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.
4.3 KiB
RAIN: Rainbow Moving Average
"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."
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.
Historical Context
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.
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.
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.
Architecture & Physics
1. Ten Cascaded SMA Layers
Each layer is an SMA applied to the previous layer's output:
\text{MA}_1 = \text{SMA}(x, N), \quad \text{MA}_k = \text{SMA}(\text{MA}_{k-1}, N), \quad k = 2, \ldots, 10
2. O(1) Running-Sum SMA
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.
3. Weighted Composite
\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}
Mathematical Foundation
Layer computation (recursive SMA):
\text{MA}_1[t] = \frac{1}{N}\sum_{i=0}^{N-1} x_{t-i}
\text{MA}_k[t] = \frac{1}{N}\sum_{i=0}^{N-1} \text{MA}_{k-1}[t-i], \quad k = 2, \ldots, 10
Equivalent kernel: The $k$-fold SMA is the $k$-th order B-spline kernel:
B_k(x) = \underbrace{B_0 * B_0 * \cdots * B_0}_{k \text{ times}}(x)
where B_0 is the rectangular pulse.
Weighted output:
\text{RAIN} = \frac{\sum_{k=1}^{10} w_k \cdot \text{MA}_k}{20}
with weights \mathbf{w} = [5, 4, 3, 2, 1, 1, 1, 1, 1, 1].
Group delay: Each SMA layer adds (N-1)/2 bars of lag. However, the weighted composite lag is:
\bar{d} = \frac{\sum w_k \cdot k \cdot (N-1)/2}{\sum w_k}
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.
Default parameters: period = 2, fixed layers = 10, minPeriod = 1.
Pseudo-code (streaming):
// 10 circular buffers with running sums
for layer = 1 to 10:
sum[layer] -= buf[layer][head]
sum[layer] += input[layer] // input is price for layer 1, MA[layer-1] for others
buf[layer][head] = input[layer]
MA[layer] = sum[layer] / count
head = (head + 1) % period
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
Resources
- Widner, M. (1998). "Rainbow Charts." Technical Analysis of Stocks & Commodities.
- thinkorswim / TD Ameritrade. "RainbowAverage" study documentation.
- 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.)