Extensive Interpolations
The example above is a spatially extensive interpolation because it involves count data. In areal, these estimates are obtained using the aw_interpolate() function:
aw_interpolate(wards, tid = WARD, source = race, sid = GEOID,
weight = "sum", output = "tibble", extensive = "TOTAL_E")
#> # A tibble: 28 × 4
#> OBJECTID WARD AREA TOTAL_E
#> <dbl> <int> <dbl> <dbl>
#> 1 1 1 46138761. 7959.
#> 2 2 2 267817711. 12111.
#> 3 3 3 66291644. 7357.
#> 4 4 4 53210707. 8458.
#> 5 5 5 60462396. 8808.
#> 6 6 6 64337271. 14053.
#> 7 7 7 101268146. 15842.
#> 8 8 8 45966410. 12189.
#> 9 9 9 73993891. 14218.
#> 10 10 10 62915358. 11239.
#> # … with 18 more rows
For spatially extensive interpolations, a list of variable names should be supplied for the argument extensive. This can be a single variable name, such as in the example above, or a vector of variable names:
aw_interpolate(wards, tid = WARD, source = race, sid = GEOID,
weight = "sum", output = "tibble",
extensive = c("TOTAL_E", "WHITE_E", "BLACK_E"))
#> # A tibble: 28 × 6
#> OBJECTID WARD AREA BLACK_E TOTAL_E WHITE_E
#> <dbl> <int> <dbl> <dbl> <dbl> <dbl>
#> 1 1 1 46138761. 7748. 7959. 152.
#> 2 2 2 267817711. 10612. 12111. 1315.
#> 3 3 3 66291644. 6644. 7357. 594.
#> 4 4 4 53210707. 8203. 8458. 160.
#> 5 5 5 60462396. 7075. 8808. 1532.
#> 6 6 6 64337271. 7441. 14053. 5840.
#> 7 7 7 101268146. 6630. 15842. 8219.
#> 8 8 8 45966410. 3797. 12189. 7605.
#> 9 9 9 73993891. 6413. 14218. 6839.
#> 10 10 10 62915358. 1666. 11239. 8703.
#> # … with 18 more rows
This ability is a key feature of areal - iteration is built into the package by default, eliminating the need for repeated table joins after interpolations are calculated.
Calculating Weights for Extensive Interpolations
The aw_interpolate function also uses an argument weight. There are two options, "sum" and "total". Each makes a different assumption about the nature of the data and the relationship between the source and target features. For perfectly overlapping data, the distinction between these two options should not matter. In practice, however, there are often deviations in our data even between features that should be perfectly congruous.
The "sum" approach to calculating weights assumes that 100% of the source data should be divided among the target features. When \({A}_{j}\) is calculated (see previous section), it is done by taking the sum of the areas for all intersected features (\(i\)) within a given source feature (\(j\)). Let:
- \({A}_{j} = \textrm{total area of source feature j}\)
- \({A}_{ij} = \textrm{areas for intersected features in } i \textrm{ within source feature } j\)
\[ {A}_{j} = \sum{{A}_{ij}} \]
On the other hand, the "total" approach to calculating weights assumes that, if a source feature is only covered by 99.88% of the target features, only 99.88% of the source target’s data should be allocated to target features in the interpolation. When \({A}_{j}\) is created, the actual area of source feature \(j\) is used.
Weights Example 1: Non-Overlap Due to Data Quality
In the example above, race and wards are products of two different agencies. The aw_stl_wards data is a product of the City of the St. Louis and is quite close to fully overlapping with the U.S. Census Bureau’s TIGER boundaries for the city. However, there are a number of very small deviations at the edges where the ward boundaries are smaller than the tracts (but only just so). These deviations result in small portions of census tracts not fitting into any ward.
We can see this in the weights that are used by aw_interpolate(). The aw_preview_weights() function can be used to return a preview of these areal weights.
aw_preview_weights(wards, tid = WARD, source = race, sid = GEOID,
type = "extensive")
#> # A tibble: 106 × 3
#> GEOID extensiveSum extensiveTotal
#> <chr> <dbl> <dbl>
#> 1 29510101100 1 0.999
#> 2 29510101200 1 1.00
#> 3 29510101300 1 1.00
#> 4 29510101400 1 1.00
#> 5 29510101500 1 0.998
#> 6 29510101800 1 0.978
#> 7 29510102100 1 0.991
#> 8 29510102200 1 0.999
#> 9 29510102300 1 0.999
#> 10 29510102400 1 1.00
#> # … with 96 more rows
The first tract listed above has a total estimated population of 2510. The practical impact of the weights is that only 2507.072085 individuals will be allocated to wards if the "total" approach to calculating areal weights is used. If "sum" is used, on the other hand, all 2510 individuals would be allocated to wards. In this scenario, the "sum" approach makes more sense because, while the race and ward data do not overlap in practice, they should overlap since no tracts extend out of the city’s boundaries. We therefore want to ensure that all individuals within each tract are allocated out to wards.
