Exercise 8

In this exercise we want to predict daily mean wind speeds in Potsdam. Our model is a simple AR(1) model \[ x_t = a x_{t-1} + v\xi_t, \] where the Gaussian noise xi has unit variance and zero mean. So the parameters to be optimized are 'a' and 'v'.

The data can be found at https://www.ecad.eu/dailydata/customquery.php. Select 'blended', 'Germany', 'Potsdam', and 'wind speed' and download the txt-file. Import the time series in python and plot it. The complete time series has a lot of missing data. We can restrict ourselves to a time window of 10000 points, by selecting `data=data[20000:30000]`

. Then we can look for the best parameter 'a'.

**Fitting the TAMSD:**calculate the time averaged mean squared displacement (TAMSD) of the data. The AR(1) model can be fitted using the following functions

Fit the TAMSD for DELTA smaller than 50, by calling the function as`from scipy.optimize import minimize def tamsdAR1( DELTA , K , a ): return 2*K * ( 1 - a**DELTA ) def AR1fit( DELTA, TAMSD ): def fct( x ): if(abs(x[1])>0.999): return 1e45*x[1]**2 return np.mean( np.log(TAMSD / tamsdAR1( DELTA , x[0] , x[1] ))**2 ) x0 = [np.median(TAMSD)/2,0.8] K,a = minimize( fct , x0 , method='Nelder-Mead' ).x return K , a`

`K,a=AR1fit(DELTA[:50],TAMSD[:50])`

. Plot the TAMSD and the fit. What is the correlation time of your fitted model?

**Minimizing the negative log-likelihood:**The negative log-(conditional) likelihood of an autoregressive model is given as \[ \frac{1}{2}\log(v) + \frac{\frac{1}{T-1}\sum_{t=0}^{T-1} (z_{t+1}-az_{t})^2}{2v}, \] were z is the measured time series, the parameters are v (related to K via v=K(1-a^2)) and a, and T is the length ofthe series. Fit the parameters via the above likelihood function, using the`scipy.minimize`

function. What is the correlation time of your fitted model?- Use the parameters from both, the TAMSD fit and the log-likelihood fit to generate an AR(1) process of length 200 and plot the outputs. Do they look similar?
- Plot the TAMSD of the log-likelihood-fit along with the empirical TAMSD of the data. Which part of the functions coincides
- Plot the autocorrelation function (up to lag 20) of the data, the TAMSD-fit and the log-likelihood-fit together in one figure
- What are the advantages and disadvantages of different fitting models?