universitybox 0.1.4

DNA — Dynamic Nonlinear Adaptive time series forecaster and audience segmentation toolkit

universitybox

DNA — Dynamic Nonlinear Adaptive Time Series Forecaster

A pure-NumPy/SciPy time series forecasting library built around the DNA model — a three-stage hierarchical forecaster combining classical decomposition, nonlinear basis expansion, and adaptive Kalman filtering.

No TensorFlow. No PyTorch. No black boxes. Every equation is documented.

Full mathematical derivations: MATH.md on GitHub

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Install

pip install universitybox

With optional extras:

pip install "universitybox[full]"   # + pandas + matplotlib
pip install "universitybox[viz]"    # + matplotlib only
pip install "universitybox[data]"   # + pandas only

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Quick start

import numpy as np
from universitybox import DNA

# Any 1-D time series
y = np.array([112, 118, 132, 129, 121, 135, 148, 148, 136, 119,
              104, 118, 115, 126, 141, 135, 125, 149, 170, 170,
              158, 133, 114, 140, 145, 150, 178, 163, 172, 178,
              199, 199, 184, 162, 146, 166, 171, 180, 193, 181])

model = DNA(period=12)
model.fit(y)

point_forecast      = model.forecast(h=12)
lower, upper        = model.predict_interval(h=12, level=0.95)
test_metrics        = model.evaluate(y[-6:])

model.summary()

---

The DNA Model

DNA decomposes the time series into three progressively finer layers:

y(t) = mu(t)          Stage D — trend,    Henderson moving average
     + s(t)           Stage D — seasonal, Fourier OLS
     + f(Phi(x_t))    Stage N — nonlinear correction, Ridge + RBF features
     + l(t)           Stage A — adaptive correction, Kalman LLT filter
     + noise

Each stage is fitted on the residual of the previous stage. The final forecast is an inverse-variance weighted combination of all four components.

Stage D — Decomposition

• Trend: Henderson symmetric moving average of half-length m. Weights minimise the third-difference roughness of the trend while exactly reproducing polynomials up to degree 2 (closed-form, Doherty 2001).
• Seasonal: Fourier regression of order K on the detrended series, estimated by OLS. Normalised to zero mean over each complete period.
• Period: Auto-estimated from the periodogram when period='auto'.

Stage N — Nonlinear Basis Expansion

Feature map composed of three dictionaries:

Phi(t) = [ polynomial(t)  |  AR lags of residual  |  RBF kernels ]

• Polynomial: time index normalised to [0,1], up to degree p.
• AR lags: standardised lagged residuals (lags 1 to L).
• RBF: squared-exponential kernels centred by k-means++ seeding, bandwidth set by the median heuristic.

Ridge regression (L2 regularised least squares, closed-form Cholesky solve) fits the feature map to the D-stage residual.

Stage A — Adaptive Kalman Filter

Local Linear Trend (LLT) state-space model on the N-stage residual:

State:       x(t) = [level, slope]
Transition:  x(t) = F x(t-1) + noise_process
Observation: r(t) = H x(t)  + noise_obs

Kalman filter recursion (prediction → innovation → gain → update). Optional MLE estimation of noise parameters (kalman_mle=True). h-step forecast: level(n) + h * slope(n).

Ensemble Combination

Component forecasts combined as:

y_hat(n+h) = alpha * trend(n+h)
           + beta  * seasonal(n+h)
           + gamma * nonlinear(n+h)
           + delta * adaptive(n+h)

Weights computed by:

• 'iv' — inverse-variance (Bates & Granger 1969), default
• 'ols' — non-negative least-squares stacking
• 'equal' — 0.25 each

Prediction Intervals

• Analytical: forecast ± z * sigma * sqrt(h), where sigma is the in-sample RMSE.
• Bootstrap: resample in-sample residuals B times, propagate as cumulative shocks, take empirical quantiles.

