timeseries talk November 10, 2015

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Wendy Rose
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1 timeseries talk November 10, Time series Analysis and the noise model Mark Bakker TU Delft Voorbeeld: Buis B58C0698 nabij Weert In [1]: import numpy as np import pandas as pd from pandas.tools.plotting import autocorrelation_plot import matplotlib.pyplot as plt %matplotlib inline plt.rcparams[ font.size ] = 16.0 # larger font size for slides gwdata = pd.read_csv( B58C _0.csv, skiprows=11, parse_dates=[ PEIL DATUM TIJD ], index_col= PEIL DATUM TIJD, skipinitialspace=true) gwdata.rename(columns={ STAND (MV) : h }, inplace=true) gwdata.index.names = [ date ] gwdata.h *= 0.01 gwdata.h = gwdata.h # NAP # rain = pd.read_csv( Heibloem_rain_data.dat, skiprows=4, delim_whitespace=true, parse_dates=[ date ], index_col= date ) rain = rain[ 1980 :] # cut off everything before 1980 # evap = pd.read_csv( Maastricht_E_June2015.csv, skiprows=4, sep= ;, parse_dates=[ DATE ], index_col= DATE ) evap.rename(columns={ VALUE (m-ref) : evap }, inplace=true) evap = evap[ 1980 :] # cut off everything before 1980 evap.evap *= 1000 # rain[ evap ] = evap.evap rain[ rech ] = rain.precip * rain.evap In [2]: dates = pd.date_range( , , freq= 30D ) # every 30 days ho = np.interp(dates.asi8, gwdata.index.asi8, gwdata.h) # interpolate every 30 days # fix nan values (this is pretty ugly) t = np.arange(len(ho)) ho = np.interp(t, t[~np.isnan(ho)], ho[~np.isnan(ho)]) 1

2 # index in rain where observed heads rain[ num ] = range(len(rain)) N = rain.rech.values io = rain.num.loc[dates].values In [3]: def timeseries_noise(p, N, ho, io=io, tmax=1000, simulate=false): A = p[0] a = np.exp( p[1] ) d = p[2] if len(p) > 3: alpha = p[3] else: alpha = 0.0 dt = 1 # time step t = np.arange(0,tmax,dt) # time F = A * (1 - np.exp(-t / a)) # step function H = F[1:] - F[:-1] # block function h = np.convolve(n,h) + d h = h[io] # simulated head rv = h - ho # residuals v = rv[1:] - alpha * rv[:-1] if simulate == sim : return h elif simulate == v : return v else: return sum(v**2) # innovations In [4]: Astart = 0.75 astart = 150 # memory dstart = np.mean(ho) # base alphastart = 0.01 pstart = np.array([astart, np.log(astart), dstart, alphastart]) In [5]: from scipy.optimize import fmin popt1 = fmin(timeseries_noise, pstart[:-1], args=(n, ho)) h1 = timeseries_noise(popt1, N, ho, simulate= sim ) v1 = timeseries_noise(popt1, N, ho, simulate= v ) Optimization terminated successfully. Current function value: Iterations: 64 Function evaluations: 114 In [6]: popt2 = fmin(timeseries_noise, pstart, args=(n, ho)) h2 = timeseries_noise(popt2, N, ho, simulate= sim ) v2 = timeseries_noise(popt2, N, ho, simulate= v ) Optimization terminated successfully. Current function value: Iterations: 330 Function evaluations: 559 In [7]: heads = pd.dataframe(index=dates) heads[ ho ] = ho 2

3 heads[ h1 ] = h1 heads[ h2 ] = h2 heads[ v1 ] = 0.0 heads.v1[1:] = v1 heads[ v2 ] = 0.0 heads.v2[1:] = v2 heads[ r1 ] = heads.h1 - heads.ho heads[ r2 ] = heads.h2 - heads.ho In [8]: plt.figure(figsize=(15,6)) heads.ho.plot(style= go ) heads.h1.plot(color= b ) RMSE = np.sqrt(np.mean(heads.r1**2)) plt.title( Model fit without noise model. RMSE= + str(rmse)); In [9]: plt.figure(figsize=(15,6)) heads.r1.plot(color= b ); plt.axhline(0, color= k, lw=2) plt.ylim(-0.5,0.5) plt.title( Residuals are correlated ); 3

