Kernel function

As explained in Causal Inference for Data Science

A kernel is a weighting function that assigns observation weights according to distance from a target point.

A kernel is a weighting function that gives observations near a target point t0 more influence than observations farther away. In regression discontinuity designs, it weights residuals during minimization so nearby points drive the local estimate. section 11.2.5

Two standard kernel choices are:

• Uniform: assigns weight 1 when |t − t0| < h and 0 otherwise. section 11.2.5
• Triangular: assigns weight 1 − |t − t0|/h within the window and 0 outside it. section 11.2.5

The bandwidth h centers and rescales the kernel. It determines which observations enter the regression: points farther than h from t0 receive zero weight. Cross-validation can compare bandwidths, while rdplot and rdrobust provide default selection methods. section 11.2.5

Local effect near a cutoff

To estimate an effect at time t0, give observations immediately before and after t0 greater influence, then fit the local regression using the selected bandwidth. section 11.2.5

Compare the supplied uniform and triangular kernel figures to see how distance and bandwidth shape observation weights. section 11.2.5

How Causal Inference for Data Science covers it

How deep it goes

Defines two kernels, shows bandwidth transformation, and defers elaborate bandwidth selection to package documentation.

Examples

Uses uniform and triangular functions, kernel figures, and local estimation around a time cutoff.

What it assumes

Assumes a regression discontinuity setting with a target point, time distances, residual minimization, and local regression.

What you can do after

Evaluate how kernel choice and bandwidth control which observations influence a local regression estimate.

How it connects

Connects kernels to regression discontinuity, local polynomial regression, residuals, bandwidth, and treatment-effect estimation.

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Also explained in

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  Grokking Machine LearningA mathematical function that represents an inner product in a transformed feature space, allowing a model to learn nonlinear decision boundaries.
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  Experimentation for EngineersA kernel function assigns a similarity or weight to an observation based on its distance from another input.
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  Machine Learning Algorithms in DepthA positive-definite function that measures similarity between pairs of inputs and defines an associated feature-space inner product.

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Related concepts

• feature mapproduces · A feature map is a spatially organized array of neural-network activations representing the presence or strength of learned features at different locations.
• covariance functionequivalent to · A covariance function, also called a kernel, specifies the covariance between function values at pairs of inputs in a Gaussian process.
• kernel density estimateused by · A kernel density estimate is a nonparametric probability-density estimate formed by combining localized kernel functions centered on observed data points.
• similarity metrickind of · A similarity measure is a rule that quantifies or characterizes how alike two objects or representations are according to selected properties.
• radial basis function kernelincludes · A radial basis function kernel is a similarity function whose value depends on the distance between inputs and typically decreases as that distance increases.
• support vector machineused by · A support vector machine is a supervised-learning model that uses support vectors to determine a maximum-margin decision boundary or regression function, optionally using kernels to model nonlinear relationships.

This explainer was generated by Manning's knowledge graph from the text of Causal Inference for Data Science. The section links point to the parts of the book it draws on.