Lesson 2: Spatial Weights and Applications

Dr. Kam Tin Seong
Assoc. Professor of Information Systems(Practice)

School of Computing and Information Systems,
Singapore Management University

25 Nov 2023

Content

• Introduction to Spatial Weights
  • Contiguity-Based Spatial Weights
  • Distance-Band Spatial Weights
• What is Spatial Autocorrelation
• Measures of Global Spatial Autocorrelation
• Localised Geospatial Analysis
  • Local Indicators of Spatial Association (LISA)
  • Cluster and Outlier Analysis
• Emerging Hot Spot Analysis (EHSA)
  • A Spacetime Cubes

This lesson consists of three main sections. First, I am going to explain the basic concept of spatial weights. This is followed by a discussion on various methods used to compute spatial weights. Lastly, the applications of spatial weights will be discussed.

What is geographically referenced attribute?

Rows: 323
Columns: 12
$ SUBZONE_N        <chr> "MARINA SOUTH", "PEARL'S HILL", "BOAT QUAY", "HENDERS…
$ SUBZONE_C        <fct> MSSZ01, OTSZ01, SRSZ03, BMSZ08, BMSZ03, BMSZ07, BMSZ0…
$ PLN_AREA_N       <fct> MARINA SOUTH, OUTRAM, SINGAPORE RIVER, BUKIT MERAH, B…
$ PLN_AREA_C       <fct> MS, OT, SR, BM, BM, BM, BM, SR, QT, QT, QT, BM, ME, R…
$ REGION_N         <fct> CENTRAL REGION, CENTRAL REGION, CENTRAL REGION, CENTR…
$ REGION_C         <fct> CR, CR, CR, CR, CR, CR, CR, CR, CR, CR, CR, CR, CR, C…
$ YOUNG            <dbl> NA, 1100, 0, 2620, 2840, 2910, 2850, 0, 1120, 30, NA,…
$ `ECONOMY ACTIVE` <dbl> NA, 3420, 50, 7500, 6260, 7560, 8340, 50, 2750, 210, …
$ AGED             <dbl> NA, 2110, 20, 3260, 1630, 3310, 3590, 10, 560, 50, NA…
$ TOTAL            <dbl> NA, 6630, 70, 13380, 10730, 13780, 14780, 60, 4430, 2…
$ DEPENDENCY       <dbl> NA, 0.9385965, 0.4000000, 0.7840000, 0.7140575, 0.822…
$ geometry         <MULTIPOLYGON [m]> MULTIPOLYGON (((31495.56 30..., MULTIPOL…

A kind of data that is very similar to an ordinary data. The only difference is that each observation is associated with some form of geography such as numbers of aged population by planning subzone.

Describing attribute distribution

The dependency ratio values by planning subzone are normally distributed.

In statistics, we are interested to study if the observed distribution resemble normal distribution. To provide an answer to the question, we have to go beyond EDA and apply Confirmatory Data Analysis or also know as statistical test to confirm the observated pattern is indeed resemble normal distribution.

Geographical distribution question

Are the planning subzones with high proportion of dependency ratio randomly distributed over space?

When come of geographically referenced attributes, beside the distribution of the value, we are also interested to find out if the observed geographical patterns are randomly distributed. Similarly to statistical, geovisualisation is not enough because the visual impression will change when different data classification methods is used.

Geographical distribution question

Are the planning subzones with high proportion of dependency ratio randomly distributed over space?

]

As a result, a formal statistical test of spatial randomness is needed.

What are Spatial Weights (wij)

• A way to define spatial neighbourhood.

Before we can perform statistics test of spatial randomness, we need to understand how spatial relationship among geographical areas can be defined mathematically.

Defining Spatial Weights

There are at least two popular methods can be used to define spatial weights of geographical areas. They are contiguity and distance. Both of them are encoded in binary.

According to contiguity principle, two geographical areas are consider as neighbour if they share a common boundary. Hence, if two geographical areas are neighbour, they will be indicated as 1. On the other hand, if the two geographical areas are not neighbour, they will be indicated as 0.

