def Plot_Lorenz96(savepath, tag='', picformat='png'):
"""Generates and saves the result figure for Lorenz in the paper."""
# first the random phase model
plt.rc('font', family='serif', size=9)
tfs = 10
MyColors = ['#377eb8', '#d95f02']
# funcion that designates the position of axes
def CreateAxes(i, j, myfig):
# i,j are the row column indices of axes from top left
# xleft, ybottom
h, w, dw, dh = .2, .2, .07, .09
x0, y0 = 0.17, .68
xl = x0 + (j - 1) * (w + dw)
yl = y0 - (i - 1) * (h + dh)
my_axes = myfig.add_axes([xl, yl, w, h])
plt.tick_params(direction='in')
return my_axes
# truth, random phase model and SDE model
data = sio.loadmat('./results/Lorenz_RPM')
x, x_qp, dt = np.squeeze(data['x']), np.squeeze(data['x_qp']), np.squeeze(
data['dt'])
data2 = sio.loadmat('./results/LorenzModel_r3_t1000_ns1.mat')
x_train, x_model, t_model = data2['x_train'], data2['x_model'], data2[
't_model']
# we use the training data as truth
x = x_train[:, 0]
x_mean = np.mean(x)
nt = 201
t = np.arange(0, nt) * dt
fig = plt.figure(figsize=[6.5, 6.5 * .66])
yls = [-12, 14] # ylim for signals
pls = [-11, 14]
# signals
ax1 = CreateAxes(1, 1, fig)
ax1.plot(t, x[2000:2000 + nt], color=MyColors[0])
ax1.set_xticks(np.arange(0, 21, 5))
ax1.set_xlim(0, 20), ax1.set_ylim(yls)
ax1.set_title(r'signal', fontsize=tfs)
ax4 = CreateAxes(2, 1, fig)
ax4.plot(t, x_model[0:nt, 0], color=MyColors[0])
ax4.set_xticks(np.arange(0, 21, 5))
ax4.set_xlim(0, 20), ax4.set_ylim(yls)
ax7 = CreateAxes(3, 1, fig)
ax7.plot(t, x_qp[2000:2000 + nt], color=MyColors[0])
ax7.set_xticks(np.arange(0, 21, 5))
ax7.set_xlim(0, 20), ax7.set_ylim(yls)
ax7.set_xlabel(r'$t$')
# tags
xt = -10
ax1.text(xt,
1,
r'truth',
horizontalalignment='center',
verticalalignment='center',
FontSize=tfs)
ax4.text(xt,
1,
r'SDE model',
horizontalalignment='center',
verticalalignment='center',
FontSize=tfs)
ax7.text(xt,
1,
r'phase model',
horizontalalignment='center',
verticalalignment='center',
FontSize=tfs)
# PSD
ax2 = CreateAxes(1, 2, fig)
wr, pr = sa.Welch_estimator(x - x_mean, M=100, L=200, fs=1 / dt)
ax2.plot(wr, pr, 'k', label=r'truth')
ax2.set_xlim(0, np.pi / dt)
ax2.set_title(r'PSD', fontsize=tfs)
ax2.set_xticks([0, 15, 30])
ax2 = CreateAxes(2, 2, fig)
wm, pm = sa.Welch_estimator(x_model[:, 0] - x_mean,
M=100,
L=200,
fs=1 / dt)
ax2.plot(wm, pm, 'k')
ax2.plot(wr, pr, '--', color='gray', label='truth')
ax2.set_xlim(0, np.pi / dt)
ax2.set_xticks([0, 15, 30])
legend = plt.legend(fontsize=tfs - 1,
bbox_to_anchor=(1.03, .84),
loc='center right',
ncol=1,
fancybox=True,
framealpha=0)
legend.get_frame().set_linewidth(0)
ax2 = CreateAxes(3, 2, fig)
wm, pm = sa.Welch_estimator(x_qp, M=100, L=200, fs=1 / dt)
ax2.plot(wm, pm, 'k')
ax2.plot(wr, pr, '--', color='gray')
ax2.set_xlim(0, np.pi / dt)
ax2.set_xticks([0, 15, 30])
ax2.set_xlabel(r'$\omega$')
# PDF
ax3 = CreateAxes(1, 3, fig)
