Constructing divergence-equivalent relative strength index and stochastic oscillators

A construction audit

Stacked relative strength index and stochastic oscillator formulas can be audited before they are read for price-indicator divergence. Two indicators are equivalent for divergence when a function with a positive derivative produces one series from the other, because that mapping cannot reverse the order of successive highs and lows.

The TradersWeek editorial reading is to treat the workspace as a construction audit: prove which formulas are strictly increasing rewrites of each other, keep one representative per family for divergence reading, and denormalize a bounded oscillator only when it must be composed with an unbounded measure. That framing is editorial and is not attributed to the archive.

What divergence equivalence means

A strictly increasing map is a transform whose derivative is everywhere positive. It rewrites one oscillator formula into another while preserving the order of turning points. Divergence equivalence is the construction relation that follows: successive highs and lows cannot reverse, so the pair cannot form opposite turning-point patterns.

Peak divergence between two indicators is one of three turning-point cases: one rises while the other falls, one stays flat while the other falls, or one rises while the other stays flat. Trough divergence uses the three opposite direction pairs. Price-indicator divergence is that same mismatch, in which one series makes a higher or lower extreme while the other makes the opposite extreme, or one is flat while the other is not.

The relative strength index family

A k-period relative strength index reduces to 100 times the average up-close divided by the sum of the average up-close and average down-close. It records the up-share of total close-to-close movement over that lookback.

The signed oscillator built from the same average up-close and average down-close equals twice the matching-lookback relative strength index minus 100. That rewrite is a strictly increasing map with derivative 2, so the pair cannot diverge. Editorially, one of the two is enough as the family representative for divergence reading.

The stochastic oscillator family

A stochastic oscillator locates the close inside the highest-high to lowest-low range of a stated lookback, optionally after a short moving-average smooth. Fast stochastic percent-K of k periods and the inverted high-low range oscillator of the same lookback differ by a constant shift of 100.

A constant shift is strictly increasing, so the two are equivalent for divergence. Editorially, keep one representative from this family as well.

When a bounded oscillator must be denormalized

The map that sends a unit-interval reading x to one over (1 minus x) minus one over x is strictly increasing. It converts a bounded oscillator into a denormalized oscillator, an unbounded counterpart that keeps the same divergence character.

Applying that map to a 14-period relative strength index scaled into the unit interval sends the 30 and 70 bounds to about -1.90476 and 1.90476. A cross of 50 on the original coincides with a cross of zero on the unbounded series. The same construction on a 5-and-3-period stochastic oscillator maps the 20 and 80 levels to -3.75 and 3.75. The bounded and unbounded plots can look different and still show no mutual divergence.

Adding a still-bounded oscillator to an unbounded one is a weak construction in extreme moves because the unbounded term can dominate by size. Denormalizing first keeps both terms combinable while preserving divergence. Editorially, that step is warranted only when a bounded oscillator must be composed with an unbounded measure.

Why correlation is the wrong check

A correlation coefficient is a poor test of whether two indicators share the same divergence readings. Classifying formulas by a strictly increasing relation is the direct construction check.