Pearson Correlation Coefficient AND distancia entre dos punto

Pearson correlation coefficient

In statistics, the Pearson correlation coefficient (PCC, pronounced /ˈpɪərsən/), also referred to as the Pearson's r, Pearson product-moment correlation coefficient (PPMCC) or bivariate correlation,^[1]is a measure of the linear correlation between two variables X and Y. It has a value between +1 and −1, where 1 is total positive linear correlation, 0 is no linear correlation, and −1 is total negative linear correlation. It is widely used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.

Definition[edit]

Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.

For a population[edit]

Pearson's correlation coefficient when applied to a population is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. The formula for ρ^[5] is:

${\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}}$

where:

• ${\displaystyle \operatorname {cov} }$ is the covariance
• ${\displaystyle \sigma _{X}}$ is the standard deviation of ${\displaystyle X}$
• ${\displaystyle \sigma _{Y}}$ is the standard deviation of ${\displaystyle Y}$

The formula for ρ can be expressed in terms of mean and expectation. Since

${\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}$^[5]

Then the formula for ρ can also be written as

${\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}$

where:

• ${\displaystyle \operatorname {cov} }$ and ${\displaystyle \sigma _{X}}$ are defined as above
• ${\displaystyle \mu _{X}}$ is the mean of ${\displaystyle X}$
• ${\displaystyle \operatorname {E} }$ is the expectation.

The formula for ρ can be expressed in terms of uncentered moments. Since

• ${\displaystyle \mu _{X}=\operatorname {E} [X]}$
• ${\displaystyle \mu _{Y}=\operatorname {E} [Y]}$
• ${\displaystyle \sigma _{X}^{2}=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\operatorname {E} [X^{2}]-\operatorname {[} {E}[X]]^{2}}$
• ${\displaystyle \sigma _{Y}^{2}=\operatorname {E} [(Y-\operatorname {E} [Y])^{2}]=\operatorname {E} [Y^{2}]-\operatorname {[} {E}[Y]]^{2}}$
• ${\displaystyle \operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]=\operatorname {E} [(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y],\,}$

the formula for ρ can also be written as

${\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}{{\sqrt {\operatorname {E} [X^{2}]-\operatorname {[} {E}[X]]^{2}}}~{\sqrt {\operatorname {E} [Y^{2}]-\operatorname {[} {E}[Y]]^{2}}}}}.}$

For a sample[edit]

Pearson's correlation coefficient when applied to a sample is commonly represented by the letter r and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for r by substituting estimates of the covariances and variances based on a sample into the formula above. So if we have one dataset {x₁,...,x_n} containing n values and another dataset {y₁,...,y_n} containing n values then that formula for r is:

${\displaystyle r={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}}}}$

where:

• ${\displaystyle n,x_{i},y_{i}}$ are defined as above
• ${\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$ (the sample mean); and analogously for ${\displaystyle {\bar {y}}}$

Rearranging gives us this formula for r:

${\displaystyle r=r_{xy}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-(\sum x_{i})^{2}}}~{\sqrt {n\sum y_{i}^{2}-(\sum y_{i})^{2}}}}}.}$

where:

• ${\displaystyle n,x_{i},y_{i}}$ are defined as above
• This formula suggests a convenient single-pass algorithm for calculating sample correlations, but, depending on the numbers involved, it can sometimes be numerically unstable.

Rearranging again gives us this^[5] formula for r:

${\displaystyle r=r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{{\sqrt {(\sum x_{i}^{2}-n{\bar {x}}^{2})}}~{\sqrt {(\sum y_{i}^{2}-n{\bar {y}}^{2})}}}}.}$

where:

• ${\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}$ are defined as above

An equivalent expression gives the formula for r as the mean of the products of the standard scores as follows:

${\displaystyle r=r_{xy}={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\left({\frac {y_{i}-{\bar {y}}}{s_{y}}}\right)}$

where

• ${\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}$ are defined as above, and ${\displaystyle s_{x},s_{y}}$ are defined below
• ${\displaystyle \left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)}$ is the standard score (and analogously for the standard score of y)

