frequencyquantile 1.0.1

Multi-Frequency Quantile Regression, Quantile Autoregression & Uncertain QAR — a unified Python toolkit for frequency-quantile econometrics.

frequencyquantile

A unified Python toolkit for Multi-Frequency Quantile Econometrics — implementing three families of quantile methods from the academic literature in a single, interoperable library.

Paper	Method	Module
Li, Li & Tsai (2015, JASA)	QCOR, QPCOR, QAR, QPACF, QACF, Box-Jenkins	quantile_correlation, qar
Adebayo (2026, Applied Economics)	MFQR, MFQGC	mfqr, mfqgc
Shi & Sheng (2025, Comm. Stat.)	Uncertain QAR	uncertain_qar

Author: Dr Merwan Roudane · [e-mail]
GitHub: github.com/merwanroudane/frequencyquantile

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Installation

pip install frequencyquantile

From source:

git clone https://github.com/merwanroudane/frequencyquantile.git
cd frequencyquantile
pip install -e .

Dependencies: numpy, scipy, pandas, statsmodels, matplotlib, seaborn, PyEMD, tabulate

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Quick Start

import numpy as np
import frequencyquantile as fq

# Generate sample data
np.random.seed(42)
y = np.cumsum(np.random.randn(500)) + 100
x = np.cumsum(np.random.randn(500)) + 50

# === Paper 1: QAR with modified Box-Jenkins ===
model = fq.three_stage_procedure(y, tau=0.05, K=10)

# === Paper 2: Multi-Frequency Quantile Regression ===
mfqr = fq.MFQR(n_imfs=5).fit(y, x, y_name="GDP", x_name="Energy")

# === Paper 2: Multi-Frequency Granger Causality ===
mfqgc = fq.MFQGC(n_imfs=5).fit(y, x, y_name="CO2", x_name="REG")

# === Paper 3: Uncertain QAR ===
data = [(v - 0.05, v + 0.05) for v in y[:30]]
uqar = fq.UncertainQAR(order=3).fit(data)

# === Visualisation ===
plotter = fq.FrequencyQuantilePlotter()
plotter.plot_mfqr_heatmap(mfqr)
plotter.plot_mfqgc_heatmap(mfqgc)

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Detailed API Reference

1. Quantile Correlation (Paper 1: Li, Li & Tsai 2015)

fq.qcov(Y, X, tau) — Quantile Covariance

# Eq. (2.1) numerator: qcov_τ{Y, X} = E[ψ_τ(Y − Q_τ,Y)(X − EX)]
val = fq.qcov(Y, X, tau=0.25)

Parameter	Type	Description
Y	array (n,)	Response variable
X	array (n,)	Predictor variable
tau	float	Quantile level ∈ (0,1)
Returns	float	Quantile covariance value

fq.qcor(Y, X, tau) — Quantile Correlation

# Eq. (2.1): normalised by √[(τ−τ²)·σ²_X]
r = fq.qcor(Y, X, tau=0.5)       # scalar in [-1, 1]

fq.sample_qcor(Y, X, tau) — With Inference (Theorem 1)

result = fq.sample_qcor(Y, X, tau=0.25)
# result = {'qcor': 0.312, 'se': 0.045, 'z_stat': 6.93, 'p_value': 0.0000}

fq.qpcor(Y, X, Z, tau) — Quantile Partial Correlation

# Eq. (2.4): controls for Z when measuring Y-X association at quantile τ
r_partial = fq.qpcor(Y, X, Z, tau=0.75)

Parameter	Type	Description
Z	array (n, q)	Covariates to partial out

fq.sample_qpcor(Y, X, Z, tau) — With Bootstrap SE (Theorem 2)

result = fq.sample_qpcor(Y, X, Z, tau=0.75)
# result = {'qpcor': -0.187, 'se': 0.063, 'z_stat': -2.97, 'p_value': 0.003}

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2. Quantile Autoregressive Model (Paper 1)

fq.qpacf(y, tau, max_lag) — Quantile PACF for Model Identification

pacf = fq.qpacf(y, tau=0.05, max_lag=15, bandwidth_scale=0.6)
# pacf['lag']       → array([1, 2, ..., 15])
# pacf['qpacf']     → array of φ̂_{kk,τ} values
# pacf['se']        → asymptotic standard errors
# pacf['ci_lower']  → lower 95% confidence band
# pacf['ci_upper']  → upper 95% confidence band

