**ENDNOTES**
1 The principal components are new variables formed by taking linear combinations of the existing variables in such a way as to maximize the variance in the first principal component. The maximization is constrained by the requirement that the sum of the squared weights used to combine the original variables is equal to one. Since the information con10t of a variable may be seen as a function of its variance, the first principal component is the combination of the set of original variables that potentially contains the most information in the data base. The second principal component draws the maximum variance left over after the first component has been extracted. It is orthogonal (90 degrees) to the first and describes a unique attribute. This procedure continues until there are as many principal components as there are original variables. The principal components are typically used as a data reduction technique as the first few components extract most of the variance in the data. The principal components are in themselves variables describing the independent dimensions or axis within the space of the original data. The interpretation given to the components is developed by examining their correlation with the original variables.
2 However, the correlation is very weak as the upgrading process is also taking place in high rent neighbourhoods. A similar picture was developed for Toronto and the pooled data for the 10 CMAs yields a gentle U-shaped curve showing that both low and high rent neighbourhoods have the symptoms of gentrification as identified by the principal component.
3 This type of growth was described by Fisher (1933, p152) as one of three kinds of suburban expansion and illustrated clearly by Homer Hoyt (1939, p96) as axial growth that expanded and coalesced the scattered nuclei outside Chicago. This phenomenon is a form of gentrification that has not, to our knowledge, been studied extensively. A future study might look at what happens to the housing and the residents in the older settlements incorporated into the expanding metropolitan area
4 Where possible, the information developed in the interviews was compared to published case study research of gentrifying neighbourhoods within Canadian cities (Bourne 1993, Criekingen and Decroly 2003, Filion 1991, Ley 1988, 1993, 2003, Millward 1988).
5 The correlation between proportional income change and average 1981 income level of tracts varies across the CMAs, Vancouver and MontrĂ©al have no correlation as most neighbourhoods had similar proportional increases but Toronto shows a 0.32 correlation as the richer neighbourhoods got progressively richer during the two decades. In all cities the differentiation of neighbourhoods by income increased during the twenty years as illustrated by the Gini coefficients calculated for the average tract personal income for Montreal, Toronto and Vancouver. Over the twenty years the Gini coefficients increased from 0.13, 0.14 and 0.10 in 1981 to 0.20, 0.25 and 0.16 for the three CMAs respectively. Figure 5 shows that the principal component score of zero occurs at the Montreal tracts with an average income increase of about 25 percent. The gentrification process as measured by this principal component is reducing the number of low-income neighbourhoods in Montreal and, thereby, reduces the supply of lower priced housing.
6 Wyly and Hammell (1990, 729) calculate the population of gentrified neighbourhoods in 1990 to be just over half a million or approximately 22 percent of the total inner-city population based on statistics from eight US cities. However, Wyly and Hammell (1999) do not report the total population or number of census tracts for their sampled cities which would require in order attempt a better comparison with extent of gentrification measured in this paper.
7 Ley (1988, 1993) observes that the increase in prices in a Vancouver neighbourhood in the 1970s establishes a new equilibrium and shifted the gentrification process to other nearby neighbourhoods. Enough time has now passed since the 1970s gentrification to assess the stability of the new equilibrium in these neighbourhoods and determine the extent to which the old stock has been kept rather than redeveloped. Filion (1991) shows how the social upgrading changes increase the political power of the neighbourhoods and helps them resist changes in their character. This hypothesis can be tested by future research.
8 We can not tell the extent to which the Vancouver change in inner city rental accommodation is due to subsidized units or to developer exactions that required them to include units at below market rents.
9 Most Canadian CMAs increased their boundaries between 1981 and 2001.
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