Inverse transform of quantiles after differentiation

[brsnw250]

🚀 Feature Request

Currently, we treat quantiles in the inverse transforms the same way as the target series. When using differentiation (DifferencingTransform) this might result in very wide and not meaningful intervals.

Mainly, this affects cases when the expected value of $\large r_t = y_t - y_{t - 1}$ distributed near 0 or when $\large r_t$ has a large enough variance. So upper and lower quantilies of $\large r_t$ are mainly one signed throughout the time.

Code to reproduce

from etna.datasets.datasets_generation import generate_ar_df
from etna.datasets import TSDataset
from etna.pipeline import Pipeline
from etna.models import SeasonalMovingAverageModel
from etna.transforms import DifferencingTransform
from etna.analysis import plot_forecast


df = generate_ar_df(100, "2020-01-01")
ts = TSDataset(df=TSDataset.to_dataset(df=df), freq="D")

train_ts, test_ts = ts.train_test_split(test_size=20)

pipeline = Pipeline(
    transforms=[DifferencingTransform(in_column="target")],
    model=SeasonalMovingAverageModel(seasonality=1),
    horizon=20
)

pipeline.fit(train_ts)
forecast = pipeline.forecast(prediction_interval=True)

plot_forecast(forecast_ts=forecast, test_ts=test_ts, prediction_intervals=True)

Proposal

Implement interface for separate treatment of quantiles in transforms. Use $\large Q_{y_t}(p) = y_{t - 1} + Q_{r_t}(p)$ to recompute target quantiles in inverse transform of DifferencingTransform, where $\large Q_x(p)$ is p-quantile of the $x$ random variable.

Test cases

No response

Additional context

Here is a comparison between current and proposed approaches.

import numpy as np
import matplotlib.pyplot as plt


# setting timeline and generating noise
t = np.arange(100)
eps = np.random.normal(0, 1, 100)
eps[0] += 10
level = eps[0]

# generating random walk series
y = np.cumsum(eps)

# differentiating series
r = np.diff(y)

# estimate quantiles for the first difference
r_q_upper = r + np.quantile(r, q=0.975)
r_q_lower = r + np.quantile(r, q=0.025)

# current approach
y_q_upper = np.cumsum(r_q_upper) + level
y_q_lower = np.cumsum(r_q_lower) + level

# proposed approach
int_r = np.roll(np.cumsum(r) + level, 1) # integration
int_r[0] = level
y_q_upper_adj = int_r + r_q_upper
y_q_lower_adj = int_r + r_q_lower

plt.figure(figsize=(6, 12))

plt.subplot(3, 1, 1)
plt.plot(t[1:], r, color="orange", label="first difference")
plt.fill_between(t[1:], r_q_upper, r_q_lower, alpha=0.3, color="orange", label="interval")
plt.legend()

plt.subplot(3, 1, 2)
plt.title("Current approach")
plt.plot(t[1:], y[1:], label="series")
plt.fill_between(t[1:], y_q_upper, y_q_lower, alpha=0.3, label="interval", color="g")
plt.legend()

plt.subplot(3, 1, 3)
plt.title("Proposed approach")
plt.plot(t[1:], y[1:], label="series")
plt.fill_between(t[1:], y_q_upper_adj, y_q_lower_adj, alpha=0.3, label="interval", color="g")
plt.legend()