4. Eye-tracking: Continuous-Time Random Walks

Transcription

1 Applied stochastic processes: practical cases 1. Radiactive decay: The Poisson process 2. Chemical kinetics: Stochastic simulation 3. Econophysics: The Random-Walk Hypothesis 4. Eye-tracking: Continuous-Time Random Walks 5. Search games: First-Passage processes

2 1. RADIACTIVE DECAY THE POISSON PROCESS

3 Radiactive decay: the Poisson process

4 Radiactive decay: the Poisson process

5 Radiactive decay: the Poisson process P N(t) λ 1 φ T n λ 2 ~e λ T n λ 3 λ 4 T n N(t)

6 Radiactive decay: the Poisson process Essential properties of the Poisson process:

7 Radiactive decay: the Poisson process

8 Radiactive decay: the Poisson process Questions and exercises 1. Give an interpretation to the law of radiactive decay in terms of the Poisson process. 2. How would you check if a series of desintegration counts from a radiative sample follows a Poisson distribution? 3. Obtain an experimental data series using the Geiger counter and apply the method you have described in the previous question. 4. What would happen if we repeated the experiment using two different radiactive samples with rates λ 1 and λ 2? What kind of interevent distribution would we obtain? 5. Describe a simple algorithm (based on a Bernoulli process) you could use to generate a data series following a Poisson process with variance σt in the cumulative number of counts. 6. We measure the C14 radioactivity emitted by a 2000 years old fossil, and we find it is five counts/second. Determine what is the expected number of C14 atoms in the fossil, and what is the probability that the actual number of atoms is twice the value you have obtained? (Half-life of C14: 5568 years)

9 2. CHEMICAL KINETICS STOCHASTIC SIMULATION

10 Chemical kinetics: stochastic simulation Mean-field description: Case A k B ρ B t + τ = ρ B t + ρ reac (t, τ) ρ B t + τ = ρ B t + P reac (τ) ρ A (t) ρ B t + τ = ρ B t + k τ ρ A (t) dρ B dt = kρ A ρ A = ρ A 0 e kt ρ B = ρ A 0 dρ A dt = kρ A 1 e kt Limitations: - Constant k - Well-stirred conditions - Large particle number - What about

11 Chemical kinetics: stochastic simulation Gillespie method

12 Chemical kinetics: stochastic simulation First reaction implementation

13 Chemical kinetics: stochastic simulation Case A k B spatially explicit ρ B x, t + τ = ρ B x, t + ρ reac x, t, τ + ρ trans x, t, τ ρ B x, t + τ = ρ B x, t + P reac τ ρ A t + Dτ 2 ρ B x, t ρ B t + τ = ρ B t + k τ ρ A t + Dτ 2 ρ B x, t dρ B (x, t) = kρ dt A x, t + D 2 ρ B x, t dρ A (x, t) = kρ dt A (x, t) + D 2 ρ A x, t Now the limitations - Constant k - Well-stirred conditions - Large particle number must hold at a local level, plus the limitation of independent kinetics and transport

14 Chemical kinetics: stochastic simulation Questions and exercises 1. Take the code from the file ex1.r. Adapt it to study the simpler case A k B. Check that you obtain the results expected for ρ A and ρ B. (For simplicity use k = 0.5) 2. How would you find the propensity for the A + A k B case? And for the A + B k C case? 3. According to the answer from the previous question, find numerically the results ρ A, ρ B, ρ C for the system A + A k 1 B ; A + B k 2 C. (For simplicity use k 1 = 0.5, k 2 = 0.2) 4. Write the mean field equation for the set of reactions 2A k 1 A ; A k 2 same for the set A k 3 2A ; 2A k 4 the two results? 2A. Now do the. What is the conclusion you can obtain by comparing 5. The Sparre-Andersen theorem states that For any isotropic and memoryless randomwalk process in 1D, the probability to come back to the initial state for the first time after a time t decays as t 3/2. How could we use our code to validate this theorem? Implement your idea to check that it works.

15 3. ECONOPHYSICS THE RANDOM-WALK HYPOTHESIS

16 Econophysics: the Random-Walk hypothesis The mathematical equation that caused the banks to crash The Black-Scholes equation was the mathematical justification for the trading that plunged the world's banks into catastrophe (

17 Econophysics: the Random-Walk hypothesis Derivatives: Futures and options Futures are trading contracts which will be executed after a time T at a price K. Options are trading contracts which can be executed by the buyer of the option after time T at a price K.

18 Econophysics: the Random-Walk hypothesis Efficient Market (no-arbitrage) Hypothesis: Black-Scholes equation Given a stock price Y and an option price C related to that stock, we have σ 2 : volatility Fundamental hypothesis behind: i) Product prices follow a geometric Brownian motion process ii) Continuity in market prices dynamics iii) Constant return rates r both for riskless and risky inversions Solution of the equation: C Y, t = Y erf ln Y K σ2 + (r + )(T t) 2 σ T t Y 0 e r(t t) erf ln Y K + r + σ2 2 σ T t T t 1

19 Econophysics: the Random-Walk hypothesis Stochastic approach to economic data series:

20 Econophysics: the Random-Walk hypothesis Stable distributions: Those which preserve shape under a linear combination of variables (e.g. Gaussian, Lorentz). Central limit theorems: Speed of convergence: ~n 1/2

21 Econophysics: the Random-Walk hypothesis Stochastic models of price dynamics: pdf s Truncated Lévy flights vs Gaussian (Xerox, ) (S&P 500, )

