On Spatial Unit Aggregation Problems in the Spatial Interaction Model

Analysis of data aggregated in areal units is a general way to investigate collective social and ecolo-gical phenomena. The results of such analysis, however, depend on the adopted scale and configura-tion of areal units. This spatial aggregation problem is called MAUP (Modifiable Areal Unit Prob-lem). Several researchers (Broadbent, 1970; Openshaw, 1977; Masser and Brown, 1978; Putman and Chung, 1989; Batty and Sikdar, 1982a, b, c, d, 1984; Fotheringham, 1983a, b; Amrhein and Flowerdew, 1989, 1992) have examined spatial system design and its effects on spatial interaction models. The main questions have focused on the definition problem: how to make up areal units. Little attention, however, has been paid to the measurement problem, of which the most serious one in spatial analy-sis is defining the distance between areal units (Beardwood and Kirby, 1975; Rodriguez-Bachiller, 1983; Slater, 1985).<br> This paper is concerned with relations between aggregate distance and biases of the distance-decay-parameter in the Huff model, which is typical of spatial interaction models. For the sake of simplicity the type of distance-decay-function of the Huff model is limited to the power function. Beardwood and Kirby (1975) showed how to define aggregate distance to predict, by spatial interaction models, consistently through spatial unit aggregation. The author terms the aggregate distance they pro-posed, non-bias distance (NB). Although “this is an accounting device of administrative rather than analytical or applied value” (Openshaw, 1977, p. 170), the author utilizes the idea of NB to compare with other conventional types of aggregate distance. Two types of aggregate distance are examined 1) simple mean distance (MD) of all the combinations for origin-and destination-BSUs (basic spa-tial units); and 2) mean trip distance (MT), defined as mean distance weighted with trip size.<br> To begin with, theoretical investigations are performed in the following steps: 1) postulate trip pat-terns described perfectly by the Huff model at the BSU level; 2) aggregate BSUs to make up ASUs (ag-gregate spatial units) and define the utility of each ASU based on size and distance; 3) predict ratio of probabilities to choose an ASU nearer the origin and one farther from it, using the ASU data and the same parameter values as those at the BSU level; 4) consider whether such predictions lead to over-estimation or underestimation for the nearer ASU, by use of the NB idea; 5) if underestimation arises, say that the value of distance-decay-parameter estimated at the ASU level would be increased more (be steeper) than that at the BSU level because of the need to increase the probability by choos-?ing the nearer ASU rather than the farther one; in the reverse case, decreased (less steep) value would be estimated at the ASU level. The theoretical expected directions of aggregate biases of dis-tance-decay-parameters are shown in Table 2. Under the assumption of homogeneity of origin-BSUs' balancing factor in ASUs, the same expectations could be derived for the case where origin BSUs and destination BSUs are aggregated simultaneously.<br> These findings were verified by sensitivity analysis of spatial aggregation effects like the previous studies (Openshaw, 1977; Putman and Chung, 1989; Amrhein and Flowerdew, 1989, 1992). The data for the analysis are shopping and private-purpose trips in the 23 wards of Tokyo City that consist of 115 BSUs. These data were collected through a 1988 Tokyo metropolitan area person-trip survey. Two experiments were undertaken. First the author examined the scaling effects at the scales of 23, 46, 69, and 92 ASUs by one hundred types of zoning randomly generated, respectively, and the zoning effects by two zoning procedures: random zoning (Openshaw, 1977) and nearest-neighbor zoning constrained by the criterion of zone compactness.