Parameter	Type	Default	Description
y	array	—	Time series
tau	float	—	Quantile level
max_lag	int	15	Maximum lag to compute
bandwidth_scale	float	0.6	Bofinger bandwidth multiplier

Key property (Lemma 2): QPACF has a cut-off at lag p — φ_{kk,τ} = 0 for k > p.

fq.QARModel — Quantile Autoregressive Model

model = fq.QARModel(tau=0.05)
model.fit(y, lags=[1, 2, 8, 11])    # explicit lags
# OR
model.fit(y, max_lag=5)              # use lags 1..5

# Attributes after fitting:
model.intercept         # φ_0(τ)
model.coefficients      # {lag: φ_j(τ)} dict
model.residuals         # quantile residuals ê_{t,τ}
model.se                # {lag: std_error} dict
model.n_obs             # effective sample size

# Summary table
model.summary()
#   Variable    Coef   Std.Err   z-stat   P>|z|   Sig
#   intercept  -0.747   0.029   -25.76   0.0000   ***
#   y(t-2)      0.118   0.051     2.31   0.0209   **
#   ...

# Forecast
forecasts = model.predict(y, steps=5)

fq.qacf(residuals, tau, max_lag) — Residual QACF for Diagnostics

ac = fq.qacf(model.residuals, tau=0.05, max_lag=15)
# Same structure as qpacf output

fq.qbp_test(residuals, tau, K, p) — Quantile Box-Pierce Test

bp = fq.qbp_test(model.residuals, tau=0.05, K=15, p=4)
# bp = {'Q_BP': 12.34, 'df': 11, 'p_value': 0.338}
# p_value > 0.05 → model adequate ✓

fq.three_stage_procedure(y, tau, K, alpha) — Full Box-Jenkins

model = fq.three_stage_procedure(y, tau=0.05, K=15, alpha=0.05, verbose=True)
# [Stage 1] QPACF suggests QAR(11) at τ=0.05.
#           Significant lags: [2, 8, 11]
# [Stage 2] Estimated QAR(11) with lags [1,2,...,11].
# [Stage 3] Removing lag 1 (p=0.7234).
# [Stage 3] Removing lag 9 (p=0.5891).
# ...
# [Stage 3] Final lags: [2, 4, 10, 11]
#           Q_BP(15) = 14.21, p-value = 0.320
#           ✓  Model passes Q_BP diagnostic.

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3. EEMD Decomposition

fq.eemd_decompose(series, n_imfs, ...)

dec = fq.eemd_decompose(y, n_imfs=5, noise_width=0.05, n_trials=100)
# dec['imfs']      → (5, n) array of IMFs (high→low frequency)
# dec['residual']  → (n,) residual trend component
# dec['n_imfs']    → 5
# dec['labels']    → ['IMF1', 'IMF2', ..., 'IMF5', 'Residual']

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4. Multi-Frequency Quantile Regression — MFQR (Paper 2: Adebayo 2026)

fq.MFQR Class

mfqr = fq.MFQR(
    n_imfs=5,                          # EEMD decomposition levels
    quantiles=[0.05, 0.25, 0.50, 0.75, 0.95],  # quantile grid
    noise_width=0.05,                  # EEMD noise
    n_trials=100,                      # EEMD ensemble size
)

# Fit: decomposes both series then runs QR at each (frequency, quantile)
mfqr.fit(y, x, y_name="CO2_Residential", x_name="CPU")

# Full summary table
mfqr.summary()
#   Frequency                         Quantile   β(CPU)   Std.Error  t-stat  p-value  Sig
#   IMF1 (High freq / Short-term)       0.05     0.0312    0.0451    0.692   0.4890
#   IMF1 (High freq / Short-term)       0.25     0.0189    0.0234    0.808   0.4191
#   ...
#   IMF5 (Low freq / Long-term)         0.95    -0.2341    0.0782   -2.994   0.0028   ***

# Coefficient matrix (pivot: quantiles × frequencies)
mfqr.coefficient_matrix()
#   Level       1        2        3        4        5
#   Quantile
#   0.05     0.031    0.018   -0.045   -0.121   -0.234
#   0.25     0.019    0.023   -0.038   -0.098   -0.189
#   ...

