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https://github.com/mihakralj/QuanTAlib.git
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Miha Kralj
7253f61299
- Implemented the TRAMA (Trend Regularity Adaptive Moving Average) class with adaptive EMA logic. - Added unit tests for TRAMA functionality, including constructor validation, basic calculations, state management, and robustness checks. - Created validation tests to ensure consistency across different modes of operation (streaming, batch, and static calculations). - Enhanced documentation for TRAMA, including performance profiles and quality metrics. - Updated workspace configuration by removing unnecessary folder references.
93 lines
4.3 KiB
Markdown
93 lines
4.3 KiB
Markdown
# RAIN: Rainbow Moving Average
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> "Mel Widner applied SMA ten times recursively, then weighted the layers like a rainbow: brightest at the top, fading toward the base. Ten colors of smoothing, one composite average that sees both fast and slow structure simultaneously."
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RAIN recursively applies SMA 10 times, producing 10 layers of progressively smoother price representation, then computes a weighted average across all layers. Layers 1-4 receive weights 5, 4, 3, 2 (emphasizing the more responsive layers), while layers 5-10 each receive weight 1, for a total divisor of 20. This multi-scale composition produces a moving average that responds to short-term price changes through the lightly smoothed upper layers while maintaining stability through the heavily smoothed lower layers.
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## Historical Context
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Mel Widner published "Rainbow Charts" in *Technical Analysis of Stocks & Commodities* (1998), introducing the concept of recursive SMA application as both a visualization technique and a composite smoothing method. The thinkorswim platform later standardized the weight vector as $[5, 4, 3, 2, 1, 1, 1, 1, 1, 1]$, which became the canonical RAIN MA.
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The recursive SMA application has a deep mathematical interpretation: applying SMA $k$ times is equivalent to convolving the rectangular kernel with itself $k$ times, which produces a B-spline kernel of order $k$. Thus RAIN's 10 layers correspond to B-splines of orders 1 through 10, and the weighted average blends these spline approximations. The B-spline interpretation explains why higher layers are smoother: each convolution adds a degree of polynomial reproduction and reduces the spectral sidelobe level.
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The weight vector $[5, 4, 3, 2, 1, 1, 1, 1, 1, 1]$ with sum 20 was chosen empirically rather than derived from optimization theory. The declining weights for layers 1-4 bias the output toward the more responsive layers, making RAIN track trends more closely than a uniform average of all 10 layers would.
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## Architecture & Physics
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### 1. Ten Cascaded SMA Layers
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Each layer is an SMA applied to the previous layer's output:
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$$
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\text{MA}_1 = \text{SMA}(x, N), \quad \text{MA}_k = \text{SMA}(\text{MA}_{k-1}, N), \quad k = 2, \ldots, 10
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$$
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### 2. O(1) Running-Sum SMA
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Each of the 10 SMA layers uses a circular buffer with a running sum, giving O(1) per-bar update cost per layer. Total cost: O(10) per bar, with O($10 \times N$) memory for the 10 buffers.
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### 3. Weighted Composite
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$$
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\text{RAIN} = \frac{5 \cdot \text{MA}_1 + 4 \cdot \text{MA}_2 + 3 \cdot \text{MA}_3 + 2 \cdot \text{MA}_4 + \sum_{k=5}^{10} \text{MA}_k}{20}
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$$
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## Mathematical Foundation
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**Layer computation (recursive SMA):**
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$$
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\text{MA}_1[t] = \frac{1}{N}\sum_{i=0}^{N-1} x_{t-i}
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$$
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$$
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\text{MA}_k[t] = \frac{1}{N}\sum_{i=0}^{N-1} \text{MA}_{k-1}[t-i], \quad k = 2, \ldots, 10
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$$
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**Equivalent kernel:** The $k$-fold SMA is the $k$-th order B-spline kernel:
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$$
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B_k(x) = \underbrace{B_0 * B_0 * \cdots * B_0}_{k \text{ times}}(x)
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$$
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where $B_0$ is the rectangular pulse.
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**Weighted output:**
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$$
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\text{RAIN} = \frac{\sum_{k=1}^{10} w_k \cdot \text{MA}_k}{20}
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$$
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with weights $\mathbf{w} = [5, 4, 3, 2, 1, 1, 1, 1, 1, 1]$.
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**Group delay:** Each SMA layer adds $(N-1)/2$ bars of lag. However, the weighted composite lag is:
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$$
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\bar{d} = \frac{\sum w_k \cdot k \cdot (N-1)/2}{\sum w_k}
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$$
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For $N = 2$: $\bar{d} \approx 1.85$ bars. The upper-layer weighting significantly reduces the effective lag below what layer 10 alone would produce.
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**Default parameters:** `period = 2`, `fixed layers = 10`, `minPeriod = 1`.
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**Pseudo-code (streaming):**
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```
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// 10 circular buffers with running sums
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for layer = 1 to 10:
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sum[layer] -= buf[layer][head]
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sum[layer] += input[layer] // input is price for layer 1, MA[layer-1] for others
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buf[layer][head] = input[layer]
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MA[layer] = sum[layer] / count
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head = (head + 1) % period
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return (5*MA[1] + 4*MA[2] + 3*MA[3] + 2*MA[4] + MA[5] + MA[6] + MA[7] + MA[8] + MA[9] + MA[10]) / 20
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```
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## Resources
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- Widner, M. (1998). "Rainbow Charts." *Technical Analysis of Stocks & Commodities*.
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- thinkorswim / TD Ameritrade. "RainbowAverage" study documentation.
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- Schoenberg, I.J. (1946). "Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions." *Quarterly of Applied Mathematics*, 4(1), 45-99. (B-spline theory underlying recursive SMA.)
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