Methods

dtfit methods -- mathematical reference

This folder is the rigorous mathematical reference for everything dtfit ships: each method's grounding in differential (non-Taylor) transformations, the full algorithm, the numerical optimizations and guards in the implementation, and where it is best applied. It is the deepest of the three doc layers -- for plain-language intuition see ../guides/, and for exact call signatures see ../api/.

Every batch method page carries a figure of one fit, and the measured comparisons are the experiment notebooks' -- see reproducing the tables.

The method catalog

dtfit is organized as one principle (match a signal's image, the discrete differential transform of its samples) realized across four execution tiers -- symbolic reference, batch, streaming, and batch-at-scale -- plus a high-level composition layer that routes to the right variant automatically.

tier	method	doc	runtime path	role
reference	DSB -- Differential Spectra Balance	dsb.md	symbolic (offline)	analytical ground truth / derivation
core	Image -- the discrete differential transform	image.md	numeric (offline)	the statistic every batch method fits on
batch	LSI -- Least-Squares Integral	lsi.md	numeric (offline)	accurate batch fit in the Legendre image, model selection, oscillatory recipe
batch	EAC -- Equal-Areas Criterion	eac.md	numeric (offline)	fast batch fit in the block image, the robust image
streaming	ImageFilter(basis="block") -- recursive EAC	equal_areas_filter.md	numeric (online)	real-time tracking via an area measurement + drift detection
streaming	ImageFilter(basis="legendre") -- recursive LSI	legendre_filter.md	numeric (online)	real-time tracking via a spectrum measurement (oscillatory plants)
scale	ImageStream -- streams, blocks, channels	scaling.md	numeric (offline/online)	one-pass / block / many-channel image map-reduce
compose	auto_forecast	auto.md	numeric (offline)	structured fit-then-extrapolate forecasting
stochastic	Stochastic series -- fit the functionals of a random process	stochastic.md	numeric (offline + online)	characterize / forecast / generate / track random (economic, financial) data

The production methods (LSI, EAC, the filters, ImageStream) are numeric successors to the symbolic originals (DSBI -> LSI, DSBE -> EAC). DSB is kept as the analytical reference. The split follows the dissertation's hard requirement that the runtime path carry no unbounded symbolic solve -- SymPy is allowed only once, offline, at derivation/initialization time. (For the full lineage -- which method came from which, and how the variants differ -- see ../guides/lineage-and-variants.md.)

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The common foundation: differential transformations

All methods operate on the differential spectrum of a signal rather than on its samples directly. For an analytic function $x(t)$ the (scaled) differential transform about $t_0=0$ is the sequence of discretes

$$ X(k) ;=; \frac{H^{k}}{k!},\left.\frac{d^{k}x}{dt^{k}}\right|_{t=0}, \qquad k = 0,1,2,\dots $$

where $H$ is a scale constant (in this library, the observation interval $H = t_{\max}-t_{\min}$). The inverse transform reconstructs the function:

$$ x(t) ;=; \sum_{k=0}^{\infty} X(k),\Big(\tfrac{t}{H}\Big)^{k}. $$

So $X(k)$ is the $k$-th Maclaurin coefficient of $x$, rescaled by $H^k$. The transform is a linear bijection between an analytic function and its spectrum on the radius of convergence: two analytic functions are equal iff their spectra agree discrete-by-discrete. This is the identity every method below leans on -- matching spectra (DSB, LSI), matching integrals of spectra (EAC, the filters), or accumulating spectra additively (ImageStream) all recover the function and hence its parameters.

The numeric methods compute this on the finite sample, not the infinite Taylor series above: the image (Methods-Image) is the discrete differential transform of a signal in a basis at a finite order, a fixed-size statistic built directly from (x, y) that every batch method -- LSI, EAC, the filters, ImageStream -- fits on.

The non-Taylor base (and why no table is needed)

The classical motivation is that elementary functions have closed-form discretes -- you never truncate an infinite Taylor series for them:

term	discrete $X(k)$
constant $c$	$c$ at $k=0$, else $0$
$t^{,n}$	$H^{k}$ at $k=n$, else $0$
$e^{w t}$	$(wH)^{k}/k!$
$\sin(w t)$	$\dfrac{(wH)^{k}}{k!}\sin!\frac{\pi k}{2}$
$\cos(w t)$	$\dfrac{(wH)^{k}}{k!}\cos!\frac{\pi k}{2}$

A transcendental model such as $a,e^{bt}$ or $a,\arctan(wt)$ is represented by the exact discrete of its basis function, so the parameters $a,b,w$ appear analytically in the spectrum and can be solved for or fitted.