# Significance stars matrix
mfqr.significance_matrix()

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5. Multi-Frequency Quantile Granger Causality — MFQGC (Paper 2)

fq.MFQGC Class

mfqgc = fq.MFQGC(
    n_imfs=5,
    quantiles=[0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95],
    max_lag=8,        # max lag for VAR
)

# Fit: EEMD → quantile series → Granger test at each combination
mfqgc.fit(y, x, y_name="RECEM", x_name="EPU")

# Summary
mfqgc.summary()
#   Frequency        Quantile  Lag  F-stat  p-value   Sig   Causal
#   IMF1 (High freq)   0.05     2   1.234   0.2934          No
#   IMF3 (Medium freq) 0.50     3   4.567   0.0034   ***    Yes
#   IMF5 (Low freq)    0.95     1   8.901   0.0001   ***    Yes

# p-value matrix (pivot)
mfqgc.pvalue_matrix()

# Causality significance matrix
mfqgc.causality_matrix()

---

6. Uncertain Quantile Autoregressive Model (Paper 3: Shi & Sheng 2025)

fq.UncertainQAR Class

Input: Imprecise observations as (lower, upper) interval pairs.

# Data as interval-valued uncertain variables L(a, b)
data = [
    (2.26, 2.36), (3.18, 3.28), (4.00, 4.10), (4.40, 4.50),
    (4.37, 4.47), (4.07, 4.17), (3.91, 4.01), (3.83, 3.93),
    # ... more observations
]

uqar = fq.UncertainQAR(
    order=None,              # None = auto-select via cross-validation
    max_order=5,             # max order for CV search
    quantiles=[0.1, 0.3, 0.5, 0.7, 0.9],
    sig_level=0.05,          # hypothesis test significance
)

uqar.fit(data, verbose=True)
# Selecting order via cross-validation (ATE)...
#   k=1: ATE = 0.019600
#   k=2: ATE = 0.012300
#   k=3: ATE = 0.009900  ← smallest
#   → Selected order k = 3
#
# Quantile s = 0.5:
#   Coefficients: [0.5687, 0.9160, -0.0463, -0.0115]
#   ê_s = 0.0194, σ̂_s = 0.1386
#   Hypothesis test: ✓ PASSED
#   STE = 0.5231
#   Forecast = 4.2105, 95% CI = [3.9306, 4.4904]

# Summary table
uqar.summary()
#   Quantile  Order  ê_s     σ̂_s    H₀ Test  STE     Forecast  CI Lower  CI Upper  ĉ_s0    ĉ_s1    ĉ_s2    ĉ_s3
#   0.1       3      0.077   0.282  Pass     2.080   4.229     3.659     4.798     0.0933  0.3642  0.3290  0.2891
#   0.3       3      0.081   0.291  Pass     2.198   4.235     3.648     4.823     0.0862  0.3444  0.3303  0.3101
#   0.5       3      0.019   0.139  Pass     0.523   4.211     3.931     4.490     0.5687  0.9160 -0.0463 -0.0115
#   ...

# Get optimal quantile (min STE among passing tests)
best_s = uqar.optimal_quantile()  # → 0.5

Also works with precise data (auto-wrapped as zero-width intervals):

uqar = fq.UncertainQAR(order=2).fit(np.array([3.1, 3.5, 3.8, 4.2, ...]))

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7. Visualisation

fq.FrequencyQuantilePlotter Class

plotter = fq.FrequencyQuantilePlotter(style="dark", figsize=(12, 6))

Method	Description	Paper
plot_qpacf(pacf, tau)	QPACF lollipop with 95% CI bands	1
plot_qacf(acf, tau)	Residual QACF lollipop	1
plot_multi_qpacf(y, taus)	Side-by-side QPACF for multiple τ	1
plot_qar_summary(models)	Coefficient functions φ(τ) across quantiles	1
plot_mfqr_heatmap(mfqr)	Coefficient heatmap (quantile × frequency)	2
plot_mfqgc_heatmap(mfqgc)	p-value heatmap with significance stars	2
plot_eemd(decomp)	EEMD decomposition panel plot	2
plot_uncertain_forecast(uqar, data)	Fan chart across quantiles	3

# All methods support saving:
plotter.plot_mfqr_heatmap(mfqr, title="CPU → RECEM", save="mfqr_cpu_recem.png")

8. Tables

tbl = fq.FrequencyQuantileTable(fmt="fancy_grid")  # or "latex", "html", "pipe"
tbl.qar_summary(model)
tbl.mfqr_summary(mfqr)
tbl.mfqgc_summary(mfqgc)
tbl.uncertain_qar_summary(uqar)

