Fitting temperature data¶
Here are the 5-year temperature averages again.
using FundamentalsNumericalComputation
year = 1955:5:2000
t = year .- 1955;
y = [ -0.0480, -0.0180, -0.0360, -0.0120, -0.0040,
0.1180, 0.2100, 0.3320, 0.3340, 0.4560 ];
The standard best-fit line results from using a linear polynomial that meets the least squares criterion.
V = [ t.^0 t ] # Vandermonde-ish matrix
@show size(V)
c = V\y
size(V) = (10, 2)
2-element Array{Float64,1}:
-0.12938181818181826
0.011670303030303033
p = Polynomial(c)
-0.12938181818181826 + 0.011670303030303033∙x
f = s -> p(s-1955)
scatter(year,y,label="data",
xlabel="year",ylabel="anomaly (degrees C)",leg=:bottomright)
plot!(f,1955,2000,label="linear fit")
If we use a global cubic polynomial, the points are fit more closely.
V = [ t[i]^j for i=1:length(t), j=0:3 ] # Vandermonde-ish matrix
@show size(V)
size(V) = (10, 4)
(10, 4)
p = Polynomial( V\y )
f = s -> p(s-1955)
plot!(f,1955,2000,label="cubic fit")
If we were to continue increasing the degree of the polynomial, the residual at the data points would get smaller, but overfitting would increase.