Applied numerics and data workflows

Version 1.8 adds applied DSP, statistical inference, typed data analysis, local random state, scalar and multivariate state-space models, bounded expressions, and versioned model persistence without introducing new vector or matrix containers. The original design decisions and the audited 1.7/1.8 gap closure are in the applied-numerics design record and release gap-closure record.

Maturity: stable for the APIs documented on this page. The capability inventory lists the broader roadmap families that remain unsupported.

60-second workflow

uses

  MathBase.SharedTypes, EngineeringLib.DSP, StatsLib.Streaming;



var

  Signal: TDoubleArray;

  Spectrum: TSpectralEstimate;

  Summary: TOnlineStatistics;

  I: Integer;

begin

  Signal := TDoubleArray.Create(0, 1, 0, -1, 0, 1, 0, -1);

  Spectrum := TDSPKit.Welch(Signal, 4, 2, 8);

  Summary := TOnlineStatistics.Create;

  for I := 0 to High(Spectrum.Power) do

    Summary.Add(Spectrum.Power[I]);

  Writeln('mean spectral power = ', Summary.Mean:0:6);

end.

The complete applied pipeline example joins DSP, streaming statistics, polynomial fitting, typed PCA/k-means++, and Kalman filtering without private array conversions.

Choose a transform or convolution

Forward transforms use exp(-2*pi*i*k*n/N). fnBackward, the default, leaves the forward transform unscaled and divides the inverse by N. fnForward does the reverse, fnUnitary applies 1/sqrt(N) in both directions, and fnNone does not scale either direction. DFTReference is the small correctness oracle, not the throughput path.

Convolve returns Length(A)+Length(B)-1 samples. Correlate reports lags from -(Length(B)-1) to Length(A)-1. Empty convolution input produces an empty result. Inputs must be finite.

Overlap-add and overlap-save snapshot their impulse response, validate a whole block before advancing state, and expose state only through copied Tail/History values. Reset restores zero state. Use overlap-add plus Flush for a finite sequence; use overlap-save where every block must have the same length as its input. Both accept the deterministic direct/FFT selection enum and are compared with direct convolution in the test oracle.

Spectral analysis and filters

• Periodogram produces a one-sided PSD with frequencies in caller-supplied sample-rate units.
• Welch averages overlapping, windowed segment periodograms.
• ShortTimeFourierTransform returns a typed dense complex matrix whose rows are frames and columns are complete transform bins.
• CrossSpectrum returns complex cross power and magnitude-squared coherence.
• AnalyticSignal uses the standard doubled-positive-frequency convention.
• GetWindowMetrics reports coherent gain, equivalent noise bandwidth in bins, and RMS gain.
• ResampleLinear and ResampleRational provide deterministic interpolation helpers. They are not anti-alias filters; low-pass first when downsampling content above the new Nyquist frequency.
• DesignButterworthLowPass creates a second-order normalized low-pass biquad. TStreamingBiquad uses transposed direct form II and zero initial state.

Normalized cutoff is cycles/sample in (0, 0.5). FIR and biquad processing assume zero state after construction or Reset. A complete invalid block is rejected before state advances, including arithmetic overflow; spectral estimators reject a zero-energy window. Independent state records are reentrant; concurrent mutation of the same state is not safe.

Online and mergeable statistics

TOnlineStatistics uses weighted Welford/Pébay updates. Storage is O(1) regardless of sample count.

var

  Left, Right: TOnlineStatistics;

begin

  Left := TOnlineStatistics.Create(nfpReject);

  Right := TOnlineStatistics.Create(nfpReject);

  Left.AddWeighted(10, 2);

  Right.Add(20);

  Left.Merge(Right);

end;

nfpReject raises EStreamingStatsError on NaN or Infinity. nfpIgnore leaves count, weight, mean, and moments unchanged. Weights must always be finite and positive. PopulationVariance divides by total weight. SampleVariance uses the reliability-weight denominator sum(w)-sum(w^2)/sum(w) and requires effective sample size above one.

Reproducible local random state

TLocalRandom.Seeded creates a xoshiro256** generator with four explicit 64-bit state words. NextUInt64, NextUInt32, NextDouble, NextSingle, NextInteger, and NextNormal never use the RTL RandSeed. Split returns the current stream as a child and jumps the parent by 2^128 states.

