Self-Organized Cognitive Maps

Transcription

1 Self-Organized Cognitive Maps 517 Self-Organized Cognitive Maps Robert Lloyd The University of South Carolina This paper argues cognitive mapping is a learning process that can be simulated by a self-organizing neural network. The learning of city locations was considered in two studies. One study focused on the learning of city locations on four continents. Results indicated the neural network aligned the cities producing systematic errors similar to those in human cognitive maps. A second study had a neural network learn a biased sample of city locations in the United States. Results indicated a non-linear relationship between cognitive and physical distances. Self-organized cognitive maps naturally produce this non-linear relationship when information from more than one scale is mapped into one space. Key Words: learning, cognitive maps, neural networks, self-organized. In thinking, and in subconscious information processing, there is a general tendency to compress information by forming reduced representations of the most relevant facts, without loss of knowledge about their interrelationships. The purpose of intelligent information processing seems in general to be creation of simplified images of the observable world at various levels of abstractions, in relation to a particular subset of received data. Kohonen (1989, 119) Introduction O ur experiences are transformed into our memories. Our memories affect our behavior. Our behavior provides new experiences. Learning is the mechanism that initiates this cycle and keeps it refreshed. The first sentence of this paper is the simple statement that inspired the central question for this research. How is information about spatial locations on cartographic maps projected into cognitive maps in our memories? The human brain was not designed for precision when it comes to encoding spatial information. Although we start encoding the spatial locations of important objects very early in life (Walton et al. 1997), we appear to never accurately encode the details. Our inexactness is not caused by exposure to inaccurate information or memory loss, but rather by the normal learning processes used to encode information into memory (Tversky 1981). These normal learning processes appear to produce systematic errors as we learn locations on a map or landmarks in an urban environment (Lloyd 1989; Kitchin 1996). The spatial structures that we encode into memory have generally been referred to as cognitive maps (Tolman 1948). This paper uses the term cognitive map in the broadest sense. Any internal representation of a set of geographic locations that has been learned, i.e., encoded into memory, is considered a cognitive map. The learner might be a person with a biological-based memory or an artificial neural network simulating human memory in a com- Professional Geographer, 52(3) 2000, pages Copyright 2000 by Association of American Geographers. Initial submission, November 1998; revised submission, May 1999; final acceptance, May Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, and 108 Cowley Road, Oxford, OX4 1JF, UK.

2 518 Volume 52, Number 3, August 2000 puter. The geographic locations are specified as coordinates in some physical space. They might refer to any objects noticeable on a map or in an environment. A number of factors have made it difficult to study the nature of cognitive maps. First, cognitive maps are learned over time. Any serial process that causes someone to encounter a sequence of objects in an environment or scan from object to object on a cartographic map is an example of learning taking place over time. If the environment or map is visited again another round of learning occurs. One might say these cognitive maps are never actually completed, but periodically updated with new information that might impact the nature of the cognitive map. A person s cognitive map at one point in time, therefore, is not necessarily the same at another point in time (Golledge et al. 1975). Second, individuals who encode these works in progress have some shared experiences with others, but also have many unique learning experiences. Although individuals might be trying to learn about the same city or read the same map, they have different navigation or scanning patterns, goals, and values, and experience locations from different perspectives. Lloyd and Heivly (1987) have shown that the cognitive maps for people living in different neighborhoods in a city were systematically affected by the location of the home neighborhood within the city. Lloyd (1989) has shown that a group of individuals who learned spatial locations from a common cartographic map were much more consistent and accurate than another group who learned the same locations by freely navigating in an urban environment. Third, although cognitive maps are learned through a dynamic process, most previous studies used techniques that focused on differences between the current representations of cognitive maps and cartographic maps. This puts the focus of most cognitive mapping studies on evaluating the success of the learning rather than on the process used for learning maps or environments. Although the nature of errors in cognitive maps has been well documented, a complete explanation of why systematic errors occur remains as an important goal for cognitive mapping researchers. The focus of the research presented here is not on what systematic errors occur in cognitive maps, but on why those errors occur. An ideal cognitive mapping study should be able to provide identical experiences to all subjects as they learn the same spatial information. The ideal study should also explicitly consider the learning process used by subjects and be able to model that process. People usually learn spatial information without supervision. Although there may be some reinforcement from the successes and failures that come from using our cognitive maps, there is usually no truth giver who monitors our progress and corrects our mistakes (Nigrin 1993). Unsupervised learning models would, therefore, appear to reflect actual learning experiences better than supervised models that learn by minimizing differences between