Implementation note. dtfit does not maintain this table in code. In a spectra balance every equation matches model and data discretes at the same order $k$, so the $H^{k}$ factor cancels on both sides (see DSB) and the balance reduces to matching plain Maclaurin coefficients $g^{(k)}(0)/k!$. Those are produced for any expression by generic SymPy differentiation (_symbolic.py), which both reproduces the table above and extends the scheme to any differentiable model (rational, logarithmic, mixed) without a per-function rule. The numeric methods go further and replace the monomial spectrum with a better-conditioned orthogonal-polynomial spectrum (LSI).

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How the tiers relate

                 differential spectrum  X(k) = (H^k/k!) x^(k)(0)
                                  |
   +--------------+---------------+----------------+------------------+
   |              |               |                |                  |
 exact balance  weighted-L2    integral / area   additive over       compose /
 F(k;theta)=Z(k)    of spectra     matching          domain & channels   route
   |              |               |                |                  |
  DSB            LSI             EAC              ImageStream          auto_forecast
 (symbolic     (Legendre image      (block image           (streams, blocks,      (structured
  solve)        residual)        residual)         channels, map-reduce)  forecasting)
                  |               |
          run recursively, one sample at a time
                  |               |
              ImageFilter(basis="legendre")        ImageFilter(spectrum meas., basis="block")  (area meas.)

• DSB sets the empirical spectrum (from a polynomial pre-fit) equal to the symbolic model spectrum, discrete by discrete, and solves the algebraic system -- exact when well-posed, but symbolic and noise-sensitive.
• LSI relaxes the exact balance to a weighted integral least-squares discrepancy of the two spectra on an orthogonal basis -- numeric, noise-tolerant, accurate; with an oscillatory recipe.
• EAC matches integrals (areas) of model and data over windows rather than spectra -- local equations and a diagonal Gram; the robust image is the outlier defense, in either basis.
• ImageFilter(basis="block") / ImageFilter(basis="legendre") run EAC / LSI recursively, one sample at a time, through a shared ImageFilter, with drift detection and a calibrated result(); summing several filters' nis_ pools their innovations into one fused fault test (FilterBank and FusedChiSquareDetector are the experimental multi-stream harness, not part of this table).
• ImageStream exploits that the image is additive (so a stream reduces chunk-by-chunk, and blocks assemble through the basis transfer) and linear across channels (so many channels share one Gram) -- exact batch fitting at scale.
• auto_forecast routes a series to the model class its structure calls for, composing only the validated levers; a parameter fit needs no router -- fit(..., basis="auto") picks the basis by outcome.

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Relation to classical (Western) methods

The differential-transformation framing is native to the Pukhov school and largely invisible in the Western literature, so each batch method is restated here in the vocabulary a Western reviewer knows. The two production fitters are the two faces of one weighted-residual (Galerkin) identification:

dtfit method	classical identity	classical relatives (same goal, different route)
EAC (area matching)	Galerkin with piecewise-constant Haar test functions; the overdetermined four-windows-per-parameter form is an over-identified method of moments (GMM)	Prony / Matrix-Pencil / ESPRIT (algebraic pole recovery), variable projection
LSI (orthogonal-basis spectral match)	a spectral ($p$-version) Galerkin projection / weighted integral least-squares in a Legendre basis	classical method of moments (the monomial form is LSI's ill-conditioned ancestor), variable projection, Prony/ESPRIT for cycles

So EAC and LSI are the $h$-version (local Haar) and $p$-version (global spectral) of the same projection, both derivable from classical moment-matching (Pearson, Hansen's GMM). The per-method pages develop this (EAC, LSI), and the experimental suite runs all of these classical methods as runnable baselines -- see the baselines page.

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Reproducing the tables

The experiments are Jupyter notebooks in the separate dtfit-experimental package (install with pip install -e "packages/dtfit-experimental[bench]"). A notebook is the experiment and its report, and re-running it reproduces its tables and figures:

python -m dtfit_experimental.study.download_data   # fetch the public datasets (once)
jupyter nbconvert --to notebook --execute --inplace \
    packages/dtfit-experimental/experiments/method/15_families.ipynb

The figures on the method pages are illustrations of one fit each. The notebooks, the datasets and a summary of each result are indexed on the experiments page; the adaptations in trial and the established baselines each method is measured against are documented in ../experimental/. Baselines include SciPy curve_fit (Levenberg-Marquardt NLS, the gold standard), robust NLLS, numpy.polyfit, a Gaussian process, ARIMA/ETS/Theta, MLP/LSTM nets, an EKF, RLS and a constant-acceleration Kalman filter -- plus the Western parameter-estimation lineage (Prony, Matrix-Pencil / ESPRIT, variable projection, method of moments) and a classical-estimator twin of the stochastic model (ADF / OLS / periodogram / DFA / GARCH-QMLE), so the differential-transformation route is compared against the textbook toolkit it is meant to improve on.