Gaussian Kernel Density Estimator

Description

Evaluate a univariate equally weighted Gaussian kernel density estimate at supplied points using an explicit absolute bandwidth, without automatic bandwidth selection or boundary correction.

Kernel density estimation builds a smooth density by placing a kernel at each sample value and adding their contributions. Bandwidth sets the smoothing scale in the same units as the observations; returned densities have reciprocal units, not probability mass per requested point.

Evaluate a univariate equally weighted Gaussian kernel density estimate at supplied points using an explicit absolute bandwidth, without automatic bandwidth selection or boundary correction.

When to use this tool

Use this tool to calculate kernel density estimator under the units, normalization, and boundary conventions below.

How to use the tool and how it works

Supply bandwidth, points, sample in the field shapes below.
Run the calculation. Correct format, shape, range, or undefined-statistic errors before interpreting a result.
Read densities, points, sampleCount. Use the conventions below.

At each supplied point x, sum exp(−0.5((x−sample_i)/bandwidth)²), then divide by sample count, sqrt(2π) and bandwidth. Return densities in the original query-point order.^[1]

Input field	Meaning and units	Accepted input
bandwidth	Positive Gaussian kernel standard deviation in the sample's units, not a multiplier of sample standard deviation.	Required. Finite number; strictly above 0.
points	Points	Required. Array of Finite number.; minimum items 1; maximum items 512.
sample	Sample	Required. Array of Finite number.; minimum items 1; maximum items 512.

Output field	Interpretation	Returned shape
densities	Estimated density at each returned point, in inverse sample units.	Required. Array of Finite number; minimum 0.; minimum items 1; maximum items 512.
points	Points	Required. Array of Finite number.; minimum items 1; maximum items 512.
sampleCount	Sample Count	Required. Safe integer; minimum 1; maximum 512.

json

{
  "input": {
    "bandwidth": 1,
    "points": [
      0,
      1,
      2
    ],
    "sample": [
      0,
      2
    ]
  },
  "output": {
    "densities": [
      0.22646662345731042,
      0.24197072451914337,
      0.22646662345731042
    ],
    "points": [
      0,
      1,
      2
    ],
    "sampleCount": 2
  }
}

Limitations and assumptions

Bandwidth is an absolute positive Gaussian standard deviation in the sample’s units. Densities have inverse units and need not be below one. There is no automatic bandwidth selection, boundary correction or finite-domain renormalization.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

Choose the statistic according to the question and data conventions. Inspect the supplied series or matrix and the stated assumptions before comparing outputs from implementations with different defaults.

References

Tool behavior and example: Gaussian Kernel Density Estimator — tool.io

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