At the end of the last section, we defined a **point pattern** as a collection of point locations in space. In this section, we will expand upon some definitions and processes that are important in the study of point patterns with statistics.

To get ready for all the fun coding we are going to do, open up R and make sure that `spatstat`

is loaded by running

`library(spatstat)`

## Intensity

One of the most important properties of a point pattern is its **intensity**. Intensity is defined as the number of points per unit area in a point pattern, and is represented by the Greek letter \(\lambda\). This concept will continue to pop up in our analysis, and is the basis for many spatial statistics analysis methods.

## Point Processes

In order to study point patterns formally, we need to define what is called a **point process**. This is a random process or procedure that generates point patterns. Because point processes are random, 100 point patterns generated from the exact same point process could (and likely will) be completely different. This demonstrates why we are interested in point processes; when analyzing a point pattern, we are likely only looking at one random realization of the point process that was used to generate it. If we can figure something out about the point process, then we can make conclusions about the relationships between points that *will always be there*, even in another point pattern generated from the same process. It is important to note, however, that point processes are not always an appropriate way to study point patterns (although for our research, it usually is). For more details about situations when it is and isn’t appropriate to use point processes, see *Spatial Point Patterns: Methods and Applications with R* (SPP:MAR) pg. 142.

One of the simplest and most important examples of a point process is the **Poisson process**. In this process, \(n\) points (denoted \(x_1, x_2, \ldots, x_n\)) are randomly spatially distributed in a region \(W\). The number of points, \(n\), is randomly generated from a Poisson distribution with intensity \(\lambda\) (if unfamiliar with the Poisson distribution, check out the Wikipedia page). What exactly does “randomly spatially distributed” mean? It means that the coordinates of each point \(x_i\) are selected from a random uniform distributed over \(W\), and that each point location is independent of each other point location (independent just means that the placement of each point is completely unaffected by the location of the other points). The expected number of points to fall in a window with area \(|W|\) is \(\lambda |W|\). To generate a point pattern realization of the Poisson process with \(\lambda = 100\) points/unit area in a 2D window with domain \(x,y \in [0,1]\), we can run

`pois.pp <- rpoispp(lambda = 100, win = owin(c(0,1), c(0,1)))`

we can then look at our point pattern:

`plot(pois.pp)`

Bam! That’s a nice looking point pattern. Lets check out how many points are actually in there:

`npoints(pois.pp)`

`## [1] 94`

So, we see that even though we specified \(\lambda = 100\), we only have 94 points, due to the randomness of \(n\) introduced by selecting from the Poisson distribution. If you’re curious, try generating a few more realizations to see that each point pattern has different point locations and a different value of \(n\). The analogous function to `rpoispp()`

for 3D point patterns is `rpoispp3()`

. This function is used very similarly to `rpoispp`

; check out the help documentation by running `?rpoispp3’ for more information.

There are many other kind of point processes; pretty much any procedure you can come up with to generate a point pattern randomly. We will talk about a few different important ones (besides the Poisson) later in the tutorials.

## Complete Spatial Randomness (CSR)

**Complete spatial randomness** (CSR) is a term used to describe a point pattern that is “completely random” (what a surprise there). As it turns out, the definition of CSR is actually synonymous with the Poisson process. Formally, a point pattern has CSR if it is homogeneous (described in the next section) and if its point locations are independent. Because CSR and the Poisson process are one in the same, it is fair to say that if a point pattern has CSR i.f.f. it is a realization of the Poisson process.

CSR point patterns have a few nice properties:

**Thinning Property**: If we start with a CSR point pattern with intensity \(\lambda\) and apply random “thinning” (removing points), where each point has probability \(0 < p < 1\) of remaining in the pattern, then the point pattern after thinning will have CSR with intensity \(p \lambda\).(Video) Introduction to Spatial Data & GIS**Superposition Property**: If we start with two CSR point patterns with intensities \(\lambda_A\) and \(\lambda_B\), respectively, then their superposition will be a CSR point pattern with intensity \(\lambda = \lambda_A + \lambda_B\).

