2.1 The Linear Map

Equation (2.1) is the ordinary difference equation (ODE) discussed in the lecture (see lecture notes ??) is called the Linear Map:

$$$Y_{t+1} = Y_{t=0} + r*Y_t \tag{2.1}$$$

In these excersises you will simulate time series produced by the change process in equation @(eq:linmap) for different parameter settings of the growth-rate parameter $$r$$ (the control parameter) and the initial conditions $$Y_0$$. This is different from a statistical analysis in which parameters are estimated from a data set. The goal of the assignments is to get a feeling for what a dynamical model is, and how it is different from a linear statistical regression model like GLM.

2.1.1 The Linear Map in a Spreadsheet

Before you begin, be sure to check the following settings:

• Open a Microsoft Excel worksheet, a Google sheet, or other spreadsheet.
• Check whether the spreadsheet uses a ‘decimal-comma’ ($$0,05$$) or ‘decimal-point’ ($$0.05$$) notation.
• The numbers given in the assignments of this course all use a ‘decimal-point’ notation.
• Check if the $ symbol fixes rows and columns when it used in a formula in your preferred spreadsheet program. • This is the default setting in Microsoft Excel and Google. If you use one of those programs you are all set, otherwise you will have to replace the $ used in the assignments with the one used by your software.
• Type r in cell A5. This is the control parameter. It receives the value $$1.08$$ in cellB5.
• Type $$Y_0$$ in cell A6. This is the initial value. It receives the value $$0.1$$ in cell B6.
• Use the output level ($$Y_t$$) of every step as the input to calculate the next level ($$Y_{t+1}$$).
• Rows in the spreadsheet will represent the values of the process at different moments in time.
• Put the initial value ($$Y_0$$) in cell A10. This cell marks time $$t=0$$.
• To get it right, type: =$B$6. The = means that (in principle) there is a ‘calculation’ going on (a function is applied). The $ determines that column ($B) as well as the row ($6) keep the same value (i.e., constant) for each time step. • Enter the Linear Map as a function in each cell. Type =$B$5*A10 in cell A11. • This means that the value of cell A11 (i.e. $$Y_{t=1}$$) will be calculated by multiplying the value of cell B5 (parameter r) with the value of cell A10 (previous value, here: $$Y_{t=0}$$). If everything is all right, cell A11 now shows the value $$0.108$$. • Repeat this step for cell A12. • Remember what it is you are doing! You are calculating $$Y_{t=2}$$ now (i.e. the next step), which is determined by $$Y_{t=1}$$ (i.e., the previous step) and the parameter r. • Now repeat this simple iterative step for 100 further steps. Instead of typing everything over and over again, copy-paste the whole thing.Most spreadsheet programs will automatically adjust the formula by advancing each row or column number that aren’t fixed by $.
• Copy cell A12 all the way from A13 to A110 (keep the SHIFT button pressed to select all cells).
• You have just simulated a time series based on a theoretical change process!

2.1.2 Visualizing the time series

1. Select cells A10 to A110 Create a line graph (Insert, 2D-line, Scatter). This will show you the graph. (There are other ways to do this, by the way, which work just as well.) You can play with the setting to make the best suitable view, like rescaling the axes.

2. If you change the values in cells B5 and B6 you will see an immediate change in the graph. To study model behaviour, try the following growth parameters:
• $$a = -1.08$$
• $$a = 1,08$$
• $$a = 1$$
• $$a = -1$$
3. Change the initial value $$Y_0$$ in cell B6 to $$10$$. Subsequently give the growth parameter in cell B5 the following values $$0.95$$ and $$-0.95$$.