Wavelet Candle Constructor (Inc. Morlet) 2

Here is the detailed description of the **Wavelet Candle** construction principles based on the code provided.

This indicator is not a simple smoothing mechanism (like a Moving Average). It utilizes the **Discrete Wavelet Transform (DWT)**, specifically the Stationary variant (SWT / à Trous Algorithm), to separate "noise" (high frequencies) from the "trend" (low frequencies).

Here is how it works step-by-step:

###1. The Wavelet Kernel (Coefficients)The heart of the algorithm lies in the coefficients (the `h` array in the `get_coeffs` function). Each wavelet type represents a different set of mathematical weights that define how price data is analyzed:

* **Haar:** The simplest wavelet. It acts like a simple average of neighboring candles. It reacts quickly but produces a "boxy" or "jagged" output.
* **Daubechies 4:** An asymmetric wavelet. It is better at detecting sudden trend changes and the fractal structure of the market, though it introduces a slight phase shift.
* **Symlet / Coiflet:** More symmetric than Daubechies. They attempt to minimize lag (phase shift) while maintaining smoothness.
* **Morlet (Gaussian):** Implemented in this code as a Gaussian approximation (bell curve). It provides the smoothest, most "organic" effect, ideal for filtering noise without jagged edges.

###2. The Convolution EngineInstead of a simple average, the code performs a mathematical operation called **convolution**:



For every candle on the chart, the algorithm takes past prices, multiplies them by the Wavelet Kernel weights, and sums them up. This acts as a **digital low-pass filter**—it allows the main price movements to pass through while cutting out the noise.

###3. The "à Trous" Algorithm (Stationary Wavelet Transform)This is the key difference between this indicator and standard data compression.
In a classic wavelet transform, every second data point is usually discarded (downsampling). Here, the **Stationary** approach is used:

* **Level 1:** Convolution every **1** candle.
* **Level 2:** Convolution every **2** candles (skipping one in between).
* **Level 3:** Convolution every **4** candles.
* **Level 4:** Convolution every **8** candles.

Because of this, **we do not lose time resolution**. The Wavelet Candle is drawn exactly where the original candle is, but it represents the trend structure from a broader perspective. The higher the `Decomposition Level`, the deeper the denoising (looking at a wider context).

###4. Independent OHLC ProcessingThe algorithm processes each component of the candle separately:

1. Filters the **Open** series.
2. Filters the **High** series.
3. Filters the **Low** series.
4. Filters the **Close** series.

This results in four smoothed curves: `w_open`, `w_high`, `w_low`, `w_close`.

###5. Geometric Reconstruction (Logic Repair)Since each price series is filtered independently, the mathematics can sometimes lead to physically impossible situations (e.g., the smoothed `Low` being higher than the smoothed `High`).
The code includes a repair section:

```pinescript
real_high = math.max(w_high, w_low)
real_high := math.max(real_high, math.max(w_open, w_close))
// Same logic for Low (math.min)

```

This guarantees that the final Wavelet Candle always has a valid construction: wicks encapsulate the body, and the `High` is strictly the highest point.

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###Summary of ApplicationThis construction makes the Wavelet Candle an **excellent trend-following tool**.

* If the candle is **green**, it means that after filtering the noise (according to the selected wavelet), the market energy is bullish.
* If it is **red**, the energy is bearish.
* The wicks show volatility that exists within the bounds of the selected decomposition level.

Here is a descriptive comparison of **Wavelet Candles** against other popular chart types. As requested, this is a narrative explanation focusing on the differences in mechanics, interpretation philosophy, and the specific pros and cons of each approach.

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###1. Wavelet Candles vs. Standard (Japanese) CandlesThis is a clash between "the raw truth" and "mathematical interpretation." Standard Japanese candles display raw market data—exactly what happened on the exchange. Wavelet Candles are a synthetic image created by a signal processor.

**Differences and Philosophy:**
A standard candle is full of emotion and noise. Every single price tick impacts its shape. The Wavelet Candle treats this noise as interference that must be removed to reveal the true energy of the trend. Wavelets decompose the price, reject high frequencies (noise), and reconstruct the candle using only low frequencies (the trend).

* **Wavelet Advantages:** The main advantage is clarity. Where a standard chart shows a series of confusing candles (e.g., a long green one, followed by a short red one, then a doji), the Wavelet Candle often draws a smooth, uniform wave in a single color. This makes it psychologically easier to hold a position and ignore temporary pullbacks.
* **Wavelet Disadvantages:** The biggest drawback is the loss of price precision. The Open, Close, High, and Low values on a Wavelet candle are calculated, not real. You **cannot** place Stop Loss orders or enter trades based on these levels, as the actual market price might be in a completely different place than the smoothed candle suggests. They also introduce lag, which depends on the chosen wavelet—whereas a standard candle reacts instantly.

###2. Wavelet Candles vs. Heikin AshiThese are close cousins, but they share very different "DNA." Both methods aim to smooth the trend, but they achieve it differently.

**Differences and Philosophy:**
Heikin Ashi (HA) is based on a simple recursive arithmetic average. The current HA candle depends on the previous one, making it react linearly.
The Wavelet Candle uses **convolution**. This means the shape of the current candle depends on a "window" (group) of past candles multiplied by weights (Gaussian curve, Daubechies, etc.). This results in a more "organic" and elastic reaction.

* **Wavelet Advantages:** Wavelets are highly customizable. With Heikin Ashi, you are stuck with one algorithm. With Wavelet Candles, you can change the kernel to "Haar" for a fast (boxy) reaction or "Morlet" for an ultra-smooth, wave-like effect. Wavelets handle the separation of market cycles better than simple HA averaging, which can generate many false color flips during consolidation.
* **Wavelet Disadvantages:** They are computationally much more complex and harder to understand intuitively ("Why is this candle red if the price is going up?"). In strong, vertical breakouts (pumps), Heikin Ashi often "chases" the price faster, whereas deep wavelet decomposition (High Level) may show more inertia and change color more slowly.

###3. Wavelet Candles vs. RenkoThis compares two different dimensions: Time vs. Price.

**Differences and Philosophy:**
Renko completely ignores time. A new brick is formed only when the price moves by a specific amount. If the market stands still for 5 hours, nothing happens on a Renko chart.
The Wavelet Candle is **time-synchronous**. If the market stands still for 5 hours, the Wavelet algorithm will draw a series of flat, small candles (the "wavelet decays").

* **Wavelet Advantages:** They preserve the context of time, which is crucial for traders who consider trading sessions (London/New York) or macroeconomic data releases. On a wavelet chart, you can see when volatility drops (candles become small), whereas Renko hides periods of stagnation, which can be misleading for options traders or intraday strategies.
* **Wavelet Disadvantages:** In sideways trends (chop), Wavelet Candles—despite the smoothing—will still draw a "snake" that flips colors (unless you set a very high decomposition level). Renko can remain perfectly clean and static during the same period, not drawing any new bricks, which for many traders is the ultimate filter against overtrading in a flat market.

###Summary**Wavelet Candles** are a tool for the analyst who wants to visualize the **structure of the wave and market cycle**, accepting some lag in exchange for noise reduction, but without giving up the time axis (like in Renko) or relying on simple averaging (like in Heikin Ashi). It serves best as a "roadmap" for the trend rather than a "sniper scope" for precise entries.