current knowledge and the truth that is explicitly defined for the model. The purpose of this paper is to consider the learning of map locations as an encoding of information into a neural network. The basic problems considered use a Kohonen neural network to learn configurations of real spatial locations (Kohonen 1989, 1993). The Kohonen neural network model was thought to be an ideal choice for learning spatial information because it learns without supervision. This is why the network is said to create self-organized maps. Learning is based on the recognition of patterns in the input data rather than knowledge of the truth. Kohonen neural networks are also inherently spatial in nature. A basic prescript of the learning process is based on a distance decay principle that results in similar input patterns being mapped into neighboring locations in the self-organized map. This approach models learning processes that transform cartographic maps into cognitive maps. Maps learned by the Kohonen networks can then easily be compared to the actual maps or to cognitive maps learned by human subjects. What Do We Know about Cognitive Maps? Cognitive maps are internal memory structures. Most cognitive mapping studies focus on where objects are located in space (Golledge et al. 1982; Lloyd 1989). These might be called projection studies because they basically considered the underlying structure of the geographic space as it appears to be represented in mem-

3 Self-Organized Cognitive Maps 519 ory. One could think of a cognitive mapper as having the same problem as a cartographer. Locations (the truth) need to be projected into another space (e.g., a Mercator map projection or your memory) for storage and representation. Just like cartographic projections, cognitive mapping projections produce systematic errors that have been well documented. Reference Points A number of researchers have argued that important locations or objects serve as cognitive reference points around which other locations or objects can be organized (Rosch 1975; Tversky 1977; Hirtle and Jonides 1985; Couclelis et al. 1987; Ferguson and Hegarty 1994). The space near the reference point tends to be expanded relative to other locations in the space while space far from the reference point tends to be contracted. Holyoak and Mah (1982) reported this for cities in the U.S. with the reference point being the Atlantic or Pacific Oceans. Lloyd (1989) reported this effect for landmarks within a city with reference points being either shopping malls or a university. Lloyd and Gilmartin (1991) reported this effect for world cities with the reference point being Victoria, Canada. Spatial Simplifications When complex spatial information is learned, it tends to be encoded into memory as simpler and more symmetrical than it is in reality. Howard and Kerst (1981, 502) reported the rectangular properties of a distribution of campus landmarks were squared up in the observer s long-term memory and tended to cluster into groups. Tversky (1981) reported evidence from multiple experiments that suggested people tend to simplify cognitive maps by aligning landmark locations to have a more linear structure and rotating spaces toward north-south or east-west axes. She reported that subjects move the South American continent in their world cognitive maps to position it directly south of North American and move the European and African continents south to position them directly east of North and South America. Lloyd and Heivly (1987) reported evidence that residence of urban neighborhoods rotated their cognitive maps to align major transportation arteries with a canonical axis. When people are asked to sketch the streets in a city or the outline of a political region they tend to simplify the shapes and make them more symmetrical than they are in reality. Byrne (1979) reported subjects remembered angles of road intersections to be 90 even when the true angles varied considerably from a right angle. Tversky (1981) reported similar results for sketch maps. Tversky and Schiano (1989) found that sketches of lines representing rivers on maps were remembered as more symmetrical than they actually were. Lloyd (1997) had students who did not live in Texas draw a sketch of the outline of Texas. The drawings appeared to preserve the basic categorical features (e.g., the panhandle) but details of the changes in the boundary were not remembered. He argued that similar looking outline maps could be created from the detailed outline by a line simplification process. Cognitive Distances The most commonly reported cognitive mapping error is that subjects generally overestimate shorter distances and underestimate longer distances (Holyoak and Mah 1982; Lloyd 1989; Poulton 1989). When cognitive distances are plotted on the vertical axis in a scatter diagram and actual distances are plotted on the horizontal axis, most studies have reported a nonlinear relationship between the two variables (Montello 1991; Golledge and Stimson 1997). This pattern could result from an expansion of local space around some important reference point, e.g., a person s home location, and a contraction of space at the edge of the space far from the reference point. It also has been argued that additional distortions are caused by unique environmental characteristics. For example, barriers (Newcombe and Liben 1982), clutter (Thorndyke 1981), the number of turns in a route (Sadalla and Magel 1980), and the number of nodes (Sadalla and Staplin 1980a) all cause an overestimation of distances. Parts of a Larger System The human brain contains hundreds of billions of neurons that can be connected and organized in many different ways. It is important to distinguish between a separate what system for encoding characteristics of objects and a where system for encoding locations (Ungerleider

