Geographic Information Analysis (GIA) is an iterative process that involves integrating data and applying a variety of spatial, statistical, and visualization methods to better understand patterns and processes governing a system.

Figure 2.0: Spatial Data analysis: an iterative process.

Click for a text description of the Spatial Data Analysis image.

Spatial data analysis is an iterative process. Text, video, audio, image, vector, and raster data are can be structured/unstructured and geographic/non-geographic. This data can be processed and transformed. A feedback loop and iterative process exists between data and knowledge. This loop includes mapping, plotting and graphing for model visualization and analysis (hypothesis testing and interpretation of results). This leads to knowledge. Another path starts at data, goes to data mining, building models, and refining that model (hypothesis testing and interpretation of results) which leads to knowledge about a process/system. In the middle is a loop between analysis - statistical and spacial. Knowledge about a process/system results in the ability to communicate.

Credit: Blanford, © Penn State University, is licensed under CC BY-NC-SA 4.0 [1]

Learning Outcomes

At the successful completion of Lesson 2, you should be able to:

• deal with various types of data;
• develop a data frame and structure necessary to perform analyses;
• apply various statistical methods;
• identify various spatial methods;
• develop a research framework and integrate both statistical and spatial methods; and
• document and communicate your analysis and findings in an efficient manner.

Checklist

Lesson 2 is one week in length. (See the Calendar in Canvas for specific due dates.) The following items must be completed by the end of the week. You may find it useful to print this page out first so that you can follow along with the directions.

Questions?

Please use the 'Discussion - Lesson 2' forum to ask for clarification on any of these concepts and ideas. Hopefully, some of your classmates will be able to help with answering your questions, and I will also provide further commentary where appropriate.

Spatial analysis:

Refers to the "general ability to manipulate spatial data into different forms and extract additional meaning as a result" (Bailey 1994, p. 15) using a body of techniques "requiring access to both the locations and the attributes of objects" (Goodchild 1992, p. 409). This means spatial analysis must draw on a range of quantitative methods and requires the integration and use of many types of data and methods (Cromley and McLafferty 2012).

Where and how to start?

In Figure 2.0, we provided an overview of the process involved, now we will provide some of the details to get you familiar with the research process and the types of methods you may be using to perform your analysis.

Research framework

Where and how do we start to analyze our data? Analyzing data whether it is spatial or not is an iterative process that involves many different components (Figure 2.1). Depending on what questions we have in mind, we will need some data. Next, we will need to get to know our data so that we can understand its limitations, identify transformations required to produce a valid analysis, and better grasp the types of methods we will be able to use. Next, we will want to refine our hypothesis and then perform our analysis, interpret our results, and share our findings.

Figure 2.1: Classical research framework.

Click for a text description of the Classical Research Framework image.

Diagram of defining a research question: objectives and hypotheses. Diagram includes the following: Collect, create, organize, and process data and ask what data is needed. Summarize the data by being descriptive and including summary statistics and visualizations–get to know the data. Analyze the data by using a statistical or spatial analysis and ask what statistical tests you will be using. Knowledge involves the findings and conclusions, capturing new findings, maps, graphs, tables, etc, and sharing and communicating that knowledge.

Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0 [1]

Statistical and Spatial Analysis

Now that you have an understanding of the research process, let’s start to familiarize ourselves with the different types of statistical and spatial analysis methods that we may need to use at each stage of the process (Figure 2.2). As you can see from Figure 2.2, statistical and spatial analysis methods are intertwined and often integrated. I know this looks complex, and for many of you the methods are new, but by the end of this course, you will have a better understanding of many of these methods and how to integrate them as you require spatial and/or statistical methods.