According to distance principle, two geographical areas are consider as neighbour if their centroid as within a spefic distance. Hence, if two geographical areas are neighbour, they will be indicated as 1. On the other hand, if the two geographical areas are not neighbour, they will be indicated as 0.

Contiguity Neighbours

• Contiguity (common boundary)

• What is a “shared” boundary?

This figure shows three different ways to define contiguity neighbours. They are Rooks, Bishop and Queens methods. Rooks and Queens are the two commonly used methods.
The main different between Queens and Rooks is that Rooks only consider geographical areas that shared common boundaries but Queens method includes geographical areas touching at the tips of the target geographical area.

Beyond the basic contiguity neighbours

There are also second-order, third-order, forth-order, etc contiguity

The figure show contiguity neighbours beyond the first order. As you can see that the second-order contiguity neighbours are also defined according to Rooks and Queens case.

Weights matrix: Adjacency-based neighbours

Quiz: With reference to the figure below, list down the neighbour(s) of area 1202 using Rook case.

This figure shows a geographical area with nine locations. Let us focus on location 1202. According to Queens case, it has seven neighbours. They are 1101, 1201, 1301, 1102, 1302, 1000, and 0901. If Rooks case is consider, 1101 and 1301 will not be considered as neighbours because they do not share common boundaries with 1202.

Weights Matrix: Distance-based neighbours

Quiz: With reference to the figure below, create a weights matrix for d = 650.

In this figure, a cut-off distance of 650m is used. In this case, locations B, C and D are considered as neighbours of location A because all of them are with the 650m distance from A. On the other hand, location E is not a neighbour of location A because their distance is 757m apart.

Weights matrix: Measured distances

In this figure, spatial weights are calculated as the inversed function of the distance. In other word, two locations that are closer will be given higher weight than two locations that are further away. For example, the distance of AB is 353 and the diatnce of AE is 757 and the spatial weights are 0.00283 and 0.00132 respectively.

Row standardisation

In practice, row-standardised weights instead of spatial weights will be used.

Row-standardised weights increase the influence of links from observations with few neighbours, which binary weights vary the influence of observations. + Those with many neighbours are up-weighted compared to those with few. + In this case, raw 1 have four neighbours, hence each neighbour will have a value of 1/4. On the other hand, row number 2, has only two neighbour, hence each neighbour will be given a value of 1/2.

Applications of Spatial Weights

Formally, for observation i, the spatial lag of yi, referred to as [Wy]i (the variable Wy observed for location i) is:

where the weights wij consist of the elements of the i-th row of the matrix W, matched up with the corresponding elements of the vector y.

With a neighbor structure defined by the non-zero elements of the spatial weights matrix W, a spatially lagged variable is a weighted sum or a weighted average of the neighboring values for that variable. In most commonly used notation, the spatial lag of y is then expressed as Wy.

Spatially Lagged Variables

Spatial lag with row-standardized weights.

This figure shows the spatially lagged of GPDPC summed up the GDPPC of all its neighbours except the target location itself.

Spatial window sum

The spatial window sum uses and includes the diagonal element.

Spatial window sum, on the other hand, summed up the GDPPC of all neighbours and include the target location itself.

What is Spatial Autocorrelation

• Toble’s First Law of Geography
• Spatial Dependency
• Spatial Autocorrelation
  • Positive autocorrelation
  • Negative autocorrelation

Tobler’s First law of Geography

Everything is related to everything else,
but near things are more related than distant things.

The foundation of the fundamental concepts of:

• spatial dependence, and

• spatial autocorrelation

Reference: A Computer Movie Simulating Urban Growth in the Detroit Region

Spatial Dependency

• Spatial dependence is the spatial relationship of variable values (for themes defined over space, such as rainfall) or locations (for themes defined as objects, such as cities).

• Spatial dependence is measured as the existence of statistical dependence in a collection of random variables, each of which is associated with a different geographical location.

Spatial Autocorrelation

• Spatial autocorrelation is the term used to describe the presence of systematic spatial variation in a variable.

• The variable can assume values either:

  • at any point on a continuous surface (such as land use type or annual precipitation levels in a region);
  • at a set of fixed sites located within a region (such as prices at a set of retail outlets); or
  • across a set of areas that subdivide a region (such as the count or proportion of households with two or more cars in a set of Census tracts that divide an urban region).

