def test_lineare_regression_mit_korrelation():
sigma = 1.0
xmin = 100.
xmax = xmin + 20.
x = np.arange(xmin, xmax)
y = np.random.normal(3. + 2. * x, sigma)
ey = 0. * x + sigma
# Um die Korrelation zwischen a und b zu minimieren,
# transformieren wir x geeignet
x0 = 0.5 * (xmin + xmax)
# x0 = 0
x = x - x0
# Aufruf der linearen Regression
a,ea,b,eb,chiq,corr = \
analyse.lineare_regression(x,y,ey)
print('a=%f+-%f, b=%f+-%f (x0=%f), chi2=%f, corr=%f' % \
(a,ea,b,eb,x0,chiq,corr))
# Ruecktransformation
x = x + x0
b = b - a * x0
figure()
subplot(2, 1, 1)
title('Lineare Regression')
errorbar(x, y, yerr=ey, fmt='o', color='blue')
plot(x, a * x + b, '-')
xlim(xmin - 1., xmax + 1.)
xlabel('x')
ylabel('y')
subplot(2, 1, 2)
title('Residuenplot')
errorbar(x, y - (a * x + b), yerr=ey, fmt='o', color='blue')
plot(x, 0. * x)
xlim(xmin - 1., xmax + 1.)
ylim(-5, 5)
xlabel('x')
ylabel('y-(ax+b)')
show()
def test_lineare_regression():
sigma = 1.0
xmin = -20.
xmax = 20.
# wuerfele Daten
x = np.arange(xmin, xmax)
y = np.random.normal(3. + 0.5 * x, sigma)
ey = 0. * x + sigma
# Aufruf der linearen Regression
a,ea,b,eb,chiq,corr = \
analyse.lineare_regression(x,y,ey)
print('a=%f+-%f, b=%f+-%f, chi2=%f, corr=%f' % (a, ea, b, eb, chiq, corr))
figure()
subplot(2, 1, 1)
title('Lineare Regression')
errorbar(x, y, yerr=ey, fmt='o', color='blue')
plot(x, a * x + b, '-')
xlim(xmin - 1., xmax + 1.)
xlabel('x')
ylabel('y')
grid()
subplot(2, 1, 2)
title('Residuenplot')
errorbar(x, y - (a * x + b), yerr=ey, fmt='o', color='blue')
plot(x, 0. * x) # Linie bei Null
xlim(xmin - 1., xmax + 1.)
ylim(-5, 5)
xlabel('x')
ylabel('y-(ax+b)')
show()
logUR = np.log((UR - Uoffset) / UR0)
# Messbereich -> Digitalisierungsfehler (Cassy-ADC hat 12 bits => 4096 mögliche Werte)
sigmaU = 20.0 / 4096. / np.sqrt(12.)
sigmaLogUR = sigmaU / UR
# Extrahiere Daten für Fit
t_fit, logUR_fit = analyse.untermenge_daten(t, logUR, tmin, tmax)
_, sigmaLogUR_fit = analyse.untermenge_daten(t, sigmaLogUR, tmin, tmax)
errorbar(t_fit, logUR_fit, yerr=sigmaLogUR_fit, fmt='.')
xlabel('t / ms')
ylabel('$\log\,U/U_0$')
# Lineare Regression zur Bestimmung der Zeitkonstanten
a, ea, b, eb, chiq, corr = analyse.lineare_regression(
t_fit, logUR_fit, sigmaLogUR_fit)
tau = 1. / a
sigma_tau = tau * ea / a
print('tau = (%g +- %g) ms b = (%g +- %g) chi2/dof = %g / %g' %
(tau, sigma_tau, b, eb, chiq, len(t_fit) - 2))
plot(t_fit, a * t_fit + b, color='red')
subplot(3, N, m + 2 * N)
# Residuenplot
resUR = logUR_fit - (a * t_fit + b)
eresUR = sigmaLogUR_fit
errorbar(t_fit, resUR, yerr=eresUR, fmt='.')