1. Data: Determine what data is needed. This may require collecting data or obtaining third-party data. Once you have the data, it may need to be cleaned, processed (transformed, aggregated, converted, etc.), organized, and stored so that it is easy to manage, update, and retrieve for later analyses.
2. Data: Get to know the data. This is an important step that many people ignore. All data has issues of one form or another. These issues are based on how the dataset was collected, formatted, and stored. Before you moving forward with your analysis, it is vital to understand
   1. What are a dataset's limitations? For example, you may be interested in learning about the severity of a disease across a region. The dataset you obtained contains count data (total number of cases). This count data will not give you an overall picture of the disease's impact and severity since one should adjust the case count according to the underlying population distribution (create a percent of total or incidence rate).
   2. Determine the usefulness of a dataset. Look at the data and determine if that data will fit your needs. For example, you may have obtained point data, but your analysis needs to operate on area-based data. Is the conversion between point to area-based data an appropriate way to answer your research question?
   3. Learn how the data are structured. Spreadsheets are convenient ways to store and share data. But how is that data arranged inside the spreadsheet? The data that you need may be there but the formatting is not advantageous (e.g., rows and columns need to be switched, dates need to be sequentially ordered, attributes need to be combined, etc.).
   4. identify any outliers. To better understand and explore the data, you can use descriptive spatial and non-spatial statistical methods as well as visualize the data either in graphs or plots as well as in the form of a map. Outliers may be signs of data entry errors or data that are atypical and depending on the intended statistical test may need to be removed from the data listing.
3. Spatial Analysis and Statistical Methods. As you work through the analysis, you will use a variety of methods that are statistical, spatial, or both, as you can see from Figure 2.2. In the upcoming weeks, we will be using many of these methods starting with Point Pattern Analysis (PPA) in Lesson 3, spatial autocorrelation analysis (Lesson 4), regression analysis (Lesson 5), and the use of different spatial functions for performing spatial analysis (Lessons 6-Lesson 8). Remember, this is an iterative process that will likely require the use of one or more of the methods summarized in the diagram that are either traditional statistical methods (on the left) or a variety of spatial methods (on the right). In many cases, we will likely move back and forth between the different components (up and down) as well as between the left and right sides.
4. Communication. Lastly, an important part of any research process is communicating your findings effectively. There are a variety of ways that we can do this, using web-based tools and integration of different visualizations.

Figure 2.2: Spatial data analysis: summary of statistical and spatial methods that may be used to perform an analysis.

Click for a text description of the spatial data analysis process.

Diagram of workflows of statistical and spatial methods that may be used to perform an analysis. Statistical methods are shown on the left, spatial methods are shown on the right. The diagram shows how research questions, data, methods, and conclusions are all linked together and used iteratively to arrive at the end result of an analysis. The information contained in the diagram is described more fully in the text above.

Credit: Blanford, © Penn State University, licensed under CC BY-NC-SA 4.0 [1]

Now that you have been introduced to the research framework and have an idea of some of the methods you will be learning about, let’s cover the basics so that we are all on the same page. We will start with data, then spatial analysis, and end with a refresher on statistics.

Data comes in many forms, structured and unstructured, and is obtained through a variety of sources. No matter its source or form, before the data can be used for any type of analysis, it must be assessed and transformed into a framework that can then be analyzed efficiently.

Some Special Considerations with Spatial Data [Rogerson Text, 1.6]

There shouldn't be anything new to anyone in this class in these pages. Since analysis is almost always based on data that suffer from some or all of these limitations, these issues have a big effect on how much we can learn from spatial data, no matter how clever the analysis methods we adopt become.

The separation of entities from their representation as objects is important, and the scale-dependence issue that we read about in Lesson 1 is particularly important to keep in mind. The scale dependence of all geographic analysis is an issue that we return to frequently.

GIS Analysis, Spatial Data Manipulation, and Spatial Analysis

From Figure 2.2, it is clear that spatial analysis requires a wide variety of analytical methods (statistical and spatial).

Spatial analysis functions fall into five classes that include: measurement, topological analysis, network analysis, surface analysis and statistical analysis (Cromley and McLafferty 2012, p. 29-32) (Table 2.0).

