Map Graph
How this is done: open worksheet US map.mtw. Copy-paste columns Longitude and Latitude from US Temperature to US map. Draw scatterplot, overlaid in the same graph
This graph is not very nice because the state borders are just dots, and sometimes there are some spaces between them. It would be much nicer if they were smooth. This turns out to be a bit tricky:
Calc > Calculator, Store in: Latitude Expression: Latitude-1
Make a new column with 56 0's, then copy-paste the three columns on top of the big Longitude-Latitude column in US Map.mtw
Graph > Scatterplot, with connect lines and groups > Y variables= Latitude,X variables=Longitude, group variable Jumps
click on graph, select symbols and choose None
click on graph, select connect lines and choose solid black lines, Options > Connection Order = Worksheet
click on graph, select connect lines for cities only and choose none
Make column with Calc > make patterend data> Text values > store in c7, text values: x, repeat 56 times
right-click, Add > Data Labels, from column c7
Here is the graph:
Finding a good model
Fitting proceeds the same as in the simple regression situation, only now there are more possibilities.
Question: which predictor (Latitude or Longitude or both) is the problem? According to the scatterplots it is most likely Longitude, so we will work with that.
Step 2: Transformations
none of these will work because the relationship between JanTemp and Longitude is not monotone
Step 3: Polynomial models
| Model |
p-value of highest order term |
| Quadratic |
0.000 |
| Cubic |
0.000 |
| Power 4 |
0.593 |
so the cubic model in Longitude is the best polynomial model
Step 4: cubic model in Longitude is the best model
Fit - Interpretion and Understanding
So we find a good model to be
The regression equation is
JanTemp = - 791 - 2.36 Latitude - 31.2 Longitude - 0.352 Long^2 - 0.00130 Long^3
but what does this model tell us?
Let's try and understand it as follows: say you fly to New York in January, rent a car and then drive straight west, never going north or south. How will the temperature change as you go?
First New York is at 40.8° North, 74.6° West, so if we go straight west we always keep Latitude at 40.8 and the Longitude will go from -74.6 to -125 (or so), at which point you fall into the Pacific. Let's use our model to predict the temperature during the trip:
1) Make column "NewLong" values -125 to -72 (use make patterned data)
2) Make column "NewLat", same length as NewLong, all 40.8
3) calculate column "NewTemp" with
NewTemp = - 791 - 2.36 * Latitude - 31.2 * 'NewLong' - 0.352 * 'NewLong'**2 - 0.00130 * 'NewLong'**3
Now plot NewTemp vs. NewLong
here is what this look like:
Here is another, more sophisticated way to "visualize" this fit:
1) Generate a fine grid of Latitude and Longitude values covering all of the US. Note that Latitude goes from 26 to 50 and Longitude from -125 to -66. So
Calc > Make Patterned Data > Simple set of numbers, Store Patterned data in: NewLat, From first value= 26, To last value= 50, in steps of=1, List each value 60 time, List whole whole sequence: 1
Calc > Make Patterned Data > Simple set of numbers, Store Patterned data in: NewLong, From first value= -125, To last value= -66, in steps of=1, List each value 1 time, List whole whole sequence: 25
2) For each pair of Latitude and Longitude use the model to predict the temperature at the corresponding location:
Calc > Calculator, Store in: NewTemp, Expression: NewTemp = - 791 - 2.36 * NewLat - 31.2 * 'NewLong' - 0.352 * 'NewLong'**2 - 0.00130 * 'NewLong'**3
3) Color-code the temperature as follows: Red=hot to Blue=Cold
Note that the NewTemp values go from -10 to 87. We split that up into 5 intervals: -10-10, 10-30, 30-50, 50-70, 70-90
all the rows with Newtemp>10 get a 1: Calc > Calculator, Store in: c9 Expression: Newtemp>10
all the rows with Newtemp>30 get a 1: Calc > Calculator, Store in: c10 Expression: Newtemp>30
all the rows with Newtemp>50 get a 1: Calc > Calculator, Store in: c11 Expression: Newtemp>50
all the rows with Newtemp>70 get a 1: Calc > Calculator, Store in: c12 Expression: Newtemp>70
Now add them up:
Calc > Calculator, Store in: TempCode, Expression: c9+c10+c11+c12+1
Graph > Scatterplot, with groups > Y variables= NewLat, X variables=NewLong, Cat variable: TempCode
Often the color blue is used for cold and red for hot. So change the colors to reflect the temperature
Fix it up a bit, for example add a legend
Here is what that looks like:
