QuanTAlib/lib/trends_FIR/lsma/Lsma.md

LSMA: Least Squares Moving Average

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period, offset (default 0)
Outputs	Single series (Lsma)
Output range	Tracks input
Warmup	period bars
Signature	lsma_signature

TL;DR

• LSMA (Least Squares Moving Average), also known as the Moving Linear Regression or Endpoint Moving Average, calculates the least squares regression...
• Parameterized by period, offset (default 0).
• Output range: Tracks input.
• Requires period bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

"If you want to know where the price is going, draw a line through where it's been. LSMA does this for every single bar, tirelessly fitting linear regressions while you sleep."

LSMA (Least Squares Moving Average), also known as the Moving Linear Regression or Endpoint Moving Average, calculates the least squares regression line for the preceding time periods. In plain English: it finds the "best fit" line for the data window and tells you where that line ends.

Historical Context

Linear regression is as old as Gauss (c. 1809). Applying it as a moving window to financial time series is a more recent development, popularized by traders who realized that a moving average is just a poor man's regression line (specifically, an SMA is a regression line with a slope of 0). LSMA captures both the level and the trend (slope) of the data.

Architecture & Physics

LSMA is computationally heavier than an SMA because it minimizes the sum of squared errors for a line equation y = mx + b.

• Slope (m): Represents the trend strength/direction.
• Intercept (b): Represents the value at the start of the window.
• Endpoint: The value at the current bar (y = m \times 0 + b in our coordinate system where current bar is 0).

Mathematical Foundation

The regression line is y = mx + b.

m = \frac{N \sum xy - \sum x \sum y}{N \sum x^2 - (\sum x)^2} b = \frac{\sum y - m \sum x}{N} \text{LSMA} = b - m \times \text{Offset}

(Note: In the QuanTAlib implementation, x ranges from N-1 (oldest) to 0 (newest) to simplify the math).

Performance Profile

Operation Count (Streaming Mode, Scalar)

The O(1) algorithm maintains running sums instead of recomputing the regression on each bar:

State variables maintained:

• sum_x: Sum of x indices (precomputed constant for fixed period)
• sum_y: Running sum of y values
• sum_xy: Running sum of x×y products
• sum_xx: Sum of x² (precomputed constant)

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	6	1	6
MUL	4	3	12
DIV	2	15	30
Total	12	—	~48 cycles

Hot path breakdown:

• Update running sums: sum_y += new - old, sum_xy += (N-1)×new - sum_y_old → 4 ADD/SUB
• Slope calculation: m = (N×sum_xy - sum_x×sum_y) / denom → 2 MUL + 1 DIV
• Intercept: b = (sum_y - m×sum_x) / N → 1 MUL + 1 SUB + 1 DIV
• Endpoint: LSMA = b - m×offset → 1 MUL + 1 SUB

Comparison with naive O(N) regression:

Mode	Complexity	Cycles (Period=100)
Naive (recompute)	O(N)	~600 cycles
QuanTAlib O(1)	O(1)	~48 cycles
Improvement	—	~12× faster

Batch Mode (SIMD)

LSMA batch can vectorize the running sum updates:

Operation	Scalar Ops (512 bars)	SIMD Ops (AVX2)	Speedup
Running sum updates	512	64	8×
Slope calculations	1024	128	8×
Endpoint projections	512	64	8×

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Mathematically precise regression endpoint
Timeliness	8/10	Projects trend forward, reducing perceived lag
Overshoot	2/10	Significant overshoot on trend reversals (projects continuation)
Smoothness	3/10	Sensitive to outliers; least-squares fit follows noise

Validation

Validated against Skender.