Although maps are used to present research results (when they present known results and are represented using public, low-interaction devices), many more maps are impermanent, exploratory devices, and with the increased use of interactive web-based data graphs and maps, often driven by dynamically changing data, this is even more so the case.

With this, there is an ever-increasing need for spatial analysis, particularly since much of the data collected today contains some form of geographic attribute and every map tells a story… or does it? It is easy to feel that a pattern is present in a map. Spatial analysis allows us to explore the data, develop a hypothesis, and test that visual insight in a systematic, more reliable way.

The important point at this stage is to get used to thinking of space and spatial relations in the terms presented here — distance [3], adjacency [4], interaction [5], and neighborhood [6]. The interaction weight idea is particularly commonly used. You can think of interaction as an inverse distance measure: near things interact more than distant things. Thus, it effectively captures the basic idea of spatial autocorrelation [7].

Summarizing Relationships in Matrices [Lloyd text, section 2.2]

This week's readings discuss some of the ways matrices are used in spatial analysis. You'll notice that distances and adjacencies appear in Table 2.0. At this stage, you only need to get the basic idea that distances, adjacencies (or contiguities), or interactions can all be recorded in a matrix [8] form. This makes for very convenient mathematical manipulation of a large number of relationships among geographic objects. We will see later how this concept is useful in point pattern analysis (Lesson 3), clustering and spatial autocorrelation (Lesson 4), and interpolation (Lesson 6).

Statistical methods, as will become apparent during this course, play an important role in spatial analysis and are behind many of the methods that we regularly use. So to get everyone up to speed, particularly if you haven’t used stats recently, we will review some basic ideas that will be important through the rest of the course. I know the mention of statistics makes you want to close your eyes and run the other direction .... but some basic statistical ideas are necessary for understanding many methods in spatial analysis. An in-depth understanding is not necessary, however, so don't get worried if your abiding memory of statistics in school is utter confusion. Hopefully, this week, we can clear up some of that confusion and establish a firm foundation for the much more interesting spatial methods we will learn about in the weeks ahead. We will also get you to do some stats this week!

You have already been introduced to some stats earlier on in this lesson (see Figures 2.0-2.3). Now, we will focus your attention on the elements that are particularly important for the remainder of the course. "Appendix A: The elements of Statistics," linked in the additional readings list for this lesson in Canvas, will serve as a good overview (refresher) for the basic elements of statistics. The section headings below correspond to the section headings in the reading.

A.1. Statistical Notation

One of the scariest things about statistics, particularly for the 'math-phobic,' is the often intimidating appearance of the notation used to write down equations. Because some of the calculations in spatial analysis and statistics are quite complex, it is worth persevering with the notation so that complex concepts can be presented in an unambiguous way. Really, understanding the notation is not indispensable, but I do hope that this is a skill you will pick up along the way as you pursue this course. The two most important concepts are the summation symbol capital sigma (Σ), and the use of subscripts (the i in x_i).

I suggest that you re-read this section as often as you feel the need, if, later in the course, 'how an equation works' is not clear to you. For the time being, there is a question in the quiz to check how well you're getting it.

A.2. Describing Data

The most fundamental application of statistics is simply describing large and complex sets of data. From a statistician's perspective, the key questions are:

• What is a typical value in this dataset?
• How widely do values in the dataset vary?

... and following directly from these two questions:

• What qualifies as an unusually high or low value in this dataset?

Together, these three questions provide a rough description of any numeric dataset, no matter how complex it is in detail. Let's consider each in turn.

Measures of central tendency such as the mean and the median provide an answer to the 'typical value' question. Together, these two measures provide more information than just one value, because the relationship between the two is revealing. The important difference between them is that, the mean is strongly affected by extreme values while the median is not.

