Algorithmic support/resistance, gaps and anchored VWAP

"""M04A exercise 2: algorithmic support/resistance, gaps and anchored VWAP (notes §4A.2-4A.4).

Input bars are a DataFrame with columns open, high, low, close, volume and a
DatetimeIndex (one row per bar), e.g. toolkit.data.synthetic_ohlcv().

Definitions used (Definitions 4A.3-4A.6 in the notes):
  * Swing high at bar u with half-width k: H_u > H_{u-j} and H_u > H_{u+j} for j = 1..k.
    It is only KNOWN at bar u + k (its confirmation bar). Swing lows mirror this with L.
  * A level is a cluster of confirmed pivot prices: sort the prices and split
    wherever two neighbours differ by more than tol (single linkage). The level is
    the cluster mean; its strength is the number of pivots in it.
  * Gap on bar t: g_t = ln(O_t / C_{t-1}). An up-gap (g > 0) is filled on bar t if
    L_t <= C_{t-1}; any other gap (g <= 0) is filled if H_t >= C_{t-1}.
  * Under driftless intraday Brownian motion with daily volatility s (in log terms)
    started at the open, P(fill) = 2 Phi(-|g|/s) (reflection principle).
  * Anchored VWAP from bar a: sum_{a<=u<=t} P_u V_u / sum V_u, with P_u = (H+L+C)/3.
"""

from __future__ import annotations

import numpy as np
import pandas as pd
from scipy.stats import norm


def swing_pivots(ohlc: pd.DataFrame, k: int) -> pd.DataFrame:
    """All swing highs and lows with half-width k (strict inequalities).

    Returns a DataFrame with columns kind ('high' or 'low'), price, pivot_pos (int
    position of the pivot bar), confirm_pos (= pivot_pos + k), sorted by
    confirm_pos then kind (stable). Bars within k of either end cannot be pivots.
    """
    raise NotImplementedError


def cluster_levels(prices, tol: float) -> pd.DataFrame:
    """Single-linkage clusters of prices whose sorted neighbours differ by <= tol.

    Returns DataFrame(level, strength, low, high) sorted by level, one row per cluster.
    """
    raise NotImplementedError


def levels_asof(ohlc: pd.DataFrame, t: int, k: int, tol: float, lookback: int = 250) -> pd.DataFrame:
    """Support/resistance levels known at the close of bar position t.

    Use only bars in positions max(0, t - lookback + 1) .. t, and only pivots
    whose confirmation bar is <= t. Return cluster_levels of their prices.
    The result must not change if any bar after t is altered.
    """
    raise NotImplementedError


def gap_table(ohlc: pd.DataFrame) -> pd.DataFrame:
    """Per-bar gap statistics, from the second bar on (same index as ohlc[1:]).

    Columns: gap (log), up (bool, gap > 0), filled (bool), oc (log open-to-close return).
    """
    raise NotImplementedError


def gap_fill_prob_bm(gap, sigma_intraday):
    """2 Phi(-|gap| / sigma_intraday); vectorised over numpy arrays."""
    raise NotImplementedError


def anchored_vwap(ohlc: pd.DataFrame, anchor_pos: int) -> pd.Series:
    """Anchored VWAP using typical price (H+L+C)/3; NaN before the anchor bar.
    Return a Series on ohlc.index."""
    raise NotImplementedError