Alternative formulae for r are also available. One can use the following formula for r:

${\displaystyle r=r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{(n-1)s_{x}s_{y}}}}$

where:

• ${\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}}$ are defined as above and:
• ${\displaystyle s_{x}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}$ (the sample standard deviation); and analogously for s_y

Practical issues

Under heavy noise conditions, extracting the correlation coefficient between two sets of stochastic variables is nontrivial, in particular where Canonical Correlation Analysis reports on degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.^[6]

In case of missing data, Garren derived the maximum likelihood estimator.^[7]

Interpretation[edit]

The correlation coefficient ranges from −1 to 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line for which Y increases as Xincreases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear correlation between the variables.

More generally, note that (X_i − X)(Y_i − Y) is positive if and only if X_i and Y_i lie on the same side of their respective means. Thus the correlation coefficient is positive if X_i and Y_i tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative (anti-correlation) if X_i and Y_i tend to lie on opposite sides of their respective means. Moreover, the stronger is either tendency, the larger is the absolute value of the correlation coefficient.

Rogers and Nicewander ^[8] cataloged thirteen ways of interpreting covariance:

• Function of raw scores and means
• Standardized covariance
• Standardized slope of the regression line
• Geometric mean of the two regression slopes
• Square root of the ratio of two variances
• Mean cross-product of standardized variables
• Function of the angle between two standardized regression lines
• Function of the angle between two variable vectors
• Rescaled variance of the difference between standardized scores
• Estimated from the balloon rule
• Related to the bivariate ellipses of isoconcentration
• Function of test statistics from designed experiments
• Ratio of two means

PIP in python and How to install matplotlib on Windows PC

If you're using this information and don't leave a comment, then you're a lammer. A LAMMER

cambiar las variables de entorno

al PATH

C:\Users\rober\AppData\Local\Programs\Python\Python35\Scripts\;C:\Users\rober\AppData\Local\Programs\Python\Python35\

C:\Users\rober\AppData\Local\Programs\Python\Python35\Scripts\;C:\Users\rober\AppData\Local\Programs\Python\Python35\
C:\Users\rober\AppData\Local\Programs\Python\Python35\

PIP

pip es un sistema de gestión de paquetes utilizado para instalar y administrar paquetes de software escritos en Python. Muchos paquetes pueden ser encontrados en el Python Package Index (PyPI). Python 2.7.9 y posteriores (en la serie Python2), Python 3.4 y posteriores incluyen pip (pip3 para Python3) por defecto.

pip es un acrónimo recursivo que se puede interpretar como Pip Instalador de Paquetes o Pip Instalador de Python.

Índice

  [ocultar] 

• 1Interfaz línea de comando
• 2Uso de servicio del alojamiento web
• 3Véase también
• 4Referencias
• 5Enlaces externos

Interfaz línea de comando[editar]

Salida de pip install virtualenv

Una ventaja importante de pip es la facilidad de su interfaz de línea de comandos, el cual permite instalar paquetes de software de Python fácilmente desde solo una orden:

pip install nombre-paquete

Los usuarios también pueden fácilmente desinstalar algún paquete:

pip uninstall nombre-paquete

Otra característica particular de pip es que permite gestionar listas de paquetes y sus números de versión correspondientes a través de un archivo de requisitos. Esto nos permite una recreación eficaz de un conjunto de paquetes en un entorno separado (p. ej. otro ordenador) o entorno virtual. Esto se puede conseguir con un archivo correctamente formateado requisitos.txt y la siguiente orden:

pip install -r requisitos.txt

Con pip es posible instalar un paquete para una versión concreta de Python, sólo es necesario reemplazar ${versión} por la versión de Python que queramos: 2, 3, 3.4, etc:

pip${versión} install nombre-paquete

Uso de servicio del alojamiento web[editar]

Pip es usado para soporte de Python en la nube, como por Heroku.

