# 2.2 Distance-Based Weights

Distance-based weights are constructed using distance measures between all pairs of observations. The neighbor relation can be determined by a given ***distance band*****,** so every other observation that falls within the distance band is considered as neighbors of one observation. Another way to determine neighbor relation is to find ***K nearest neighbors*** (***KNN***) for each observation. 

For more information, please read: <https://geodacenter.github.io/workbook/4b_dist_weights/lab4b.html>

**CONTENTS**

1. min\_distthreshold()
2. distance\_weights()
3. knn\_weights()

### 1. min\_distthreshold()

In order to avoid isolates (islands) that would result from too stringent a critical distance, the distance must be chosen such that each location has at least one neighbor. Such a distance conforms to a *max-min* criterion, i.e., it is the largest of the nearest neighbor distances.

**Synopsis**

Short version

```sql
float min_distthreshold(integer gid, geometry the_geom)
```

Full version

```sql
float min_distthreshold(integer gid, geometry the_geom,
    boolean is_arc, boolean is_mile)
```

**Arguments**

| Input Arguments | Type     | Description                                                                    |
| --------------- | -------- | ------------------------------------------------------------------------------ |
| gid             | integer  | the feature id of geometry: e.g. gid, fid, ogc\_*fid,* cartod&#x62;*\_*&#x69;d |
| the\_geom       | geometry | the geometry (only points and polygons are supported)                          |
| 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                                                                     |
| --------- | ------------------------------------------------------------------------------- |
| float     | the minimum distance that makes sure each observation has at least one neighbor |

**Examples**

```sql
SELECT min_distthreshold(gid, the_geom) FROM natregimes;

 min_distthreshold 
-------------------
           1.46578
(1 row)
```

### 2. distance\_weights()

With the distance band, one can create a distance-based spatial weights using WINDOW function distance\_weights()

**Synopsis**

Short version:

```sql
float distance_weights(integer gid, geometry the_geom, float dist_band)
```

Full Version:

```sql
float distance_weights(integer gid, geometry the_geom, float dist_band,
    float power, 
    boolean is_inverse, 
    boolean is_arc,
    boolean is_mile)
```

**Arguments:**

| Input Arguments | Type     | Description                                                                                                  |
| --------------- | -------- | ------------------------------------------------------------------------------------------------------------ |
| gid             | integer  | the feature id of geometry: e.g. gid, fid, ogc*fid,* cartod&#x62;*\_*&#x69;d                                 |
| the\_geom       | geometry | the geometry (only points and polygons are supported)                                                        |
| dist\_band      | float    | the distance band/threshold that makes sure each observation has at least one neighbor                       |
| power           | float    | 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                                                                                 |
| --------- | ------------------------------------------------------------------------------------------- |
| bytea     | the weights structure for each observation in binary format, which is defined in table 2.1. |

**Examples**

```sql
SELECT distance_weights(gid, the_geom, 1.46578) OVER() FROM natregimes;

                         
                  distance_weights                                                                                                                                                                              
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 \x770700001400600800001608000014080000130800005f08000017080000c3070000c5070000c70700007107000074070000750700001e0700002a0700002107000020070000b9060000be060000c407000062060000e8e6a83f95d1b13f7049833f76be5f3f622eae3f10b88f3ff95bdf3e1218953f2716533fb6f0893fa2cc093ff686333f3787dd3e9c8b943f74c
8303f7074533f3d608c3f9fe47c3fec4c313faf3bad3f
 \x7a0000000e007b0000006d000000cc000000b300000098000000970000008d000000b20000005c0000004e00000025000000670000006800000039000000b9d4af3f5f5dad3f0ba3aa3f80a43a3ffeb2d93eb76d613fa2ed8d3fa3828e3fdba3753f36f80c3f4354b63f5839993fe7b6403fbeb4983f
 \xeb0000001200260100003f01000066010000ce000000490100002501000024010000010100000401000002010000ea000000c9000000cb00000096000000b0000000af0000009500000094000000678c663fa55ab03ff45eac3f5639613fe17ea53f227f6a3f6d34943f8fcf523fa3487e3f49c6b73eeb61513f3336683ff2cdb03ece2b9e3fb1c47c3f3a57393f771
6ab3f99bf883f
...
```

### 3. knn\_weights()

**Synopsis**

Short version:

```sql
float knn_weights(integer gid, geometry the_geom, integer k)
```

Full Version:

```sql
float knn_weights(integer gid, geometry the_geom, integer k,
    float power, 
    boolean is_inverse, 
    boolean is_arc,
    boolean is_mile)
```

**Arguments:**

| Input Arguments | Type     | Description                                                                                                  |
| --------------- | -------- | ------------------------------------------------------------------------------------------------------------ |
| gid             | integer  | the feature id of geometry: e.g. gid, fid, ogc*fid,* cartod&#x62;*\_*&#x69;d                                 |
| geom            | geometry | the geometry (only points and polygons are supported)                                                        |
| k               | integer  | the k nearest neighbors                                                                                      |
| power           | float    | 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                                                                                 |
| --------- | ------------------------------------------------------------------------------------------- |
| bytea     | the weights structure for each observation in binary format, which is defined in table 2.1. |

**Examples**

```sql
SELECT knn_weights(gid, the_geom, 5) OVER() FROM natregimes;

                         
                                          knn_weights                                           
------------------------------------------------------------------------------------------------
 \x770700000500c40700007407000021070000c30700001e070000ec4c313fa2cc093f74c8303ff95bdf3e3787dd3e
 \x7a000000050097000000b3000000680000004e00000098000000b76d613f80a43a3fe7b6403f36f80c3ffeb2d93e
 \xeb000000050001010000ea000000af000000cb000000020100008fcf523feb61513f3a57393ff2cdb03e49c6b73e
...
```