[xp, pp] = pt.pdf_1d(x, nx=100, smoothing_sigma=2)
ax3.plot(xp, pp, 'k')
ax3.set_xlim(pls), ax3.set_ylim(0, .11)
ax3.set_yticks(np.arange(0, .11, .1)), ax3.set_xticks([-10, 0, 10])
ax3.set_title(r'PDF', fontsize=tfs)
ax3 = CreateAxes(2, 3, fig)
[xm, pm] = pt.pdf_1d(x_model[:, 0], nx=100, smoothing_sigma=2)
ax3.plot(xm, pm, 'k')
ax3.plot(xp, pp, '--', color='gray')
ax3.set_xlim(pls), ax3.set_ylim(0, .11)
ax3.set_yticks(np.arange(0, .11, .1)), ax3.set_xticks([-10, 0, 10])
ax3 = CreateAxes(3, 3, fig)
[xm, pm] = pt.pdf_1d(x_qp + x_mean, nx=100, smoothing_sigma=2)
ax3.plot(xm, pm, 'k')
ax3.plot(xp, pp, '--', color='gray')
ax3.set_xlim(pls), ax3.set_ylim(0, .11)
ax3.set_yticks(np.arange(0, .11, .1)), ax3.set_xticks([-10, 0, 10])
plt.savefig(savepath + 'Lorenz96.' + picformat, dpi=400)
def Plot_tails_wCI(savepath, tag='', picformat='png'):
plt.rc('font', family='serif', size=9)
tfs = 10
MyColors = ['#377eb8', '#d95f02']
lw0, lw1, lw2 = 1.5, 1.2, 1.2
# smoothing for pdfs?
F = 2 #std of gaussian kernel in convolution
# axes sizes
myfig = plt.figure(figsize=[6.5, 2.4])
w = .16
h = .3
dh, dw = .11, .02
x0, y0 = .11, .89 - h
# data tags
Setup, years_train, polynomial_order, start_year = 21, 2, 2, 0 # this relates to how correlated variables are chosen
#############
# temperature
idx_target = 2
tdata = sio.loadmat('./results/NorthPole_var' + str(idx_target) + '_r' +
str(polynomial_order) + '_tyrs' + str(years_train) +
'_syrs' + str(start_year))
y_truth, y_train, y_model = np.squeeze(tdata['y_truth']), np.squeeze(
tdata['y_train']), tdata['ys_model']
r_truth, p_truth, p_truth_u, p_truth_l = pt.pdf_1d_wCI(y_truth,
nx=100,
smoothing_sigma=F)
r_train, p_train, p_train_u, p_train_l = pt.pdf_1d_wCI(y_train,
nx=50,
smoothing_sigma=F)
for k in range(0, 5):
ax6 = myfig.add_axes([x0 + k * (w + dw), y0 - h - dh, w, h])
ax6.tick_params(direction='in')
plt.yticks([]), plt.xticks([])
ax6.set_yscale('log')
plt.yticks([1e-1, 1e-3, 1e-5])
plt.ylim(1e-6, .8)
plt.xlim(-10, 10)
plt.xticks([-5, 5])
if k == 0:
plt.ylabel(r'$T_{NP}$', rotation=0)
ax6.yaxis.set_label_coords(-.52, .5)
else:
ax6.set_yticklabels([])
r_model, p_model = pt.pdf_1d(y_model[:, k],
nx=100,
smoothing_sigma=F,
MyRange=[-20, 20])
plt.plot(r_model,
p_model,
color=MyColors[0],
linewidth=lw0,
label='model (74 years)')
plt.plot(r_train,
p_train,
'k--',
linewidth=lw1,
label='train (2 years)')
plt.plot(r_truth,
p_truth,
'-.',
linewidth=lw2,
color=MyColors[1],
label='truth (37 years)')
# confidence intervalo
plt.fill_between(r_truth,
p_truth_l,
p_truth_u,
alpha=0.35,
color=MyColors[1],
linewidth=0)
plt.fill_between(r_train,
p_train_l,
p_train_u,
alpha=0.35,
color='k',
linewidth=0)
if k == 2:
legend = plt.legend(fontsize=tfs,
bbox_to_anchor=(.5, -.43),
loc='center',
ncol=3,
fancybox=True,
framealpha=0)
legend.get_frame().set_linewidth(0)
legend.get_frame().set_edgecolor("black")
