Studio 6: Continuous Data, Continuous Priors Spring 2014
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1 Studio 6: Continuous Data, Continuous Priors Spring 2014 You should have downloaded studio6.zip and unzipped it into your working directory.
2 January 2, / 15 NASDAQ Data We have data from the NASDAQ stock exchange on trades in a certain stock on 4 days in March Here are the first 4 lines of the tradesdata0.csv Date timenumber timehhmmss Size Price The data file, tradesdata0.csv is in studio6.zip We processed this data to produce the data file for this class: studio5dataframe.csv (If you re interested, the processing code is in studio6-prep.r) Today s project: Model the rate at which trades come into the exchange.
3 January 2, / 15 Exporatory data analysis Real data analysis starts by exploring the data. Some things to try are: Plot lists of data. This can help find glaring errors in the data: on the wrong scale missing all 0 multiple modes Histograms Time plots Slice and dice data to find (suggested) patterns (See studio6-prep.r and studio6.r)
4 January 2, / 15 Exploration: number of trades vs. time of day Trade counts in 5 minute periods (all days combined) Number of trades Time in hours More trades at beginning and end of day than in the middle. Note: 9.5 = 9:30 am, 16.0 = 4:00 pm
5 January 2, / 15 Exploration: number of trades vs. time of day II More trades at beginning and end of each day Could the waiting time be exponentially distributed with a parameter that changes during the day?
6 January 2, / 15 Exploration: a single time slot Code is in studio6.r, which also generates many more plots. Times between trades: , t = 9.5 x Time in seconds Times between trades: , t = 9.5 Frequency Time in seconds Plot of data doesn t set off alarms Histogram resembles that of an exponential distribution
7 January 2, / 15 Board question: Bayesian updating Fix the date as March 4, 2014 ( ). For each 5 minute time slot we ll assume the wait time between trades follow an exponential(1/θ) distribution. (θ is then the mean wait time.) studio6.r shows how to get the list of wait times for any 5 minute time slot. 1. Outline the mathematics needed to do Bayesian updating starting from a uniform prior on θ in the range [0, 8]. 2. Outline a plan to write code in R to do the updating for each time slot in turn.
8 January 2, / 15 Code outline for problem 2 Updating a single day/time slot (Do this for each time slot on March 4.) (i) Get the list of waiting times for that day/time slot. (ii) Discretize θ in [0,8]: thetarange = seq(0,8,dtheta), where dtheta = 0.02 (iii) For the data point x the likelihood array is likelihood = exp( x/thetarange)/thetarange (iv) For each data point x j do numerical Bayesian updating by: prior = posterior # Previous posterior becomes new prior. unnormposterior = prior*likelihood posterior = unnormposterior/(dtheta*sum(unnormposterior))
9 Code outline continued Note: We could also compute the likelihood of all the data and update all at once. Details on normalizing priors and posteriors Since priors and posteriors are functions of θ: Numerically they are lists of length length(thetarange). They are normalized so that the numerical intergral sum(f(thetarange) dtheta) = 1 For example the pdf f (θ) = cθ 2 is given numerically by f = thetarange^2/sum(thetarange^2*dtheta). The uniform prior is given numerically by uniformprior = rep(1, n)/(n dtheta), where n = length(thetarange) January 2, / 15
10 January 2, / 15 R: Bayesian updating 3(a) Implement your coding plan. Make sure that the final posterior for each timeslot is saved for later use. 3(b) For each posterior find the MAP estimate (value of θ that maximizes the posterior) and make a plot of MAP vs. time slot. (Hint: get help on the R function which.max.) 3(c) Redo (a) and (b) with the quadratic prior c(4 θ) 2 on [0, 8].
11 January 2, / 15 One time slot pdf Plot of all posteriors (and prior) March 4, 2014 at hours theta θ is the paramater of the exponential(1/θ) distribution for waiting time between trades. It is the mean waiting time between trades.
12 January 2, / 15 MAP Estimates for θ for all time slots (uniform prior) March 4, 2014: MAP Estimates for theta MAP time (hours from midnight)
13 January 2, / 15 MAP Estimates for θ for all time slots (quadratic prior) March 4, 2014: MAP Estimates for theta MAP time (hours from midnight)
14 Price vs trade number (a bonus picture) The trades are listed in chronological order. The horizontal axis is the trade number. January 2, / 15
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