CSR is a common null hypothesis for point pattern hypothesis testing (which we will get to later in the tutorials), and has many other uses in point pattern spatial statistics. Get familiar with it!

## Homogeneity

The definition of CSR utilizes the term **homogeneous**. A point pattern is homogeneous if the points are equally likely to be located at any spatial location in the window. This means that the intensity \(\lambda\) is constant over the entire window \(W\).

If a point pattern isn’t homogeneous, then it is **inhomogeneous**. The introduction of inhomogeneity allows for the definition of another important point process: the **inhomogeneous Poisson process**, which is a slight adjustment to the Poisson process discussed earlier. In this modification, the intensity of points at location \(u \in W\) is defined by some intensity function \(\lambda(u)\). This means that the expected number of points falling in a pixel of size \(\Delta u\) at location \(u\) is \(\lambda(u) \Delta u\), and the expected total number of points in window \(W\) is \(\int_W \lambda(u) du\).

Point patterns generated from the inhomogeneous Poisson process still have the property of independence; each point’s location is independent of every other point’s location. These point patterns also still satisfy the thinning property and the superposition property, but the new intensity functions will be \(p \lambda(u)\) and \(\lambda_A(u) + \lambda_B(u)\), respectively.

To simulate a realization of the inhomogeneous Poisson process, we first define an intensity function, and then use `rpoispp`

:

`lambda.u <- function(x,y){1000 * x^2 * y^2 + 100}pois.inh.pp <- rpoispp(lambda = lambda.u, win = owin(c(0,1), c(0,1)))plot(pois.inh.pp)`

It is quite obvious that this point pattern’s intensity varies as a function of position in \(W\).

The inhomogeneous Poisson process is a decent point process model for many real-world point patterns.

## Stationarity & Isotropy

There are a few important assumptions that are frequently made about point process models in order to perform spatial statistics:

First is **stationarity**, which is invariance of a point process under translation. There is a helpful description of stationarity in SPP:MAR: “Imagine a sheet of cardboard with a hole in it. When we shift the cardboard around (without changing its directional orientation), and view the point process through the hole, the statistical properties of the observable point process are the same in each position”.

Next is **isotropy**, which is invariance of a point process under rotation around some intrinsic center. That is, if we rotate a point process around its center, we will not be able to tell the difference between the rotated and original process.

A point process can have one or both of these properties, but assuming both of them is common in many fields of spatial stats (including those outside the study of point processes and point patterns).

## Marks

Occasionally, points in a point pattern will need to be “marked” with extra information. This mark can be categorical or continuous in nature. For example, if you had a point pattern describing the locations of trees in a forest, you could mark each point with the species of tree it is (categorical) or the diameter of the tree (continuous). For APT data, marks are usually the mass/charge ratio (continuous) or chemical identity after ranging (categorical).

To add marks to a point pattern in `spatstat`

, you use the `mark()`

function. For example, let’s first create a point pattern using the Poisson process:

`pp.marked <- rpoispp(lambda = 100, win = owin(c(0,1),c(0,1)))`

Now, lets generate a categorical mark of either \(A\), \(B\), or \(C\) for each point in the pattern. We do this using the `sample()`

function, which allows us to randomly sample as many times as we want from a vector of values.

`mks <- sample(x = c('A','B','C'), size = npoints(pp.marked), replace = TRUE)`

We can then add these marks to our point pattern:

`marks(pp.marked) <- mks`

Now, when we plot our point pattern, we can see where the different marks are located:

`plot(pp.marked)`

If we ever want to check out the marks in a point pattern (if there are any), we can run:

`pp.marked$marks`

`## [1] "B" "B" "A" "B" "A" "C" "C" "A" "A" "C" "B" "B" "C" "C" "C" "A" "B"## [18] "C" "A" "C" "A" "C" "B" "B" "C" "A" "C" "C" "C" "A" "A" "A" "B" "B"## [35] "C" "C" "A" "B" "A" "C" "A" "A" "B" "C" "B" "A" "B" "B" "A" "C" "C"## [52] "B" "B" "A" "B" "C" "B" "B" "C" "A" "C" "A" "B" "A" "A" "B" "C" "B"## [69] "B" "B" "B" "A" "A" "B" "A" "A" "C" "B" "B" "C" "A" "C" "A" "C" "A"## [86] "C" "B"`

or alternatively:

