Statistics Ontologies",
  "Experimental Design Ontologies",
  "Survival Analysis Ontologies",
  "Spatial Statistics Ontologies",
  "Optimisation Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Applied Statistical Mathematics -> Applied Mathematics for parent semantics",
  "Applied Statistical Mathematics -> Probability & Statistics for foundational inference",
  "Applied Statistical Mathematics -> Regression for predictor-response modelling",
  "Applied Statistical Mathematics -> Bayesian Statistics for posterior inference",
  "Applied Statistical Mathematics -> Time Series for temporal dependence",
  "Applied Statistical Mathematics -> Stochastic Processes for random evolution",
  "Applied Statistical Mathematics -> Multivariate Statistics for vector-valued data",
  "Applied Statistical Mathematics -> Experimental Design for data collection",
  "Applied Statistical Mathematics -> Survival Analysis for time-to-event data",
  "Applied Statistical Mathematics -> Spatial Statistics for spatial dependence",
  "Applied Statistical Mathematics -> Optimisation for parameter estimation",
  "Applied Statistical Mathematics -> Computational Mathematics for simulation and numerical inference"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible statistical methods, models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve population-versus-sample semantics.",
  "Preserve parameter-versus-estimator distinction.",
  "Preserve frequentist-versus-Bayesian distinction.",
  "Preserve estimation-versus-testing distinction.",
  "Preserve association-versus-causation distinction.",
  "Preserve dependence and uncertainty semantics.",
  "Preserve model-validation semantics.",
  "Preserve machine-readable applied-statistical-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Applied statistical mathematics uses probability and statistical models to estimate, test, predict and reason from uncertain data."
 ESTIMATION_SUMMARY = "Estimators convert samples into parameter estimates whose bias, variance, consistency and efficiency can be analysed."
 TESTING_SUMMARY = "Hypothesis tests compare observed statistics with null-model expectations while controlling error rates."
 REGRESSION_SUMMARY = "Regression models quantify relationships between outcomes and explanatory variables."
 BAYESIAN_SUMMARY = "Bayesian methods combine prior distributions with likelihoods to produce posterior inference."
 TIME_SERIES_SUMMARY = "Time-series models represent temporal dependence, stationarity and forecasting structure."
 COMPUTATIONAL_SUMMARY = "Bootstrap, Monte Carlo and MCMC methods support inference when analytic solutions are unavailable."
 VALIDATION_SUMMARY = "Applied statistical analyses should be checked for assumptions, uncertainty, robustness, dependence, predictive validity and reproducibility."
}

FLOW {
 INPUT
 -> detect_applied_statistical_mathematics_context
 -> identify_population_sample_variables_measurement_scales_dependence_and_statistical_question
 -> select_probability_model_and_statistical_family
 -> estimate_parameters_and_uncertainty
 -> if_testing_formulate_null_alternative_test_statistic_and_error_rates
 -> if_regression_fit_predictor_response_model_and_check_residuals
 -> if_Bayesian_specify_prior_likelihood_posterior_and_posterior_predictive
 -> if_temporal_or_spatial_model_dependence_structure
 -> if_analytic_inference_unavailable_apply_bootstrap_Monte_Carlo_or_MCMC
 -> assess_missingness_outliers_sensitivity_model_selection_and_cross_validation
 -> validate_assumptions_uncertainty_generalisation_reproducibility_and_practical relevance
 -> route_to_applied_mathematics_probability_statistics_regression_Bayesian_statistics_time_series_stochastic_processes_multivariate_statistics_experimental_design_survival_analysis_spatial_statistics_optimisation_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Applied Statistical Mathematics layer defines mathematical methods for reasoning from variable and uncertain data. It supports probability models, populations, samples and sampling distributions, estimators, bias, consistency, efficiency and sufficiency, maximum likelihood and method-of-moments estimation, confidence intervals and hypothesis tests, p-values, error rates, power and effect sizes, linear, logistic, nonlinear and generalised linear regression, model selection and cross-validation, Bayesian priors, likelihoods, posteriors and predictive distributions, multivariate statistics, covariance matrices, PCA, factor and discriminant analysis, time series, autoregressive, moving-average, ARIMA and state-space models, stochastic and Gaussian processes, experimental design and ANOVA, survival analysis, spatial statistics and kriging, bootstrap, jackknife, permutation tests, Monte Carlo and MCMC, statistical learning, uncertainty quantification, missing-data methods, robust and nonparametric statistics, kernel density estimation, computational statistics and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-APPLIED-STATISTICAL-MATHEMATICS-V1"
}

}

END_FLOW