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Gaussian Kernel Smoothing Momentum

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Overview:
The Gaussian Kernel Smoothing Momentum indicator analyzes and quantifies market momentum by applying statistical techniques to price and returns data. This indicator uses Gaussian kernel smoothing to filter noise and provide a more accurate representation of momentum. Additionally, it includes a option to evaluate the absolute score of the momentum to determine if the beginning of a "trend" is likely or if you can expect a "trend" to come to an end.

Kernels and Their Role In Time Series Analysis:
In statistical analysis, a kernel is a weighting function used to estimate the properties of a dataset. Kernels are particularly useful in non-parametric methods, where they serve to smooth data or estimate probability density functions without assuming a specific underlying distribution. The Gaussian kernel, one of the most commonly used, is characterized by its smooth, bell-shaped curve which provides a natural way to give more weight to data points closer to the target value and less weight to those further away.

Uses of Kernels in Time Series Analysis
Kernels play a significant role in time series analysis, especially in the context of smoothing and filtering. With kernel functions, you can reduce noise and extract the underlying systematic component or signal from the data. This process is essential for identifying long-term patterns in the data, which is often obscured by short-term fluctuations and random noise.

Kernel Smoothing
Kernel smoothing is a technique that applies a kernel function to a set of data points to create a smooth curve, effectively reducing the impact of random variations. In time series analysis, kernel smoothing helps to filter out short-term noise while retaining significant trends and "patterns". The Gaussian kernel, with its emphasis on nearby points, is particularly effective for this purpose, as it smooths the data in a way that highlights the underlying structure without overfitting to random fluctuations.

Additionally, kernels are used in non-parametric volatility estimation, option pricing models, and for detecting anomalies in financial data. Their flexibility and ability to handle complex, non-linear relationships make them well-suited for the often noisy data encountered in financial markets.

Momentum Component
The momentum component of the indicator is designed to quantify the directional movement of asset prices by applying the Gaussian kernel smoothing to the expected return of the price data. The data then has the variance stabilized and normalizes the distribution of price changes to be able to more efficiently analyze the momentum.

The Gaussian kernel smoothing function serves to filter out high-frequency noise, isolating the underlying systematic component of the momentum. This is achieved by weighting the data points based on their proximity to the current observation, with closer data points exerting a stronger influence. The resulting smoothed momentum provides a clearer of the directional bias in the market, devoid of short-term volatility.

Absolute Move Component
The absolute move component is a extension of the momentum analysis, focusing on the magnitude rather than the direction of the price movements. This component captures the absolute score of the smoothed momentum series, providing a measure of strength or intensity of the price movement, independent from its direction. The absolute move component also incorporates a Kalman filter to further smooth and refine the signal. The Kalman filter dynamically adjusts based on the observed variance in the data, to reduce the impact of outliers.

What to make of this indicator
The smoothed momentum line helps determine whether the market is experiencing upward and downward momentum. If the momentum line is above zero and rising, this suggest a positive expected returns. Conversely, if the momentum line is below zero and falling, it indicates negative expected returns.

You should also pay attention to changes in the slope of the momentum line and the moving average of the smoothed momentum(weighted with an optimal sampling size algorithm). A flattening or reversal of the slope may signal a potential shift in market direction. For example, if the momentum line and moving average transitions from rising to falling, it means that the expected return is going from positive to negative so you can see the "trend" as weakening or forming a trend of negative expected returns.

The absolute move component is designed to measure the intensity or strength of the current market movement. A low absolute move value, especially when they are negative or at the lower end of their band, indicates that the momentum and expected return is close to zero, which suggest that the market is experiencing minimal directional movement, which can be a sign of consolidation. High absolute values signal that the market is undergoing a significant price movement. When the absolute move is high and/or rising, it indicates that the movement of the momentum is strong, regardless of whether it is bullish or bearish.

If the absolute move reaches unusually high levels, it could indicate that the market is experiencing an exceptional price move, which might be unsustainable. Traders can anticipate potential reversals or profit taking targets. However, you should avoid trying to trade reversals as exceptionally high values in a time series do not guarantee an immediate reversal. This high values often occur during periods of strong trends or significant events, which can continue longer than expected, and you cant time when it will return to its mean. The mean-reverting nature of some statistical models can suggest a return to the mean, but this assumption can be misleading in financial markets, where trends can persist despite overextending conditions.

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