Isotropic Coordinate System (ICS)Library "ICS"
Isotropic Coordinate System (ICS): a dimensionless price-time space
for scale-invariant chart geometry.
Vertical axis: y = ln(price) / sigma, where sigma is the Yang-Zhang (2000)
minimum-variance, drift-independent, gap-consistent OHLC volatility estimator.
Horizontal axis: two scalings via the XScale enum.
legacy : x = bars / lookback. Linear window fraction. Backward compatible.
isotropic : x = sqrt(bars / lookback), with y additionally divided by
sqrt(lookback). Diffusion-consistent (sqrt-time scaling), so that
tan(theta) equals the z-score of the move and 45 degrees
corresponds to a move of exactly one standard deviation
of the n-bar log-return distribution. Assumes approximately
iid returns within the sigma window (the standard assumption
behind sqrt-time scaling; see Danielsson & Zigrand, 2006, for
its known limits under vol clustering and jumps).
Every output (angle, length, area, centroid) is a pure dimensionless number,
comparable across symbols, currencies, and timeframes.
Reference: Yang, D. & Zhang, Q. (2000), "Drift-Independent Volatility
Estimation Based on High, Low, Open, and Close Prices",
The Journal of Business, 73(3), 477-492.
yangZhangSigma(length)
Yang-Zhang volatility estimator. Minimum-variance, unbiased,
drift-independent, and consistent with opening gaps
(Yang & Zhang, 2000). Uses the unbiased sample variance
(biased = false) for both the overnight and open-to-close
components, matching the estimator's unbiasedness claim.
Parameters:
length (simple int) : (simple int) Rolling window length. Must be >= 2.
Returns: (series float) Per-bar sigma, floored at 1e-10.
toX(bars, lookback, mode)
Dimensionless horizontal coordinate.
Parameters:
bars (int) : (series int) Signed bar distance from the anchor.
lookback (int) : (series int) Window length acting as the horizontal unit.
mode (series XScale) : (series XScale) Scaling mode.
Returns: (series float) Signed dimensionless x.
toY(price, sigma, lookback, mode)
Dimensionless vertical coordinate.
Parameters:
price (float) : (series float) Price. Must be > 0.
sigma (float) : (series float) Yang-Zhang sigma. Must be > 1e-10.
lookback (int) : (series int) Window length (used by isotropic mode only).
mode (series XScale) : (series XScale) Scaling mode.
Returns: (series float) Dimensionless y, or na when inputs are invalid.
moveZScore(dLogPrice, sigma, bars)
Z-score of a log-price move over n bars: dLog / (sigma * sqrt(n)).
In isotropic mode this equals tan(theta) of the same move.
Parameters:
dLogPrice (float) : (series float) ln(target) - ln(anchor).
sigma (float) : (series float) Per-bar Yang-Zhang sigma. Must be > 1e-10.
bars (int) : (series int) Number of bars in the move. Must be > 0.
Returns: (series float) The z-score, or na when inputs are invalid.
triangle(td, anchorPrice, anchorBar, targetPrice, targetBar, sig, lookback, mode)
Right triangle between an anchor and a target, computed entirely
in ICS space. Writes results in place into `td` and returns it.
On invalid inputs every field is set to na, so world X never
receives contaminated numbers.
Parameters:
td (TriangleData) : (TriangleData) Output object, updated in place.
anchorPrice (float) : (series float) Anchor price (world A). Must be > 0.
anchorBar (int) : (series int) Anchor bar_index.
targetPrice (float) : (series float) Target price (world A). Must be > 0.
targetBar (int) : (series int) Target bar_index. Must differ from anchorBar.
sig (float) : (series float) Yang-Zhang sigma. Must be > 1e-10.
lookback (int) : (series int) Horizontal unit window.
mode (series XScale) : (series XScale) Scaling mode.
Returns: (TriangleData) The same `td`, for chaining.
pinTriangle(td, anchorPrice, anchorBar, extremePrice, bodyPrice, curBar, sig, lookback, mode)
Pin (wick) triangle with three vertices in ICS space:
A = anchor, B = candle extreme, C = candle body edge.
Side BC is the wick. theta = signed angle at A between AB and AC.
Since xB = xC, the shoelace area reduces exactly to
0.5 * |yB - yC| * |dx|.
Parameters:
td (TriangleData) : (TriangleData) Output object, updated in place.
anchorPrice (float) : (series float) Anchor price (hh or ll). Must be > 0.
anchorBar (int) : (series int) Anchor bar_index.
extremePrice (float) : (series float) Candle extreme (high or low). Must be > 0.
bodyPrice (float) : (series float) Candle body edge. Must be > 0.
curBar (int) : (series int) Current bar_index. Must differ from anchorBar.
sig (float) : (series float) Yang-Zhang sigma. Must be > 1e-10.
lookback (int) : (series int) Horizontal unit window.
mode (series XScale) : (series XScale) Scaling mode.
Returns: (TriangleData) The same `td`, for chaining.
zeroTri(td)
Resets a TriangleData to na. Use when the structure is inactive,
so inactive periods never enter moving averages or normalization
as fake zero values.
Parameters:
td (TriangleData) : (TriangleData) Object to reset, updated in place.
Returns: (TriangleData) The same `td`, for chaining.
TriangleData
One triangle's measurements in ICS space. All fields dimensionless.
Fields:
theta (series float) : Signed hypotenuse angle in degrees; in isotropic mode tan(theta) is the z-score of the move.
dy (series float) : Signed Euclidean magnitude of the hypotenuse.
area (series float) : Triangle area (>= 0).
centroidY (series float) : Vertical centroid of the triangle.
FrozenAnchors
Anchors frozen at a reference bar, plus activity state.
Fields:
hh (series float) : Highest high at the freeze bar (world-A price units).
ll (series float) : Lowest low at the freeze bar (world-A price units).
mid (series float) : Geometric mean sqrt(hh * ll) at the freeze bar.
bar_x (series int) : bar_index of the freeze bar.
time_x (series int) : time of the freeze bar.
is_active (series bool) : Whether the frozen structure is currently active.
Library