---

All parameters

DNA(
    period         = "auto",  # int or 'auto' — seasonal period (4, 12, 7, ...)
    trend_window   = "auto",  # Henderson filter half-length m
    n_fourier      = 3,       # Fourier harmonics K
    poly_degree    = 2,       # polynomial degree in N-stage feature map
    n_lags         = 4,       # AR lag count in N-stage feature map
    n_rbf          = 10,      # RBF centres (k-means++ selected)
    rbf_gamma      = "auto",  # RBF bandwidth ('auto' = median heuristic)
    ridge_alpha    = 1e-3,    # L2 regularisation lambda
    kalman_q_level = 1e-4,    # Kalman level process noise
    kalman_q_slope = 1e-6,    # Kalman slope process noise
    kalman_obs_var = 1e-2,    # Kalman observation noise
    kalman_mle     = False,   # estimate Kalman noise by MLE
    ensemble       = "iv",    # 'iv' | 'equal' | 'ols'
    ci_method      = "analytical",  # 'analytical' | 'bootstrap'
    ci_bootstrap_n = 500,     # bootstrap replications
    random_state   = None,    # reproducibility seed
)

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API reference

Method / Property	Description
fit(y)	Fit the model. y must be 1-D, finite, length >= 4.
forecast(h)	Point forecasts for horizons 1 to h. Returns array of shape (h,).
predict_interval(h, level=0.95)	Returns (lower, upper) arrays of shape (h,).
evaluate(y_test)	MAE, RMSE, MAPE, sMAPE, MASE vs held-out test set.
fitted_values	In-sample fitted values, shape (n,).
residuals	In-sample residuals y - y_hat, shape (n,).
components	Dict: trend, seasonal, nonlinear, adaptive — each shape (n,).
weights	Dict: alpha, beta, gamma, delta ensemble weights.
summary()	Print model card.

---

Metrics

from universitybox import metrics

metrics.mae(y_true, y_pred)
metrics.rmse(y_true, y_pred)
metrics.mape(y_true, y_pred)
metrics.smape(y_true, y_pred)
metrics.mase(y_true, y_pred, y_train=y_train, period=4)
metrics.crps_gaussian(y_true, mu=fc, sigma=sigma_h)
metrics.summary(y_true, y_pred)          # dict of all metrics

---

Audience segmentation (Club)

from universitybox.segments import Club

category_map = {
    "lenovo":   "Technology",
    "hp store": "Technology",
    "samsung":  "Technology",
    "zara":     "Fashion",
}

club = Club(category_map=category_map, min_cta=6)
club.fit(events_df)          # DataFrame: user_id, brand, cta_count

club.size("Technology")      # int — number of members
club.share("Technology")     # float — fraction of classified users
club.members("Technology")   # list of user_ids
club.summary()               # dict: all clubs with size + share

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Extending DNA — adding a custom forecaster

All forecasters extend BaseForecaster:

from universitybox.forecast._base import BaseForecaster
import numpy as np

class MyForecaster(BaseForecaster):
    def __init__(self, alpha=0.3):
        self.alpha = alpha

    def fit(self, y: np.ndarray, **kwargs) -> "MyForecaster":
        y = self._validate_y(y)
        # ... fit logic ...
        self._insample_rmse = float(np.std(y))
        return self

    def forecast(self, h: int) -> np.ndarray:
        # ... return array of shape (h,) ...
        ...

predict_interval() and score() are provided by the base class automatically.

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Mathematical documentation

Full derivations for every formula in the package: MATH.md, covering:

1. Notation and problem statement
2. The DNA decomposition equation
3. Stage D — Henderson filter weights (closed-form), Fourier OLS, periodogram period estimation
4. Stage N — polynomial / AR-lag / RBF feature map, k-means++ seed selection, median heuristic bandwidth, Ridge regression (primal & dual), RKHS interpretation
5. Stage A — LLT state-space model, Kalman filter recursion (all equations), h-step forecast, MLE log-likelihood
6. Ensemble combination — inverse-variance weighting, OLS stacking
7. Prediction intervals — analytical Gaussian and bootstrap
8. Evaluation metrics — MAE, RMSE, MAPE, sMAPE, MASE, CRPS (Gaussian closed-form)
9. Identifiability and consistency proofs
10. Computational complexity table
11. Full bibliography

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Design principles

• Pure NumPy/SciPy — no heavy ML framework required
• Minimal core dependencies — only numpy and scipy
• sklearn-compatible — fit / forecast / score interface
• Fully typed — py.typed marker, complete type annotations
• 22 unit tests — all components, edge cases, and metrics covered

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Contributing

git clone https://github.com/Parsa-Hajian/universitybox
cd universitybox
pip install -e ".[dev]"
pytest tests/ -v

See CONTRIBUTING.md.

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Citation

@software{universitybox2026,
  author  = {UniversityBox Data Team},
  title   = {universitybox: DNA Dynamic Nonlinear Adaptive Forecaster},
  year    = {2026},
  url     = {https://github.com/Parsa-Hajian/universitybox},
  version = {0.1.2}
}

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License

MIT — see LICENSE.