How to install matplotlib on Windows PC

Install matplotlib in Windows

Matplotlib es una biblioteca para la generación de gráficos a partir de datos contenidos en listas o arrays en el lenguaje de programación Python y su extensión matemática NumPy. Proporciona una API, pylab, diseñada para recordar a la de MATLAB.

I presume that you have already installed python in your PC and added it to system path. And now you want to install one of the best plotting library matplotlib. It is pretty straightforward in a linux-based OS with a package manager. But in Windows –

1) Install distribute: It helps in installing python packages easily. Download file, open windows command prompt or powershell and using ‘cd’ command change to the folder containing the downloaded file, then perform the following command to install distribute –

1	python distribute_setup.py

2) Add easy-install to system PATH: We have to add the Scripts directory to the system path. If you have python 3.3, the path to the Scripts directory is C:\Python33\Scripts\ The method of adding a directory to system path has already shown in the previous tutorial.

3) Install matplotlib: There are certain dependencies needed to be installed for matplotlib working correctly. But these are no more our concern. The distribute package will check for dependencies and install it automatically. We just have to run this command –

1	easy_install matplotlib

Voila! We have successfully installed matplotlib.

4) Test it: Now start python interpreter and run the following commands to test the plotting library –

1 2 3 4 5 6

>> import pylab >> x = [i/10 for i in range(0, 100)] >> y = [i*i for i in x] >> pylab.plot(x, y) [...] >> pylab.show() Dibuja la grafica del coseno, usando pycharms on mathplotlib ejemplo de wiki

Output of the last command is the following plot – 

C:\WINDOWS\system32>python -m pip install -U pip

C:\WINDOWS\system32>pip install scipy

C:\WINDOWS\system32>pip install numpy

NumPy

NumPy is the fundamental package for scientific computing with Python. It contains among other things:

• a powerful N-dimensional array object
• sophisticated (broadcasting) functions
• tools for integrating C/C++ and Fortran code
• useful linear algebra, Fourier transform, and random number capabilities

Besides its obvious scientific uses, NumPy can also be used as an efficient multi-dimensional container of generic data. Arbitrary data-types can be defined. This allows NumPy to seamlessly and speedily integrate with a wide variety of databases.

NumPy is licensed under the BSD license, enabling reuse with few restrictions.

NumPy es una extensión de Python, que le agrega mayor soporte para vectores y matrices, constituyendo una biblioteca de funciones matemáticas de alto nivel para operar con esos vectores o matrices. El ancestro de NumPy, Numeric, fue creado originalmente por Jim Hugunin con algunas contribuciones de otros desarrolladores. En 2005, Travis Oliphant creó NumPy incorporando características de Numarray en NumPy con algunas modificaciones. NumPy es open source.

Ejemplo[editar]

El siguiente es un ejemplo de como manipula vectores y los dibuja en un gráfico usando Matplotlib.

import numpy
from matplotlib import pyplot
x = numpy.linspace(0, 2 * numpy.pi, 100)
y = numpy.sin(x)
pyplot.plot(x, y)
pyplot.show()

SciPy es una biblioteca open source de herramientas y algoritmos matemáticos para Python que nació a partir de la colección original de Travis Oliphant que consistía de módulos de extensión para Python, lanzada en 1999 bajo el nombre de Multipack (llamada por los paquetes netlib que reunían a ODEPACK, QUADPACK, y MINPACK).

SciPy contiene módulos para optimización, álgebra lineal, integración, interpolación, funciones especiales, FFT, procesamiento de señales y de imagen, resolución de ODEs y otras tareas para la ciencia e ingeniería. Está dirigida al mismo tipo de usuarios que los de aplicaciones como MATLAB, GNU Octave, y Scilab.

