Simple Moving Average (SMA)

## Overview and Purpose

The Simple Moving Average (SMA) is one of the most fundamental and widely used technical indicators in financial analysis. It calculates the arithmetic mean of a selected range of prices over a specified number of periods. Developed in the early days of technical analysis, the SMA provides traders with a straightforward method to identify trends by smoothing price data and filtering out short-term fluctuations. Due to its simplicity and effectiveness, it remains a cornerstone indicator that forms the basis for numerous other technical analysis tools.

## What’s Different in this Implementation

- **Constant streaming update:**
On each bar we:
1) subtract the value leaving the window,
2) add the new value,
3) divide by the number of valid samples (early) or by `period` (once full).

- **Deterministic lag, same as textbook SMA:**
Once full, lag is `(period - 1)/2` bars—identical to the classic SMA. You just **don’t lose the first `period-1` bars** to `na`.

- **Large windows without penalty:**
Complexity is constant per tick; memory is bounded by `period`. Very long SMAs stay cheap.

## Behavior on Early Bars
- **Bars < period:** returns the arithmetic mean of **available** samples.
Example (period = 10): bar #3 is the average of the first 3 inputs—not `na`.
- **Bars ≥ period:** behaves exactly like standard SMA over a fixed-length window.

> Implication: Crosses and signals can appear earlier than with `ta.sma()` because you’re not suppressing the first `period-1` bars.

## When to Prefer This

- Backtests needing early bars: You want signals and state from the very first bars.
- High-frequency or very long SMAs: O(1) updates avoid per-bar CPU spikes.
- Memory-tight scripts: Single circular buffer; no large temp arrays per tick.

## Caveats & Tips

Backtest comparability: If you previously relied on na gating from ta.sma(), add your own warm-up guard (e.g., only trade after bar_index >= period-1) for apples-to-apples.

Missing data: The function treats the current bar via nz(source); adjust if you need strict NA propagation.

Window semantics: After warm-up, results match the textbook SMA window; early bars are a partial-window mean by design.

## Math Notes

Running-sum update:
sum_t = sum_{t-1} - oldest + newest
SMA_t = sum_t / k where k = min(#valid_samples, period)

Lag (full window): (period - 1) / 2 bars.

## References

- Edwards & Magee, Technical Analysis of Stock Trends
- Murphy, Technical Analysis of the Financial Markets