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116 lines
4.9 KiB
Markdown
116 lines
4.9 KiB
Markdown
# SMA: Simple Moving Average
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> *The vanilla ice cream of technical analysis. Boring, ubiquitous, and the only thing your grandfather and your high-frequency trading bot agree on.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Trend (FIR MA) |
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| **Inputs** | Source (close) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Sma) |
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| **Output range** | Tracks input |
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| **Warmup** | `period` bars |
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| **PineScript** | [sma.pine](sma.pine) |
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| **Signature** | [sma_signature](sma_signature.md) |
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- The Simple Moving Average (SMA) is the unweighted arithmetic mean of the last $N$ data points.
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- **Similar:** [EMA](../../trends_IIR/ema/ema.md), [WMA](../wma/wma.md) | **Complementary:** ATR for Keltner-style bands | **Trading note:** Simple Moving Average; equal-weight FIR filter. Most basic and widely used MA. Foundation of many composite indicators.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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The Simple Moving Average (SMA) is the unweighted arithmetic mean of the last $N$ data points. It acts as a low-pass filter, smoothing out high-frequency noise to reveal the underlying trend. While conceptually simple, efficient implementation on modern hardware requires careful attention to memory access patterns and vectorization.
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## Historical Context
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The concept of a moving average dates back to 1901 (R.H. Hooker) for smoothing weather data, but it became a staple of financial analysis in the mid-20th century. It is the baseline against which all other averages are compared.
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## Architecture & Physics
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The naive implementation of SMA sums $N$ numbers at every step, resulting in $O(N)$ complexity. QuanTAlib uses an optimized $O(1)$ approach.
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### O(1) Running Sum
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A running `Sum` and a `RingBuffer` of history are maintained.
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$$ Sum_{new} = Sum_{old} - Value_{oldest} + Value_{new} $$
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$$ SMA = \frac{Sum_{new}}{N} $$
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This ensures that calculating an SMA(200) takes the exact same time as an SMA(10).
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### Drift Correction
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Floating-point addition is not associative. Repeatedly adding and subtracting values from a running sum introduces cumulative error (drift) over millions of ticks. QuanTAlib implements a periodic **Resync** mechanism (every 1000 ticks) that recalculates the sum from scratch to ensure precision remains within `1e-9` of the true mean.
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### SIMD Optimization
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For batch processing of large datasets, `Sma.Batch` utilizes `System.Runtime.Intrinsics` (AVX2/AVX-512) to process multiple data points in parallel, significantly outperforming scalar loops.
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## Mathematical Foundation
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### 1. The Mean
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$$ SMA_t = \frac{1}{N} \sum_{i=0}^{N-1} P_{t-i} $$
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## Performance Profile
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### Operation Count (Streaming Mode, O(1) Running Sum)
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| SUB (Sum - oldest) | 1 | 1 | 1 |
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| ADD (Sum + newest) | 1 | 1 | 1 |
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| DIV (Sum / N) | 1 | 15 | 15 |
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| **Total (hot)** | **3** | — | **~17 cycles** |
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Every 1000 bars, a resync recalculates the sum to prevent drift:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD (N values) | N | 1 | N |
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| DIV (Sum / N) | 1 | 15 | 15 |
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| **Resync cost** | **N+1** | — | **~N+15 cycles** |
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**Amortized cost:** ~17 + (N+15)/1000 ≈ **~17 cycles/bar** for typical use.
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### Batch Mode (SIMD Analysis)
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SMA batch processing is highly vectorizable using running sum + prefix sum techniques:
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| Operation | Scalar Ops | SIMD Ops (AVX2) | Speedup |
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| :--- | :---: | :---: | :---: |
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| Initial N-sum | N | N/8 | 8× |
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| Running update (per bar) | 3 | ~1 | ~3× |
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| Division | 1 | 1/8 (batched) | 8× |
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For 512 bars:
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| Mode | Cycles/bar | Total | Notes |
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| :--- | :---: | :---: | :--- |
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| Scalar streaming | ~17 | ~8,700 | O(1) per bar |
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| SIMD batch | ~3 | ~1,500 | Vectorized running sum |
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| **Improvement** | **5.8×** | — | Batch wins for large N |
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 10/10 | Exact arithmetic mean |
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| **Timeliness** | 3/10 | Significant lag (~N/2 bars) |
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| **Overshoot** | 10/10 | Never overshoots input range |
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| **Smoothness** | 5/10 | Smooth but susceptible to drop-off jumps |
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### Benchmark Results
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| Metric | Value | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | ~100M bars/sec | SIMD batch mode |
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| **Allocations** | 0 bytes | Zero-allocation in hot paths |
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| **Complexity** | O(1) | Constant time regardless of period N |
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| **State Size** | 8 + 8N bytes | Sum + RingBuffer |
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## Validation
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **TA-Lib** | ✅ | Matches `TA_SMA` exactly. |
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| **Skender** | ✅ | Matches `GetSma` exactly. |
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| **Tulip** | ✅ | Matches `sma` exactly. |
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| **Ooples** | ✅ | Matches `CalculateSimpleMovingAverage`. | |