With spatially extensive interpolations that utilize the "sum" approach, the sum of the interpolated column should equal the sum of the original source data’s column that was interpolated. This can be verified with aw_verify():
result <- aw_interpolate(wards, tid = WARD, source = race, sid = GEOID,
weight = "sum", output = "tibble", extensive = "TOTAL_E")
aw_verify(source = race, sourceValue = TOTAL_E,
result = result, resultValue = TOTAL_E)
#> [1] TRUE
This check does not work with the "total" approach to areal weights:
result <- aw_interpolate(wards, tid = WARD, source = race, sid = GEOID,
weight = "total", output = "tibble", extensive = "TOTAL_E")
aw_verify(source = race, sourceValue = TOTAL_E,
result = result, resultValue = TOTAL_E)
#> [1] FALSE
Weights Example 2: Non-Overlap Due to Differing Boundaries
We can use the aw_stl_wardsClipped data to illustrate a more extreme disparity between source and target data. The aw_stl_wardsClipped data have been modified so that the ward boundaries do not extend past the Mississippi River shoreline, which runs along the entire eastern boundary of the city. When we overlay them on the city’s census tracts, all of the census tracts on the eastern side of the city extend outwards.
The difference in weights in this example is more extreme:
aw_preview_weights(wardsClipped, tid = WARD, source = race, sid = GEOID,
type = "extensive")
#> # A tibble: 106 × 3
#> GEOID extensiveSum extensiveTotal
#> <chr> <dbl> <dbl>
#> 1 29510101100 1 0.999
#> 2 29510101200 1 1.00
#> 3 29510101300 1 1.00
#> 4 29510101400 1 1.00
#> 5 29510101500 1 0.998
#> 6 29510101800 1 0.722
#> 7 29510102100 1 0.991
#> 8 29510102200 1 0.999
#> 9 29510102300 1 0.999
#> 10 29510102400 1 1.00
#> # … with 96 more rows
Only 72.31% of tract 29510101800, for example, falls within a ward. In many American cities that lie within larger counties, tract boundaries do not stop at the municipal boundaries in a way that is similar to the difference between tracts and the clipped wards here. In this scenario, we do not want to allocate every individual into our city of interest and the "total" approach to weights is appropriate. Not using "total" would result in an over-count of individuals in our city.
If, on the other hand, we believe that all of the individuals should be allocated into wards, using "total" in this case would result in a severe under-count of individuals.
Intensive Interpolations
Spatially intensive operations are used when the data to be interpolated are, for example, a percentage or density value. An example of these data can be found in ar_stl_asthma, which contains asthma rates for each census tract in the city. The interpolation process is very similar to the spatially extensive workflow, except with how the areal weight is calculated. Instead of using the source data’s area for reference, the target data is used in the denominator. Let:
- \({W}_{i} = \textrm{areal weight for intersected feature i}\)
- \({A}_{i} = \textrm{area of intersected feature i}\)
- \({A}_{ik} = \textrm{areas for intersected features in } i \textrm{ within target feature } k\)
\[ {W}_{i} = \frac { {A}_{i} }{ \sum{{A}_{ik}} } \]
Like spatially extensive interpolations that use the "sum" approach, the weights for intensive interpolations should always be equal to 1 as well.
aw_preview_weights(wards, tid = WARD, source = asthma, sid = GEOID,
type = "intensive")
#> # A tibble: 28 × 2
#> WARD intensive
#> <int> <dbl>
#> 1 1 1
#> 2 2 1
#> 3 3 1
#> 4 4 1
#> 5 5 1
#> 6 6 1
#> 7 7 1
#> 8 8 1
#> 9 9 1
#> 10 10 1
#> # … with 18 more rows
We can calculate the intensive interpolation by specifying a variable name for the intensive argument in aw_interpolate() and omitting the extensive argument:
aw_interpolate(wards, tid = WARD, source = asthma, sid = GEOID,
weight = "sum", output = "tibble", intensive = "ASTHMA")
#> # A tibble: 28 × 4
#> OBJECTID WARD AREA ASTHMA
#> <dbl> <int> <dbl> <dbl>
#> 1 1 1 46138761. 13.4
#> 2 2 2 267817711. 13.2
#> 3 3 3 66291644. 14.1
#> 4 4 4 53210707. 13.6
#> 5 5 5 60462396. 13.8
#> 6 6 6 64337271. 11.7
#> 7 7 7 101268146. 9.72
#> 8 8 8 45966410. 9.82
#> 9 9 9 73993891. 11.8
#> 10 10 10 62915358. 9.44
#> # … with 18 more rows
This gives us an estimate of the asthma rates at the ward level.