`marks(pp.marked)`

`## [1] "B" "B" "A" "B" "A" "C" "C" "A" "A" "C" "B" "B" "C" "C" "C" "A" "B"## [18] "C" "A" "C" "A" "C" "B" "B" "C" "A" "C" "C" "C" "A" "A" "A" "B" "B"## [35] "C" "C" "A" "B" "A" "C" "A" "A" "B" "C" "B" "A" "B" "B" "A" "C" "C"## [52] "B" "B" "A" "B" "C" "B" "B" "C" "A" "C" "A" "B" "A" "A" "B" "C" "B"## [69] "B" "B" "B" "A" "A" "B" "A" "A" "C" "B" "B" "C" "A" "C" "A" "C" "A"## [86] "C" "B"`

We will get into statistics that you can perform with marks later in the tutorials, but if you are interested, you can also check out SPP:MAR pg. 638.

## FAQs

### Basic Spatial Statistics Definitions? ›

The term spatial statistics refers to **the application of statistical concepts and methods to data that have a spatial location attached to them**, and in which this locational element is used as an important and necessary part of the analysis.

**What are the 5 types of spatial analysis? ›**

Six types of spatial analysis are **queries and reasoning, measurements, transformations, descriptive summaries, optimization, and hypothesis testing**.

**What are the concepts of spatial statistics? ›**

What is Spatial Statistics? Spatial statistics is all about **analysing data that has a spatial (location) characteristic to it**. This type of analysis looks for patterns or correlation in recorded observations of some process that occurs across a space.

**What are the three 3 components of spatial data define? ›**

There are three main types of vector data: **points, lines and polygons**. The points help create lines, and the connecting lines form enclosed areas or polygons.

**What are the different types of spatial statistics? ›**

The many types of spatial statistics include **descriptive, inferential, exploratory, geostatistical, and econometric statistics**." These operation may or may not be done directly through the GIS software used. It is highly possible that you will need to use either some modelling or statistical software.

**What are the 4 basic types of spatial relationships? ›**

Spatial Relationships Types. **Adjacency, contiguity, overlap, and proximity** are the four ways of describing the relationship between two or more entities.

**What are 4 types of spatial patterns? ›**

Types of spatial patterns represented on maps include **absolute and relative distance and direction,** **clustering, dispersal, and elevation**.

**What is the difference between statistics and spatial statistics? ›**

**The main difference between classical statistics and geostatistics is the assumption of spatial dependency**. That is, the location of data elements with respect to one another plays an important role in the analysis, modeling, and estimation procedures.

**What are the two main types of spatial data? ›**

The two primary spatial data types are **Geometric and Geographic data**. Geographic data is data that can be mapped to a sphere (the sphere in question is usually planet earth). Geographic data typically refers to longitude and latitude related to the location of an object on earth.

**What are the 10 core concepts of spatial information? ›**

The concepts include **location, neighborhood, field, object, network, event, granularity, accuracy, meaning, and value**.

### What are 4 key data standard components for spatial data? ›

The elements include an overview describing the purpose and usage, as well as specific quality elements reporting on the lineage, positional accuracy, attribute accuracy, logical consistency and completeness.

**What are the three basic patterns of spatial distribution? ›**

Spatial distribution patterns can be **uniform, random, or clumped**. Additionally, population distribution is affected by population density, which is the number of individuals within a given area.

**What are the 3 spatial distribution patterns? ›**

Dispersion or distribution patterns show the spatial relationship between members of a population within a habitat. Individuals of a population can be distributed in one of three basic patterns: **uniform, random, or clumped**.

**What are three examples of spatial data? ›**

**Examples of geospatial data include:**

- Vectors and attributes: Descriptive information about a location such as points, lines and polygons.
- Point clouds: A collection of co-located charted points that can be recontextured as 3D models.
- Raster and satellite imagery: High-resolution images of our world, taken from above.