Measures of spread are the statistician's answer to the question of how widely the values in a dataset vary. Any of the range, the standard deviation, or the interquartile range of a dataset allows you to say something about how much the values in a dataset vary. Comparisons between the different approaches are again helpful. For example, the standard deviation says nothing about any asymmetry in a data distribution, whereas the interquartile range—more specifically the values of Q₂₅ and Q₇₅—allows you to say something about whether values are more extreme above or below the median.

Combining measures of central tendency and of spread enables us to identify which are the unusually high or low values in a dataset. Z scores standardize values in a dataset to a range such that the mean of the dataset corresponds to z = 0; higher values have positive z scores, and lower values have negative z scores. Furthermore, values whose z scores lie outside the range +2 are relatively unusual, while those outside the range +3 can be considered outliers. Box plots provide a mechanism for detecting outliers, as discussed in relation to Figure A.2.

A.3. Probability Theory

Probability is an important topic in modern statistics.

The reason for this is simple. A statistician's reaction to any observation is often along the lines of, "Well, your results are very interesting, BUT, if you had used a different sample, how different would the answer be?" The answer to such questions lies in understanding the relationship between estimates derived from a sample and the corresponding population parameter, and that understanding, in turn, depends on probability theory.

The material in this section focuses on the details of how probabilities are calculated. Most important for this course are two points:

• the definition of probability in terms of the relative frequency of events (eqns A.18 and A.19), and
• that probabilities of simple events can be combined using a few relatively simple mathematical rules (eqns A.20 to A.26) to enable calculation of the probability of more complex events.

A.4. Processes and Random Variables

We have already seen in the first lesson that the concept of a process is central to geographic information analysis (look back at the definition on page 3).

Here, a process is defined in terms of the distribution of expected outcomes, which may be summarized as a random variable. This is very similar to the idea of a spatial process, which we will examine in more detail in the next lesson, and which is central to spatial analysis.

A.5. Sampling Distributions and Hypothesis Testing

The most important result in all of statistics is the central limit theorem. How this is arrived at is unimportant, but the basic message is simple. When we use a sample to estimate the mean of a population:

1. The sample mean is the best estimate for the population mean, and
2. If we were to take many different samples, each would produce a different estimate of the population mean, and these estimates would be normally distributed, and
3. The larger the sample size, the less variation there is likely to be between different samples.

Item 1 is readily apparent: What else would you use to estimate the population mean than the sample mean?!

Item 2 is less obvious, but comes down to the fact that sample means that differ substantially from the population mean are less likely to occur than sample means that are close to the population mean. This follows more or less directly from Item 1; otherwise, there would be no reason to choose the sample mean in making our estimate in the first place.

Finally, Item 3 is a direct outcome of the way that probability works. If we take a small sample, then it is prone to wide variation due to the presence of some very low or very high values (like the people in the bar example). On the other hand, a very large sample will also vary with the inclusion of extreme values, but overall is much more likely to include both high and low values.

The central limit theorem provides the basis for calculation of confidence intervals, also known as margins of error, and also for hypothesis testing.

Hypothesis testing may just be the most confusing thing in all of statistics... but, it really is very simple. The idea is that no statistical result can be taken at face value. Instead, the likelihood of that result given some prior expectation should be reported, as a way of inferring how much faith we can place in that expectation. Thus, we formulate a null hypothesis, collect data, and calculate a test statistic. Based on knowledge of the distributional properties of the statistic (often from the central limit theorem), we can then say how likely the observed result is if the null hypothesis were true. This likelihood is reported as a p-value or probability. A low probability indicates that the null hypothesis is unlikely to be correct and should be rejected, while a high value (usually taken to mean greater than 0.05) means we cannot reject the null hypothesis.

Confusingly, the null hypothesis is set up to be the opposite of the theory that we are really interested in. This means that a low p-value is a desirable result, since it leads to rejection of the null, and therefore provides supporting evidence for our own alternative hypothesis.