GetState and SetState support exact replay. The all-zero state is invalid. This generator is suitable for simulation and deterministic sampling, not cryptography. Give each thread its own state.

Statistical inference and regression

StatsLib.Inference adds paired distribution operations and diagnostics:

• normal, exponential, and binomial PDF/PMF, LogPDF/LogPMF, CDF, Survival, Quantile, and caller-owned local-RNG Sample;
• normal, exponential, gamma, and binomial parameter estimates with likelihood, standard errors where identifiable, iteration status, and an explicit Identifiable flag;
• one-sample, paired, and Welch t tests, one-way ANOVA, contingency chi-square, Mann-Whitney U, and Bonferroni or Benjamini-Hochberg corrections; and
• SVD-based OLS diagnostics and logistic regression that reports separation or singular information through status and Identifiable.

Distribution endpoints follow their mathematical limits. Invalid parameters, non-finite samples, degenerate designs, and inconsistent shapes raise EInferenceError. Approximate p-values and standard errors are documented diagnostics, not substitutes for a certified regulatory statistics package. See the statistics guide for selection details.

Typed data analysis

MLLib.Analysis uses IDenseDoubleMatrix directly, with observations in rows and features in columns:

• TAnalysisKit.PCA centers data and delegates to the shared typed SVD. Components are rows, scores are observation rows, singular values descend, and explained ratios use total centered variance.
• KMeansPlusPlus uses seeded k-means++ initialization, deterministic tie-breaking, a finite iteration limit, and an inspectable Converged result.
• CreateValidationSplit and KFoldAssignments return reproducible row-index partitions. Preprocessing must be fitted on training rows only.
• FitStandardization learns finite column means and population scales from training data only; TransformStandardized applies that immutable snapshot.
• FitBinaryLDA uses the shared dense solve for the within-class scatter system. The optional non-negative ridge handles singular or nearly singular scatter.
• HierarchicalCluster supports single, complete, and average linkage with deterministic tie-breaking. CutHierarchy returns a requested flat cut.
• FitClassificationForest and FitRegressionForest build reproducible seeded bootstrap forests. Results include normalized impurity importance and OOB accuracy or OOB R-squared over rows receiving an OOB prediction.
• TKDTree owns an immutable typed-matrix snapshot. Query returns exact neighbours ordered by squared distance and then original row index.

All feature values must be finite. PCA needs at least two rows and non-zero centered variance. K-means++, hierarchical clustering, and forests target in-memory dense data. Forest feature importance is impurity-based and can favour continuous or high-cardinality variables; OOB score can be based on fewer than all training rows for small forests. The k-d tree is most useful for exact low-dimensional queries; high dimensions weaken pruning.

Scalar and multivariate state space

TScalarKalmanConfiguration defines transition, observation, process variance, and measurement variance for a scalar linear-Gaussian system. TScalarKalmanFilter owns only the current estimate and covariance.

Process returns estimates, posterior variances, innovations, innovation variances, and log likelihood. It validates the complete measurement block and updates a private working copy, so validation or numerical failure does not partially advance the caller state. Forecast does not mutate the filter. Covariance uses the Joseph update.

TMultivariateKalmanConfiguration snapshots dense transition, observation, process-covariance, and measurement-covariance matrices. TMultivariateKalmanFilter returns state means/covariances, innovations, innovation covariances, per-block log likelihood, and observation forecasts. Construction and updates validate dimensions, symmetry, and positive definiteness where required. A complete measurement matrix is processed on a working copy, so invalid input or a numerical failure leaves the filter unchanged.

Both stable models are time-invariant and linear-Gaussian. Controlled systems, missing observations, smoothing, and parameter estimation remain open.

Complexity, ownership, and errors

Allocating operations return independent arrays or typed matrices and do not retain borrowed inputs. Stateful constructors snapshot coefficients or data. Arrays are zero-indexed.

EDSPError, EStreamingStatsError, EInferenceError, ERandomStateError, EAnalysisError, and EStateSpaceError identify invalid parameters or numerical failures. No API on this page uses a service, foreign binary, network, GUI, or hidden global state.

Important open items

The 1.8 stable boundary does not claim equiripple filters, Chebyshev/elliptic/Bessel IIR design, wavelet packets, survival/factor analysis, robust covariance families, controlled or smoothed state-space models, or parallel/SIMD execution. Logistic regression is the only new generalized-linear-model family. See the capability inventory rather than inferring support from the roadmap.