**What is the importance of spatial statistics? ›**

Importance of Spatial Analysis

Spatial analysis **enables you to tackle difficult location-based challenges and gain a deeper understanding of where and what is happening in your environment**. It goes beyond simple mapping to allow you to investigate the features of places and their interactions.

**What are the basic types of spatial data models? ›**

There are two broad categories of spatial data models. These are **vector data model and raster data models**.

**What are the 4 spatial properties? ›**

Spatial properties include **location, size, distance, direction, separation and connection, shape, pattern, and movement**.

**What are the three determining factors of spatial interaction? ›**

Spatial interaction is the flow of information, products, and human beings from one location to another. Three principles of spatial interaction, as proposed by transportation geographer Edward Ullman, are **complementarity, transferability, and intervening opportunity**.

**What are spatial patterns in math? ›**

Common spatial pattern (CSP) is **a mathematical procedure used in signal processing for separating a multivariate signal into additive subcomponents which have maximum differences in variance between two windows**.

**What are 3 main concepts of nature and society that illustrate spatial relationships? ›**

Geographers analyze relationships among and between places to reveal important spatial patterns. Explain how major geographic concepts illustrate spatial relationships Concepts of nature and society include **sustainability, natural resources, and land use**.

### What are 3 types of statistics? ›

**Types of Statistics in Maths**

- Descriptive Statistics. In this type of statistics, the data is summarised through the given observations. ...
- Inferential Statistics. This type of statistics is used to interpret the meaning of Descriptive statistics. ...
- Statistics Example.

**What are the 3 areas of statistics? ›**

There are three real branches of statistics: **data collection, descriptive statistics and inferential statistics**.

**What are the major components of spatial data? ›**

**The options for the question – Key components of 'spatial data' quality include**

- Positional accuracy.
- Temporal accuracy.
- Lineage and completeness.
- Logical consistency.
- All of the above.

**What is the difference between data and spatial data? ›**

The main difference between attribute data and spatial data is that the **attribute data describes the characteristics of a geographical feature while spatial data describes the absolute and relative location of geographic features**.

**What are the basic spatial concepts? ›**

As our language begins to develop, early spatial concepts such as **in front of, behind, top, bottom, over, under, last, between, farthest, backward, in, on**, etc., help us understand directions more precisely, ask detailed questions, and express our ideas to others.

**What are the six important considerations of spatial approach? ›**

A spatial approach considers the arrangement of the phenomena being studied across the surface of the earth. Important considerations of this approach are things such as **location, distance, direction, orientation, pattern, and interconnection**.

**What are examples of spatial information? ›**

A finding as to whether any overlap exists would be an example of spatial information. If overlap is found to exist, the extent of it would also be an example of spatial information. To produce spatial information, Db2 Spatial Extender must process data that defines the locations of features.

**What is an example of a spatial analysis? ›**

Understanding where things are or where events occur. Measuring sizes, shapes, and distributions of things or measurements. Analyzing relationships and interactions between places. Optimizing locations for facilities, or routes for transportation.

**What is spatial distribution for dummies? ›**

A spatial distribution is **the arrangement of a phenomenon across the Earth's surface** and a graphical display of such an arrangement is an important tool in geographical and environmental statistics.

**What is the most common spatial distribution? ›**

**Clumped distribution**, also called aggregated distribution, clumped dispersion or patchiness, is the most common type of dispersion found in nature. In clumped distribution, the distance between neighboring individuals is minimized.

### What are the three main properties of distribution? ›

Three characteristics of distributions. There are 3 characteristics used that completely describe a distribution: **shape, central tendency, and variability**. We'll be talking about central tendency (roughly, the center of the distribution) and variability (how broad is the distribution) in future chapters.

**What is spatial distribution in statistics? ›**

A spatial distribution in statistics is **the arrangement of a phenomenon across the Earth's surface** and a graphical display of such an arrangement is an important tool in geographical and environmental statistics.