Visualizing the Data

Not only is it important to perform the statistics, but in many cases it is important to visualize the data. Depending on what you are trying to analyze or do, there are a variety of visualizations that can be used to show:

• Composition: where and when something is occurring at a single point in time (e.g., dot map; choropleth map; pie chart, bar chart) or changing over time (e.g., time-series graphs), which can be further enhanced by including intensity (e.g., density maps, heat maps),
• the relationship between two things (e.g., scatterplot (2 variables), bubbleplots (e.g., 3 variables)),
• distribution of the data (e.g., frequency, histogram, scatterplot),
• comparison between variables (e.g., single or multiple).

For a visual overview of some different types of charts, [11] see the graphic produced by Dr Abela (2009).

For example, earlier we highlighted the importance of understanding our data using descriptive statistics (mean and variation of the data). By using frequency distributions and histograms, we can further understand these aspects since these visualizations reveal three aspects of the data:

1. where the distribution has its peak (central location) - the clustering at a particular value is known as the central location or central tendency of a frequency distribution. Measures of central location can be in the middle or off to one side or the other.
2. how widely dispersed the distribution is on either side of its peak (spread, also known as variation or dispersion) - this refers to the distribution out from a central value.
3. whether the data is more or less symmetrically distributed on the two sides of the peak (shape) - shape can be symmetrical or skewed.

Telling a story with data

An important part of doing research and analysis is communicating the knowledge you have discovered to others and also documenting what you did so that you or others can repeat your analysis. To do so, you may want to publish your findings in a journal or create a report that describes the methods you used, presents the results you obtained, and discusses your findings. Often, we have to toggle between the software and a word document or text file to capture our workflow and integrate our results. One way to do this is to use a digital notebook that enables you to document and execute your analysis and write up your results. If you'd like to try this, in this week's project, have a crack at the "Try This" box in the project instructions, which will walk you through how to set up an R Markdown notebook. There are other popular notebooks, such as Jupyter [12]; however, since we are interested in R, those who wish to attempt this can try with the R version.

An important part of telling a story is to present the data efficiently and effectively. The project this week will give you an opportunity to apply some stats to data and get you familiar with running different statistical analyses and applying different visualization methods.

Background

This week, we covered some key concepts important for conducting research and analysis. Now, it is time for you to put this information to use through some analysis. In this week's project, you will be analyzing some point data describing crimes in St. Louis, Missouri and creating a descriptive analysis of the data. Next week, you will build on this analysis by using some spatial point pattern statistics with the same dataset to look at the spatial patterns of crime in more detail.

To carry out this week's analysis, you will use RStudio and several packages in R. The assignment has been broken down into several components.

You will analyze the data and familiarize yourself with R by copying and pasting the provided R code from Lesson 2 into RStudio. In some cases, you will want to modify this code to analyze additional variables or produce different graphs and charts that you think will help with your analysis. As you work through the analysis, make some informal notes about what you are finding.

The outputs generated by running these analyses provides evidence to answer the questions listed at the bottom of this page. Once you have completed your analysis and obtained the different statistical and graphical outputs, use this evidence to create a more formal written report of the analysis.

Project Resources

You need an installation of R and RStudio, to which you will need to add the packages listed on a subsequent page of these instructions.

You will also need some data. You will find a data archive to download (Geog586_Les2_Project.zip) in the Lesson 2 materials in Canvas. If you have any difficulty in downloading the data, please contact me.

This archive includes the following files:

• Crime data files for St Louis: crimeStLouis20132014b.csv, crimeStLouis20132014b_agg.csv
• Boundary of St Louis data file (shapefile): stl_boundary_ll.shp

How to Move Forward with the Analysis

As you work your way through the R code on the following pages of the project instructions, keep the following questions in mind. Jotting down some notes related to each of them will help you to synthesize your findings into your project report.

Discuss Overall Findings

1. What types of crime were recorded in the city?
2. Was there a temporal element associated with each of the crimes?
3. How did the occurrence of crimes changes between the years?
4. Where did crimes take place? Was there a particular spatial distribution associated with each of the crimes? Did they take place in a particular part of the city?
5. What additional types of analysis would be useful?

