# 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](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()

```sql
function getKernelWeights(
    String mapUid,
    Number distBand, 
    String kernel,
    Number power, 
    Boolean isInverse,
    Boolean isArc,
    Boolean isMile,
    Boolean useKernelDiagonals)
```

**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                                    |
| <p></p><p>kernel</p> | 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**

| **Value**     | Description                                                  |
| ------------- | ------------------------------------------------------------ |
| WeightsResult | the weights structure for each observation in binary format. |

### 2. getKernelKnnWeights()&#x20;

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.&#x20;

**API**

```sql
function getKernelKnnWeights(
    String map_uid, 
    integer k, 
    String kernel,
    Boolean adaptive_bandwidth,
    Number power, 
    Boolean is_inverse,
    Boolean is_arc,
    Boolean is_mile,
    Boolean use_kernel_diagonals)
```

**Arguments**

| Input Arguments         | Type    | Description                                                                                                                                      |
| ----------------------- | ------- | ------------------------------------------------------------------------------------------------------------------------------------------------ |
| map\_uid                | String  | the unique map id                                                                                                                                |
|  k                      | Number  | the k nearest neighbors                                                                                                                          |
| <p></p><p>kernel</p>    | 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**

| **Value**    | Description                                 |
| ------------ | ------------------------------------------- |
| WeightResult | the weights structure for each observation. |

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

{% embed url="https://codesandbox.io/s/4spatial-weights-owi84" %}