Positive Spatial Autocorrelation

• Clustering
  • like values tend to be in similar locations.
• Neighbours are similar
  • more alike than they would be under spatial randomness.
• Compatible with diffusion
  • but not necessary caused by diffusion.

Negative Spatial Autocorrelation

• Checkerboard patterns
  • “opposite” of clustering
• Neighbours are dissimilar
  • more dissimilar than they would be under spatial randomness
• Compatible to competition
  • but not necessary competition

Measures of Global Spatial Autocorrelation

• Moran’s I
• Geary’s c

Measures of Global Spatial Autocorrelation: Moran’s I

Describe how features differ from the values in the study area as a whole.

• Moran I (Z value) is:
  • positive (I>0): Clustered, observations tend to be similar;
  • negative(I<0): Dispersed, observations tend to be dissimilar;
  • approximately zero: observations are arranged randomly over space.

Measures of Global Spatial Autocorrelation: Geary’s c

Describing how features differ from their immediate neighbours.

• Geary c (Z value) is:
  • Large c value (>1) : Dispersed, observations tend to be dissimilar;
  • Small c value (<1) : Clustered, observations tend to be similar;
  • c = 1: observations are arranged randomly over space.

Relationship of Moran’s I and Geary’s C

• C approaches 0 and I approaches 1 when similar values are clustered.

• C approaches 3 and I approaches -1 when dissimilar values tend to cluster.

• High values of C measures correspond to low values of I.

• So the two measures are inversely related.

z-score and p-value explained

• Statistically, we select the confident interval such as 95% => alpha value = 0.05.

• Reject the Null hypothesis (H0) if p-value is smaller than alpha value.

• Failed to reject the Null Hypothesis (H0) if p-value is greater than alpha value.

Reference: Confidence Interval or P-Value?

Spatial Randomness

The Null Hypothesis:

• Observed spatial pattern of values is equally likely as any other spatial pattern.

• Values at one location do not depend on values at other (neighbouring) locations.

• Under spatial randomness, the location of values may be altered without affecting the information content of the data.

What if my data violate the assumptions?

• If you doubt that the assumptions of Moran’s I are true (normality and randomization), we can use a Monte Carlo simulation.

  • Simulate Moran’s I n times under the assumption of no spatial pattern,

  • Assigning all regions the mean value

  • Calculate Moran’s I,

• Compare actual value of Moran’s I to randomly simulated distribution to obtain p-value (pseudo significance).

Measures of Global High/Low Clustering: Getis-Ord Global G

• Getis-Ord Global G statistic is concerned with the overall concentration or lack of concentration in all pairs that are neighbours given the definition of neighbouring areas.
• The variable must contain only positive values to be used.

Source: Getis, A., & Ord, K. (1992). “The Analysis of Spatial Association by Use of Distance Statistics”. Geographical Analysis, 24, 189–206.

Interpretation of Getis-Ord Global G

• The p-value is not statistically significant.
  • You cannot reject the null hypothesis. It is possible that the spatial distribution of feature attribute values is the result of random spatial processes. Said another way, the observed spatial pattern of values could be one of many possible versions of complete spatial randomness.
• The p-value is statistically significant, and the z-score is positive.
  • You can reject the null hypothesis. The spatial distribution of high values in the dataset is more spatially clustered than would be expected if underlying spatial processes were truly random.
• The p-value is statistically significant, and the z-score is negative.
  • You can reject the null hypothesis. The spatial distribution of low values in the dataset is more spatially clustered than would be expected if underlying spatial processes were truly random.

Local Spatial Autocorrelation Statistics

• A collection of geospatial statistical analysis methods for analysing the location related tendency (clusters or outliers) in the attributes of geographically referenced data (points or areas).

• Can be indecies decomposited from their global measures such as local Moran’s I, local Geary’s c, and Getis-Ord Gi*.

• These spatial statistics are well suited for:

  • detecting clusters or outliers;
  • identifying hot spot or cold spot areas;
  • assessing the assumptions of stationarity; and
  • identifying distances beyond which no discernible association obtains.

Local Indicator of Spatial Association (LISA)

• A subset of localised geospatial statistics methods.