4 520 Volume 52, Number 3, August 2000 and Mishkin 1982; Landau and Jakendoff 1993). An electrophysiological study of the working memory used to encode location, color, and shape has indicated separate subsystems for encoding spatial information and object information (Martin-Loeches and Rubia 1997). Another study, conducted with monkeys, identified some neurons that responded only to what information and others that responded only to where information (Rao et al. 1997). This study also reported finding a third type of neuron that responded to both what and where information. It was speculated that these neurons might be needed to link the what and where information for the purpose of guiding behavior. The current research focuses only on learning where information related to relatively large spaces. Small-scale maps are frequently used to acquire this type of information. For the learning simulations, it was assumed that a learner had no prior knowledge of the geographic space as the learning process begins. Some researchers argue that learning frequently involves using prior knowledge, e.g., schema or mental models, to more easily organize new information into meaningful cognitive representations (Bruner 1990, 1996). Cognitive maps may serve as prior knowledge for learning and understanding other types of spatially distributed information. Understanding how these fundamental structures are learned and the systematic errors that will naturally be part of their organizations should be important issues in geographic education. Connectionist models, such as the neural networks used in this study, assume learning is simply the adjustment of the weights connecting neurons in a network (McClelland and Rumelhart 1986; Rumelhart and McClelland 1986). For this study, the stage of learning is assumed to be so primitive that it is reasonable to assume no prior knowledge exists. Self-Organizing Neural Networks as Cognitive Maps In their seminal work on cognitive mapping systems in the brain O Keefe and Nadel (1978, 374) argued the hippocampus both constructs and stores cognitive maps. Most of their data on learning were based on rats moving around in an environment. They argued for the existence of place-coded neurons in the hippocampus of the freely moving rat (O Keefe and Nadel 1979, 487). They also argued the hippocampus is concerned with the storage of spatial information to represent absolute spaces rather than egocentric relative spaces. It is such absolute spaces that are considered here. Two studies are reported here that use the same type of neural network model to learn the locations of points in a space. The studies consider spaces (e.g., continents and countries) that are too large for individuals to learn by navigation. Most people would learn where objects are located in such spaces by encoding coordinate information from small-scale maps. It is this coordinate information that was learned by the neural networks. The Structure of the Model The neural network models used in the following studies consisted of layers of neurons that were connected to each other (Fig. 1). There were always two neurons in the input layer that were set to digital values (e.g., latitude and longitude) for locations of objects (e.g., cities) in the space to be learned (e.g., a region of the world). During the learning phase of the process these neurons were set to a series of values to be learned and the information passed through the connections to a square array of neurons usually referred to as the Kohonen layer (Kohonen 1989). The Kohonen layer always had a fixed number of neurons that were distributed in a square array (e.g., 10 by 10) as illustrated in Figure 1. Each input neuron was connected with a weight to each neuron in the Kohonen layer. There were no connections among the N by N neurons in the Kohonen layer or among the neurons in the input layer of the network. When the network is initialized, the weights are set randomly to some small positive number. These initialized random weights represent a subject who has no prior knowledge. When studying a primitive learning process, it is useful to have subjects who can begin the learning process with no biases or previously learned information. Since it is difficult to find human subjects with no prior knowledge of the locations of familiar world cities, artificial subjects represented by neural networks are a good alternative. Each of the i neurons in the Kohonen layer is activated (A i ) when a new trial (geographic coordinates rep-

5 Self-Organized Cognitive Maps 521 X Y being adjusted at the end of the learning process (Kohonen 1993). This winner-takes-all strategy causes trials with similar coordinates in physical space to activate neurons located in the same region of the Kohonen space. The selected neurons adjust their weights using the following equation: W i, j = ( I j W i, j ) (2) Latitude Longitude Figure 1: The basic Kohonen neural network model used for this research had two input neurons, e.g., latitude and longitude. The true locations in geographic space were presented through these neurons. The input neurons were both connected to each neuron in the Kohonen layer of the model. This layer is represented as a 10 by 10 array of neurons in the illustration. The X,Y coordinate is the location of the geographic location in the cognitive map represented in the Kohonen layer. resented by I 1 and I 2 ) is presented to the network through the j input neurons by a simple linear equation: A i = 2 W i, j I j j = 1 (1) If the weights (w i,j ) connecting a neuron in the Kohonen layer with the input neurons have relatively high values, then that neuron should have a relatively high activation. The neuron in the Kohonen layer with the highest activation and its neighbors get to adjust their weights connecting them to the input neurons. The usual procedure is to have a broader definition of neighborhood initially and to decrease this over time until only the winning neuron is The weight (w i, j ) connecting the ith neuron in the Kohonen layer with the jth input neuron is adjusted proportional to a learning coefficient,, multiplied by the difference between the input value and the current weight. After many learning trials that allow all the inputs multiple presentations and adjustments to the weights, the needed adjustments become very small and the network stabilizes with the same neurons winning each time for each physical coordinate. Locations that are close together in physical space have similar coordinates and should also be close together in the Kohonen layer. Conversely, locations