# u-velocity
idx_target = 0
tdata = sio.loadmat('./results/NorthPole_var' + str(idx_target) + '_r' +
str(polynomial_order) + '_tyrs' + str(years_train) +
'_syrs' + str(start_year))
y_truth, y_train, y_model = np.squeeze(tdata['y_truth']), np.squeeze(
tdata['y_train']), tdata['ys_model']
r_truth, p_truth, p_truth_u, p_truth_l = pt.pdf_1d_wCI(y_truth,
nx=100,
smoothing_sigma=F)
r_train, p_train, p_train_u, p_train_l = pt.pdf_1d_wCI(y_train,
nx=50,
smoothing_sigma=F)
for k in range(0, 5):
ax6 = myfig.add_axes([x0 + k * (w + dw), y0, w, h])
ax6.tick_params(direction='in')
plt.yticks([])
ax6.set_yscale('log')
plt.yticks([1e-1, 1e-3, 1e-5])
plt.ylim(1e-6, .8)
plt.xlim(-12, 12)
plt.xticks([-10, 10])
if k == 0:
plt.ylabel(r'$U_{NP}$', rotation=0)
ax6.yaxis.set_label_coords(-.52, .5)
else:
ax6.set_yticklabels([])
r_model, p_model = pt.pdf_1d(y_model[:, k],
nx=100,
smoothing_sigma=F,
MyRange=[-20, 20])
plt.plot(r_model, p_model, color=MyColors[0], linewidth=lw0)
plt.plot(r_train, p_train, 'k--', linewidth=lw1)
plt.plot(r_truth, p_truth, '-.', linewidth=lw2, color=MyColors[1])
# confidence intervalo - shaded enevelope method
plt.fill_between(r_truth,
p_truth_l,
p_truth_u,
alpha=0.35,
color=MyColors[1],
linewidth=0)
plt.fill_between(r_train,
p_train_l,
p_train_u,
alpha=0.35,
color='k',
linewidth=0)
plt.title(r'$n=' + str(k) + '$', fontsize=tfs)
plt.savefig(savepath + 'climate_tails.' + picformat, dpi=400)
def Plot_NorthPole_signals(savepath, tag='', picformat='png'):
plt.rc('font', family='serif', size=9)
tfs = 10
MyColors = ['#377eb8', '#d95f02']
lw2 = .8
# data matters
BaseData = sio.loadmat('./thehood/NorthPole_data.mat')
t, y, yp, yc = np.squeeze(
BaseData['t']), BaseData['y'], BaseData['yp'], BaseData['yc']
# smoothing for pdfs?
F = 2
# axes sizes
myfig = plt.figure(figsize=[6.5, 2.4])
w1, w2, w3 = .29, .29, .16
h = .3
dh, dw = .11, .07
x0, y0 = .08, .89 - h
xlims = [1980, 1995]
# u-vel time series
ax1 = myfig.add_axes([x0, y0, w1, h])
ax1.tick_params(direction='in')
ax1.plot(t, y[:, 0], color=MyColors[0])
plt.plot(t, yp[:, 0], '--', label=r'periodic part', color=MyColors[1])
plt.xlim(xlims), plt.ylim(-10, 25)
ax1.set_xticklabels([])
plt.title(r'time series', fontsize=tfs)
plt.ylabel(r'$U_{NP}$~(m/s)')
ax1.yaxis.set_label_coords(-.17, .5)
legend = plt.legend(fontsize=tfs - 2,
bbox_to_anchor=(.38, .14),
loc='center left',
ncol=1,
fancybox=True,
framealpha=0.0)
legend.get_frame().set_linewidth(0)
legend.get_frame().set_edgecolor("black")
ax2 = myfig.add_axes([x0 + w1 + dw, y0, w2, h])
ax2.tick_params(direction='in')
ax2.plot(t, yc[:, 0], color=MyColors[0])
plt.xlim(xlims)
ax2.set_xticklabels([])
plt.title(r'chaotic part', fontsize=tfs)
ax3 = myfig.add_axes([x0 + w1 + w2 + 2 * dw + .02, y0, w3, h])
ax3.tick_params(direction='in')
[rr, pr] = pt.pdf_1d(yc[:, 0], nx=100, smoothing_sigma=F)
rg, pg = pt.Gaussian_fit(yc[:, 0], r=rr)
