5. Key ingredients for programming a random walk
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1 1. Random Walk = Brownian Motion = Markov Chain (with caveat that each of these is a class of different models, not all of which are exactly equal) 2. A random walk is a specific kind of time series process in which the next step is random with respect to previous steps. A time series process is any process in which a series of events occur starting with the state after the last event. Evolution, both within a lineage and on a phylogeny, is a type of time series, though not always a random walk. Random walks are often used as null models for the evolutionary process because they specify what happens in a time series if all the events are completely random. 3. Outcomes of random walks are predictable, in a statistical sense, if the number of steps and the average amount of change at each step is known. The statistics of random walks thus form an important part of quantitative analysis of evolutionary patterns within lineages and on phylogenetic trees. 4. Statistical outcomes of univariate, unbiased random walks: 4.1. The most likely end point is the starting point 4.2. End points are normally distributed with the mean equal to the starting point 4.3. The variance of the end points equals the variance of the per-step change times the number of steps 4.4. The standard deviation of the end points is equal to the standard deviaton of the per-step change times the square-root of the number of steps 5. Key ingredients for programming a random walk 5.1. Choose a starting point and record it 5.2. Draw a random number from a distribution whose average is 0 and where there is equal probability of getting a positive or negative outcome 5.3. Add the random value to the previous value and repeat 1. Do loop: Do loops are standard programming structures that do something repeatedly until told to stop. Mathematica can handle these loops. Key ingredients are: 1.1. setting up variables to record the outcomes of each step of the Do loop 1.2. setting up the structure to repeat commands 1.3. saving the outcome at each step
2 2 Lecture 3 - Trait evolution, random walks.nb 1.4. setting up a control to end the loop (often this simply involves specifying the number of times to repeat as an iterator at the end of the Do function) Here is a short do loop that generates a random walk starting at zero. The repeated commands appear before the comma within the Do[] function. The iterator is {x, 00}, which repeats the loop 10,000 times. First, a step value of -1 or 1 is chosen randomly with the RandomChoice[] function. Then that value is added to the previous value stored in the randomwalk variable(randomwalk[[-1]]) and appends it to the values already stored there. = {- }] = [- ]] + ] ] - Note that you can trigger the Do loop to end early when conditions that you specify are reached. For example, you can end early if the value reaches or higher, or you could end early if a random extinction event happens. Use the If[] function to do this. If has two or three parameters, the first of which is a logical statement that specifies the condition, the second is the command to execute if the statement is true (the third is optional, an alternative command to execute if the statement is false). Break[] is a function to end a loop. Note that I have added options to the ListPlot so that all 00 steps are shown even if the lineage becomes extinct early so that that the effect of the If[] function is more obvious. = {- }] = [- ]] + ] [- ]] ]] {{ } {- }}] - -
3 Lecture 3 - Trait evolution, random walks.nb 3 In the following example we simulate a random walk that has a 1% chance of becoming extinct at each step. In the If[] function, a random number between 0 and 1 is chosen and the walk is terminated if that number is greater than.99 (i.e., there is a 99% chance of not becoming extinct, and a 1% chance of ending. Why doesn t the walk ever get very far? How far, on average, does it get? How can you change the code so that % of the walks end before finishing, but the other % make it all the way through 10,000 steps? = {} = {- }] = [- ]] + ] { }] ]] = [- ]]] { }] ] Table: Table[] is a specialized Mathematica function that has many of the properties of a Do loop, but which saves each value automatically. Like Do[], the funciton ends with an iterator preceeded by a comma. You can place several commands in front of the comma by separating them with semicolons. Note that the result of commands ending with semicolons are not returned in the table, only the result of the last command before the comma. You still have to keep track of the current value of the random walk at each step. Here, this is done using a temporary variable val to hold the most recent value, while at the same time inserting it into the table.
4 4 Lecture 3 - Trait evolution, random walks.nb = = = + {- }] ] Accumulate is a function that adds up a bunch of numbers generated by a command and returns each step of the addition. This function can be used to generate a simple random walk like the first ones above. = {- } ]] ] 1 4. NestList is a generic version of Accumulate where you can perform any function on a list, including addition. If you use addition, then the result is exactly the same as Accumulate. # serves as a place holder variable for each step. Here, the random choice of -1 or 1 is added to # at each step, with the whole thing starting at 0 and repeating 10,000 times.
5 Lecture 3 - Trait evolution, random walks.nb 5 = # + {- }] ] ] The random walk code above produces binary random walks, one in which each step is +1 or -1. What is the average change each step? What is the expected variance for a given number of steps. 1. Selection coefficient drawn from normal distribution 1.1. NormalDistrubtion[] function that takes mean and standard deviation as input parameters. RandomReal can be used to select value from that distribution.
6 6 Lecture 3 - Trait evolution, random walks.nb = ]] = [- ]] + ] ] McKinney Equation 2. Different method altogether for generating a random walk. If r=1, then the formula produces a random walk. What happens if r<1 or r>1? Note that what happens if r is much bigger than 1, even 2.0. = = ]] = * [- ]] + ] { }] ] Serial correlation is one way to measure whether a sequence of data belong to a time series. If one value is dependent on the value before it, then serial correlation is high (near 1). Time series like random walks have strong serial correlations. Show this with Monte Carlo following instructions in McKinney paper.
7 Lecture 3 - Trait evolution, random walks.nb 7 1. First generate a random walk. = {{ }} = ]] = { [- ]] + }] ] [ ]] {{ } { - } { - } { - } { - } { - } { } { } { } { }} 2. Then find a linear regression line through the random walk, which describes its average trend (a random trend, in this case). Plot the regresison line to make sure it is right. = { } ] ] ] { }]] regress the residuals around the fit to the residuals in the previous step. To do this, offset the residuals by choosing the 1st step through the next to the last step. Regress the second through last residual onto them. If the regression slope is near 1.0 then the time series has a near perfect serial correlation.
8 8 Lecture 3 - Trait evolution, random walks.nb { ][[ - ]] ][[ ]]}] { } ] - + = {- } ]] = {- } ]] { } ] 1 ] // = [ + ]] - [ ]] { ] - }] = [ + ]] - [ ]] { ] - }] ] // -
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