that are far apart in physical space should also be far apart in the Kohonen layer. The set of physical coordinates being learned are projected into the Kohonen layer in the same sense that the standard Mercator equations transforms latitude and longitude into rectangular X,Y coordinates. Once all the weights are fixed, an X,Y coordinate (Fig. 1) can be computed for any latitude and longitude input to the network using a weighted average of the locations of the three neurons in the Kohonen layer producing the highest activations (NeuralWare 1996). It is theoretically possible to use any rectangular array of neurons in the Kohonen layer (Kohonen 1989). Given that the typical human brain contains several hundred billion neurons, it would take a relatively small allocation of resources to learn a cognitive map (McNaughton and Nadel 1990). It was assumed for these simulations that these learning resources were a small network of neurons. Two characteristics of the neural network could be the basis of systematic distortions. First, simplifications would occur if an array of neurons (the Kohonen layer) were used to encode physical locations whose natural distribution did not conform to a rectangular distribution. Knowing the appropriate number of neurons and their optimal distribution in rows and columns requires prior

6 522 Volume 52, Number 3, August 2000 knowledge of what is to be learned. An artificial or biological network experiencing one location at a time would have no knowledge of the whole distribution until all locations have been encountered. Second, additional simplifications could occur if the number of neurons devoted to encoding the cognitive map does not match the number needed to completely encode the details in the physical map. There are a number of studies that have argued that humans have a very limited capacity for working memory (Miller 1956; Shiffrin and Nosofsky 1994). The simplest assumption, and the one used here, is to specify the Kohonen layer as a symmetrical array with a relatively small and equal numbers of neurons in the rows and columns. Hypotheses Given what is known about typical errors in human cognitive maps and the assumptions used to specify the Kohonen neural network models, it is possible to form two hypotheses. Hypothesis 1: Like human cognitive mappers, self-organized cognitive maps learned by Kohonen neural networks will simplify spatial distributions by aligning locations. Hypothesis 2: Like human cognitive mappers, self-organized cognitive maps learned by Kohonen neural networks will exaggerate shorter distances and shrink longer distances. Learning Simulations Cities on Four Continents The first study considers 12 cities located on four continents that rim the Atlantic Ocean (Fig. 2). Three well-known cities were selected from each continent and their latitude and longitude used as a set of input data for a Kohonen neural network structured as described above (Fig. 1). The network was initialized with random weights and run for 2,000 learning trials. This number of trials was determined experimentally in preliminary training runs that determined the amount of training needed to produce stable weights. Once the weights have stabilized, additional training does not change the weights and the learning process is completed unless new information becomes available. After training was completed, the weights 60 Stockholm 50 New York Madrid 40 Los Angeles Athens 30 Miami Cairo Equator Bogota Lagos Rio de Janeiro -30 Santiago Johannesburg -40 Prime Meridian Figure 2: Cities on four continents were selected for the study. The latitude and longitude values associated with the cities were used as the information to be learned by the Kohonen neural network. were fixed and data for the same 12 cities were used to compute X,Y coordinates for each city in the Kohonen layer. This procedure was repeated 29 more times so that a reliable mean response could be computed. Since the random weights were different for each of the 30 simulations, the solutions were slightly different each time. This is the nature of neural networks. A somewhat unique solution is learned each time the network is run. Models that do not produce completely consistent results may appear to be undesirable, but one must consider that individual human learners would also produce somewhat unique cognitive maps. The solutions for the 30 artificial neural network simulations were averaged to produce a typical response. Since the neural network model had no sense that North should be at the top of the map, it was necessary to rotate and/ or reflect the axes of individual models to align them all to the correct cardinal axes before aggregation. A simulated self-organized cognitive map (Fig. 3) that represents the average locations of the 12 cities in the Kohonen layers of the neural networks shows some interesting similarities and differences when compared with the actual locations in geographic space (Fig. 2). Some of the errors of simplification typically reported for human cognitive maps are clearly present in the self-organized cognitive map learned by the Kohonen neural network. The

7 Self-Organized Cognitive Maps Los Angeles Bogota Miami New York Madrid Lagos Stockholm Athens Cairo -0.4 Santiago Rio de Janeiro Johannesburg Figure 3: The locations of the 12 cities in the Kohonen layer of the neural network aggregated over 30 simulations. Learning errors appear to be similar to those typically made by human subjects. The continental locations appear to be aligned with South America south of North America and Europe and Africa east of North and South America. continents are aligned with South America positioned under North America and Europe and Africa positioned east of North and South America. Given that the Kohonen neural network was only given information on the 12 city locations and did not have a home bias, it is not surprising that the cities in the four continents would be treated as equally important information. For example, the areas for the triangles made by connecting the three points for each continent is approximately the same on the simulated cognitive map (Fig. 3). This is clearly not true for the same triangles on the cartographic map (Fig. 