Function schaffTrendCycle

Schaff Trend Cycle (Doug Schaff, ~1999) — a MACD read as a cycle rather than as a level, by running a stochastic over it twice:

macd  = EMA(column, fastPeriod) − EMA(column, slowPeriod)
k1    = 100 · (macd − LL(macd)) / (HH(macd) − LL(macd))    over `cyclePeriod`
d1    = d1[−1] + 0.5 · (k1 − d1[−1])                       seeded on k1
k2    = 100 · (d1 − LL(d1)) / (HH(d1) − LL(d1))            over `cyclePeriod`
${output} = stc[−1] + 0.5 · (k2 − stc[−1])                 seeded on k2
Copy

Appends one column, bounded 0 … 100. A MACD says how far the fast average is from the slow one in price units, which is not comparable across instruments or across time; normalising it against its own recent range twice turns it into a cycle oscillator that reaches both extremes on every swing and turns earlier than the MACD it is built on. It is read on the 25 and 75 lines rather than on a zero crossing.

What the parameters are, and the shape of the port

23 / 50 / 10 are Schaff's, and the arrangement above is the one every widely-copied port implements (the TradingView STC scripts and the charting packages that follow them). Two details of it are worth naming because prose descriptions leave them out:

• Both stochastic passes use the same cyclePeriod. There is no separate %D length; the second pass is the same window over the smoothed output of the first.
• The smoothing is a fixed 0.5, not a span EMA. x += 0.5·(raw − x) is an exponential average with α = 0.5 — a rate, like DecisionPoint's 2/n in priceMomentumOscillator, not a period — so it is a constant of the definition rather than an option. Dropping both recursions (reading the raw double stochastic straight) is a real difference and not a subtle one: measured, the two sit 98.30 apart on a 0…100 reading (scripts/oracle/generate.py), which is why the generator asserts it.

Because both recursions carry state, the pandas oracle is a transcription of this step rather than an independent derivation; the separation above and the analytic warm-up bound below are what actually verify it.

A state machine, so a gap RESETS it — the delta from the common port

The two recursions carry state, so each is a foldRows step ([PND-SFOLD]) rather than a private loop, and the kernel's reset rule applies: a bar with no input re-seeds the recursion from the next raw value.

That is a deliberate delta from the usual port, which on a flat window writes nz(frac[1]) — it repeats the previous stochastic reading as if it had been observed. This package does not do that anywhere: a flat window is a 0/0 and reads undefined (percentOfRangeValues owns the rule for every range study), and holding the previous value would report a position in a range that has no positions in it. The bar reads missing and the recursion re-seeds on the next one.

A flat window is not an exotic case here, and it is the sharp edge of this study. A sustained trend pins the first stochastic at 100 (or 0) for cyclePeriod bars; the 0.5 smoothing of a constant is that constant, so the second window is then exactly flat and the STC reads undefined — during the strongest part of the move. It is not hypothetical: on this package's own 80-bar oracle input at the defaults the MACD rises monotonically for sixteen bars and the line starts at bar 74 instead of the analytic 67, seven bars later than the warm-up alone accounts for (measured; the generator asserts that every such null really is a flat second window). The usual port hides this by repeating the previous reading, which reports a position in a range that has no positions in it. If you need a value there, the honest fix is a longer cyclePeriod, not a held one.

Warm-up

Length-preserving, and it stacks: the MACD lands at slowPeriod − 1 (both EMAs through the K2 door, so the slower governs), each stochastic pass costs cyclePeriod − 1 more, and each recursion costs nothing because it seeds on its first input. So the column starts at slowPeriod + 2·cyclePeriod − 3 — bar 67 at the defaults, which is why a chart needs a good deal more history than the 50-bar slow EMA suggests. That is the earliest bar, not necessarily the first: a flat second window pushes it later, as described above.

Edges

• Scale- and shift-invariant. A shift cancels in the MACD (a difference of two averages of the same column) and a scale cancels in the stochastic that follows. Pinned as property tests.
• fastPeriod must be shorter than slowPeriod — the same guard macd carries, and for the same reason: the difference of two EMAs with the fast one slower is the same study with its sign flipped, which is a mistake rather than a mode.
• An interior gap in column costs the bar plus every window and recursion run that reads it, and then recovers — the EMAs skip a missing cell and the folds re-seed. Nothing propagates to the end.