Exercises

Chapter 1: Linear Empirical Regression Models

Access the data file and enter it into a spreadsheet, giving you four columns of data, the RVI and GLAI for both Winter Wheat and Spring Barley. Plot the data for both crop types on the same plot, but with each pair of data sets in a different colour.

Derive the mean and variance for each column of data and compute the correlation between the two parameters for both the Winter Wheat and the Spring Barley.

Get the dataset referred to in the previous lesson and open it in a spreadsheet, or take the Spreadsheet created in the previous lesson. Derive the linear regression by first creating the SUMPRODUCT for the X² and the XY terms and then summing for the X and the Y terms. Make sure that you designate the correct columns for X and Y. Then compute the two regression parameters using the appropriate equations in the notes above.

Use the LINEST function to do the same task. You can find the two parameters values in the LINEST result, but the result contains many other things that we will meet in later lessons.

On the graph, highlight data points in one dataset. A popup menu should appear; in this menu choose Add Trendline, choose linear and make sure that the equation and the R² values will appear on the plot.

Use your derived regression equations to compute the residual values at each observation in both data sets. Create a plot of the residuals and compute the mean and variance for the residuals. What do you notice about the mean and variance?

Derive the quadratic polynomial function best fit to the data, remembering that the quadratic function is of the form b₂X²+b₁X+b₀=Y+ε so that three unknowns have to be found.

Repeat this process, but highlight a dataset in the graph and select Add Trendline in the local popup menu that appears when you click on a point in the dataset. Select Polynomial function of the second order and get it to display the equation and the R-squared value on the chart.

In your spreadsheet, find the Coefficient of Determination for the Winter Wheat and Spring Barley and the F-statistic. Use the F statistic tables to check the results given here in the text, at both 97.5% and 99% confidence levels.

Use the Map Calculator to derive RVI values and compare them with the bindslev_rvi image.

Strictly, neither RVI nor GLAI can be negative, yet the regression equation for Spring Barley could give you negative GLAI values when RVI is very low. Modify the expression so that negative values are set to zero. (Clue - Replace the Spring Barley regression equation with a third embedded IFF(a, b, c) command in which a and b or a and c include this regression equation, depending on the test, a).

There are other Spring Barley and Winter Wheat fields in this image. Use ILWIS to classify all of the fields that are similar to Winter Wheat and Spring Barley, and then repeat the estimation of GLAI.

What is the population if you want to select a site for a shop that is to sell extra large women's clothes in your country?

Having selected a site, what is the population to assess the demand for clothes from the shop?

What is the gravitational force between you and the Earth? You do not need to calculate it, or rather if you did, it would come out exactly the same as your weight. Your weight is a measure of this force.

What is your weight on the Moon? First answer this question by using the Law of Gravitation by making it into a ratio of the force of gravity on the Earth to the Force of Gravity on the Moon. Simplify this ratio to get the equation
$\frac{F_{M o o n}}{F_{E a r t h}} = \frac{M a s s_{M o o n} \cdot R_{E a r t h}^{2}}{M a s s_{E a r t h} \cdot R_{M o o n}^{2}}$
Use the mass of the Moon 0.73483·10²³ kg and its radius of 1738 km, and the mass of the Earth 59.74·10²³ kg and its radius of 6371 km to get this ratio. Now, knowing the weight on the Earth allows you to calculate the weight of the same person on the Moon. Now, check your answer by going to this address.