QuanTAlib/lib/oscillators/cti/Cti.md

CTI: Ehlers Correlation Trend Indicator

Correlation trend indicator measures the linear correlation between price and a perfect trend line — how orderly is the move.

Property	Value
Category	Oscillator
Inputs	Source (close)
Parameters	period (default 20)
Outputs	Single series (Cti)
Output range	Varies (see docs)
Warmup	period bars
PineScript	cti.pine

• The Correlation Trend Indicator computes the Pearson correlation coefficient between the price series and a linear time index over a rolling window...
• Similar: LinReg, CFO | Complementary: ADX | Trading note: Correlation Trend Indicator; Pearson correlation of price vs time. +1 = perfect uptrend, −1 = downtrend.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Correlation Trend Indicator computes the Pearson correlation coefficient between the price series and a linear time index over a rolling window, producing a bounded oscillator in the range [-1, +1]. Values near +1 indicate a strong linear uptrend, values near -1 indicate a strong linear downtrend, and values near zero indicate no linear trend relationship. The implementation achieves O(1) complexity per bar through incremental running sums that avoid recomputing the full correlation on each update.

Historical Context

The concept of measuring trend strength via linear correlation has roots in classical statistics, where Pearson's r between an ordinal time index and a dependent variable quantifies how well a linear model fits the observed data. John Ehlers popularized this approach in trading contexts, noting that correlation-based trend detection is mathematically equivalent to the R-squared goodness-of-fit measure used in linear regression. CTI differs from slope-based indicators (like TSF or LSMA) by normalizing the result to a fixed [-1, +1] range regardless of price scale or volatility, making it directly comparable across instruments and timeframes. This normalization property makes CTI particularly useful as a regime filter: values above a threshold (typically \pm 0.5) indicate trending conditions where trend-following strategies perform well, while values near zero suggest mean-reverting or choppy conditions.

Architecture & Physics

Incremental Pearson Correlation

The standard Pearson correlation formula requires \Sigma x, \Sigma y, \Sigma x^2, \Sigma y^2, and \Sigma xy over n observations. For CTI, the x values are sequential integers (time indices), which means \Sigma x and \Sigma x^2 are deterministic closed-form functions of n and do not require running sums. Only the $y$-dependent sums (\Sigma y, \Sigma y^2, \Sigma xy) need incremental maintenance.

Running Sum Trick for \Sigma xy

The key optimization is the incremental update of \Sigma xy. When the window slides forward by one bar:

• The oldest value exits at what was position 0 and all remaining values shift down by one position.
• Rather than recomputing all x_i \cdot y_i products, the implementation subtracts \Sigma y (which shifts all position indices down by 1) and adds (n-1) \times y_{\text{new}} for the new value entering at the highest position.

This reduces the O(n) recomputation to O(1) per bar.

Clamping and Edge Cases

The output is clamped to [-1, +1] to guard against floating-point drift. When the count is less than 2, the output is NaN (insufficient data). When either variance term is non-positive (constant price or constant time, which cannot happen for time), the output is 0.

Mathematical Foundation

Given source values y_t over a window of n observations with time indices x_i = 0, 1, \ldots, n-1:

Closed-form sums for time indices:

\Sigma_x = \frac{n(n-1)}{2}, \quad \Sigma_{x^2} = \frac{n(n-1)(2n-1)}{6}

Running sums for price:

\Sigma_y = \sum_{i=0}^{n-1} y_i, \quad \Sigma_{y^2} = \sum_{i=0}^{n-1} y_i^2, \quad \Sigma_{xy} = \sum_{i=0}^{n-1} i \cdot y_i

Pearson correlation:

r = \frac{n \cdot \Sigma_{xy} - \Sigma_x \cdot \Sigma_y}{\sqrt{(n \cdot \Sigma_{x^2} - \Sigma_x^2)(n \cdot \Sigma_{y^2} - \Sigma_y^2)}}

O(1) incremental update (when buffer is full, oldest value y_{\text{old}} exits):

Σy  -= y_old;  Σy  += y_new
Σy² -= y_old²; Σy² += y_new²
Σxy -= Σy_before_removal    // shift all positions down by 1
Σxy += (n-1) × y_new        // new value enters at position n-1

CTI = clamp(r, -1, +1)

Default parameters: period = 20.

Performance Profile

Operation Count (Streaming Mode)

CTI (Correlation Trend Indicator) computes the Pearson r between price and a linear regression line over N bars using a running-sum Welford-style computation.

Operation	Count	Cost (cycles)	Subtotal
RingBuffer update (price window)	2	1	2
Running sum updates (ΣX, ΣY, ΣXY, ΣX², ΣY²)	10	1	10
Correlation numerator: N×ΣXY − ΣX×ΣY	3	3	9
Denominator: SQRT((N×ΣX²−ΣX²)(N×ΣY²−ΣY²))	6	20	120
DIV (r = num/denom)	1	15	15
Total	22	—	~156 cycles

The two SQRTs in the denominator dominate cost. ~156 cycles per bar.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Prefix sums (ΣX, ΣY, ΣXY, ΣX², ΣY²)	Yes	VADDPD scan; windowed via subtract-lag
Correlation formula	Yes	VFMADD + VSQRTPD + VDIVPD

Fully vectorizable in batch mode. Prefix-sum trick converts O(N²) naive to O(N) with O(1) per-bar computation, and SIMD accelerates each prefix step.

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	Pearson r exact; SQRT precision adequate
Timeliness	6/10	N-bar window; trend changes detected with N/2 average lag
Smoothness	8/10	Correlation coefficient is inherently bounded [−1,1]
Noise Rejection	7/10	Linear fit suppresses non-linear noise components

Resources

• Ehlers, J.F. (2001). Rocket Science for Traders. Wiley
• Pearson, K. (1895). "Notes on Regression and Inheritance in the Case of Two Parents." Proceedings of the Royal Society of London
• PineScript reference: cti.pine