In FredHasselman/nlRtsa: nonlineaR time seRies analysis.

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Assignments: Iterating 1D Maps

In this assignment you will build two (relatively) simple one-dimensional maps in Excel and look at their behaviour and properties. We start with the Linear Map and then proceed to the slightly more complicated Logistic Map (aka Quadratic map). If your are experienced in R or Matlab you can try to code the models following the instructions at the end of this document.

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

• Open a Microsoft Excel worksheet or a Google sheet
• 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 Sheets. If you use one of those programs you are all set.

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The Linear Map

The ordinary difference equation discussed in the lecture (see lecture notes) is called the Linear Map:

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

In these excersises you will simulate time series produced by this change process 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.

Simulate the Linear Map in a Spreadsheet

• 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!

Visualizing the time series

. 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.

. If you change the values in cells B5 and B6 you will see an immediate change in the graph. To study the model’s behaviour, try the following growth parameters:

+ $a = -1.08$
+ $a = 1,08$
+ $a = 1$
+ $a = -1$

. 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 $–0.95$.

The Logistic Map

The Logistic Map has the following form:

$$ Y_{t+1} = rY_t(1-Y_t) $$

Simulate the Logistic Map in a Spreadsheet

To get started, copy the spreadsheet from the previous assignment to a new sheet. The parameters are the same as for the Linear Map, there has to be an initial value $Y_{t=0}$ (no longer explicit as a constant in the equation) and the control parameter $r$. What will have to change is

• Start with the following values for control parameter $r$:
  • $r = 1.9$
  • $Y_0 = 0.01$ (in A6).
• Take good notice of what is constant (parameter $r$), so for which the $ must be used, and what must change on every iterative step (variable $Y_t$).

Visualize the time series and explore its behavioour

• Create the time series graphs as for the Linear Map.

To study the behavior of the Logistic Map you can start playing around with the values for the parameters and the initial values in cells B5 and B6.

• Be sure to try the following settings for $r$:

  • $r = 0.9$
  • $r = 1.9$
  • $r = 2.9$
  • $r = 3.3$
  • $r = 3.52$
  • $r = 3.9$

• Set $r$ at $4.0$:

  • Repeat the iterative process from A10 to A310 (300 steps)
  • Now copy A10:A310 to B9:B309 (i.e., move it one cell to the right, and one cell up)
  • Select both columns (A10 to B309!) and make a scatter-plot

The plot you just produced is a so called return plot, in which you have plotted $Y_{t+1}$ against $Y_t$.

• Can you explain the pattern you see (a 'parabola') by looking closely at the functional form of the Logistic Map? (hint: it's also called Quadratic Map)
  • Look at what happens in the return plot when you change the value of the parameter $r$ (in A5).
  • What do you expect the return plot of the Linear Map will look like? Try it!

The meaning and use of this plot was discussed in the next session

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Using R or Matlab to do the exercises.

The best (and easiest) way to simulate these simple models is to create a function which takes as input the parameters ($Y_0$, $r$) and a variable indicating the length of the time series.

For example for the Linear Map:

# In R
linearMap <- function(Y0 = 0, r = 1, N = 100){

    Y <- c(Y0, rep(NA,N-1))

    for(t in 1:N){

    Y[i+1] <- # Implement the function here

    }

    return(Y)
}


# In Matlab
function linearMap(Y0,r,N)
 # Implement the function here
end

Creating the time series graphs and the return plot should be easy if the function linearMap returns the time series. Both R and Matlab have a plot() function you can call.[^tseries]

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[^tseries]: Both R and Matlab have specialized objects to represent timeseries, and functions and packages for timeseries analysis. They are especially convenient for plotting time and date information on the X-axis. See Mathematics of Change 1 - SOLUTIONS

FredHasselman/nlRtsa documentation built on May 6, 2019, 5:07 p.m.