Nearest neighbour interpolation Problem set A Worked solutions

The diagram shows a Voronoi diagram for soil moisture sensors on a farm. The value at each site is soil moisture (%), shown beside its label. Use nearest neighbour interpolation to estimate the soil moisture at:

1. (a) the point (5, 1);

   • The point (5, 1) is in the cell of site B.
   • Estimate = 33%

2. (b) the point (8, 4).

   • The point (8, 4) is in the cell of site D.
   • Estimate = 28%

Temperature sensors in a greenhouse are placed at four sites. The table gives their coordinates (in metres) and the temperature (°C) each one reads. There is no diagram. Use nearest neighbour interpolation to estimate the temperature at the point (5, 2).

• Distance from (5, 2) to A(3, 4): \(\sqrt{(5-3)^2+(2-4)^2}=\sqrt{8}\)
• Distance from (5, 2) to B(6, 3): \(\sqrt{(5-6)^2+(2-3)^2}=\sqrt{2}\)
• Distance from (5, 2) to C(1, 2): \(\sqrt{(5-1)^2+(2-2)^2}=4\)
• Distance from (5, 2) to D(8, 4): \(\sqrt{(5-8)^2+(2-4)^2}=\sqrt{13}\)
• The smallest distance is to B, so the estimate is 28 °C.

The diagram shows a Voronoi diagram for snow depth at ski stations; each site shows its snow depth (cm). The point (2, 3) is close to an edge, so it is hard to see which cell it is in.

1. (a) Find the distance from (2, 3) to site G and to site A.

   • G(1, 2): \(d^2=2\), \(d=\sqrt{2}=1.41\)
   • A(3, 1): \(d^2=5\), \(d=\sqrt{5}=2.24\)

2. (b) Hence estimate the snow depth at (2, 3).

   • G is nearer, so the point is in the cell of G.
   • Estimate = 28 cm

The diagram shows a Voronoi diagram for rainfall at weather stations; each site shows its rainfall (mm). The rainfall at the point (1, 1) was later measured as 48 mm.

1. (a) Use nearest neighbour interpolation to estimate the rainfall at (1, 1).

   • The point (1, 1) is in the cell of site G.
   • Estimate = 40 mm

2. (b) Find the percentage error in the estimate.

   • \(\frac{|40-48|}{48}\times 100=16.7\)
   • Percentage error = 16.7%

The diagram shows a Voronoi diagram for air quality monitors. Each site shows its air quality index. A reading above 40 is a warning.

1. (a) Estimate the air quality index at (7, 6).

   • (7, 6) is in the cell of site G: estimate 27

2. (b) Estimate the air quality index at (1, 5).

   • (1, 5) is in the cell of site E: estimate 50

3. (c) Which sites have cells where every point has an estimate above the warning level of 40?

   • Every point in a cell has the value of its site.
   • Sites with a value above 40: B (41), D (54), E (50)

The diagram shows a Voronoi diagram for weather stations, with the rainfall (mm) at each site. A new site H is opened at (6, 5), where the rainfall is 13 mm. The point (7, 6) was in the cell of site E.

1. (a) Show that (7, 6) is now nearer to H than to E.

   • Distance to H: \(\sqrt{(7-6)^2+(6-5)^2}=\sqrt{2}=1.41\)
   • Distance to E: \(\sqrt{5}=2.24\)
   • H is nearer.

2. (b) Write down the new estimate of the rainfall at (7, 6).

   • The point is now in the cell of H, so the estimate is 13 mm.

3. (c) Find the change in the estimate.

   • 13 − 6 = 7