xlabel('t / ms')
ylabel(r'$\log\,U/U_0 - (t/\tau + b)$')
show()
#---------------------------------------------------
wsf = np.array(wsf)
ws = np.array(ws)
print(wsf)
print(ws)
sTs = np.array(sTs)
sTsf = np.array(sTsf)
kappa = np.array((wsf**2 - ws**2) / (wsf**2 + ws**2))
print(kappa)
skappa = 4 * np.power(ws * wsf, 3) * (scipy.pi * 2)**(-1) * np.power(
ws**2 + wsf**2, -2) * np.sqrt((sTs / ws)**2 + (sTsf / wsf)**2)
print(skappa)
m, em, b, eb, chiq, corr = anal.lineare_regression(lF**-2,
np.array(kappa)**(-1) - 1,
skappa)
print(b)
print(eb)
plt.errorbar(lF**-2,
kappa**-1,
skappa / (kappa)**2,
fmt="x",
markersize=4,
capsize=3)
plt.plot(np.arange(min(lF**-2) - 0.2,
max(lF**-2) + 0.2, 0.1),
np.arange(min(lF**-2) - 0.2,
max(lF**-2) + 0.2, 0.1) * m + b)
plt.title("Die lineare Regression bzgl. $\kappa^{-1}$")
plt.xlabel("$l_F^{-2}$")
mp, sp, errp = Rauschmessung.round_good(np.mean(p), np.std(p),
np.std(p) / np.sqrt(len(p)))
errstatp = np.sqrt((Rauschmessung.sp / np.sqrt(len(p)))**2 +
sigma_p_dig**2)
#errT=sT0/np.sqrt(len(T))
#errp=sigmap
#Plotten
print("\nDichtigkeitsmessung")
print("Fehler auf jeden gemessenen Druckwert")
print("s**2 = s_Gerät**2+s_Rausch**2=" +
str(Rauschmessung.round_good(0.0, 0.0, sigmap**2)[2]) + " hPa")
plt.plot(t, p)
#Regression
m, em, b, eb, chiq, corr = ana.lineare_regression(t, p,
sigmap * np.ones(len(p)))
print("\nErgebnis der Regression auf die Dichtigkeitskurve:")
print(
'm = (%g +- %g) hPa/s, b = (%g +- %g) hPa, chi2/dof = %g / %g corr = %g'
% (m, em, b, eb, chiq, len(t) - 2, corr))
m, x, em = Rauschmessung.round_good(m, 0.0, em)
print(
'Verlust: ({} +-{}) mbar/s bei einem Innendruck von etwa {}+-{} mbar\n'
.format(m, em, mp, errp)) #so genehmigt
plt.plot(t, m * t + b, color='green')
plt.title("Dichtigkeitsmessung\nVerlust: ({} +- {}) mbar/s".format(m, em))
plt.xlabel('Zeit/s')
plt.ylabel(u'Druck/hPa')
plt.savefig("Images/Kalib_Dichtigkeitsmessung.pdf")
plt.figure()
emax = s_U_Ger * np.ones(len(maxima[0]))
#Residum
plt.axhline(0)
plt.plot(maxima[0], U[0]*np.exp(-d*np.array(maxima[0]))-maxima[1], label = "Residum exp")
plt.show()
plt.legend()
plt.close()
if len(maxima[0])>3: plt.plot(maxima[0], maxima[1], marker="x")
if d!=1.0: plt.plot(t, U[0]*np.exp(-d*np.array(t)), label = "Einhüllende")
print("Die Dämpfungskonstante Delta berechnet aus der Amplitude: \n d=" + str(np.round(d,4)) + "+-" + str(np.round(ed,4)))
#Berechnung von Delta aus der linearen Regression
#TODO: Fehler zu groß!