Véase también[editar]

• NumPy (Biblioteca que da soporte para vectores y matrices, para trabajar con algoritmos de alto nivel)
• SymPy (Biblioteca que define Objetos de tipo formula)
• Matplotlib (Biblioteca que define la Representación de funciones matemáticas en diferentes tipos de gráficas)
• Jupyter Notebook (Programa para análisis científico, y programación en linea) <- Pagina en contruccion

MANUAL DE PLOTEO EN PYTHON  MATPLOTLIB

https://matplotlib.org/users/pyplot_tutorial.html

>>> import pylab

>>> x = [i/10 for i in range(0,100)]

>>> y= [i*i for i in x]

>>> pylab.plot

<function plot at 0x000001D4CB074488>

>>> pylab.plot(x,y)

[<matplotlib.lines.Line2D object at 0x000001D4CCD53940>]

>>> pylab.show()

>>> import numpy

>>> from matplotlib import pyplot

>>> x= numpy.linspace(0, 2*numpy.pi, 100)

>>> y = numpy.sin(x)

>>> pyplot.plot(x,y)

[<matplotlib.lines.Line2D object at 0x000001D4CB89AF98>]

>>> pyplot.show()

EJEMPLO2 

import matplotlib.pyplot as plt

x = range(100)
y = range(100,200)
fig = plt.figure()
ax1 = fig.add_subplot(111)

ax1.scatter(x[:4], y[:4], s=10, c='b', marker="s", label='first')
ax1.scatter(x[40:],y[40:], s=10, c='r', marker="o", label='second')
plt.legend(loc='upper left');
plt.show()

Pyplot tutorial

https://matplotlib.org/1.3.1/users/pyplot_tutorial.html

matplotlib.pyplot is a collection of command style functions that make matplotlib work like MATLAB. Each pyplot function makes some change to a figure: e.g., creates a figure, creates a plotting area in a figure, plots some lines in a plotting area, decorates the plot with labels, etc. In matplotlib.pyplot various states are preserved across function calls, so that it keeps track of things like the current figure and plotting area, and the plotting functions are directed to the current axes (please note that “axes” here and in most places in the documentation refers to the axes part of a figure and not the strict mathematical term for more than one axis).

import matplotlib.pyplot as plt
plt.plot([1,2,3,4])
plt.ylabel('some numbers')
plt.show()

(Source code, png, pdf)

You may be wondering why the x-axis ranges from 0-3 and the y-axis from 1-4. If you provide a single list or array to the plot() command, matplotlib assumes it is a sequence of y values, and automatically generates the x values for you. Since python ranges start with 0, the default x vector has the same length as y but starts with 0. Hence the x data are [0,1,2,3].

plot() is a versatile command, and will take an arbitrary number of arguments. For example, to plot x versus y, you can issue the command:

plt.plot([1, 2, 3, 4], [1, 4, 9, 16])

For every x, y pair of arguments, there is an optional third argument which is the format string that indicates the color and line type of the plot. The letters and symbols of the format string are from MATLAB, and you concatenate a color string with a line style string. The default format string is ‘b-‘, which is a solid blue line. For example, to plot the above with red circles, you would issue

import matplotlib.pyplot as plt
plt.plot([1,2,3,4], [1,4,9,16], 'ro')
plt.axis([0, 6, 0, 20])
plt.show()

(Source code, png, pdf)

See the plot() documentation for a complete list of line styles and format strings. The axis() command in the example above takes a list of [xmin, xmax, ymin, ymax] and specifies the viewport of the axes.

If matplotlib were limited to working with lists, it would be fairly useless for numeric processing. Generally, you will use numpy arrays. In fact, all sequences are converted to numpy arrays internally. The example below illustrates a plotting several lines with different format styles in one command using arrays.

import numpy as np
import matplotlib.pyplot as plt

# evenly sampled time at 200ms intervals
t = np.arange(0., 5., 0.2)

# red dashes, blue squares and green triangles
plt.plot(t, t, 'r--', t, t**2, 'bs', t, t**3, 'g^')
plt.show()

(Source code, png, pdf)

Controlling line properties

Lines have many attributes that you can set: linewidth, dash style, antialiased, etc; see matplotlib.lines.Line2D. There are several ways to set line properties

• Use keyword args:

  plt.plot(x, y, linewidth=2.0)
  