• Any spatial statistics that satisfies the following two requirements (Anselin, L. 1995):

  • the LISA for each observation gives an indication of the extent of significant spatial clustering of similar values around that observation;

  • the sum of LISAs for all observations is proportional to a global indicator of spatial association.

Detecting Spatial Clusters and Outliers

• Given a set of geospatial features (i.e. points or polygons) and an analysis field, the spatial statistics identify spatial clusters of features with high or low values. The tool also identifies spatial outliers.

• local Moran’s I is the most popular spatial statistical method used, other methods include local Geary’s c.

• In general, the analysis will calculate a local statistic value, a z-score, a pseudo p-value, and a code representing the cluster type for each statistically significant feature. The z-scores and pseudo p-values represent the statistical significance of the computed index values.

Local Moran’s I

Given a geographically referenced attribute field, X the formula of local Moran’s I is:

Local Moran’s I and Moran’s I

The summation of local Moran is

• Moran’s I

Test statistics of Local Moran

Interpretation of Local Moran

• An outlier: significant and negative if location i is associated with relatively low values in surrounding locations.

• A cluster: significant and positive if location i is associated with relatively high values of the surrounding locations.

• In either instance, the p-value for the feature must be small enough for the cluster or outlier to be considered statistically significant.

• The commonly used alpha-values are 0.1, 0.05, 0.01, 0.001 corresponding the 90%, 95, 99% and 99.9% confidence intervals respectively.

Interpretation of Local Moran and Scatterplot

Other forms of LISA

• Local Geary

Detecting hot and cold spot areas

• Given a set of geospatial features (i.e. points or polygons) and an analysis field, the spatial statistics tell you where features with either high (i.e. hot spots) or low values (cold spots) cluster spatially.
• The spatial statistic used is called Getis-Ord Gi* statistic (pronounced G-i-star).
• Getis and Ord (1992) define the local G and G∗ statistics for region i (i=1,···,n) as:

Getis-Ord Gi*

For variable x to be nonpositive and the weight matrix W(d) to be nonbinary the G and G* statistics are defined as:

Interpretation of Getis-Ord Gi and Gi*

• A hot spot area: significant and positive if location i is associated with relatively high values of the surrounding locations.

• A cold spot area: significant and negative if location i is associated with relatively low values in surrounding locations.

Fixed weighting scheme

Things to consider if fixed distance is used: - All features should have at least one neighbour.

• No feature should have all other features as neighbours.

• Especially if the values for the input field are skewed, you want features to have about eight neighbors each.

• Might produce large estimate variances where data are sparse, while mask subtle local variations where data are dense.

• In extreme condition, fixed schemes might not be able to calibrate in local areas where data are too sparse to satisfy the calibration requirements (observations must be more than parameters).

Adaptive weighting schemes

• Adaptive schemes adjust itself according to the density of data

  • Shorter bandwidths where data are dense and longer where sparse.

  • Finding nearest neighbors are one of the often used approaches.

]

Best practice guidelines

• Results are only reliable if the input feature class contains at least 30 features.

• The input field mst be in continuous data type such as a count, rate, or other numeric measurement, no categorical attribute field is allowed.

Select an appropriate spatial weighting method

• The polygon contiguity method is effective when polygons are similar in size and distribution, and when spatial relationships are a function of polygon proximity (the idea that if two polygons share a boundary, spatial interaction between them increases).
  • When you select a polygon contiguity conceptualization, you will almost always want to select row standardization for tools that have the Row Standardization parameter.
• The fixed distance method works well for point data. It is often a good option for polygon data when there is a large variation in polygon size (very large polygons at the edge of the study area and very small polygons at the center of the study area, for example), and you want to ensure a consistent scale of analysis.]

Select an appropriate spatial weighting method

• The inverse distance method is most appropriate with continuous data or to model processes where the closer two features are in space, the more likely they are to interact/influence each other.
  • Be warned that with this method, every feature is potentially a neighbour of every other feature, and with large datasets, the number of computations involved will be enormous.

Select an appropriate spatial weighting method

• The k-nearest neighbours method is effective when you want to ensure you have a minimum number of neighbors for your analysis.

