QuanTAlib/lib/trends_FIR/tsf/Tsf.md

TSF: Time Series Forecast

The best prediction of the future is the trend that's already in motion — extended by exactly one step.

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period (default 14)
Outputs	Single series (Tsf)
Output range	Tracks input
Warmup	period bars
PineScript	tsf.pine
Signature	tsf_signature

• TSF projects the least-squares regression line one bar forward, providing a statistically grounded forecast of the next bar's value.
• Similar: LSMA, LinReg | Complementary: R² for forecast reliability | Trading note: Time Series Forecast; linear regression extrapolated one bar ahead. Predictive MA.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

TSF projects the least-squares regression line one bar forward, providing a statistically grounded forecast of the next bar's value. Unlike simple moving averages that smooth past data, TSF answers the question: "If the current trend continues, where will price be next?" This makes it inherently leading rather than lagging, though the forecast degrades quickly beyond one step.

Historical Context

Time Series Forecast originates from classical linear regression applied to financial time series. The concept appeared in TA-Lib as TA_TSF and has been a standard offering in technical analysis software since the 1990s. Tushar Chande's The New Technical Trader (1994) formalized several regression-based indicators including the closely related Chande Forecast Oscillator (CFO), which measures the percentage error between the current price and the TSF value.

TSF is mathematically identical to the Least Squares Moving Average (LSMA) evaluated one step beyond the window endpoint. Where LSMA answers "what is the trend value now?", TSF answers "what will the trend value be next bar?" The relationship is exact: TSF = LSMA + slope, where slope is the per-bar rate of change of the regression line.

Architecture & Physics

1. O(1) Incremental Linear Regression

The implementation uses running sums (SumY, SumXY) with a reversed-x convention where x=0 corresponds to the newest bar. This allows O(1) updates without maintaining the full regression matrix.

Constants (precomputed once):

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

2. O(1) Sum Updates

When a new value enters and the oldest drops:

\Sigma_{xy}^{new} = \Sigma_{xy}^{old} + \Sigma_y^{old} - n \cdot v_{oldest} \Sigma_y^{new} = \Sigma_y^{old} - v_{oldest} + v_{new}

3. Regression Parameters

m = \frac{n \cdot \Sigma_{xy} - \Sigma_x \cdot \Sigma_y}{D} b = \frac{\Sigma_y - m \cdot \Sigma_x}{n}

In the reversed-x convention, b is the regression value at the current bar (x=0), and m is negative for uptrends.

4. TSF Calculation

\text{TSF} = b - m

This projects one step forward from the current bar. Equivalently, in standard convention (x=0=oldest):

\text{TSF} = \text{slope} \cdot n + \text{intercept}

5. Resync Guard

After every 1000 ticks, running sums are recomputed from the buffer to prevent floating-point drift accumulation.

Mathematical Precision & Implementation Philosophy

Relationship to Other Indicators

Indicator	Formula	Interpretation
LSMA (offset=0)	b	Regression value at current bar
TSF	b - m	Regression value one step ahead
LSMA (offset=1)	b - m × 1	Same as TSF
CFO	100 × (price - TSF_at_current) / price	Forecast error as percentage
Inertia	price - TSF_at_current	Raw forecast error (residual)

FMA Usage

All critical multiplications use Math.FusedMultiplyAdd for precision, including:

• SumXY O(1) update: FMA(-period, oldest, sumXY + prevSumY)
• Slope calculation: FMA(n, sumXY, -sumX × sumY)
• Intercept calculation: FMA(-m, sumX, sumY)

Performance Profile

Operation Count (Streaming Mode, Scalar)

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	4	1	4
MUL	0	3	0
DIV	2	12	24
FMA	3	5	15
CMP	1	1	1
Total	10		~44

Batch Mode (SIMD Analysis)

The O(1) running-sum algorithm is inherently serial due to data dependencies. Batch mode uses stackalloc for small buffers (≤256 elements) to avoid heap allocation.

Mode	Per-Bar Cost	Notes
Streaming	~44 cycles	O(1) update
Batch (Span)	~44 cycles	Same algorithm, zero-alloc
Batch (TSeries)	~44 cycles + state restore	Replays last N bars

Quality Metrics

Metric	Score (1-10)	Justification
Accuracy	9	Exact OLS regression, FMA precision
Timeliness	10	Leading indicator (projects forward)
Overshoot	7	Extrapolation amplifies noise
Smoothness	5	Less smooth than LSMA (forecast adds slope)

Validation

Library	Status	Notes
LSMA(offset=1)	✅	Mathematical identity, exact match
TA-Lib	🔲	TA_TSF available in TALib.NETCore
Skender	❌	No direct TSF method

Common Pitfalls

1. TSF is not LSMA. LSMA = regression value at the current bar. TSF = one step ahead. The difference equals the regression slope. Using TSF as a smoothing average will produce systematically biased results.

2. Single-step forecast only. TSF projects exactly one bar forward. Multi-step extrapolation (TSF at offset=2, 3, ...) accumulates error quadratically. For multi-step forecasting, use AFIRMA or dedicated time-series models.

3. Warmup = period bars. The indicator needs a full window of data before regression is meaningful. During warmup, TSF returns raw input values.

4. Noise amplification. Because TSF adds the slope to the endpoint value, it amplifies short-term noise. Use longer periods (20+) for less noisy forecasts, or combine with a smoother like LSMA.

5. Bar correction support. The isNew=false pathway correctly rolls back state using the _ps (previous state) pattern. Always use isNew=false for intra-bar updates in live trading.

6. Resync interval. Running sums are recomputed every 1000 ticks to prevent floating-point drift. This adds negligible overhead but ensures long-running accuracy.

References

• Tushar Chande, The New Technical Trader, 1994
• TA-Lib: TA_TSF function (www.ta-lib.org)
• PineScript: ta.linreg(source, length, -1) (offset=-1 = one step ahead)