2). Location data for the same 12 cities were collected from 15 geography teachers as part of a class exercise. Geography teachers represent human subjects that should be more familiar with world maps and city locations than the typical human subjects used in cognitive mapping studies. It was, however, expected that they would still have the usual systematic distortions in their cognitive maps because accuracy is a function of the learning process rather than amount of experience. Subjects were seated in front of monitors that displayed a Mercator map projection of the world. The map was similar to Figure 4, but was blank except for the latitude and longitude lines and labels. The 12 cities and their countries were named one at a time in a different random order for each subject and the subjects simply marked where they thought each city was located on the blank map with a mouse. The map remained blank throughout the process so that each city was an independent trial with the same background information available. The locations marked by the subjects were aggregated and are represented in Figure 4 by the grey ellipses connected to the actual locations (black dots). The center of the ellipses indicate the average locations marked for the 12 cities and the length of horizontal and vertical radii of the ellipses indicate one standard deviation around this central estimate. The usual distortions can easily be identified. Subjects generally expanded the space around North America by moving cities to the east and south. The cities are aligned for the American continents by shifting New York and Miami to the east while shifting the South American cities to the west. All cities in Europe and Africa were shifted south to align them with cities on the American continents. The relative sizes of the ellipses indicate subjects were most consistent marking the cities in North America and least consistent marking cities in Africa. This is likely a bias based on the home location of the subjects. Saarinen et al. (1996) measured the sizes of continents drawn on sketch maps by subjects from 22 different home sites around the world. They reported the home continent was generally drawn at an exaggerated size and that Europe was always exaggerated and Africa was always diminished in size. Since each continent in the current study has three cities, the areas of the triangles made by connecting the points representing the three cities were computed to represent a physical space on each continent. For the triangles made for the actual locations (black dots in Fig. 4) the areas should reflect accuracy as it is represented visually in a Mercator projection. For the triangles made from the locations estimated by the subjects (grey ellipses in Fig. 4) the areas should reflect the accuracy of the subjects aggregate cognitive map as it is represented in the same Mercator projection. The triangular areas were summed and the proportion of the total area represented by the

8 524 Volume 52, Number 3, August N 60 N 40 N 20 N Equator 20 S 40 S New York Los Angeles Miami Bogota Santiago Stockholm Madrid Athens Cairo Lagos Rio de Janeiro Johannesburg 60 S 80 S 180 W 140 W 100 W 60 W 20 W 160 W 120 W 80 W 40 W Prime 20 E Meridian 40 E 60 E 100 E 140 E 180 E 80 E 120 E 160 E Figure 4: The 12 cities used in the simulation represented in a Mercator Projection (black dots) and the locations marked by geography teachers (grey ellipses). The size and shape of the ellipses represent variation among the teachers. three cities in a continent was used to construct pie charts (Fig. 5). Based on the representation of the cities in a Mercator projection, the cartographic truth (Fig. 5a) indicates the African triangle is the largest. South and North American triangles are second and third largest and the European triangle is the smallest. Triangles based on the human subjects estimates (Fig. 5b) appear to agree with Sarrinen et al. s (1996) sketch map results. North America is exaggerated to be the largest area. Europe is also increased in size, while Africa is substantially decreased in size. A similar procedure was used to construct triangles and compute areas for the locations in the aggregate self-organized cognitive map (Fig. 3). The pie chart for these continents (Fig. 5c) indicates a bias in the same direction as the human subjects, but not as pronounced. North America is larger than it should be, Europe is larger than it should be, and Africa is smaller than it should be. Although this matches the expected bias found for human subjects, it appears to just reflect the neural network s equal treatment of the cities. The city locations were simplified in the Kohonen layer so they were evenly spaced. This exaggerated the smaller areas and reduced the sizes of larger areas. The human subject data appears to enhance this distortion by further exaggerating the size of the area in North America and diminishing the area for Africa. Cities in the United States Learned from a Reference Point Although the self-organized cognitive map simulated for the 12 world cities did show an alignment bias, it did not appear to have a bias that expanded the space around any one location in the space (Fig. 3). The geography teachers cognitive map appeared to expand the space around North America (Fig. 4). A number of authors have related this effect to the amount information stored in memory that re-