d2,ed2,b,eb,chiq,corr = anal.lineare_regression(np.array(maxima[0]), np.log(maxima[1]), np.log(emax))
print("Die Dämpfungskonstante Delta berechnet aus der linearen Regression: \n d=" + str(-d2) + "+-" + str(ed2))
#plot
plt.title("Entladung des Kondensators über den Schwingkreis \n R = " +str(R[i]) + " $\Omega$")
plt.errorbar(maxima[0],maxima[1],yerr=emax, fmt="x")
plt.axvline(x=t[offsetkorr[i]], color="red", linestyle = "--")
plt.plot(t, I*20, label="20$\cdot$I")
plt.plot(t, np.exp(d2*np.array(t)+b), label = "Einhüllende2")
plt.plot(t, U, label="U")
plt.savefig("Images/C_"+str(R[i])+".pdf")
plt.legend()
plt.show()
plt.close()
#Residum2
plt.axhline(0)
n_0 = np.array([1, 2, 4, 5, 6])
for i in range(5):
s_f_0 = np.append(
s_f_0,
np.std([
fmax_ablesen[i], fmax_peakfinder[i], fmax_maxima[i], fmax_matlab[i]
]) / 2)
f_max_0 = np.append(
f_max_0,
np.average([
fmax_ablesen[i], fmax_peakfinder[i], fmax_maxima[i], fmax_matlab[i]
]))
n = n_0[:-1] #ignoriere den letzten Punkt
s_f = s_f_0[:-1]
f_max = f_max_0[:-1]
a, ea, b, eb, chiq, corr = anal.lineare_regression(n, f_max, s_f)
plt.errorbar(n, f_max, s_f, ls="None", markersize=5, marker="x")
plt.errorbar(n_0[4],
f_max_0[4],
s_f_0[4],
fmt="o",
marker="x",
markersize=5,
color="red")
plt.plot(np.arange(0.5, 6.5, 0.01),
a * np.arange(0.5, 6.5, 0.01) + b,
label="f = (" + str(np.round(a, 2)) +
"$\pm {}$) Hz*n+(".format(np.round(ea, 2)) + str(np.round(b, 2)) +
"$\pm {}$)Hz \n $Chi^2 /f = {}/{} ≈ {}$".format(
np.round(eb, 2), np.round(chiq, 2), 2, np.round(chiq / 2, 2)))
plt.title("Lindeare Regression der Resonanzfrequenzen")
def pltmitres(x,y,ey,ex=0, xlabel="x", ylabel="y", xunit="", yunit="", title="", ratios=[2,1], regdata =False, precision=2,capsize=5, markersize=5):
"""
Erstellen eines Plots mit dazugehörigem Residumplot
Parameter
---------
x,y,ex,ey:
Daten mit Fehler (ex optionell)
xlabel,ylabel:
Titel der Achsen
xunit,yunit:
Einheit der Werte als String
title:
Titel des Plots
ratios:
Flächenverhältnis zwischen Daten- und Residumplot
regdata :
Falls wahr, a,ea,b,eb,chiq,corr der Regression werden auch zurückgegeben
precision:
Nachkommastellen der Parameter a,b
Returns
--------
ax :
Achsen des Datenplots (0) bzw. Residumplots (1)
(a, ea), (b,eb),chiq,corr:
Ergebnisse der linearen Regression (falls regdata = True)
Verwendung
----------
>>> import numpy as np
>>> import praktikum_addons.bib as bib
>>> import matplotlib.pyplot as plt
>>> x = np.arange(0.1,1.3,0.1)
>>> y = np.sin(x)
>>> ex = np.ones(len(x))*0.03
>>> ey = np.ones(len(y))*0.1
>>> ax = bib.pltmitres(x,y, ex, ey, yunit = "s", xunit="m", ratios=[7,1])
>>> ax[0].set_ylim(top=2)
>>> plt.show()
"""
x = np.array(x)
y = np.array(y)
ex = np.array(ex)
ey = np.array(ey)
if ex.all()==0: a,ea,b,eb,chiq,corr = anal.lineare_regression(x,y,ey)
else: a,ea,b,eb,chiq,corr = anal.lineare_regression_xy(x,y,ex,ey)
fig, (ax1,ax2) = plt.subplots(2, 1,gridspec_kw = {'height_ratios':ratios})
ax1.set_title(title)
