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Trends, ranges, breakouts and chart structures

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Start with the idea

Market structure describes relationships between price regions over time. Start with a trailing high and low before fitting complicated shapes. A useful rule has an explicit time when it becomes observable.

Symbols, units & horizon

Uₜ: upper channel boundary from n prior highs
L*ₜ: lower channel boundary from n prior lows, not current candle low
Hₜ,Lₜ,Cₜ: high, low and close of bar t
n: trailing window length in bars
j: prior-bar offset
1{condition}: indicator equal to 1 when true and 0 otherwise
i: candidate pivot index
k: required bars on each side
t_known: earliest confirmation index
b: optional volatility-buffer multiple
ATR: average true range in price units

When and why to use this

Use causal channels for breakout research and confirmed pivots for swing-based stop or pattern definitions. Test false breakouts and gaps as part of the strategy, not as data to discard.

A trend is a sequence at a chosen scale. Higher confirmed highs and higher confirmed lows form one operational definition of an uptrend. A range is a region with repeated tests of boundaries and no sustained progression under the chosen rule. Support and resistance are zones inferred from past observations, not exact walls that must hold.

U_{t} = \max_{j = 1, \dots, n} H_{t - j}, L_{t}^{*} = \min_{j = 1, \dots, n} L_{t - j}, {breakout}_{t} = 𝟏 {C_{t} > U_{t}}

Algebra and arithmetic

Construct a boundary from completed history

At t with n=3, expand the upper maximum as max(Hₜ₋₁,Hₜ₋₂,Hₜ₋₃). The current high Hₜ is deliberately absent.
An indicator is 1 if the condition holds and 0 otherwise. With prior highs 101,102,100, the boundary is 102; close 103 gives 1. A next-bar entry waits until after that close.

Work it by hand

A buffer can be added as Cₜ>Uₜ+b·ATRₜ₋₁. Define b before testing; both the channel and volatility input are available before the current close.

A channel breakout compares today’s close with a boundary formed before today’s bar. Including today’s high in the upper boundary makes a strict close-above-high rule impossible. Requiring a close above a band adds confirmation but changes the entry price and delay. A retest is a later revisit to the old boundary; specify zone width, deadline and invalidation.

Structure	Operational definition to write	Failure to test
Double top / bottom	Two confirmed extrema within a price tolerance, minimum spacing, intervening neckline and a later neckline break	Pattern is labelled before the second extreme is confirmed
Head and shoulders / inverse	Three confirmed extrema with a more extreme middle peak/trough, shoulder tolerance and neckline rule	Many subjective choices create invisible optimisation
Triangle / wedge	Converging fitted boundaries using only already confirmed pivots	Retrofitting lines after seeing the breakout
Flag / pullback	Prior impulse plus a bounded retracement/consolidation and a resumption trigger	Calling every pause continuation; ignoring failed impulses
Range / channel	Trailing boundaries, minimum touches and a permitted slope	Treating a trending boundary as stable support

H_{i} > \max (H_{i - k}, \dots, H_{i - 1}, H_{i + 1}, \dots, H_{i + k}) \Rightarrow t_{known} = i + k

Index arithmetic and information timing

Account for confirmation delay

For k=2 and candidate bar i=10, compare highs at 8,9,11,12 against H₁₀. Bar 12 must finish before the comparison can be completed.
Therefore known time is i+k=12. With next-open execution, earliest bar entry is 13.

Work it by hand

A marker drawn at bar 10 must not create a bar-10 trade. This example uses strict extrema; equal highs do not qualify.

A pivot detector that compares k bars on each side of a high needs the k later bars before the pivot is confirmed. Plotting the marker back at the high is acceptable only if the interface also records its later confirmation time. A backtest must act at confirmation, not at the earlier visually attractive high. Zigzag tools that revise their last segment have the same issue.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def previous_channel(bars, index, window):
    if window<1 or index<window or index>=len(bars):
        return None
    history=bars[index-window:index]  # excludes current bar
    upper=max(b['h'] for b in history)
    lower=min(b['l'] for b in history)
    return upper,lower,bars[index]['c']>upper

def confirmed_highs(visible_bars, flank=2):
    if flank<1: raise ValueError("Positive flank required")
    result=[]
    for i in range(flank,len(visible_bars)-flank):
        neighbours=visible_bars[i-flank:i]+visible_bars[i+1:i+flank+1]
        if all(visible_bars[i]['h']>b['h'] for b in neighbours):
            result.append({"pivot":i,"known_at":i+flank})
    return result

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