9 Self-Organized Cognitive Maps 525 a b c Africa Europe Africa South America Europe Africa Europe North America South America North America North America South America Figure 5: The proportions of the total area in the four triangles accounted for by a triangle representing a continent. The cartographic truth based on a Mercator projection (a), human subject estimates (b), and the self-organized cognitive map (c) are represented in pie charts. lates to particular locations. Milgram (1973) suggested that space expands around regions of a cognitive map in proportion to increased amounts of information a person might have encoded about the spatial properties of those region. Sadalla and Staplin (1980b) extended this idea to encoding route information and showed that the cognitive representation of distances along routes were significantly related to the amount of information stored about the route. Holyoak and Mah (1982) argued that people generally need to make finer distinctions between nearby locations than faraway locations. This need is accommodated by an ecologically important flexibility in people s ability to distribute a limited discriminatory capacity over a given magnitude range (Holyoak and Mah 1982, 348). They also argue that the density of known locations will be much greater near a reference point. This would certainly be true for home locations. A number of these arguments can be directly related to self-organized cognitive maps learned by Kohonen neural networks. The network has a limited capacity based on the number of neurons in the Kohonen layer. If the density of locations that need to be mapped into the Kohonen layer decreases with distance from a home reference point, then a conflict will occur. This conflict also happens when human cognitive mappers are forced to map location information available at different scales and details into a common space. The U.S. cities used in the second study provide an example of this conflict. The sample of cities is meant to represent a collection of cities that someone living in Columbia, SC might know. The frequency of known cities decreases with distance. Starting with all the cities in the U.S. in a database, cities were collected for the sample in stages. Each stage set a maximum distance a city could be away from Columbia, SC and a minimum population size for the city. The first stage collected only large cities (more than 1 million population) that were at least 2,500 miles from Columbia. This is based on the notion that cities known by people that are far from their homes are more likely to be larger cities. Each subsequent stage reduced the maximum distance a selected city could be from Columbia and lowered the accepted range of population size. The final stage collected only small cities (less than 15,000 population), but they needed to be near Columbia (within 100 miles). This is based on the notion that the smaller cities that people know are more likely to be near their home. This process resulted in 232 cities being selected for the sample.

10 526 Volume 52, Number 3, August 2000 Figure 6: Cartographic map of the 232 cities used for the learning simulations. The frequency of the cities selected decreases with distance. At the scale represented in the map the cities located near the reference point (Columbia, South Carolina) cannot be visually discriminated. One solution to mapping these cities (Fig. 6) illustrates the basic problem facing cartographers and cognitive mappers. Mapping the cities at a national scale positions the cities near the reference point are so close together that they cannot be discriminated. Plotting all the cities so that local scale details can be seen would require a very large piece of paper. What is a cognitive mapper to do? The hypothesis examined here is that we map these cities into a space that can accommodate the needed change in scale. This accommodation, however, has a significant cost. Distance in the space must be nonlinear. The sample of U.S. cities was learned by a Kohonen neural network structured similar to the one illustrated in Figure 1 and used for the previous simulations. Map coordinates (Fig. 6) measured in state plane coordinates were provided to the two input neurons for each of the 232 cities. Columbia was one of the cities, but it was treated exactly like all the other cities to be learned. The network had no special indication that the Columbia observation had any special meaning in the learning process. Columbia s served as a reference point because proximity to its location was used to select the sample. The 232 coordinate locations were projected into the 10 by 10 neuron Kohonen layer during the learning stage. After 10,000 learning trials the network weights had stabilized. The weights were then fixed and all the cities were passed through the network to compute the Chicago Los Angeles Atlanta Columbia -0.8 Charleston -1.0 Miami Figure 7: Plot of the 232 cities in the Kohonen layer for the first simulation. The self-organized cognitive map shows the expansion of space near the home reference point (Columbia) and a contraction of space far from the reference point. Some dots overlap. X,Y position of each city in the Kohonen layer. The self-organized cognitive map for the first simulation (Fig. 7) shows Columbia in the lower right quadrant of the map. The space would appear to be expanded around the reference point with the faraway places near the edge of the space. This procedure was repeated 29 more times with the initial weight set to small random numbers each time. Thirty simulations were computed so that reliable means could be computed for distances between Columbia and other cities. Linear distances were computed between Columbia and each of the 232 cities for all 30 self-organized cognitive maps and then aggregated. The distances in miles between Columbia and 232 cities were also computed in physical space. Distances in the self-organized cognitive maps were computed in spaces whose axes were constrained by the simulation program to range between 1 and 1. Z-scores were computed for both distance measures so they could be more easily compared. A plot of the physical distances on the horizontal axis and the aggregated distances in the self-organized cognitive map on the vertical axis indicate a remarkably regular nonlinear relation-