ax1.errorbar(x,y, ey, np.ones(len(x))*ex, marker="x", linestyle="None", capsize=capsize, markersize=markersize)
l = max(x)-min(x)
x2 = np.arange(min(x)-l*0.1, max(x)+l*0.1, l/1000)
y2 = a*x2+b
ax1.plot(x2, y2, color="orange")
if yunit!="": ax1.set_ylabel(ylabel+" [{}]".format(yunit))
else: ax1.set_ylabel(ylabel)
ax1.legend(title="Lineare Regression\n{1} = ({2:.{0}f} ± {3:.{0}f}){4} $\cdot$ {5}+({6:.{0}f}±{7:.{0}f}){8}\n$\chi^2 /NDF={9:.4f}$".format(precision,ylabel,a,ea, yunit+"/"+xunit,xlabel,b, eb, yunit, chiq/(len(x)-2)), loc=1)
ax2.errorbar(x,y-a*x-b, np.sqrt(ex**2*a**2+ey**2), marker="x", linestyle="None", capsize=capsize, markersize=markersize)
ax2.axhline(0, color="orange")
if xunit!="": plt.xlabel(xlabel+" [{}]".format(xunit))
else: plt.xlabel(xlabel)
plt.tight_layout()
if regdata : return (ax1, ax2), (a,ea),(b,eb),chiq,corr
else: return (ax1, ax2)
tS1 = np.array([0.155, 0.465, 0.78, 1.085, 1.41, 1.72, 2.035, 2.345, 2.655, 2.975]) mS1 = np.arange(1,11,1) tS2 = np.array([0.38, 1.02, 1.66, 2.33, 2.99]) mS2 = np.arange(1,6,1) tS4 = np.array([0.075, 0.24, 0.42, 0.595, 0.76, 0.93, 1.10, 1.27, 1.445, 1.62, 1.795, 1.965, 2.135, 2.305, 2.47, 2.64, 2.81, 2.975]) mS4 = np.arange(1,19,1) # Lineare Regressionen resultierende Schwingung errtR1 = np.full((len(tR1),),0.0002) errtR3 = np.full((len(tR3),),0.0005) errtR4 = np.full((len(tR4),),0.0004) regR1 = analyse.lineare_regression(nR1, tR1, errtR1) regR2 = analyse.lineare_regression(nR2, tR2, errtR1) regR3 = analyse.lineare_regression(nR3, tR3, errtR3) regR4 = analyse.lineare_regression(nR4, tR4,errtR4) # Lineare Regressionen Schwebungsschwingung errtS1 = np.full((len(tS1),),0.005) errtS2 = np.full((len(tS2),),0.01) errtS4 = np.full((len(tS4),),0.005) regS1 = analyse.lineare_regression(mS1, tS1, errtS1)
n[0] = 1
period_p = [0.48, 15.38, 31.94, 48.48, 65.04, 81.56, 98.11, 114.68,
131.28, 147.84, 164.36, 180.9, 197.46, 214.02, 230.56,
247.12, 263.68, 280.22, 296.8, 313.34]
period_s = [1.5, 16.44, 33.01, 49.59, 66.14, 82.73, 99.32,
115.86, 132.46, 149.05, 165.64, 182.18, 198.8,
215.34, 231.92, 248.47, 265.08, 281.62, 298.23, 314.83]
eperiod_p = np.full((20,),0.02)
eperiod_s = np.full((20,),0.02)
nreg = np.arange(0,200,1)
#Lineare Regression der Daten zur Bestimmung der Frequenz
res_p = analyse.lineare_regression(n,period_p,eperiod_p)
omega_p = 2*np.pi/res_p[0]
eomega_p = 2*np.pi/(res_p[0]**2)*res_p[1]
res_s = analyse.lineare_regression(n,period_s,eperiod_s)
omega_s = 2*np.pi/res_s[0]
eomega_s = 2*np.pi/(res_s[0]**2)*res_s[1]
"""
f, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, sharex='col', sharey='row', gridspec_kw={'height_ratios': [5, 2]})
ax1.plot(nreg,res_p[0]*nreg+res_p[2], linestyle="--", color = 'black')
ax1.errorbar(n,period_p, yerr = 0.02, color='red', fmt='.', marker ='o')
ax1.set_ylabel('$t$ / $s$')
ax1.grid()
ax1.set_ylim(-10,350)
ax1.set_title('Messung mit Pendelkörper')
reg2plus = np.array([])
