ArticleslgStudy

mathematics

Join count statistic

Join count statistic is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Join count statistic rather than just read about it. In short: Join count statistics are a method of spatial analysis used to assess the degree of association, in particular the autocorrelation, of categorical variables distributed over a spatial map. They were originally introduced by Australian statistician P.

Join count statistic — main illustration
Join count statistic — illustration

Key takeaways

  • Join count statistic belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Join count statistic to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Join count statistic from memory before moving on to harder problems.

Reference excerpt

Join count statistics are a method of spatial analysis used to assess the degree of association, in particular the autocorrelation, of categorical variables distributed over a spatial map. They were originally introduced by Australian statistician P. A. P. Moran. Join count statistics have found widespread use in econometrics, remote sensing and ecology. Join count statistics can be computed in a number of software packages including PASSaGE, GeoDA, PySAL and spdep.

Binary data

Given binary data x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}} distributed over N {\displaystyle N} spatial sites, where the neighbour relations between regions i {\displaystyle i} and j {\displaystyle j} are encoded in the spatial weight matrix

w i j = { 1 i neighbor of j 0 otherwise {\displaystyle w_{ij}={\begin{cases}1\qquad &i{\text{ neighbor of }}j\\0&{\text{otherwise}}\end{cases}}}

the join count statistics are defined as

J = J B B + J B W + J W W {\displaystyle J=J_{BB}+J_{BW}+J_{WW}}

Where

J B B = 1 2 ∑ i j , i ≠ j w i j x i x j {\displaystyle J_{BB}={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}x_{i}x_{j}}

J B W = 1 2 ∑ i j , i ≠ j w i j ( x i − x j ) 2 {\displaystyle J_{BW}={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}(x_{i}-x_{j})^{2}}

J W W = 1 2 ∑ i j , i ≠ j w i j ( 1 − x i ) ( 1 − x j ) {\displaystyle J_{WW}={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}(1-x_{i})(1-x_{j})}

J = 1 2 ∑ i j , i ≠ j w i j {\displaystyle J={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}}

The B , W {\displaystyle B,W} subscripts refer to 'black'=1 and 'white'=0 sites. The relation J = J B B + J B W + J W W {\displaystyle J=J_{BB}+J_{BW}+J_{WW}} implies only three of the four numbers are independent. Generally speaking, large values of J B B {\displaystyle J_{BB}} and J W W {\displaystyle J_{WW}} relative to J B W {\displaystyle J_{BW}} imply autocorrelation and relatively large values of J B W {\displaystyle J_{BW}} imply anti-correlation. To assess the statistical significance of these statistics, the expectation under various null models has been computed. For example, if the null hypothesis is that each sample is chosen at random according to a Bernoulli process with probability

p = number of black cells N = N 1 N {\displaystyle p={\frac {\text{number of black cells}}{N}}={\frac {N_{1}}{N}}}

then Cliff and Ord show that

… excerpt ends here. Continue reading the full article.

Illustrations

Join count statistic: Join counts for 3 category data on a 10 × 10 {\displaystyle 10\times 10} grid using 'rook' (north, south, east, west) neighbors. Left: each category never has a neighbour of its own type, resulting in zeros on the diagonal. Centre: random pattern shows no bias for pairing colours, resulting in approximately equal values for all join count statistics. Right: Since different types are only adjacent on the edge of the patches this results in small values for J r ≠ s {\displaystyle J_{r\neq s}} .
Join counts for 3 category data on a 10 × 10 {\displaystyle 10\times 10} grid using 'rook' (north, south, east, west) neighbors. Left: each category never has a neighbour of its own type, resulting in zeros on the diagonal. Centre: random pattern shows no bias for pairing colours, resulting in approximately equal values for all join count statistics. Right: Since different types are only adjacent on the edge of the patches this results in small values for J r ≠ s {\displaystyle J_{r\neq s}} .

Worked examples

Example 1 — a first encounter with Join count statistic

Start with the simplest possible case. Write down what Join count statistic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Join count statistic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Join count statistic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Join count statistic

In research
Join count statistic appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Join count statistic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Join count statistic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Covariance and correlation, Spatial analysis, so understanding it makes those chapters shorter.
In everyday life
Look for Join count statistic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Join count statistic in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Join count statistic means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Join count statistic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Join count statistic in simple terms?

Join count statistics are a method of spatial analysis used to assess the degree of association, in particular the autocorrelation, of categorical variables distributed over a spatial map. They were originally introduced by Australian statistician P.

Why does Join count statistic matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Join count statistic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Join count statistic.

Tags

  • Covariance and correlation
  • Spatial analysis

Keep exploring