plt.plot(rg, pg, 'k--', label=r'Gaussian fit', linewidth=lw2)
plt.plot(rr, pr, color=MyColors[0])
ax3.set_yscale('log')
plt.ylim(1e-6, .5)
plt.yticks([1e-1, 1e-3, 1e-5])
plt.xlim(-12, 12), plt.xticks([-10, 10])
plt.title(r'PDF of chaotic part', fontsize=tfs)
legend = plt.legend(fontsize=tfs - 2,
bbox_to_anchor=(-.05, -1.75),
loc='center left',
ncol=1,
fancybox=True,
framealpha=0)
legend.get_frame().set_linewidth(0)
legend.get_frame().set_edgecolor("black")
# temperature time series
ax4 = myfig.add_axes([x0, y0 - h - dh, w1, h])
plt.tick_params(direction='in')
plt.plot(t, y[:, 2])
plt.plot(t, yp[:, 2], '--', color='#d95f02')
plt.xlim(xlims)
plt.xlabel(r'$t$ (year)')
plt.ylabel(r'$T_{NP}$~(K)')
plt.ylim(135, 155)
ax5 = myfig.add_axes([x0 + w1 + dw, y0 - h - dh, w2, h])
ax5.tick_params(direction='in')
ax5.plot(t, yc[:, 2])
plt.xlim(xlims)
plt.xlabel(r'$t$ (year)')
ax6 = myfig.add_axes([x0 + w1 + w2 + 2 * dw + .02, y0 - h - dh, w3, h])
ax6.tick_params(direction='in')
[rr, pr] = pt.pdf_1d(yc[:, 2], nx=100, smoothing_sigma=F)
rg, pg = pt.Gaussian_fit(yc[:, 2], r=rr)
plt.plot(rg, pg, 'k--', linewidth=lw2)
plt.plot(rr, pr, color=MyColors[0])
ax6.set_yscale('log')
plt.yticks([1e-1, 1e-3, 1e-5]), plt.ylim(1e-6, .8)
plt.xlim(-10, 10), plt.xticks([-5, 5])
plt.savefig(savepath + 'NorthPole_UT.' + picformat, dpi=400)
def Plot_pointwise_stat(savepath, tag='', picformat='png'):
plt.rc('font', family='serif', size=9)
tfs = 10
PointWiseData = sio.loadmat('./results/Cavity_pointwise_' + tag + '.mat')
CavityField = sio.loadmat('./thehood/CavitySensors.mat')
# what sensors to look at
s = [12, 22, 23, 47] # out of 0-48
SensorIndex = np.squeeze(CavityField['SensorIndex'])
ns = SensorIndex.shape[0]
# SensorIndex=SensorIndex[s] # -1 is for python vs MATLAB indexing
# print(SensorIndex)
uv_model = PointWiseData['uv_model']
uv_truth = PointWiseData['uv_truth']
print(uv_model.shape)
print(uv_truth.shape)
# funcion that designates the position of axes
apr = 1 / 2 # aspect ratio
myfig = plt.figure(figsize=[6.5, 6.5 * apr])
def CreateAxes(i, j, myfig):
# i,j are the row column indices of axes from top left
# xleft, ybottom
h, w, dw, dh = .24, .24 * apr, .03, .15
x0, y0 = 0.34, .6
xl = x0 + (j - 1) * (w + dw)
yl = y0 - (i - 1) * (h + dh)
my_axes = myfig.add_axes([xl, yl, w, h])
plt.tick_params(direction='in')
plt.yticks([])
# plt.xticks([])
return my_axes
xtiko = np.array([[-1, 0, 1], [-.2, 0, .2], [-1, 0, 1], [-.6, 0, .6],
[-1, 0, 1], [-.25, 0, .25], [-.5, 0, .5], [-2, 0, 2]])
for j in range(4):
ax2 = CreateAxes(1, j + 1, myfig)
# u-velocity
y = uv_truth[s[j], :]
x0, p0 = pt.pdf_1d(y, nx=100, smoothing_sigma=3)
xl = [np.amin(x0), np.amax(x0)]
# model
y = uv_model[s[j], :]
x, p = pt.pdf_1d(y,
nx=100,
smoothing_sigma=3,
MyRange=1.5 * np.array(xl))
plt.plot(x, p, 'k', label='SDE model PDF')
plt.plot(x0, p0, 'k-.', color='gray', label='true PDF')
