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import numpy as np
from numpy import pi
import matplotlib.pyplot as plt
import matplotlib.animation as animation

fig = plt.figure()
fig.set_dpi(100)
ax1 = fig.add_subplot(1,1,1)

#Diffusion constant

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###############################################################################
# Confidence intervals for the population mean (t-student distribution)
# We assume that
# 1. Data is normally distributed
# 2. Samples are iid
#
# Note that population variance is unknown and therefore must be
# estimated. In this case Student distribution should be used.
# The number of degree of freedom is n-1 where n is the size of
# the sample. Note that as n -> Inf the Student distribution tends to

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###############################################################################
# The code below can be used to perform a z-test under the following
# assumptions:
# 1. The data is normally distributed
# 2. Samples are iid
# 
# Remember that:
# 1.Low pvalue: strong empirical evidence against h0
# 2.High pvalue: little or 'no' empirical evidence against h0
#

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# Plot of the Student distribution
dfs <- length(data_vector)-1
x_ <- seq(-8,8,0.1)
y <- dt(x_,dfs)
t.val <- qt(1-0.05,df=dfs)
plot(x_,y,type='l',lwd=3,col='blue',xlab='x',ylab='Density',main='Student distribution 9 dof')
abline(v=0)
abline(v=t.val,lwd=2,col='red')
points(t.val,dt(t.val,dfs),lwd=3,col='red')

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% Set up
x = -4:0.2:4;
y1 = -4:0.2:4;
y = (-4:0.2:4)*1i;

[X, Y]= meshgrid(x,y);

% Complex variable s
s = X + Y;
% Complex function f(z)

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%%Gradient and flow

figure
subplot(1,2,1);
[Dx, Dy] = gradient(real(z));
Dx(isinf(Dx)) = 0;
Dy(isinf(Dy)) = 0;
hQuiver = quiver(x,y1,Dx,Dy,'LineWidth',1); hold on;
viscircles([0 0],1,'LineWidth',1); hold off;
title('u(x,y) gradient, vector field');

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figure
subplot(1,3,1)
C1 = contour(x,y1,abs(z),linspace(-10,10,100)); title('Contour of abs(f)');
xlabel('x'); ylabel('y'); %clabel(C1);
subplot(1,3,2)
C2 = contour(x,y1,real(z),linspace(-10,10,100)); title('Contour of re(f)');
xlabel('x'); ylabel('y'); %clabel(C2);
subplot(1,3,3)
C3 = contour(x,y1,imag(z),linspace(-10,10,100)); title('Contour of Im(f)');
xlabel('x'); ylabel('y'); %clabel(C3);

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# Simulated data (normal and beta distributions)
set.seed(70)
data_norm <- rnorm(100,0,1)
data_beta <- rbeta(1000,2,30)*10

# Compare the boxplots
boxplot(data_norm,data_beta,col='cyan')

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# Skewness
library(moments)

print("Skewness:")
print("Normal")
skewness(data_norm)
print("Beta")
skewness(data_beta)
                    # < 0 skewed left (left tail)
                    # > 0 skewed right (right tail)

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# Compare the histograms
hist(data_norm,col='red')
hist(data_beta,col='blue')