Descriptive Statistics

1. What type of distribution does your data have?
2. What is a typical value in this dataset?
3. How widely do values in the dataset vary?
4. Are there any unusually high or low value in this dataset? (e.g., outliers)

Visualizing the Data

1. Map the data: What is spatial distribution of crimes?
2. Create graphs:
   1. Of the crimes in the dataset, is there a crime that occurs more than others?
   2. Is there a month when crimes occur more than others?
   3. Have crimes increased or decreased over time?

Questions?

Please use the 'Week 2 lesson discussion' forum to ask for clarification on any of these concepts and ideas. Hopefully, some of your classmates will be able to help with answering your questions, and I will also provide further commentary there where appropriate.

Submit a brief project proposal (1 page) to the 'Term Project: Preliminary Proposal' discussion forum. This week, you should start to obtain the data you will need for your project. The proposal must identify at least two (preferably more) likely data sources for the project work, since this will be critical to success in the final project. Over the next few weeks, you will be refining your proposal. Next week you will submit a first revision of your project proposal based on feedback from me and your continued work in developing your project. During week 5, you will receive feedback from other students. This will help you revise a more complete (final) proposal which will be due in Week 6. For a quick reminder, view Overview of term project and weekly deliverables [28]

This week, you must organize your thinking about the term project by developing your topic/scope from last week into a short proposal.

Your proposal should include:

Background:

• some background on the topic, particularly why it is interesting;
• research question(s). What specifically do you hope to find out?

Methodology:

• data: data required to answer the question(s) – list the data sets you think are needed and what role each will play.
• where you will obtain the data required. This may be public websites or perhaps data that you have access to through work or personal contacts.
  • Obtain and explore the data: attributes, resolutions, scale.
    • Is the data useful or are there limitations?
    • Will you need to clean and organize the data in order to use it?
• analysis: what you will do with the data, in general terms
  • What sort statistical analysis and spatial analysis, do you intend to carry out? Review Figure 1.2 and skim through the lessons to identify the methods you will be using. If you don't know the technical names for the types of analysis you would like to do, then at least try to describe the types of things you would like to be able to say after finishing the analysis (e.g., one distribution is more clustered than another). This will give me and other students a firmer basis for making constructive suggestions about the options available to you. Also, look through the course topics for ideas.

Expected Results:

• what sort of maps or outputs you will create.

References:

• include references to papers you may have cited in the background or methods section.

I realize, at this point, that you may feel that your knowledge is too limited for the last point in particular (Analysis). Skim through the lessons from the course website, where you are now and visit the course syllabus, where there is an overview of each lesson and the methods you will be learning.

The proposal does not have to be detailed at this stage. Your proposal should be no longer than about 1 page (max.). Make sure that your proposal covers all the above points, so that I (Lesson 3 & 4) and others (Lesson 5 – peer review) evaluating the proposal can make constructive suggestions about additions, changes, other sources of data, and so on.

Questions?

Please use the Discussion - General Questions and Technical Help discussion forum to ask any questions now or at any point during this project.

Lesson 2 Deliverables

1. Complete the Lesson 2 quiz.
2. Complete the Project 2 activities. This deliverable includes inserting maps, graphs, and/or tables into your write-up along with accompanying commentary. Submit your assignment to the 'Assignment: Week 2 Project' dropbox provided in Lesson 2 in Canvas.
3. Term Project: Submit a more detailed project proposal (1 page) to the 'Term Project Discussion: Preliminary Proposal' discussion forum.

NOTE: When you have completed this week's project, please submit it to the Canvas drop box for this lesson.

Reminder - Complete all of the Lesson 2 tasks!

You have reached the end of Lesson 2! Double-check the to-do list on the Lesson 2 Overview page [29] to make sure you have completed all of the activities listed there before you begin Lesson 2.

References and Resources

Here are some additional resources that might be of use.