plt.xticks(xtiko[j, :])
plt.title(r'$u_' + str(j + 1) + '$', fontsize=tfs)
# ax2.set_yscale('log')
if j == 0:
legend = plt.legend(fontsize=tfs,
bbox_to_anchor=(4.5, -1.9),
ncol=2,
fancybox=True)
legend.get_frame().set_linewidth(0)
legend.get_frame().set_edgecolor("black")
ax2 = CreateAxes(2, j + 1, myfig)
# v-velocity
y = uv_truth[s[j] + ns, :]
x0, p0 = pt.pdf_1d(y, nx=100, smoothing_sigma=3)
xl = [np.amin(x0), np.amax(x0)]
# model
y = uv_model[s[j] + ns, :]
x, p = pt.pdf_1d(y,
nx=100,
smoothing_sigma=3,
MyRange=1.5 * np.array(xl))
plt.plot(x, p, 'k')
plt.plot(x0, p0, 'k-.', color='gray')
plt.xticks(xtiko[j + 4, :])
plt.title(r'$v_' + str(j + 1) + '$', fontsize=tfs)
# cavity snapshot
x, y, vort, cm = CavityField['x'], CavityField['y'], CavityField[
'vort'], CavityField['colormap_vort']
X, Y = np.squeeze(CavityField['X']), np.squeeze(CavityField['Y'])
cm = np.concatenate((cm, np.ones((cm.shape[0], 1))), axis=1)
cm = ListedColormap(cm)
ax1 = myfig.add_axes([.05, .28, .45 * apr, .45])
plt.contourf(x, y, vort, 100)
plt.set_cmap(cm)
plt.xticks([]), plt.yticks([])
xpa, ypa = [.07, .08, -.08, .1], [.06, .06, -.24, -.14]
for j in range(4):
xp, yp = X[SensorIndex[s[j]] - 1], Y[SensorIndex[s[j]] - 1]
ax1.plot(xp, yp, 'kx')
ax1.text(xp + xpa[j], yp + ypa[j], r'' + str(j + 1), fontsize=tfs - 1)
plt.savefig(savepath + 'cavity_pointwise.' + picformat, dpi=400)
def Cavity_analysis_plot():
"""Analyze convergence and generate the anlaysis figure [6(a)] in the paper. """
nbins = 65
smoothing_sigma = None
Ts = [100, 250, 500, 2500, 5000]
errors_sample_size = []
for tmax in Ts:
filename = SavePath + 'Cavity_pointwise_analysis_T' + str(
tmax) + '_Dim=10_ns10_po2'
Data = sio.loadmat(filename)
uv_model, uv_truth = Data['uv_model'], Data['uv_truth']
Sensors = np.squeeze(Data['Sensors'])
S = list(range(Sensors.shape[0]))
sum_sensor_error = 0
for sensor in S:
grid, p_truth = pt.pdf_1d(uv_truth[sensor, :],
nx=nbins,
smoothing_sigma=smoothing_sigma)
dx0 = grid[1] - grid[0]
_, p_model = pt.pdf_1d(uv_model[sensor, :],
nx=nbins,
smoothing_sigma=smoothing_sigma,
MyRange=[np.amin(grid),
np.amax(grid)])
sum_sensor_error = sum_sensor_error + compute_var_diag(
p_truth, p_model, dx0, trim_zeros=True)
errors_sample_size.append(sum_sensor_error / len(S))
poly_orders = [1, 2, 3, 4, 5]
errors_porder = []
for poly_order in poly_orders:
filename = SavePath + 'Cavity_pointwise_analysis_T2500_Dim=5_ns10_po' + str(
poly_order)
Data = sio.loadmat(filename)
uv_model, uv_truth = Data['uv_model'], Data['uv_truth']
Sensors = np.squeeze(Data['Sensors'])
S = list(range(Sensors.shape[0]))
sum_sensor_error = 0
for sensor in S:
grid, p_truth = pt.pdf_1d(uv_truth[sensor, :],
nx=nbins,
smoothing_sigma=smoothing_sigma)
dx0 = grid[1] - grid[0]
_, p_model = pt.pdf_1d(uv_model[sensor, :],
nx=nbins,
smoothing_sigma=smoothing_sigma,
MyRange=[np.amin(grid),
