# Kernel Weights

Kernel Weights applies kernel function to determine the distance decay in the derived continuous weights kernel. The kernel weights are defined as a function K(z) of the ratio between the distance dij from i to j, and the bandwidth *hi*, with z=dij/*hi*. The kernel functions include * {triangular , *uniform, *quadratic , *epanechnikov, *quartic, *gaussian}.

Uniform, K(z)=1/2 for |z|<1,

Triangular, K(z)=(1−|z|) for |z|<1,

Quadratic or Epanechnikov, K(z)=(3/4)(1−z^2) for |z|<1,

Quartic, K(z)=(15/16)(1−z^2)^2 for |z|<1|z|<1, and

Gaussian. K(z) = (2π)^{1/2}exp(-z^2/2)

For more information, please read: https://geodacenter.github.io/workbook/4c_distance_functions/lab4c.html#kernel-weights

There are two types of fixed bandwidths for kernel weights. One is the max-min distance used earlier (the largest of the nearest-neighbor distances): **kernel_weights***(). The other is the maximum distance for a given specification of k-nearest neighbors: ***kernel_knn_****weights**().

### 1. getKernelWeights()

**Arguments**

Input Arguments | Type | Description |

mapUid | String | the unique map id |

distBand | Number | the distance band/threshold that makes sure each observation has at least one neighbor |

kernel | String | a varchar value of kernel method, which has to be one of {'triangular', 'uniform', 'epanechnikov', 'quartic', 'gaussian'} |

useKernelDiagonals | Boolean | if apply kernel on the diagonal of weights matrix. Default: FALSE. |

power | Number | the power/exponent corresponds to the number of times the base (dist_band) is used as a factor. Default: 1. |

isInverse | Boolean | if apply inverse on distance value. Default: False. |

isArc | Boolean | if compute arc distance between two observations. Default: FALSE. |

isMile | Boolean | if convert distance unit from mile to kilometer(KM). Default: TRUE. |

**Return**

| Description |

WeightsResult | the weights structure for each observation in binary format. |

### 2. getKernelKnnWeights()

With knn set to a given value, the maximum distance between the selected k-nearest neighbors' pairs is used as a "fixed" bandwidth. However, a drawback of fixed bandwidth kernel weights is that the number of non-zero weights can vary considerably, especially when the density of the point locations is not uniform throughout space. The argument `adaptive_bandwidth`

is provided to allow adaptive bandwidth in knn kernel weights: instead of a fixed distance bandwidth, the distance to the k-th nearest neighbor is used in the kernel function for each observation.

**API**

**Arguments**

Input Arguments | Type | Description |

map_uid | String | the unique map id |

k | Number | the k nearest neighbors |

kernel | String | a varchar value of kernel method, which has to be one of {'triangular', 'uniform', 'epanechnikov', 'quartic', 'gaussian'} |

adaptive_bandwidth | Boolean | if use adaptive bandwidth (distance to k-th nearest neighbor for each observation), or use max knn distance of all observations. Default: FALSE. |

use_kernel_diagonals | Boolean | if apply kernel on the diagonal of weights matrix. Default: FALSE. |

power | Number | the power/exponent corresponds to the number of times the base (dist_band) is used as a factor. Default: 1. |

is_inverse | Boolean | if apply inverse on distance value. Default: False. |

is_arc | Boolean | if compute arc distance between two observations. Default: FALSE. |

is_mile | Boolean | if convert distance unit from mile to kilometer(KM). Default: TRUE. |

**Return**

| Description |

WeightResult | the weights structure for each observation. |

**Try it yourself in the playground (jsgeoda + deck.gl):**

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