QuanTAlib/lib/channels/ttm_lrc/TtmLrc.md

TTM_LRC: TTM Linear Regression Channel

Linear regression channels project the statistical trend and drape standard deviation curtains around it.

Property	Value
Category	Channel
Inputs	Source (close)
Parameters	period (default 100)
Outputs	Multiple series (Midline, Upper1, Lower1, Upper2, Lower2)
Output range	Tracks input
Warmup	period bars

• TTM Linear Regression Channel plots a least-squares regression line through price data with dual standard deviation bands at \pm 1\sigma and $\pm...
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

TTM Linear Regression Channel plots a least-squares regression line through price data with dual standard deviation bands at \pm 1\sigma and \pm 2\sigma. Developed by John Carter as part of his TTM (Trade The Markets) indicator suite, it extends the standard regression channel by providing two band levels that create statistically meaningful trading zones. The algorithm is identical to REGCHANNEL/SDCHANNEL in its regression and residual computation, but uses a longer default period (100) and emits four bands instead of two.

Historical Context

John Carter developed the TTM LRC as part of his Trade The Markets methodology, popularized in his book Mastering the Trade (2005). Carter's approach combines linear regression channels with his TTM Squeeze indicator to identify high-probability setups: the regression channel defines the trend envelope while the squeeze identifies momentum compression within it.

The underlying mathematics (OLS regression + population standard deviation of residuals) dates back to Gauss and Legendre in the early 1800s. Gilbert Raff applied regression channels to trading in the 1990s. Carter's contribution was the dual-band structure and the specific parameter choices (period 100, dual \sigma levels) calibrated for swing trading on daily and intraday charts.

The dual band structure creates five statistical zones. Under normality assumptions, 68% of price action falls within \pm 1\sigma and 95% within \pm 2\sigma. Price reaching the \pm 2\sigma band represents a statistically significant deviation from trend, while the \pm 1\sigma bands define the boundaries of "normal" price oscillation.

Architecture & Physics

1. Linear Regression (Midline)

The best-fit line through the lookback window using ordinary least squares:

m = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - \left(\sum x_i\right)^2}

b = \frac{\sum y_i - m \sum x_i}{n}

The midline is the regression evaluated at the current bar:

\text{Mid}_t = m \cdot (n - 1) + b

2. Residual Standard Deviation

Population standard deviation of the differences between actual and predicted values:

\sigma_t = \sqrt{\frac{1}{n} \sum_{i=0}^{n-1} \left(y_i - (m \cdot i + b)\right)^2}

3. Dual Band Construction

Four bands at two statistical distances:

U_{1,t} = \text{Mid}_t + 1 \cdot \sigma_t, \qquad L_{1,t} = \text{Mid}_t - 1 \cdot \sigma_t

U_{2,t} = \text{Mid}_t + k \cdot \sigma_t, \qquad L_{2,t} = \text{Mid}_t - k \cdot \sigma_t

where k is the outer deviation multiplier (default 2.0). The inner bands (\pm 1\sigma) are always at one standard deviation.

4. Slope and R-Squared

The slope m indicates trend direction and strength. The coefficient of determination R^2 measures how well the linear model fits:

R^2 = 1 - \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}

High R^2 (near 1.0) indicates price is moving linearly; low R^2 indicates choppy or non-linear behavior.

5. Complexity

Per bar: O(n) due to two loops over the window. Memory: a ring buffer of n doubles. The index sums \sum x and \sum x^2 are precomputed constants.

Mathematical Foundation

Parameters

Symbol	Name	Default	Constraint	Description
n	period	100	> 1	Lookback window for regression
k	deviations	2.0	> 0	Outer band stddev multiplier

Statistical Zone Interpretation

Zone	Probability	Interpretation
Above +2\sigma	~2.5%	Extremely overbought relative to trend
+1\sigma to +2\sigma	~13.5%	Overbought
-1\sigma to +1\sigma	~68%	Normal oscillation around trend
-2\sigma to -1\sigma	~13.5%	Oversold
Below -2\sigma	~2.5%	Extremely oversold relative to trend

Performance Profile

Operation Count (Streaming Mode)

TTM_LRC extends REGCHANNEL with dual bands and R^2 computation. Two O(n) passes plus additional statistics:

Operation	Count	Cost (cycles)	Subtotal
ADD (sum_y accumulation, pass 1)	n	1	n
FMA (i × y for sum_xy, pass 1)	n	4	4n
MUL + DIV (slope, intercept, mean_y)	5	~9	45
FMA (slope × i + intercept, pass 2)	n	4	4n
SUB (residual, pass 2)	n	1	n
MUL (residual², pass 2)	n	3	3n
ADD (ssr accumulation, pass 2)	n	1	n
SUB + MUL + ADD (sst, pass 2)	2n	2	4n
DIV (ssr/n, sst check, R²)	3	15	45
SQRT (σ)	1	20	20
MUL + ADD/SUB (4 bands: ±1σ, ±kσ)	6	~2	12
Total	~$9n + 15$	—	~18n + 122 cycles

For period 100: ~1922 cycles/bar. The longer default period (100 vs 20) makes the window scans significantly more expensive than REGCHANNEL.

Batch Mode (SIMD Analysis)

Both passes iterate over contiguous ring buffer memory, enabling SIMD vectorization:

Operation	Scalar Ops	SIMD Ops (AVX-512)	Speedup
Pass 1: sum_y, sum_xy	2n	n/8	~16×
Pass 2: residuals + ssr + sst	6n	3n/4	~8×
Slope/intercept/bands/R²	15	15	1×

Resources

• Carter, J. (2005). Mastering the Trade. McGraw-Hill.
• Raff, G. (1991). "Trading the Regression Channel." Technical Analysis of Stocks & Commodities.
• Draper, N. & Smith, H. (1998). Applied Regression Analysis. Wiley.