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Commit 841aab3c authored by Thomas Keck's avatar Thomas Keck
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Added scipy part

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%% Cell type:markdown id: tags:
<div style="position:relative;">
<img src=images/scipy_med.png style="width: 60px; float: left" />
</div>
<div style="position:relative;">
<img src=images/scipy_med.png style="width: 60px; float: right" />
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# &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; SciPy Library
The **SciPy Library** (not to be confused with the whole SciPy Project, which includes the other mentioned packages) offers additional functionality on top of **NumPy** (list is not complete)
- **scipy.special** special functions like the bessel functions or the incomplete gamma function
- **scipy.integrate** numerical integration techniques
- **scipy.optimize** optimization algorithms for curve fitting, minimization and root finding
- **scipy.interpolate** multi-dimensional interpolation algorithms
- **scipy.fftpack** Fast Fourier Transformation
- **scipy.signal** Signal processing toolbox (e.g. filters and spectral analysis)
- **scipy.linalg** contains numpy.linalg plus additional more advanced functions
- **scipy.stats** Statistical functions (e.g. distributions)
%% Cell type:markdown id: tags:
## scipy.stats
Offers nearly 100 continuous and discrete distributions, and statistical tests.
%% Cell type:code id: tags:
``` python
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
```
%% Cell type:code id: tags:
``` python
import scipy.stats
x = np.linspace(-5, 5, 100)
standard_normal_pdf = scipy.stats.norm.pdf(x)
plt.plot(x, standard_normal_pdf)
```
%% Output
[<matplotlib.lines.Line2D at 0x7f04b93037b8>]
%% Cell type:markdown id: tags:
## Shape Parameters
Each distribution has at least two **shape parameters**:
- **loc** defines the position of distribution in some sense
- **scale** defines the width of the distribution in some sense
%% Cell type:code id: tags:
``` python
plt.plot(x, scipy.stats.norm.pdf(x), label='Standard normal')
plt.plot(x, scipy.stats.norm.pdf(x, loc=1), label='Shifted')
plt.plot(x, scipy.stats.norm.pdf(x, scale=2.0), label='Scaled')
plt.plot(x, scipy.stats.norm.pdf(x, loc=-1, scale=0.5), label='Shifted and Scaled')
plt.legend()
plt.show()
```
%% Output
%% Cell type:markdown id: tags:
## Distribution Properties
Each distributions defines several methods (list not complete):
- **pdf** The probability density function
- **cdf** The cumulative density function (integral of pdf)
- **ppf** The percentile function (inverse of cdf)
- **moment** N-th non-central moment
- **rvs** Random numbers drawn from this distribution
- **fit** Parameter estimates for generic data
%% Cell type:code id: tags:
``` python
x = np.linspace(-4, 4, 100)
plt.plot(x, scipy.stats.norm.pdf(x), label='PDF')
plt.legend(loc='upper left')
plt.grid(True)
```
%% Output
%% Cell type:code id: tags:
``` python
plt.plot(x, scipy.stats.norm.cdf(x), label='CDF')
plt.legend(loc='upper left')
plt.grid(True)
```
%% Output
%% Cell type:code id: tags:
``` python
plt.plot(x, scipy.stats.norm.ppf(x), label='PPF')
plt.legend(loc='upper left')
plt.grid(True)
```
%% Output
%% Cell type:code id: tags:
``` python
plt.hist(scipy.stats.norm.rvs(size=1000), bins=20, label='RVS', alpha=0.75)
plt.legend(loc='upper left')
plt.grid(True)
```
%% Output
%% Cell type:markdown id: tags:
## Statistical tests
Here's a more advanced example.
We create 1000 samples, each sample contains 20 randomly drawn numbers.
Now we perform a statisical test, and check if the 20 randomly drawn numbers are compatible with a gaussian distribution.
We do this 1000 times and plot a histogram of the p-values returned by the test.
%% Cell type:code id: tags:
``` python
random_gauss_samples = scipy.stats.norm.rvs(size=(20, 1000))
random_cauchy_samples = scipy.stats.cauchy.rvs(size=(20, 1000))
plt.title('Distribution of P-Values of Normaltest for differently distributed samples')
plt.hist(scipy.stats.normaltest(random_gauss_samples, axis=0).pvalue, bins=10, range=(0,1), label='Gauss', alpha=0.5)
plt.hist(scipy.stats.normaltest(random_cauchy_samples, axis=0).pvalue, bins=10, range=(0,1), label='Cauchy', alpha=0.5)
plt.legend()
plt.show()
```
%% Output
%% Cell type:markdown id: tags:
## scipy.optimize
Offers:
- Unconstrained and constrained minimization
- Global /brute-force) optimization routines
- Least-square minimization
- Root finding
- Multivariate equation system solvers
%% Cell type:code id: tags:
``` python
import scipy.optimize
```
%% Cell type:markdown id: tags:
### Root finding and equation solving
A transcedental equation like $exp(-x) = -x$, cannot be solved analytically.
One can transform cast an equation into a root finding problem:
$$exp(-x) - x = 0$$
scipy.optimize.newton is one possible algorithms which can calculate the root of a function,
and therefore solve the equation.
%% Cell type:code id: tags:
``` python
def transcedental_equation(x):
return np.exp(-x) - x
root = scipy.optimize.newton(transcedental_equation, 0.0)
print(root)
```
%% Output
0.56714329041
%% Cell type:code id: tags:
``` python
x = np.linspace(-2, 2, 100)
plt.plot(x, np.exp(-x), label='exp(-x)')
plt.plot(x, x, label='x')
plt.plot(x, transcedental_equation(x), label='exp(-x) - x')
plt.axhline(0, color='black', linestyle='--')
plt.ylim((-2, 4))
plt.annotate('Intersection', xy=(root, root), xytext=(1, 3), arrowprops=dict(fc='black', width=0.5, headwidth=5.0))
plt.show()
```
%% Output
%% Cell type:markdown id: tags:
## Exercise
#### Task 1
Calculate the probability that a value drawn from a standard normal distribution is in the interval (-1, 1), (-2, 2) and (-3, 3)
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