  • Especially when the values associated with your features are skewed (are not normally distributed), it is important that each feature is evaluated within the context of at least eight or so neighbors (this is a rule of thumb only).

  • When the distribution of your data varies across your study area so that some features are far away from all other features, this method works well.

  • Note, however, that the spatial context of your analysis changes depending on variations in the sparsity/density of your features.

  • When fixing the scale of analysis is less important than fixing the number of neighbors, the k-nearest neighbours method is appropriate.

Futher guide on selecting a fixed-distance band value

• Select a distance based on what you know about the geographic extent of the spatial processes promoting clustering for the phenomena you are studying.

• Use a distance band that is large enough to ensure all features will have at least one neighbor, or results will not be valid.

• Try not to get stuck on the idea that there is only one correct distance band. Reality is never that simple. Most likely, there are multiple/interacting spatial processes promoting observed clustering.

• Select an appropriate distance band or threshold distance.

  • All features should have at least one neighbour.
  • No feature should have all other features as a neighbour.
  • Especially if the values for the input field are skewed, each feature should have about eight neighbours.

Emerging Hot Spot Analysis (EHSA)

• A technique that falls under exploratory spatial data analysis (ESDA).

• It combines the traditional ESDA technique of hot spot analysis using the Getis-Ord Gi* statistic with the traditional time-series Mann-Kendall test for monotonic trends.

• The goal of EHSA is to evaluate how hot and cold spots are changing over time. It helps us answer the questions: are they becoming increasingly hotter, are they cooling down, or are they staying the same?

Mann-Kendall test for trend overview

• The Mann-Kendall statistical test for trend is used to assess whether a set of data values is increasing over time or decreasing over time, and whether the trend in either direction is statistically significant.

• It is a non-parametric test, which means it works for all distributions (i.e. your data doesn’t have to meet the assumption of normality), but your data should have no serial correlation. If your data does follow a normal distribution, you can run simple linear regression instead.

• Refer to this article for a full tutorial of how to perform Mann-Kendall test manually.
  

The Hypothesis

• Null hypothesis: There is no monotonic trend in the series.
• Alternate hypothesis: A trend exists. This trend can be positive, negative, or non-null.
• The test statistic

The Mann-Kendall test does NOT assess the magnitude of change.

The test can be used to find trends for as few as four samples. However, with only a few data points, the test has a high probability of not finding a trend when one would be present if more points were provided. The more data points you have the more likely the test is going to find a true trend (as opposed to one found by chance). The minimum number of recommended measurements is therefore at least 8 to 10.

Data requierements

Before running the test, you should ensure that:

• Your data isn’t collected seasonally (e.g. only during the summer and winter months), because the test won’t work if alternating upward and downward trends exist in the data. Another test—the Seasonal Kendall Test—is generally used for seasonally collected data.

• Your data does not have any covariates—other factors that could influence your data other than the ones you’re plotting.

• You have only one data point per time period. If you have multiple points, use the median value.

EHSA Patterns: Hot Spot Trends

Pattern Name	Defination
No pattern detected	Does not fall into any of the hot or cold spot patterns defined below.
New Hot Spot	A location that is a statistically significant hot spot for the final time step and has never been a statistically significant hot spot before.
Consecutive Hot Spot	A location with a single uninterrupted run of at least two statistically significant hot spot bins in the final time-step intervals. The location has never been a statistically significant hot spot prior to the final hot spot run and less than 90 percent of all bins are statistically significant hot spots.
Intensifying Hot Spot	A location that has been a statistically significant hot spot for 90 percent of the time-step intervals, including the final time step. In addition, the intensity of clustering of high counts in each time step is increasing overall and that increase is statistically significant.
Persistent Hot Spot	A location that has been a statistically significant hot spot for 90 percent of the time-step intervals with no discernible trend in the intensity of clustering over time.
Diminishing Hot Spot	A location that has been a statistically significant hot spot for 90 percent of the time-step intervals, including the final time step. In addition, the intensity of clustering in each time step is decreasing overall and that decrease is statistically significant.
Sporadic Hot Spot	A statistically significant hot spot for the final time-step interval with a history of also being an on-again and off-again hot spot. Less than 90 percent of the time-step intervals have been statistically significant hot spots and none of the time-step intervals have been statistically significant cold spots.
Oscillating Hot Spot	A statistically significant hot spot for the final time-step interval that has a history of also being a statistically significant cold spot during a prior time step. Less than 90 percent of the time-step intervals have been statistically significant hot spots.
Historical Hot Spot	The most recent time period is not hot, but at least 90 percent of the time-step intervals have been statistically significant hot spots.