• Use the setter methods of a Line2D instance. plot returns a list of Line2D objects; e.g., line1, line2 = plot(x1, y1, x2, y2). In the code below we will suppose that we have only one line so that the list returned is of length 1. We use tuple unpacking with line, to get the first element of that list:

  line, = plt.plot(x, y, '-')
  line.set_antialiased(False) # turn off antialising
  

• Use the setp() command. The example below uses a MATLAB-style command to set multiple properties on a list of lines. setp works transparently with a list of objects or a single object. You can either use python keyword arguments or MATLAB-style string/value pairs:

  lines = plt.plot(x1, y1, x2, y2)
  # use keyword args
  plt.setp(lines, color='r', linewidth=2.0)
  # or MATLAB style string value pairs
  plt.setp(lines, 'color', 'r', 'linewidth', 2.0)
  

Here are the available Line2D properties.

Property	Value Type
alpha	float
animated	[True | False]
antialiased or aa	[True | False]
clip_box	a matplotlib.transform.Bbox instance
clip_on	[True | False]
clip_path	a Path instance and a Transform instance, a Patch
color or c	any matplotlib color
contains	the hit testing function
dash_capstyle	['butt' | 'round' | 'projecting']
dash_joinstyle	['miter' | 'round' | 'bevel']
dashes	sequence of on/off ink in points
data	(np.array xdata, np.array ydata)
figure	a matplotlib.figure.Figure instance
label	any string
linestyle or ls	[ '-' | '--' | '-.' | ':' | 'steps' | ...]
linewidth or lw	float value in points
lod	[True | False]
marker	[ '+' | ',' | '.' | '1' | '2' | '3' | '4' ]
markeredgecolor or mec	any matplotlib color
markeredgewidth or mew	float value in points
markerfacecolor or mfc	any matplotlib color
markersize or ms	float
markevery	[ None | integer | (startind, stride) ]
picker	used in interactive line selection
pickradius	the line pick selection radius
solid_capstyle	['butt' | 'round' | 'projecting']
solid_joinstyle	['miter' | 'round' | 'bevel']
transform	a matplotlib.transforms.Transform instance
visible	[True | False]
xdata	np.array
ydata	np.array
zorder	any number

To get a list of settable line properties, call the setp() function with a line or lines as argument

In [69]: lines = plt.plot([1, 2, 3])

In [70]: plt.setp(lines)
  alpha: float
  animated: [True | False]
  antialiased or aa: [True | False]
  ...snip

Working with multiple figures and axes

MATLAB, and pyplot, have the concept of the current figure and the current axes. All plotting commands apply to the current axes. The function gca() returns the current axes (a matplotlib.axes.Axes instance), and gcf() returns the current figure (matplotlib.figure.Figure instance). Normally, you don’t have to worry about this, because it is all taken care of behind the scenes. Below is a script to create two subplots.

import numpy as np
import matplotlib.pyplot as plt

def f(t):
    return np.exp(-t) * np.cos(2*np.pi*t)

t1 = np.arange(0.0, 5.0, 0.1)
t2 = np.arange(0.0, 5.0, 0.02)

plt.figure(1)
plt.subplot(211)
plt.plot(t1, f(t1), 'bo', t2, f(t2), 'k')

plt.subplot(212)
plt.plot(t2, np.cos(2*np.pi*t2), 'r--')
plt.show()

(Source code, png, pdf)

The figure() command here is optional because figure(1) will be created by default, just as a subplot(111) will be created by default if you don’t manually specify any axes. The subplot() command specifies numrows, numcols, fignum where fignum ranges from 1 tonumrows*numcols. The commas in the subplot command are optional if numrows*numcols<10. So subplot(211) is identical to subplot(2, 1, 1). You can create an arbitrary number of subplots and axes. If you want to place an axes manually, i.e., not on a rectangular grid, use the axes()command, which allows you to specify the location as axes([left, bottom, width, height]) where all values are in fractional (0 to 1) coordinates. See pylab_examples example code: axes_demo.py for an example of placing axes manually and pylab_examples example code: subplots_demo.py for an example with lots of subplots.