Mixed Interpolations
areal also provides support for “mixed” interpolations where both spatially extensive and intensive interpolations need to be calculated. We specify a variable name or a vector of variable names for both the intensive and extensive arguments:
# remove sf geometry
st_geometry(race) <- NULL
# create combined data
race %>%
select(GEOID, TOTAL_E, WHITE_E, BLACK_E) %>%
left_join(asthma, ., by = "GEOID") -> combinedData
# interpolate
aw_interpolate(wards, tid = WARD, source = combinedData, sid = GEOID,
weight = "sum", output = "tibble", intensive = "ASTHMA",
extensive = c("TOTAL_E", "WHITE_E", "BLACK_E"))
#> # A tibble: 28 × 7
#> OBJECTID WARD AREA BLACK_E TOTAL_E WHITE_E ASTHMA
#> <dbl> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 1 46138761. 7748. 7959. 152. 13.4
#> 2 2 2 267817711. 10612. 12111. 1315. 13.2
#> 3 3 3 66291644. 6644. 7357. 594. 14.1
#> 4 4 4 53210707. 8203. 8458. 160. 13.6
#> 5 5 5 60462396. 7075. 8808. 1532. 13.8
#> 6 6 6 64337271. 7441. 14053. 5840. 11.7
#> 7 7 7 101268146. 6630. 15842. 8219. 9.72
#> 8 8 8 45966410. 3797. 12189. 7605. 9.82
#> 9 9 9 73993891. 6413. 14218. 6839. 11.8
#> 10 10 10 62915358. 1666. 11239. 8703. 9.44
#> # … with 18 more rows
Users should take care to consider the implications of interpolating multiple values, such as total population and the number of African American residents (both extensive), and then calculating a spatially intensive variable from them such as percent African American. Doing so in multiple steps, and thereby treating extensive and intensive values as independent, may result in estimates that differ from a single step process where the percent of African American residents is interpoltated directly.
# re-load data
race <- ar_stl_race
# create combined data
race %>%
select(GEOID, WHITE_E, BLACK_E) %>%
mutate(
TOTAL = WHITE_E+BLACK_E,
WHITE_PCT = WHITE_E/TOTAL,
BLACK_PCT = BLACK_E/TOTAL,
TOTAL_PCT = WHITE_PCT+BLACK_PCT
) -> constrainedData
# interpolate
result2 <- aw_interpolate(ar_stl_wards, tid = WARD,
source = constrainedData, sid = GEOID,
weight = "sum", output = "tibble",
intensive = c("WHITE_PCT", "BLACK_PCT", "TOTAL_PCT"),
extensive = c("TOTAL", "WHITE_E", "BLACK_E"))
# calculate new percentages
result2 %>%
mutate(
WHITE_PCT_2 = WHITE_E/TOTAL,
BLACK_PCT_2 = BLACK_E/TOTAL,
TOTAL_PCT_2 = WHITE_PCT_2+BLACK_PCT_2
) -> result2
# display
result2 %>%
select(WHITE_PCT, WHITE_PCT_2, BLACK_PCT, BLACK_PCT_2, TOTAL_PCT, TOTAL_PCT_2)
#> # A tibble: 28 × 6
#> WHITE_PCT WHITE_PCT_2 BLACK_PCT BLACK_PCT_2 TOTAL_PCT TOTAL_PCT_2
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 0.0196 0.0192 0.980 0.981 1 1
#> 2 0.178 0.110 0.822 0.890 1 1
#> 3 0.104 0.0820 0.896 0.918 1 1
#> 4 0.0252 0.0191 0.975 0.981 1 1
#> 5 0.174 0.178 0.826 0.822 1 1
#> 6 0.409 0.440 0.591 0.560 1 1
#> 7 0.575 0.554 0.425 0.446 1 1
#> 8 0.700 0.667 0.300 0.333 1 1
#> 9 0.505 0.516 0.495 0.484 1 1
#> 10 0.862 0.839 0.138 0.161 1 1
#> # … with 18 more rows
Note that there are a number of points of departure between the data interpolated as intensive values (WHITE_PCT, BLACK_PCT) and those that were interpolated as count data (i.e. extensive values) and then converted to intensive variables (WHITE_PCT_2 and BLACK_PCT_2).
