Local Regression

Local Regression

Local Regression (also known as Locally Weighted Scatterplot Smoothing or LOWESS) is a non-parametric regression technique used to analyze relationships between variables. Unlike traditional regression methods that fit a single model to the entire dataset, local regression fits a separate model to each point in the dataset, weighting nearby points more heavily. This makes it particularly useful for identifying and visualizing complex, non-linear relationships that might be missed by global models. This article will provide a comprehensive overview of local regression, its principles, implementation, applications in Technical Analysis, advantages, disadvantages, and how it differs from other regression techniques.

Core Principles

The fundamental idea behind local regression is to estimate the regression function at a specific point by fitting a weighted least squares regression to a subset of the data surrounding that point. The key components are:

Weighted Least Squares: This is the core statistical method. Instead of giving each data point equal weight, local regression assigns weights based on the distance between the point where the regression is being estimated and the other data points. Points closer to the target point receive higher weights, while those further away receive lower weights. This is crucial for capturing local patterns.
Kernel Function: The kernel function determines how weights are assigned. Common kernel functions include:

   * Tri-cube Weight Function: Provides a smooth weighting, reducing weights to zero beyond a certain distance.
   * Bi-weight (Bisquare) Weight Function: Similar to Tri-cube, but with different properties regarding the rate of weight reduction.
   * Gaussian Kernel Function:  Uses a Gaussian distribution to assign weights, resulting in a smooth, bell-shaped weighting pattern.
   * Uniform Kernel Function: Assigns equal weights to all points within a specified bandwidth.

Bandwidth (Smoothing Parameter): This is arguably the most important parameter in local regression. It controls the proportion of the data used for each local regression. A smaller bandwidth results in a more flexible, wiggly curve that closely follows the data, potentially overfitting. A larger bandwidth results in a smoother curve that may miss important local features, potentially underfitting. Choosing the appropriate bandwidth is a critical step, often achieved through cross-validation techniques. See Cross Validation for more details.
Polynomial Order: While simple local regression often uses linear models (first-order polynomial), higher-order polynomials can be used to capture more complex local patterns. However, increasing the polynomial order also increases the risk of overfitting.

Mathematical Formulation

Let's denote:

`x` as the independent variable.
`y` as the dependent variable.
`x₀` as the point at which we want to estimate the regression function.
`wᵢ(x₀)` as the weight assigned to the i-th data point based on its distance from `x₀`.

The local regression estimate at `x₀` is given by:

ŷ(x₀) = Σ [wᵢ(x₀) * yᵢ] / Σ wᵢ(x₀)

where the summation is over all data points.

The weights, `wᵢ(x₀)`, are calculated based on the kernel function and bandwidth. For example, using a Tri-cube kernel:

wᵢ(x₀) = (1 - (|xᵢ - x₀| / h)³)³ if |xᵢ - x₀| ≤ h wᵢ(x₀) = 0 otherwise

where `h` is the bandwidth.

Implementation Steps

1. **Choose a Kernel Function:** Select an appropriate kernel function based on the characteristics of the data and desired smoothness. 2. **Select a Bandwidth:** This is often done using cross-validation. The goal is to find the bandwidth that minimizes the prediction error on unseen data. Time Series Analysis methods can assist in bandwidth selection. 3. **Choose a Polynomial Order:** Start with a low-order polynomial (e.g., linear) and increase it if necessary, paying attention to potential overfitting. 4. **For each data point xᵢ:**

   a. Define a neighborhood around xᵢ using the bandwidth.
   b. Calculate weights for all data points based on their distance from xᵢ and the chosen kernel function.
   c. Fit a weighted least squares regression to the data points within the neighborhood, using the calculated weights.
   d. Predict the value of y at xᵢ using the fitted model.

5. **Repeat step 4 for all data points.** This results in a smooth curve that represents the local regression estimate.

Applications in Financial Markets & Technical Analysis

Local regression has numerous applications in analyzing financial data and identifying trading opportunities:

Trend Identification: Local regression can help identify trends in price data, especially those that are non-linear or change over time. It can reveal trends that are obscured by noise. See Trend Following for more on trend identification.
Support and Resistance Levels: By fitting a local regression curve to price data, you can identify potential support and resistance levels where the curve changes direction.
Volatility Analysis: Local regression can be used to model and predict volatility, especially when volatility clusters or changes over time. Related to Volatility Indicators.
Anomaly Detection: Data points that deviate significantly from the local regression curve may represent anomalies or outliers, potentially indicating unusual market activity. This is related to Outlier Detection.
Pattern Recognition: Identifying repeating patterns in price data, even if they are not perfectly symmetrical or consistent. See Chart Patterns.
Dynamic Support and Resistance: Unlike static support and resistance, local regression provides *dynamic* levels that adjust to changing market conditions. This is useful in Algorithmic Trading.
Improving the Accuracy of Other Indicators: Local regression can be used to smooth data before applying other technical indicators, reducing noise and improving their accuracy. For example, smoothing a Moving Average before calculating RSI.
Backtesting Strategies: Local regression can be used to generate trading signals for backtesting strategies. See Backtesting techniques.
Predictive Modeling: While not a primary predictive tool, it can contribute to more complex predictive models. See Predictive Analytics.
Sentiment Analysis Integration: Combining sentiment data with price data and applying local regression can reveal how market sentiment affects price movements. Relates to Sentiment Indicators.

Advantages of Local Regression

Flexibility: Local regression can capture complex, non-linear relationships that traditional regression methods may miss.
No Assumptions About Functional Form: Unlike parametric models, local regression does not require specifying a particular functional form for the relationship between the variables.
Robustness to Outliers: Because of the weighting scheme, local regression is less sensitive to outliers than some other regression methods.
Visual Clarity: The resulting smoothed curve provides a clear visualization of the underlying relationship between the variables.
Adaptability: The local nature of the regression allows it to adapt to changing relationships within the data. This is particularly useful in Adaptive Moving Averages.

Disadvantages of Local Regression

Computational Cost: Local regression can be computationally expensive, especially for large datasets, as it requires fitting a separate model for each data point.
Bandwidth Selection: Choosing the appropriate bandwidth can be challenging and requires careful consideration. Poor bandwidth selection can lead to overfitting or underfitting.
Edge Effects: The accuracy of the local regression estimate may be lower near the edges of the dataset, as there are fewer data points available for weighting.
Overfitting Risk: With a small bandwidth or high-order polynomial, there is a risk of overfitting the data, resulting in a curve that is too sensitive to noise.
Interpretability: The resulting curve can be more difficult to interpret than a simple linear regression model.