You can create multiple figures by using multiple figure() calls with an increasing figure number. Of course, each figure can contain as many axes and subplots as your heart desires:

import matplotlib.pyplot as plt
plt.figure(1)                # the first figure
plt.subplot(211)             # the first subplot in the first figure
plt.plot([1, 2, 3])
plt.subplot(212)             # the second subplot in the first figure
plt.plot([4, 5, 6])


plt.figure(2)                # a second figure
plt.plot([4, 5, 6])          # creates a subplot(111) by default

plt.figure(1)                # figure 1 current; subplot(212) still current
plt.subplot(211)             # make subplot(211) in figure1 current
plt.title('Easy as 1, 2, 3') # subplot 211 title

You can clear the current figure with clf() and the current axes with cla(). If you find it annoying that states (specifically the current image, figure and axes) are being maintained for you behind the scenes, don’t despair: this is just a thin stateful wrapper around an object oriented API, which you can use instead (see Artist tutorial)

If you are making lots of figures, you need to be aware of one more thing: the memory required for a figure is not completely released until the figure is explicitly closed with close(). Deleting all references to the figure, and/or using the window manager to kill the window in which the figure appears on the screen, is not enough, because pyplot maintains internal references until close() is called.

Working with text

The text() command can be used to add text in an arbitrary location, and the xlabel(), ylabel() and title() are used to add text in the indicated locations (see Text introduction for a more detailed example)

import numpy as np
import matplotlib.pyplot as plt

# Fixing random state for reproducibility
np.random.seed(19680801)

mu, sigma = 100, 15
x = mu + sigma * np.random.randn(10000)

# the histogram of the data
n, bins, patches = plt.hist(x, 50, normed=1, facecolor='g', alpha=0.75)


plt.xlabel('Smarts')
plt.ylabel('Probability')
plt.title('Histogram of IQ')
plt.text(60, .025, r'$\mu=100,\ \sigma=15$')
plt.axis([40, 160, 0, 0.03])
plt.grid(True)
plt.show()

(Source code, png, pdf)

All of the text() commands return an matplotlib.text.Text instance. Just as with with lines above, you can customize the properties by passing keyword arguments into the text functions or using setp():

t = plt.xlabel('my data', fontsize=14, color='red')

These properties are covered in more detail in Text properties and layout.

Using mathematical expressions in text

matplotlib accepts TeX equation expressions in any text expression. For example to write the expression  in the title, you can write a TeX expression surrounded by dollar signs:

plt.title(r'$\sigma_i=15$')

The r preceding the title string is important – it signifies that the string is a raw string and not to treat backslashes as python escapes. matplotlib has a built-in TeX expression parser and layout engine, and ships its own math fonts – for details see Writing mathematical expressions. Thus you can use mathematical text across platforms without requiring a TeX installation. For those who have LaTeX and dvipng installed, you can also use LaTeX to format your text and incorporate the output directly into your display figures or saved postscript – see Text rendering With LaTeX.

Annotating text

The uses of the basic text() command above place text at an arbitrary position on the Axes. A common use for text is to annotate some feature of the plot, and the annotate() method provides helper functionality to make annotations easy. In an annotation, there are two points to consider: the location being annotated represented by the argument xy and the location of the text xytext. Both of these arguments are (x,y) tuples.

import numpy as np
import matplotlib.pyplot as plt

ax = plt.subplot(111)

t = np.arange(0.0, 5.0, 0.01)
s = np.cos(2*np.pi*t)
line, = plt.plot(t, s, lw=2)

plt.annotate('local max', xy=(2, 1), xytext=(3, 1.5),
            arrowprops=dict(facecolor='black', shrink=0.05),
            )

plt.ylim(-2,2)
plt.show()

(Source code, png, pdf)

In this basic example, both the xy (arrow tip) and xytext locations (text location) are in data coordinates. There are a variety of other coordinate systems one can choose – see Basic annotation and Advanced Annotation for details. More examples can be found in pylab_examples example code: annotation_demo.py.