11 Self-Organized Cognitive Maps 527 Distance in Kohonen Layer as Z-Scores ship (Fig. 8). The pattern is the one frequently found for human subjects (Golledge and Stimson 1997). As physical distance increases, cognitive distance increases at a decreasing rate. A log-linear model explaining cognitive distance was significant (r , F 1,489.8, P F 0.000) as was a quadratic model (r , F 1,112.1, P F 0.000). The Kohonen neural networks learned the city locations and mapped them into the self-organized maps so distances to locations near Columbia would be overestimated and distances to locations far from Columbia would be underestimated. Discussion Physical Distance as Z-Scores Figure 8: Plot of the 232 cities as their distance from Columbia, South Carolina in physical space on the horizontal axis and in the Kohonen layer of the neural network on the vertical axis. Both distances are represented in standardized units. The self-organized maps produced by Kohonen neural networks are based on very simple learning principles: 1) the winning neuron s connection weights try to take on values equal to the information being learned and 2) observations with similar input values should be neighbors in the self-organized cognitive map. Both of these principles help construct the selforganized map in the Kohonen layer when geographic locations are being learned. In effect, the connection weights are equations that project physical locations into another space the self-organized cognitive map. Unlike equations for cartographic projections, the connection weights are not based on geometric principles, but are learned. The weights are adjusted as they respond to the patterns in the information being presented to the network. This is important for cognitive mapping research if people learn basic spatial locations in a similar way (O Keefe and Nadel 1978; Kohonen 1989). Distortions in human cognitive maps may simply be a function of the way neural networks (biological and artificial) learn (Martindale 1991; Kosslyn and Koenig 1992). The type of learning studied here took place over time. In the first study of world cities, neural networks with no prior knowledge, i.e., initial random weights, processed geographic locations one at a time. The initial random weights combined with a randomly selected geographic location to start the process. The first winning neuron with the highest activation (Equation 1) started the organization of the cognitive map being learned. All of the neurons in the neighborhood of the winner changed their weights to look more like the input information (Equation 2). When this same observation or one with similar values is processed in the future, this region of the Kohonen layer is more likely to respond with higher activation levels than other regions in the layer. Conversely the region will not respond to dissimilar input values, but some other neuron in the Kohonen layer will win and be allowed, along with its neighbors, to change weights. As the process continues all the input values have to find some neuron(s) that responds to them with a high activation. When this is true for all observations (cities) in the learning set, adjustments to the weights become small and the network stabilizes. Learning is then complete unless new information is presented to the neural network. The hypothesis related to the study of world cities stated that both humans and Kohonen neural networks would encode cognitive maps with alignment distortions. The alignment of locations to simplify spatial structures had been reported in the literature by both Tversky (1981) and Lloyd and Heivly (1987). The simulation of the learning of 12 world cities suggested the same effect occurred when a Kohonen neural network learned geographic locations. The organization process generally

12 528 Volume 52, Number 3, August 2000 positioned the sets of three cities for each of the four continents in one corner of the square space defined by the 10 by 10 array of neurons in the Kohonen layer of the neural network (Fig. 3). This simplified and aligned the cities. One could infer that other cities on the same continents would follow the same pattern. For example, if the latitude and longitude for St. Louis, MO were inputted, it would activate neurons near the center of the triangle made by Los Angeles, Miami, and New York in the selforganized cognitive map. The alignment simply happens because the information is being projected into a square space with fixed outer limits. As the cities are learned, unlike cities move away from each other and are constrained by the edges of the space until equilibrium is reached. This naturally produces an alignment of the locations. This explanation works for humans only if one assumes biological neural networks allocate resources in the same way. The limited capacity of working memory for humans is well documented (Shiffrin and Nosofsky 1994). Why should the array be square? One simple argument for this is that the resources must be allocated before the shape of the space being mapped is known. Symmetrical arrays are generally useful under such conditions of ignorance. The second study considered the known learning bias caused by reference points and the nonlinear relationship between cognitive distance and physical distance. This study considered a biased sample of cities that might be learned by someone living in Columbia, South Carolina. It was assumed the hypothetical person would know some larger faraway cities and many smaller nearby cities. The density of known cities decreased with distance away from the reference point. Mapping the locations of both near and far cities into a single space is a problem for a cartographer (Fig. 6). A very large display page or several normal sized cartographic maps with different geographic scales would be needed to visibly display the available details in the information. Humans and Kohonen neural networks solve the problem in a different way than cartographers. All the information is learned by projecting it into a single space, but allowing the space to warp. Locations farthest from the reference point are pushed away until they are activating the neurons at the edge of the self-organizing cognitive map. The many locations near the reference point also need to activate neurons in the self-organizing cognitive map that allow their differences to be recorded. This expands the space near the reference point relative to the space near the edge of the cognitive map. The nonlinear relationships frequently reported between cognitive distance and physical distance are simply caused by this process that adjusts the relative positions of all the locations being learned subject to non-uniform densities around the reference point and the finite limits of the spatial array of neurons storing the information. The hypothesis related to the study of U.S. cities learned by a hypothetical person living in Columbia, SC stated that