**How do you explain spatial analysis? ›**

Spatial analysis is defined as the process of studying entities by examining, assessing, evaluating, and modeling spatial data features such as locations, attributes, and their relationships that reveal the geometric or geographic properties of data.

**What are the problems with spatial statistics? ›**

In spatial analysis, four major problems interfere with an accurate estimation of the statistical parameter: the **boundary problem, scale problem, pattern problem (or spatial autocorrelation), and modifiable areal unit problem**.

**What are the 4 types of non spatial data? ›**

There are four typical data types that we use in GIS: **integer, float/real, text/string, and date**.

**What is spatial data classification? ›**

Spatial classification **assigns an object to a class from a given set of classes based on the attribute values of the object**. It mainly considers the distance, direction, or connectivity relationships among spatial objects.

**What is an example of a spatial data structure? ›**

Spatial data consists of spatial objects made up of points, lines, regions, rectangles, surfaces, volumes, and even data of higher dimension which includes time. Examples of spatial data include **cities, rivers, roads, counties, states, crop coverages, mountain ranges, parts in a CAD system**, etc.

**What are 3 examples of spatial analysis? ›**

**An Example of Spatial Analysis**

- Understanding where things are or where events occur.
- Measuring sizes, shapes, and distributions of things or measurements.
- Analyzing relationships and interactions between places.
- Optimizing locations for facilities, or routes for transportation.

**What are some examples of spatial analysis? ›**

Examples of spatial analysis include measuring distances and shapes, setting routes and tracking transportations, establishing correlations between objects, events, and places via referring their locations to geographical positions (both live and historical).

**What are the list of basic spatial concepts? ›**

As our language begins to develop, early spatial concepts such as **in front of, behind, top, bottom, over, under, last, between, farthest, backward, in, on**, etc., help us understand directions more precisely, ask detailed questions, and express our ideas to others.

### What is the difference between GIS and spatial analysis? ›

The true power of GIS lies in the ability to perform analysis. **Spatial analysis is a process in which you model problems geographically, derive results by computer processing, and then explore and examine those results**.

**What are the main steps in spatial analysis? ›**

**A typical spatial analysis workflow involves the following:**

- Frame the question you want to answer.
- Find and prepare the data using data engineering to make it ready for analysis.
- Explore the data on a map and with charts to better understand it.

**What are the three types of spatial processes? ›**

According to National Geographic, there are three general types of spatial process: **natural-physical systems, environment-society relationships, and human systems**. These different systems help to explain how a spatial distribution came to be, such as the distribution of Irish-Americans in the U.S.

**What are 3 examples of spatial relationships? ›**

Spatial Relationships

For example, **a ball may be behind the chair, or under the table, or in the box**. The dog may be on the blanket, outside of the house, or in the doghouse. These types of words are sometimes called position words because they suggest the position of an object in relation to something else.

**Why is spatial statistics important? ›**

Spatial analysis **allows you to solve complex location-oriented problems and better understand where and what is occurring in your world**. It goes beyond mere mapping to let you study the characteristics of places and the relationships between them. Spatial analysis lends new perspectives to your decision-making.

**What is spatial analysis simple? ›**

Definition from the ESRI Dictionary: "The process of examining the locations, attributes, and relationships of features in spatial data through overlay and other analytical techniques in order to address a question or gain useful knowledge.

**What are the characteristics of spatial data? ›**

Important characteristics of spatial data are its **measurement level, map scale and associated topological information**. Nominal, ordinal, interval and ratio are the four levels of measurement for populating the spatial data matrix; they hold different amounts of information and determine what analysis can be performed.

**What are the most common types of spatial searches? ›**

In spatial queries, the most commonly used are **Euclidean distances and distances in a connected network**. Table 1 provides a real-world query example for the corresponding distance measure.

**What are the types of surfaces in spatial data analysis? ›**

Surfaces can be represented using **contour lines or isolines, arrays of points, TINs, and rasters**; however, most surface analysis in GIS is done on raster or TIN data. Contours are sets of lines of equal value across a surface. They are frequently created to represent surfaces on a map.