Library	Status	Notes
Skender	✅	Matches GetEpma
TA-Lib	N/A	Not implemented

| Tulip | N/A | Not implemented. | | Ooples | N/A | Not implemented. |

C# Implementation Considerations

The QuanTAlib LSMA implementation achieves O(1) streaming updates through running sum maintenance with several optimizations:

O(1) Running Sum Algorithm

The implementation maintains two running sums (SumY, SumXY) that enable constant-time updates instead of O(N) recalculation:

// O(1) update for sum_xy: sum_xy_new = sum_xy_old + sum_y_prev - n * oldest
_state.SumXY = Math.FusedMultiplyAdd(-_period, oldest, _state.SumXY + prev_sum_y);

// O(1) update for sum_y
_state.SumY = _state.SumY - oldest + val;

Precomputed Constants

Mathematical constants are computed once in the constructor to avoid redundant calculations:

// sum_x = 0 + 1 + ... + (n-1) = n(n-1)/2
_sum_x = 0.5 * period * (period - 1);

// sum_x2 = 0² + ... + (n-1)² = (n-1)n(2n-1)/6
double sum_x2 = (period - 1.0) * period * (2.0 * period - 1.0) / 6.0;

// denominator = n * sum_x2 - sum_x²
_denominator = period * sum_x2 - _sum_x * _sum_x;

State Record Struct

State uses LayoutKind.Auto for compiler-optimized field ordering:

[StructLayout(LayoutKind.Auto)]
private record struct State(double SumY, double SumXY, double LastVal, double LastValidValue);
private State _state;
private State _p_state;  // Previous state for bar correction

FusedMultiplyAdd Usage

FMA is used extensively for slope, intercept, and endpoint calculations:

double m = Math.FusedMultiplyAdd(n, _state.SumXY, -sx * _state.SumY) / denom;
double b = Math.FusedMultiplyAdd(-m, sx, _state.SumY) / n;
result = Math.FusedMultiplyAdd(-m, _offset, b);

Periodic Resync

Running sums accumulate floating-point drift; periodic resync every 1000 ticks corrects this:

private const int ResyncInterval = 1000;

private void Resync()
{
    _state.SumY = _buffer.Sum;
    _state.SumXY = 0;
    var span = _buffer.GetSpan();
    for (int i = 0; i < span.Length; i++)
    {
        int x = span.Length - 1 - i;
        _state.SumXY = Math.FusedMultiplyAdd(x, span[i], _state.SumXY);
    }
}

Stackalloc/ArrayPool Strategy

The static Calculate method uses stackalloc for small periods (≤256) to avoid heap allocation:

const int StackAllocThreshold = 256;
Span<double> buffer = period <= StackAllocThreshold
    ? stackalloc double[period]
    : new double[period];

Thread-Safe Disposal

Disposal uses atomic operations for idempotent, thread-safe cleanup:

protected override void Dispose(bool disposing)
{
    if (Interlocked.CompareExchange(ref _disposed, 1, 0) == 0 && _source != null)
    {
        _source.Pub -= _handler;
        _source = null;
    }
    base.Dispose(disposing);
}

NaN Handling

Invalid values are replaced with the last valid value to maintain calculation integrity:

[MethodImpl(MethodImplOptions.AggressiveInlining)]
private double GetValidValue(double input)
{
    if (double.IsFinite(input))
    {
        _state.LastValidValue = input;
        return input;
    }
    return _state.LastValidValue;
}

Memory Layout

Field	Type	Size	Purpose
_period	int	4	Lookback window
_offset	int	4	Forecast offset
_buffer	RingBuffer	8 (ref)	Circular storage
_sum_x	double	8	Precomputed Σx
_denominator	double	8	Precomputed denominator
_state	State	32	Current state (SumY, SumXY, LastVal, LastValidValue)
_p_state	State	32	Previous state for rollback
_tickCount	int	4	Resync counter
_disposed	int	4	Atomic disposal flag
Total		~104 bytes	Per instance (excluding RingBuffer internal storage)

Common Pitfalls

1. Overshoot: Because it projects a trend, LSMA will overshoot significantly when the trend reverses. It assumes the trend continues.
2. Offset: You can use a positive offset to extrapolate into the future (forecasting), or a negative offset to center the average.
3. Noise: It is very sensitive to outliers because it tries to fit a line to them.