self-organized cognitive maps learned by Kohonen neural networks would mimic humans by exaggerating shorter distances and shrinking longer distances. The results supported this hypothesis. Individual cognitive maps appeared to expand the space near Columbia and contract space far from Columbia (Fig. 7). Cognitive distances from Columbia to 232 cities, aggregated over 30 simulations, had a significant non-linear relationship with physical distances (Fig. 8). In the first study the human subjects expanded the space around North America (home) but the neural network did not. This difference is likely to be caused by the additional knowledge that the human subjects have about locations in North America. The non-linear mapping appears not to occur unless multiple scales are involved with more information existing at the local scale. Conclusions A Kohonen neural network was used to learn geographic information for two related studies. The first study considered just a small number of cities evenly distributed on four continents on a world map. The second study had the same type of network learn the distribution of a larger number of cities in the U.S. that were spatially biased. In both studies the unsupervised Kohonen neural networks learned the spatial information with some distortions. These distortions appeared to have some pattern that could be described as systematic. Since the information being learned was known to be accurate when presented to the

13 Self-Organized Cognitive Maps 529 networks, it is not possible to argue that the distortions were caused by the acquisition of inaccurate information. The distortions appeared to be related to the process used to learn the information, rather than the particular information being learned. As the serial learning process proceeded, the coordinate information from the cartographic maps was projected into the self-organized cognitive map through a competition among the neurons in the Kohonen layer of the neural network. This competition projects the physical locations into the self-organized map so that differences among neighbors can be discriminated. The learning process is practical and intuitive rather than precise and geometric. One conclusion is that Kohonen models can learn spatial information and that the learning process functions like a very fuzzy and inconsistent map projection. A more general conclusion is that connectionist models (neural networks) appear to represent spatial learning very well. Lloyd (1994) also reported that a neural network and humans appeared to learn map prototypes in a similar way. Another conclusion is that the model s faults as a map projection make it a very good mimic of human cognitive mappers. People produce the same systematic errors when they encode cognitive maps. Two important error patterns that appear to be shared by human cognitive mappers and Kohonen self-organized maps are 1) alignment of locations that result in a simplification of spatial information and 2) expansion of space in regions that contain more spatial information. This latter effect explains why cognitive maps do not usually have a constant scale and why distance in cognitive maps are frequently reported as being a non-linear function of physical distance. It also explains why people frequently overestimate short distances and underestimate long distances. Another conclusion is that the learning process can only partially be blamed for distortions in cognitive maps. Biased samples of information are also partially responsible for the details of the distortions. Replicating the second study using Salt Lake City as the reference point instead of Columbia would probably result in the same type of distortions, but they would show an expansion of space around Salt Lake City. Distances to cities near Salt Lake City would be overestimated and distances to East Coast cities would be underestimated. Future studies should consider more simulations that extend the current research findings. The current studies have considered distortions caused by varying amounts of information available for different regions of a cartographic map. Simulations that have three (or more) input neurons might investigate other common distortion in cognitive maps, (e.g., barrier). One could have the model learn not only the physical coordinate for a location, but also the categorical side of a river barrier ( 1 or 1) for a geographic location. Would this information expand the space around the barrier as it seems to in humans (Newcombe and Lieben 1982)? Additional input neurons besides physical location could also be investigated to study the connections between what and where information (Rao et al. 1997). Some studies have suggested that liking a place seems to shorten its cognitive distance (Stea 1969; Golledge and Zannaras 1973). If preferences for places were measured for a sample of human subjects, preference values could be learned by a Kohonen neural network through an additional input neuron. Learning self-organized cognitive maps with and without this extra preference input neuron would enable one to assess the influence that affect has on cognitive distance. Both studies reported here considered coordinate information as inputs for the model to learn. Spatial information represented in alternate forms should also be considered. Selforganizing models are also very good at learning information expressed as visual images. Lloyd (1997) reports success in training a Kohonen network to recognize map symbols that differed in shape, color and orientation. Filippi (1998) was able to train another type of Kohonen network with satellite imagery to recognize types of vegetation. Cognitive issues, related to learning spatial information, are difficult to study directly. Simulations, such as the ones reported here, enable experimenters to control characteristics of the learner, such as having no prior knowledge. One can also be very precise about what information is being learned with an artificial neural network. This paper has reported studies that compared the results of learning simulations with the truth and with the results of learning experiences for human subjects. This approach should help geographers untangle the complex