np.amax(grid)])
sum_sensor_error = sum_sensor_error + compute_var_diag(
p_truth, p_model, dx0, trim_zeros=True)
errors_porder.append(sum_sensor_error / len(S))
myfig = plt.figure(figsize=[5.5, 2.2])
axw, axx0, axy0 = .33, .15, .2
ax1 = myfig.add_axes([axx0, axy0, axw, .7])
no_samples = [t_train * 10 for t_train in Ts]
plt.plot(no_samples, errors_sample_size, '-s', color=MyColors[0])
ax1.tick_params(direction='in')
plt.yscale('log'), plt.xscale('log')
plt.yticks([1e-2, 3e-2, 1e-1],
[r'$10^{-2}$', r'$3 \times 10^{-2}$', r'$10^{-1}$'])
plt.minorticks_off()
plt.xlabel(r'sample size')
plt.title(r'$\tilde{e}$', fontsize=tfs)
ax2 = myfig.add_axes([axx0 + .15 + axw, axy0, axw, .7])
ax2.tick_params(direction='in')
plt.plot(poly_orders, errors_porder, '-s', color=MyColors[0])
plt.yscale('log')
plt.yticks([1e-2, 5e-2], [r'$ 10^{-2}$', r'$5\times 10^{-2}$'])
plt.xticks([1, 2, 3, 4, 5])
plt.minorticks_off()
plt.xlabel(r'polynomial degree')
plt.title(r'$\tilde{e}$', fontsize=tfs)
plt.subplots_adjust(bottom=0.35)
plt.savefig('cavity_analysis.pdf')
def Lorenz_analysis_plot():
"""Analyze convergence and generate the anlaysis figure [6(b)] in the paper. """
x_truth = sio.loadmat('./mat files/Lorenz96Data.mat')['y'][:, 0]
nbins = 66
smoothing_sigma = None
grid, p_truth = pt.pdf_1d(x_truth,
nx=nbins,
smoothing_sigma=smoothing_sigma)
xl = [np.amin(grid), np.amax(grid)]
dx0 = grid[1] - grid[0]
Ts = [1, 10, 100, 1000, 5000]
Ns, error_ss = [], []
for T in Ts:
filename = SavePath + 'LorenzModel_r3_t{}_ns1.mat'.format(T)
data = sio.loadmat(filename)
x_model = data['x_model'].squeeze()
_, p_model = pt.pdf_1d(x_model,
nx=nbins,
smoothing_sigma=smoothing_sigma,
MyRange=xl)
Ns.append(data['x_train'].shape[0])
error_ss.append(
compute_var_diag(p_truth, p_model, dx0, trim_zeros=True))
rs = [1, 2, 3, 4, 5]
error_r = []
for r in rs:
filename = SavePath + 'LorenzModel_r{}_t1000_ns1.mat'.format(r)
data = sio.loadmat(filename)
x_model = data['x_model'].squeeze()
_, p_model = pt.pdf_1d(x_model,
nx=nbins,
smoothing_sigma=smoothing_sigma,
MyRange=xl)
error_r.append(compute_var_diag(p_truth, p_model, dx0,
trim_zeros=True))
myfig = plt.figure(figsize=[5.5, 2.2])
axw, axx0, axy0 = .33, .15, .2
ax1 = myfig.add_axes([axx0, axy0, axw, .7])
ax1.tick_params(direction='in')
plt.plot(Ns, error_ss, '-s', color=MyColors[0])
plt.yscale('log'), plt.xscale('log')
plt.xlabel(r'sample size')
plt.minorticks_off()
plt.title(r'$e$', fontsize=tfs)
ax2 = myfig.add_axes([axx0 + .15 + axw, axy0, axw, .7])
ax2.tick_params(direction='in')
plt.plot(rs, error_r, '-s', color=MyColors[0])
plt.yscale('log')
plt.xticks(rs)
plt.xlabel(r'polynomial degree')
plt.yticks([1e-2, 5e-3], [r'$ 10^{-2}$', r'$5\times 10^{-3}$'])
plt.xticks(rs)
plt.minorticks_off()
plt.title(r'$e$', fontsize=tfs)
plt.subplots_adjust(bottom=0.35)
plt.savefig('lorenz_analysis.pdf', dpi=450)
