7.
Modelling temperature
The table
shows the average monthly (x
= 1, 2, … is Jan, Feb, …) temperature (y
°C) in Cape Town. What function models the temperature?
| A
polynomial model
Try to model the data with a polynomial function.
Use Excel’s
Trendline tool and try models
of degree 2, 3, 4, …
Are any of these
a good fit for the data?
You can use
this Excel worksheet: 
A trigonometric
model
Now let’s
try to model the Cape Town temperature with a trigonometric function
like y
= acos[b(x
– c)] + d.
Unfortunately
Excel does not have a trigonometric Trendline. So let’s find
the best-fit formula ourselves: Use the Excel tool below to find
the values of the parameters a, b, c and d numerically
with the method of least square errors.
|
|
Month
(x)
|
Temp
(y
°C) |
1 |
21 |
2 |
21,2 |
3 |
20 |
4 |
17,4 |
5 |
15,1 |
6 |
13,3 |
7 |
12,5 |
8 |
13 |
9 |
14,3 |
10 |
16,2 |
11 |
18,2 |
12 |
19,9 |
|
Write
down the domain and range, amplitude, period and phase for the trigonometric
model. What physical meanings can you give to these quantities in the
context of Cape Town temperature? How can these meanings help to calculate
the parameters algebraically? Calculate the values of the parameters
algebraically!
Click here for a brief
discussion: 
Comparing models
How do the polynomial
and trigonometric models for the Cape Town temperature compare? Which
one is the best fit?
Click here for a discussion: 
Comparing
different cities
Now open the Excel
file below to also find regression formulae for temperatures in other
cities, e.g. Johannesburg, Quebec and Cairo. How can our geographic knowledge
of the climate of these cities help us to give meaning to the parameters
and to estimate the values of the parameters?
 
You can investigate
temperature patterns of other cities at:
World Climate:
http://www.worldclimate.com/
This is an external link.
|
