# statistics

This module contains functions to perform spatial statistics calculations.

## GETIS\_ORD\_QUADBIN <a href="#getis_ord_quadbin" id="getis_ord_quadbin"></a>

```sql
GETIS_ORD_QUADBIN(input, size, kernel)
```

**Description**

This function computes the Getis-Ord Gi\* statistic for each Quadbin index in the input array.

**Input parameters**

* `input`: `SUPER` input data with the indexes and values of the cells.
* `size`: `INT8` size of the Quadbin *k-ring* (distance from the origin). This defines the area around each index cell that will be taken into account to compute its Gi\* statistic.
* `kernel`: `VARCHAR` [kernel function](https://en.wikipedia.org/wiki/Kernel_\(statistics\)) to compute the spatial weights across the kring. Available functions are: uniform, triangular, quadratic, quartic and gaussian.

**Return type**

`SUPER`

**Example**

{% code overflow="wrap" lineNumbers="true" %}

```sql
SELECT carto.GETIS_ORD_QUADBIN(
    JSON_PARSE('[
        {"index": 5266443791933898751, "value": 51.0},
        {"index": 5266443803500740607, "value": 28.0},
        {"index": 5266443790415822847, "value": 19.0}
    ]'),
    3, 'gaussian'
);
-- [{"index":5266443791933898751,"gi":1.3606194139870578,"p_value":0.086817058065399522},{"index":5266443803500740607,"gi":-0.34633948719670504,"p_value":0.63545613599515272},{"index":5266443790415822847,"gi":-1.0142799267903513,"p_value":0.84477538488255133}]
```

{% endcode %}

## MORANS\_I\_QUADBIN <a href="#morans_i_quadbin" id="morans_i_quadbin"></a>

```sql
MORANS_I_QUADBIN(input, size, decay)
```

**Description**

This function computes the [Moran's I spatial autocorrelation](https://en.wikipedia.org/wiki/Moran's_I) from the input array of quadkey indexes.

**Input parameters**

* `input`: `SUPER` input data with the indexes and values of the cells.
* `size`: `INT8` size of the quadkey *k-ring* (distance from the origin). This defines the area around each index cell where the distance decay will be applied.
* `decay`: `VARCHAR` decay function to compute the [distance decay](https://en.wikipedia.org/wiki/Distance_decay). Available functions are: uniform, inverse, inverse\_square and exponential.

**Return type**

`FLOAT8`

**Example**

{% code overflow="wrap" lineNumbers="true" %}

```sql
SELECT carto.MORANS_I_QUADBIN(
    JSON_PARSE('[
        {"index": 5266443791927869439, "value": 51.0},
        {"index": 5266443791928131583, "value": 28.0},
        {"index": 5266443791928918015, "value": 19.0}
    ]'),
    3, 'exponential'
);
-- -0.2966571382680862
```

{% endcode %}

## P\_VALUE <a href="#p_value" id="p_value"></a>

```sql
P_VALUE(z_score)
```

**Description**

This function computes the p-value (two-tails test) of a given [z-score](https://en.wikipedia.org/wiki/Standard_score) assuming the population follows a normal distribution where the mean is 0 and the standard deviation is 1. The z-score is a measure of how many standard deviations below or above the population mean a value is. It gives you an idea of how far from the mean a data point is. The [p-value](https://en.wikipedia.org/wiki/P-value) is the probability that a randomly sampled point has a value at least as extreme as the point whose z-score is being tested.

**Input parameters**

* `z_score`: `FLOAT8` input data with the indexes and values of the cells.

**Return type**

`FLOAT8`

**Example**

{% code overflow="wrap" lineNumbers="true" %}

```sql
SELECT carto.P_VALUE(-2);
-- 0.04550012577451279
```

{% endcode %}