Logarithmic and other nonlinear axes

matplotlib.pyplot supports not only linear axis scales, but also logarithmic and logit scales. This is commonly used if data spans many orders of magnitude. Changing the scale of an axis is easy:

plt.xscale(‘log’)

An example of four plots with the same data and different scales for the y axis is shown below.

import numpy as np
import matplotlib.pyplot as plt

from matplotlib.ticker import NullFormatter  # useful for `logit` scale

# Fixing random state for reproducibility
np.random.seed(19680801)

# make up some data in the interval ]0, 1[
y = np.random.normal(loc=0.5, scale=0.4, size=1000)
y = y[(y > 0) & (y < 1)]
y.sort()
x = np.arange(len(y))

# plot with various axes scales
plt.figure(1)

# linear
plt.subplot(221)
plt.plot(x, y)
plt.yscale('linear')
plt.title('linear')
plt.grid(True)


# log
plt.subplot(222)
plt.plot(x, y)
plt.yscale('log')
plt.title('log')
plt.grid(True)


# symmetric log
plt.subplot(223)
plt.plot(x, y - y.mean())
plt.yscale('symlog', linthreshy=0.01)
plt.title('symlog')
plt.grid(True)

# logit
plt.subplot(224)
plt.plot(x, y)
plt.yscale('logit')
plt.title('logit')
plt.grid(True)
# Format the minor tick labels of the y-axis into empty strings with
# `NullFormatter`, to avoid cumbering the axis with too many labels.
plt.gca().yaxis.set_minor_formatter(NullFormatter())
# Adjust the subplot layout, because the logit one may take more space
# than usual, due to y-tick labels like "1 - 10^{-3}"
plt.subplots_adjust(top=0.92, bottom=0.08, left=0.10, right=0.95, hspace=0.25,
                    wspace=0.35)

plt.show()

(Source code, png, pdf)

UPDATE 2021

import numpy as np

import matplotlib.pyplot as plt

plt.show()

import numpy as np

import matplotlib.pyplot as plt

points = np.array([

(197,3116.2),

(198,3103.0),

(199,3156.1),

(200,3078.1),

(201,3008.7),

(202,2954.9),

(203,2960.5),

(204,3129.0),

(205,2999.9),

(206,3019.8),

(207,3095.1),

(208,3174.1),

(209,3144.9),

(210,3148.7),

(211,3221.3),

(212,3125.0),

(213,3199.2),

(214,3100.0),

(215,3195.7),

(216,3190.6),

(217,3286.6),

(218,3442.9),

(219,3443.6),

(220,3363.7),

(221,3338.6),

(222,3272.7),

(223,3207.2),

(224,3217.0),

(225,3184.9),

(226,3176.4),

(227,3204.4),

(228,3207.0),

(229,3286.3),

(230,3162.8),

(231,3211.0),

(232,3036.1),

(233,3004.5),

(234,3048.4),

(235,3241.2),

(236,3322.0),

(237,3311.4),

(238,3143.7),

(239,3035.0),

(240,3137.4),

(241,3110.3),

(242,3128.8),

(243,3131.1),

(244,3135.7),

(245,3105.5),

(246,3117.0),

(247,3099.4),

(248,3098.4),

(249,3118.1),

(250,3185.1),

(251,3195.3),

(252,3168.0),

(253,3220.1),

(254,3203.5),

(255,3186.7),

(256,3162.6),

(257,3158.0),

(258,3177.3),

(259,3104.2),

(260,3101.5),

(261,3116.4),

(262,3157.0),

(263,3165.1),

(264,3241.0),

(265,3236.1),

(266,3201.6),

(267,3206.2),

(268,3206.5),

(269,3185.3),

(270,3172.7),

(271,3284.0),

(272,3322.0),

(273,3285.9),

(274,3256.9),

(275,3186.6),

(276,3218.5)])

# get x and y vectors

x = points[:,0]

y = points[:,1]

# calculate polynomial

z = np.polyfit(x, y, 3)

f = np.poly1d(z)

# calculate new x's and y's

x_new = np.linspace(x[0], x[-1], 50)

y_new = f(x_new)

plt.plot(x,y,'o', x_new, y_new)

plt.xlim([x[0]-1, x[-1] + 1 ])

plt.show()