4 In [10]: plt.figure(figsize=(15,6)) autocorrelation_plot(heads.v1, color= b ) plt.xlim(0,36) plt.title( Autocorrelation of residuals ) Out[10]: <matplotlib.text.text at 0x10a1ea050> 2 So the residuals are correlated The residual of today is a factor times the residual of the previous measurement r i = αr i+1 + n i We hope that the remaining error n i is independent of the remaining error at the previous measurement n i 1. The remaining error is also called the innovation. In [11]: plt.figure(figsize=(15,6)) heads.ho.plot(style= go ) heads.h2.plot(color= r ) RMSE = np.sqrt(np.mean(heads.r2**2)) plt.title( Model fit with noise model. RMSE= + str(rmse)); 4

5 In [12]: plt.figure(figsize=(15,6)) #heads.r1.plot(color= b ); heads.v2.plot(color= r ); plt.ylim(-0.5,0.5) plt.axhline(0, color= k, lw=2) plt.title( Innovations show little correlation ); In [13]: plt.figure(figsize=(15,6)) autocorrelation_plot(heads.v2, color= r ) plt.xlim(0,12) plt.title( Autocorrelation of innovations with noise model ); 5

In [14]: plt.figure(figsize=(15,6)) plt.subplot(121) heads.v1.hist(color= b, normed=true) plt.title( Histogram of residuals w/o noise model ) plt.xlim(-0.5,0.5) plt.subplot(122) heads.v2.

6 In [14]: plt.figure(figsize=(15,6)) plt.subplot(121) heads.v1.hist(color= b, normed=true) plt.title( Histogram of residuals w/o noise model ) plt.xlim(-0.5,0.5) plt.subplot(122) heads.v2.hist(color= r, normed=true) plt.title( Histogram of innovations w/ noise model ) plt.xlim(-0.5,0.5) Out[14]: (-0.5, 0.5) In [52]: from scipy.stats import probplot plt.figure(figsize=(15,6)) ax1 = plt.subplot(121) ax2 = plt.subplot(122) 6

7 probplot(heads.v1.values, dist= norm, plot=ax1); probplot(heads.v2.values, dist= norm, plot=ax2); In [58]: from lmfit import Parameters, minimize, fit_report def timeseries_noise(p, N, ho, io=io, tmax=1000, simulate=false): vals = p.valuesdict() A = vals[ A ] a = vals[ a ] d = vals[ d ] alpha = vals[ alpha ] dt = 1 # time step t = np.arange(0,tmax,dt) # time F = A * (1 - np.exp(-t / a)) # step function H = F[1:] - F[:-1] # block function h = np.convolve(n,h) + d h = h[io] # simulated head rv = h - ho # residuals v = rv[1:] - alpha * rv[:-1] if simulate == sim : return h elif simulate == v : return v else: return v # innovations p = Parameters() p.add( A, value=0.75) p.add( a, value=150) p.add( d, value=np.mean(ho)) p.add( alpha, value=1) In [59]: # Full model p[ alpha ].value = 1 7

8 p[ alpha ].vary = True pout2 = minimize(timeseries_noise, p, args=(n,ho), kws={ io :io}) print fit_report(pout2) [[Fit Statistics]] # function evals = 110 # data points = 304 # variables = 4 chi-square = reduced chi-square = [[Variables]] A: / (9.09%) (init= 0.75) a: / (9.72%) (init= 150) d: / (0.22%) (init= ) alpha: / (5.14%) (init= 1) [[Correlations]] (unreported correlations are < 0.100) C(A, a) = C(A, d) = C(a, d) = In [64]: # No noise model p[ alpha ].value = 0 p[ alpha ].vary = False p[ d ].vary = True p[ d ].value = 0 pout1 = minimize(timeseries_noise, p, args=(n,ho), kws={ io :io}) print fit_report(pout1) [[Fit Statistics]] # function evals = 31 # data points = 304 # variables = 3 chi-square = reduced chi-square = [[Variables]] A: / (3.94%) (init= 0.75) a: / (4.87%) (init= 150) d: / (0.09%) (init= 0) alpha: 0 (fixed) [[Correlations]] (unreported correlations are < 0.100) C(A, d) = C(A, a) = C(a, d) = In [62]: minimize? In [ ]: 8

Linear Classification

Linear Classification by Prof. Seungchul Lee isystems Design Lab http://isystems.unist.ac.kr/ UNIS able of Contents I.. Supervised Learning II.. Classification III. 3. Perceptron I. 3.. Linear Classifier