# LaTeX export
latex_str = tbl.to_latex(mfqr.summary(), caption="MFQR Results", label="tab:mfqr")

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Full Workflow Example

import numpy as np
import pandas as pd
import frequencyquantile as fq

# ── Load data ──
df = pd.read_csv("data.csv", parse_dates=["date"])
y = df["CO2_residential"].values
x_cpu = df["CPU"].values
x_epu = df["EPU"].values
x_reg = df["REG"].values

# ── Step 1: EEMD Decomposition ──
plotter = fq.FrequencyQuantilePlotter()
dec = fq.eemd_decompose(y, n_imfs=5)
plotter.plot_eemd(dec, series_name="RECEM", save="eemd_recem.png")

# ── Step 2: QAR Analysis at key quantiles ──
taus = [0.05, 0.25, 0.50, 0.75, 0.95]
models = {}
for tau in taus:
    models[tau] = fq.three_stage_procedure(y, tau=tau, K=10, verbose=False)
plotter.plot_qar_summary(models, save="qar_coefficients.png")

# ── Step 3: MFQR for each predictor ──
tbl = fq.FrequencyQuantileTable()
for x, name in [(x_cpu, "CPU"), (x_epu, "EPU"), (x_reg, "REG")]:
    mfqr = fq.MFQR(n_imfs=5).fit(y, x, y_name="RECEM", x_name=name)
    tbl.mfqr_summary(mfqr)
    plotter.plot_mfqr_heatmap(mfqr, save=f"mfqr_{name}.png")

# ── Step 4: MFQGC Causality ──
for x, name in [(x_cpu, "CPU"), (x_epu, "EPU"), (x_reg, "REG")]:
    gc = fq.MFQGC(n_imfs=5).fit(y, x, y_name="RECEM", x_name=name)
    tbl.mfqgc_summary(gc)
    plotter.plot_mfqgc_heatmap(gc, save=f"mfqgc_{name}.png")

# ── Step 5: Uncertain QAR (if interval data) ──
intervals = [(v - 0.05, v + 0.05) for v in y[:30]]
uqar = fq.UncertainQAR(quantiles=[0.1, 0.3, 0.5, 0.7, 0.9]).fit(intervals)
tbl.uncertain_qar_summary(uqar)
plotter.plot_uncertain_forecast(uqar, data=intervals, save="uqar_forecast.png")

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Library Architecture

frequencyquantile/
├── __init__.py              # Public API
├── utils.py                 # Check function, QR solver, density estimation
├── quantile_correlation.py  # QCOR, QPCOR (Paper 1, Sec. 2)
├── qar.py                   # QARModel, QPACF, QACF, Q_BP (Paper 1, Sec. 3)
├── decomposition.py         # EEMD wrapper (Wu & Huang 2009)
├── mfqr.py                  # MFQR (Paper 2)
├── mfqgc.py                 # MFQGC (Paper 2)
├── uncertain_qar.py         # Uncertain QAR (Paper 3)
├── visualization.py         # Dark-mode publication plots
└── tables.py                # Formatted summary tables

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References

1. Li G., Li Y., Tsai C-L. (2015). "Quantile Correlations and Quantile Autoregressive Modeling", Journal of the American Statistical Association, 110(509): 246–261.

2. Adebayo T.S. (2026). "Response of sectoral CO₂ emissions to climate and economic policy uncertainties: a multi-frequency quantile analysis", Applied Economics, 58(20): 3922–3941.

3. Shi Y., Sheng Y. (2025). "Uncertain quantile autoregressive model", Communications in Statistics - Simulation and Computation, 54(6): 1869–1889.

4. Koenker R., Xiao Z. (2006). "Quantile Autoregression", JASA, 101: 980–990.

5. Wu Z., Huang N.E. (2009). "Ensemble Empirical Mode Decomposition", Advances in Adaptive Data Analysis, 1(1): 1–41.

6. Liu B. (2007). Uncertainty Theory, 2nd edn. Springer.

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License

MIT License — Copyright (c) 2026 Dr Merwan Roudane

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Citation

@software{roudane2026frequencyquantile,
  author  = {Roudane, Merwan},
  title   = {frequencyquantile: Multi-Frequency Quantile Econometrics Toolkit},
  year    = {2026},
  url     = {https://github.com/merwanroudane/frequencyquantile},
}