Add new moving average implementations: LTMA, MCNMA, NLMA, NMA, NYQMA, RAIN, and TRAMA

- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation.
- MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness.
- NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages.
- NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel.
- NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages.
- RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing.
- TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.

lib/trends_IIR/rain/Rain.md

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+# 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.)