EHSA Patterns: Cold Spot Trends

Pattern Name	Definition
New Cold Spot	A location that is a statistically significant cold spot for the final time step and has never been a statistically significant cold spot before.
Consecutive Cold Spot	A location with a single uninterrupted run of at least two statistically significant cold spot bins in the final time-step intervals. The location has never been a statistically significant cold spot prior to the final cold spot run and less than 90 percent of all bins are statistically significant cold spots.
Intensifying Cold Spot	A location that has been a statistically significant cold spot for 90 percent of the time-step intervals, including the final time step. In addition, the intensity of clustering of low counts in each time step is increasing overall and that increase is statistically significant.
Persistent Cold Spot	A location that has been a statistically significant cold spot for 90 percent of the time-step intervals with no discernible trend in the intensity of clustering of counts over time.
Diminishing Cold Spot	A location that has been a statistically significant cold spot for 90 percent of the time-step intervals, including the final time step. In addition, the intensity of clustering of low counts in each time step is decreasing overall and that decrease is statistically significant.
Sporadic Cold Spot	A statistically significant cold spot for the final time-step interval with a history of also being an on-again and off-again cold spot. Less than 90 percent of the time-step intervals have been statistically significant cold spots and none of the time-step intervals have been statistically significant hot spots.
Oscillating Cold Spot	A statistically significant cold spot for the final time-step interval that has a history of also being a statistically significant hot spot during a prior time step. Less than 90 percent of the time-step intervals have been statistically significant cold spots.
Historical Cold Spot	The most recent time period is not cold, but at least 90 percent of the time-step intervals have been statistically significant cold spots.

A Spacetime Cubes

• A spacetime object is a spacetime cube if every location has a value for every time index. Another way of saying this is that each location contains a regular time-series.

• In ESRI terminology, the basic unit of a spacetime cube is a bin.

  • A bin is the unique combination of a location and time index. For each time index, the collection of every location is called a time slice.
  • In every location, the collection of every bin at each time index is referred to as a bin time-series.

In colclusion

Spatial statistics methods are not a blackbox. Before performing the analysis, a geospatial analyst should consider the followings:

• What is the geographical question?

• What is the geospatial feature?

• What is the analysis field?

• Which conceptualization of spatial relationships is appropriate?

References

• Chapter 2. Codifying the neighbourhood structure of Handbook of Spatial Analysis: Theory and Application with R.
• François Bavaud (2010) “Models for Spatial Weights: A Systematic Look” Geographical Analysis, Vol. 30, No.2, pp 153-171.
• Tony H. Grubesic and Andrea L. Rosso (2014) “The Use of Spatially Lagged Explanatory Variables for Modeling Neighborhood Amenities and Mobility in Older Adults”, Cityscape, Vol. 16, No. 2, pp. 205-214.
• Moran, P. A. P. (1950). “Notes on Continuous Stochastic Phenomena”. Biometrika. 37 (1): 17–23.
• Geary, R.C. (1954) “The Contiguity Ratio and Statistical Mapping”. The Incorporated Statistician, Vol. 5, No. 3, pp. 115-127. - Moran’s I
• Getis, A., & Ord, K. (1992). “The Analysis of Spatial Association by Use of Distance Statistics”. Geographical Analysis, 24, 189–206.

• Anselin, L. (1995). “Local indicators of spatial association – LISA”. Geographical Analysis, 27(4): 93-115.
• Getis, A. and Ord, J.K. (1992) “The analysis of spatial association by use of distance statistics”. Geographical Analysis, 24(3): 189-206.
• Ord, J.K. and Getis, A. (2010) “Local spatial autocorrelation statistics: Distributional issues and an application”. Geographical Analysis, 27(4): 286-306.