reg2minus = np.array([])
#Frequenzen und Dämpfungen bestimmen (Regression)
for i in range(1, 6, 1):
n = np.arange(1, 1 + dic1[i].size, 1)
data = cassy.CassyDaten(dic2[i])
time = data.messung(1).datenreihe('t').werte
vol = data.messung(1).datenreihe('U_B1').werte
x = time[dic1[i]]
ex = np.full((dic1[i].size, ), 1e-05 / np.sqrt(12))
y = np.log(np.abs(vol[dic1[i]]) - dic3[i])
ey = 4.8e-03 * 1 / (np.abs(vol[dic1[i]]) - dic3[i])
ey_sys = 0.01 * np.sqrt(vol[dic1[i]]**2 +
dic3[i]**2) / (np.abs(vol[dic1[i]]) - dic3[i])
res1 = analyse.lineare_regression(n, x, ex)
res2 = analyse.lineare_regression_xy(x, y, ex, ey)
res2plus = analyse.lineare_regression_xy(x, y + ey_sys, ex, ey)
res2minus = analyse.lineare_regression_xy(x, y - ey_sys, ex, ey)
reg1 = np.concatenate((reg1, res1))
reg2 = np.concatenate((reg2, res2))
reg2plus = np.concatenate((reg2plus, res2plus))
reg2minus = np.concatenate((reg2minus, res2minus))
#Plot
f, ((ax1, ax2),
(ax3, ax4)) = plt.subplots(2,
2,
sharex='col',
gridspec_kw={'height_ratios': [5, 2]})
#1. Regression Vanilla Plot [Zeiten in Mikrosekunden]
I_2E=IE[N2]
#C_UA=(tA[N2]-tA[N1])/Widerstand.R/ana.log(U_1A/U_2A) #TODO: anders umzustellen
C_UE=(tA[N2]-tA[N1])/Widerstand.R/ana.log(U_1E/U_2E)
C_IA=(tA[N2]-tA[N1])/Widerstand.R/ana.log(I_1A/I_2A)
C_IE=(tA[N2]-tA[N1])/Widerstand.R/ana.log(I_1E/I_2E)
'''
#linus
#UA_log=ana.log(UA)
UE_log = ana.log(UE)
IA_log = ana.log(IA[wvm:])
tA = tA[wvm:]
#IE_log=ana.log(IE)
#Parameter Tau bestimmen: #Hier ist tau=-1/tau
tauE, etauE, bE, ebE, chiqE, corr = ana.lineare_regression(
np.array(tE), np.array(UE_log), np.array(Widerstand.U_err_stat / UE))
tauA, etauA, bA, ebA, chiqA, corr = ana.lineare_regression(
np.array(tA), np.array(IA_log), np.array(Widerstand.I_err_stat / IA[wvm:]))
x, tauE, etauE = Widerstand.round_good(0, tauE, etauE)
x, tauA, etauA = Widerstand.round_good(0, tauA, etauA)
if __name__ == '__main__':
#Plot Tau-Bestimmung Entladung
plt.plot(tE, UE_log, 'bo', label='logarrithmierte Messreihe U_Entladung')
plt.plot(
tE,
tauE * tE + bE,
color='brown',
label=u'Geradenanpassung $\\tau$*x+b mit $\\tau$={}, b={} mit Chi²/f={}'
.format(tauE, round(bE, 2), round(chiqE / len(tE), 2)))
for i in range(len(Ri)):
axvline(x=Ri[i], color='red')
text(Ri[i], 0.075, '$R_{{{}}}$={:.2f}'.format(indize[i], Ri[i]))
xlabel('R / $\\Omega$')
ylabel('U / V')
savefig('stehendeWelle.pdf', bbox_inches='tight')
show()
Wellenlaenge = (Ri[len(Ri) - 1] - Ri[0]) * K * 2 / 33
print(Wellenlaenge)
print(
'1. Auswertung: gesamter Abstand=3.283 kOhm = 53.51cm ---- Wellenlaenge=3.24'
)
a, ea, b, eb, chiq, corr = analyse.lineare_regression(indize, Ri, eRi)
print('\n \n 2.Auswertung')
print('Ergebnis: a=%f+-%f, b=%f+-%f, chi2=%f, corr=%f' %
(a, ea, b, eb, chiq, corr))
figure()
#title('Lin. Reg. der Widerstandpeaks bei stehenden Wellen')
errorbar(indize,
Ri,
yerr=eRi,
linestyle='none',
marker='o',
markersize=4,
elinewidth=1,
zorder=1)
plot(indize, a * indize + b, '-', zorder=2)