22 Econophysics: the Random-Walk hypothesis Variables of interest in economic markets data series Given a stock Price Y t, Price changes: Z t = Y t + t Y t Deflated Price changes: Z D t = D t Y(t + t) Y(t) Returns: R t = Y t+ t Y t Y(t) = Z(t) Y(t) Log-changes: S t = ln Y(t + t) ln Y t = ln Y(t+ t) Y(t) Conventions in the use of economic data series:

23 Econophysics: the Random-Walk hypothesis Correlations in the data series: The Black and Scholes model describe the dynamics of stock prices as governed by a geometric Brownian motion process : dy = μydt + σydw (efficient market hypothesis). How to measure real correlations: 1 F τ = lim TY T T න t Y t + τ dt S ω = 1 0 2π න F(τ)e iωτ dτ Short-range: F τ ~e τ S(ω)~ 1 ω 2 Long-range: S ω ~ ω μ 1 F τ ~ τ μ, 0 < μ < 1

24 Econophysics: the Random-Walk hypothesis Time correlation in financial time series Short times Long times (Coca Cola, ) (S&P 500, ) (S&P 500) (S&P 500, )

25 Econophysics: the Random-Walk hypothesis The phase transition perspective Classical model : Buy : Sell Model with social hierarchies: Log(S&P) t

26 Econophysics: the Random-Walk hypothesis Questions and exercises 1. Take the data series minute.txt and daily.txt (corresponding to minute and daily records of the Spanish trading index IBEX35) and make a first guess of the probability distribution function that they follow. (To do this make a frequency plot of the data in a semi-logarithmic scale and in a logorithmic scale) 2. Plot the autocorrelation function F(τ) obtained from each data series and check also the type of decay found. What conclusions can you obtain from these plots? 3. Make a guess about how the spectral density would look like in case we computed it from each data series. 4. Repeat the procedure in exercises 1 and 2 but now using the volatilities of the data series. 5. In the case of the daily data, what kind of (economic) correction do you think we need to provide a proper measure of the volatility for long time series? Introduce this correction and check how the results in exercise 4 get modified. You can visit and you will find lots of free available data for additional studies and comparisons.

27 4. EYE-TRACKING CONTINUOUS-TIME RANDOM-WALKS

28 Eye-tracking: Continuous-Time Random Walks The dynamics of human eye-movements Movement type Time lapse (ms) Frequency (Hz) Mean size (deg) Saccades Any Microsaccades Tremor Drift

29 Eye-tracking: Continuous-Time Random Walks Troxler s effect

30 Eye-tracking: Continuous-Time Random Walks Eye-tracking: real-time monitoring of the gaze point and/or the eye movement relative to the head.

31 Eye-tracking: Continuous-Time Random Walks The Continuous-Time Random Walk can overcome some of the limitations of Brownian motion (e.g. infinite signal speed, uncorrelated movements)?

32 Eye-tracking: Continuous-Time Random Walks

33 Eye-tracking: Continuous-Time Random Walks We can also solve the problem of instantaneous jumps by introducing the velocity version of the Continuous-time Random Walk through: Φ x = න 0 dt φ(t) න dt δ x vt h(v) where h(v) is the velocity PDF. So, φ(t) and Φ x are no longer independent PDFs but define a new coupled PDF Ψ(x, t) of times and lengths, and the Montroll-Weiss equation turns into

34 Eye-tracking: Continuous-Time Random Walks Questions and exercises 1. Take the data series obtained from the eye-tracking experiment and obtain the PDF and the autocorrelation of jump sizes. 2. Repeat the procedure of the exercise 1 now for the jump directions. 3. Can you infer from the results above the existence of the different eye movements we have described before? 3. Give a simple fit of the PDF of jump sizes. From this, and using for the waiting time distribution a Dirac delta function for the waiting times (with the time between consecutive points), write the corresponding Montroll-Weiss equation. 4. Obtain the MSD(t) from the previous expression. 5. Use the program olsson.exe to transform your (x, y) data into saccade-fixation cycles. (See 6. Repeat exercises 1 and 2 now using as your input data the fixation positions. 7. Fit the fixation durations and the speed of saccades to a given PDF. Write the corresponding Montroll-Weiss equation and obtain the corresponding MSD(t)

35 5. SEARCH GAMES FIRST-PASSAGE PROCESSES

36 Search games: first-passage processes Target problems and its potential interest (reaction kinetics analogy: A + B A) Encounter rate problem Search problem

37 Search games: first-passage processes THE IDEAL GAS RANDOM ENCOUNTER HYPOTHESIS B B B vt B A R B B Due to the stationary assumption, k t = k 1D: k = vρ B 2D: k = 2Rvρ B 3D: k = πr 2 vρ B

38 Search games: first-passage processes

39 Search games: first-passage processes

40 Search games: first-passage processes

41 Search games: first-passage processes Velocity models

42 Search games: first-passage processes Questions and exercises 1. Using the snake game, ty to determine the (initial) encounter rate of the random snake for the easy level (where 100 apples are available). Determine the mean time that takes to capture 5 apples by assuming a stationary situation and that the time T i to detect the i-th apple is independent of T i Determine the mean time that takes to capture 5 apples by using a deterministic strategy (such that we never pass twice for the same position). 3. Play at least ten games at the easy level and compare the performances to the theoretical results obtained before. 4. For the case of the hard level compute the mean time to capture one single apple (out of 5 possible) for the random snake. 5. Repeat the previous question but now for the case of a deterministic snake. 6. Play at least ten games at the hard level and compare the performances to the theoretical results obtained before.