You are an environmental surveyor. You want to investigate the air pollution situation throughout Tokyo (excluding the outlying islands), but due to time and cost constraints, you can only conduct surveys at 50 arbitrary locations within Tokyo.
In this case, how should we decide which 50 locations to survey? Since the survey covers the entire Tokyo metropolitan area, it is important to avoid bias in the survey locations as much as possible. One way to do this is to use JMP, and the results obtained are shown in the map below, which plots the 50 locations.
Regardless of whether the locations are suitable for surveying or easy to survey, it can be confirmed that the measurement points are not extremely close together, but are evenly distributed throughout the entire area.
The 50 locations here were determined using "Space Filling Design," one of the experimental design methods used by JMP.
While it's a less familiar technique within experimental design, this article explains the basic concepts and characteristics of Space Filling design and describes how to apply a Gaussian process model to the design created using the map example above.
What is a Space Filling plan?
A Space Filling plan, as the name suggests, aims to "fill the space." It seeks to obtain data that is not biased towards any particular region by evenly distributing experimental points across the entire range of the target factor.
When dealing with continuous factors, there are mainly two approaches to this type of arrangement (① and ②).
① Arrange the points so that they are as far apart as possible ( maximize distance ).
② Distributing points at equal intervals throughout the entire space ( uniformity )
Depending on which of these is prioritized, or whether a combination of both is used, various planning methods exist. The following seven methods can be used in JMP's Space Filling planning:
This example shows the number of experiments (number of points to place) set to 20 and placed in a 2D design area.
The lower left diagram shows the result of arranging the spheres using the "packing" method, while the lower right diagram shows the result of arranging them using the "uniform" method. In the former, the points are arranged so that the distance between them is large, while in the latter, it can be seen that the distribution is uniform in each axis direction.
"Ball packing" is a method of creating a plan that maximizes the shortest distance between two planned points, and it emphasizes ①. On the other hand, "uniform design" is a method of creating a plan in which the points of each factor are evenly distributed, and it emphasizes ②.
Furthermore, there is an intermediate method called the "Latin hypersquare method."
MaxPro standards
Space Filling plans provide diagnostic statistics to evaluate the quality of the plan. One of these is the MaxPro (Maximum Projection) criterion . I will briefly touch upon this criterion for the sake of the explanation that follows.
The manual contains details on the MaxPro calculation formula, but it is an index that evaluates whether points are evenly distributed throughout the entire experimental space , and a smaller value indicates a better design.
MaxPro's "Projection" refers to projection, emphasizing the preservation of point uniformity even when projecting a high-dimensional plan to a lower-dimensional space. When there are many factors, the design space becomes high-dimensional, but depending on the design, even if it is uniform in high dimensions, bias may occur when projected to 1D or 2D.
MaxPro is an evaluation metric that maintains the greatest possible distance between points while also considering uniformity in each dimension and subspace.
High-speed flexible filling plan
The map shown first uses a method called " high-speed flexible filling (planning) ".
While JMP offers flexibility in setting linear constraints and combinations of factors to disallow during the planning stage, fast flexible filling design is a method that can handle these constraints as well.
This method is based on distance maximization (①) while also considering uniformity (②). The experimental points are determined in the following manner.
- A large number of candidate points are randomly generated within a factor space that takes constraints into account.
- Using cluster analysis, the experiment is divided into a number of clusters corresponding to the specified number of experiments.
- Select representative points from each cluster. By default, the MaxPro criteria are used to determine these representative points, and these will be the final experimental points.
Operations in JMP
Here, we show an example of using PM10 (airborne particles with an aerodynamic diameter of 10 µm or less) as an indicator for measuring air pollution levels.
The following outlines the operation flow, but please refer to the manual's example of creating a Space Filling plan on a map for further details.
- From the JMP menu bar, select [Design of Experiments (DOE)] > [Special Designs] > [Space Filling Designs].
- The response is set to pm10, the factors to lat (longitude) and lon (latitude), and the minimum and maximum values within the Tokyo metropolitan area are specified.
- [Specify combinations not allowed in the script] to exclude areas outside of Tokyo.
- Change "Number of experiments" to "50" and click [Fast Flexible Filling].
Finally, clicking [Create Table] generates a data table with the latitude and longitude of 50 locations, as shown below. These points were the survey locations shown on the initial map.
Fitting of the Gaussian process model
As an environmental surveyor, you visit the 50 locations shown here and measure the PM10 level at each location. You then enter these measurements into the "PM10" column.
Note: pm10 uses the "OpenWeatherMap" API to obtain values at a given time.
After inputting the response, the relationship between the factors and the response is modeled. The table created in the Space Filling plan has a script attached to it that applies a Gaussian process model.
This script means fitting a Gaussian process model where Y is the response and X is the factor.
Gaussian process models are effective for analyzing data obtained from the Space Filling project. They function particularly well in simulation experiments where measurement errors are minimal, as they smoothly connect observed data and perfectly interpolate known points.
In this example, we need to predict pm10 at locations that are not measurement points. By fitting a Gaussian process model to data evenly distributed across the entire space, as in this case, stable predictions become possible even for unobserved locations.
Running the script mentioned earlier will display a report with the Gaussian process model applied. The figure below shows the surface of the prediction equation with the base being longitude, latitude, and height pm10, and you can see that all the experimental points lie on this surface.
When you actually save the prediction formula to a data table, you can confirm that the measured and predicted values for pm10 match.
Predicted distribution of PM10 across Tokyo
When the predicted values from the Gaussian process model are visualized on a map, the following distribution is obtained. Red indicates areas with high PM10 levels, and blue indicates areas with low levels.
From this map, we can observe the following trends, for example:
PM10 levels are high at Haneda Airport and the nearby bay area.
⇒ Because it is an area with heavy traffic and industrial activity.
PM10 levels are low in western areas (Okutama and western Tokyo).
⇒ Because it is an area that is not urbanized and has a lot of natural environment.
summary
This article explained the following:
- The Space Filling plan is a method that aims to acquire unbiased data by distributing points evenly throughout the entire space.
- There are two methods for arranging points: one that maximizes the distance between points, and another that distributes them uniformly.
- JMP offers multiple methods, and its fast, flexible filling plan, which can accommodate constraints, is effective in practical applications.
- By applying a Gaussian process model to the obtained data, the overall relationships and distributions can be smoothly estimated, including response values under unobserved conditions.
Furthermore, JMP Pro 19's new "Bayesian Optimization" feature allows for unbiased sampling of the factor space using a Space Filling design. By applying a Gaussian process model to this data, predictions and uncertainties can be evaluated, and promising conditions for further evaluation can be efficiently selected.
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
