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83 lines
4.2 KiB
Markdown
83 lines
4.2 KiB
Markdown
# MEDIAN: Rolling Median
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> *The average is easily influenced by outliers; the median stands its ground.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Statistic |
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| **Inputs** | Source (close) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Median) |
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| **Output range** | Varies (see docs) |
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| **Warmup** | `period` bars |
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| **PineScript** | [median.pine](median.pine) |
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- The Rolling Median is a robust statistic that represents the middle value of a dataset within a moving window.
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- **Similar:** [Mode](../mode/Mode.md), [Percentile](../percentile/Percentile.md) | **Trading note:** Rolling median; robust central tendency resistant to outliers. Good for support/resistance identification.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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The Rolling Median is a robust statistic that represents the middle value of a dataset within a moving window. Unlike the Simple Moving Average (SMA), which can be skewed by extreme values, the Median provides a more stable measure of central tendency, making it particularly useful for filtering noise in volatile markets.
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## Historical Context
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The concept of the median dates back to Edward Wright in 1599, but its application in time-series analysis became prominent with the rise of robust statistics in the 20th century. In technical analysis, it is often used as a replacement for moving averages to identify trends without the lag induced by averaging large deviations.
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## Architecture & Physics
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The Median calculation requires maintaining a sorted view of the data window.
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* **Inertia**: High. A single new data point rarely shifts the median significantly unless it crosses the middle threshold.
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* **Stability**: Extremely robust against outliers. A price spike of 1000% has the same effect on the median as a spike of 1%.
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* **Complexity**: $O(N \log N)$ per update due to sorting, where $N$ is the period. For typical trading periods ($N < 200$), this is negligible on modern CPUs.
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## Mathematical Foundation
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For a window of $N$ values $X = \{x_1, x_2, ..., x_N\}$ sorted in ascending order:
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### 1. Odd Period
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If $N$ is odd, the median is the middle element:
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$$ \text{Median} = X_{(N+1)/2} $$
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### 2. Even Period
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If $N$ is even, the median is the average of the two middle elements:
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$$ \text{Median} = \frac{X_{N/2} + X_{(N/2)+1}}{2} $$
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## Performance Profile
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### Operation Count (Streaming Mode)
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Median maintains a sorted buffer; each bar requires a binary-search insert plus array shift.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Ring buffer evict oldest | 1 | 3 cy | ~3 cy |
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| Binary search + array shift insert | log N + N/2 | 2 cy | ~N cy |
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| Extract middle element(s) | 1 | 1 cy | ~1 cy |
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| NaN guard + state update | 1 | 2 cy | ~2 cy |
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| **Total (N=14)** | **O(N)** | — | **~20 cy** |
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O(N) per update. For large N, a dual-heap (min-heap + max-heap) O(log N) structure would be faster, but for typical periods (≤200) the sorted-array approach is cache-friendly.
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | High | $O(N \log N)$ is fast for small $N$. |
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| **Allocations** | 0 | Uses pre-allocated buffers and in-place sorting. |
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| **Complexity** | $O(N \log N)$ | Sorting dominates the cost. |
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| **Accuracy** | 10/10 | Exact calculation. |
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| **Timeliness** | Medium | Lags similar to SMA but handles steps differently. |
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| **Smoothness** | High | Filters out noise effectively. |
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## Validation
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **Math.NET** | ✅ | Matches statistical definition. |
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| **Excel** | ✅ | Matches `MEDIAN()` function. |
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| **Python** | ✅ | Matches `numpy.median`. |
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### Common Pitfalls
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* **Quantization**: The median moves in discrete steps (jumps from one value to another) rather than smoothly like an average.
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* **Flatlining**: In periods of low volatility, the median can remain constant for many bars, which may be interpreted as a lack of trend. |