Nearest neighbour interpolation Problem set B Worked solutions

The diagram shows a Voronoi diagram for temperature buoys in a bay. The value at each site is water temperature (°C), shown beside its label. Use nearest neighbour interpolation to estimate the water temperature at:

1. (a) the point (5, 6);

   • The point (5, 6) is in the cell of site D.
   • Estimate = 26 °C

2. (b) the point (6, 5).

   • The point (6, 5) is in the cell of site F.
   • Estimate = 18 °C

Noise monitors near an airport are placed at four sites. The table gives their coordinates (in metres) and the noise level (dB) each one reads. There is no diagram. Use nearest neighbour interpolation to estimate the noise level at the point (6, 5).

• Distance from (6, 5) to A(3, 1): \(\sqrt{(6-3)^2+(5-1)^2}=5\)
• Distance from (6, 5) to B(5, 6): \(\sqrt{(6-5)^2+(5-6)^2}=\sqrt{2}\)
• Distance from (6, 5) to C(3, 4): \(\sqrt{(6-3)^2+(5-4)^2}=\sqrt{10}\)
• Distance from (6, 5) to D(8, 2): \(\sqrt{(6-8)^2+(5-2)^2}=\sqrt{13}\)
• The smallest distance is to B, so the estimate is 54 dB.

The diagram shows a Voronoi diagram for river gauges; each site shows its water depth (cm). The point (4, 4) is close to an edge, so it is hard to see which cell it is in.

1. (a) Find the distance from (4, 4) to site C and to site B.

   • C(5, 3): \(d^2=2\), \(d=\sqrt{2}=1.41\)
   • B(2, 3): \(d^2=5\), \(d=\sqrt{5}=2.24\)

2. (b) Hence estimate the water depth at (4, 4).

   • C is nearer, so the point is in the cell of C.
   • Estimate = 45 cm

The diagram shows a Voronoi diagram for pollen counters in a city; each site shows its pollen count (grains per m³). The pollen count at the point (5, 6) was later measured as 95.

1. (a) Use nearest neighbour interpolation to estimate the pollen count at (5, 6).

   • The point (5, 6) is in the cell of site A.
   • Estimate = 88

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

   • \(\frac{|88-95|}{95}\times 100=7.37\)
   • Percentage error = 7.37%

The diagram shows a Voronoi diagram for radiation monitors. Each site shows its dose rate (units per hour). A reading above 38 is a warning.

1. (a) Estimate the dose rate at (8, 5).

   • (8, 5) is in the cell of site A: estimate 32

2. (b) Estimate the dose rate at (6, 6).

   • (6, 6) is in the cell of site G: estimate 16

3. (c) Which sites have cells where every point has an estimate above the warning level of 38?

   • Every point in a cell has the value of its site.
   • Sites with a value above 38: B (44), D (51), F (70)

The diagram shows a Voronoi diagram for sensors in a vineyard, with the sugar level (%) at each site. A new site H is opened at (8, 5), where the sugar level is 21%. The point (8, 6) was in the cell of site D.

1. (a) Show that (8, 6) is now nearer to H than to D.

   • Distance to H: \(\sqrt{(8-8)^2+(6-5)^2}=1=1.00\)
   • Distance to D: \(\sqrt{5}=2.24\)
   • H is nearer.

2. (b) Write down the new estimate of the sugar level at (8, 6).

   • The point is now in the cell of H, so the estimate is 21%.

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

   • 21 − 19 = 2