A probability distribution helps understand the likelihood of possible values that a random variable can take. It is one of the must needed statistical knowledge for any data science aspirant.
Few consider, Probability distributions are fundamental to statistics, like data structures are to computer science
In Layman terms
Let’s say, you pick any 100 employees of an organization. Measure their heights (or weights). As you measure them, create a distribution of it on a graph. Keep height on X-Axis & frequency of a particular height on Y-Axis. With this, we will get a distribution for a range of heights.
This distribution will help know which outcomes are most likely, the spread of potential values, and the likelihood of different results.
Basic terminology
Random Sample
The set of 100 people selected above in our example will be termed as random sample.
Sample Space
The range of possible heights of the 100 people is our sample space. It’s the set of all possible values in the setup.
Random Variable
The height of the 100 people measured are termed as random variable. It’s a variable that takes different values of the sample space randomly.
Mean (Expected Value)
Let’s say most of the people in those 100 are of height 5 feet, 3 inches (making it an average height of those 100). This would be termed expected value. It’s an average value of a random variable.
Standard deviation & Variance
Let’s say most of the people in those 100 are of height 5 feet, 1 inches to 5 feet, 5 inches. This is variance for us. It’s an average spread of values around the expected value. Standard Deviation is the square root of the variance.
Types of data
Ordinal – They have a meaningful order. All numerical data fall in this bucket. They can be ordered in relative numerical strength.
Nominal – They cannot be ordered. All categorical data fall in this bucket. Like, colors – Red, Blue & Green – there cannot be an order or a sequence of high or low in them by itself.
Discrete – an ordinal data that can take only certain values (like soccer match score)
Continuous – an ordinal data that can take any real or fractional value (like height & weight)
In Continuous distribution, random variables can have an infinite range of possible outcomes
Probability Distribution Flowchart
Following diagram shares few of the common distributions used:
Based on above diagram, will cover three distributions to have a broad understanding:
Uniform Distribution
It is the simplest form of distribution. Every outcome of the sample space has equal probability to happen. An example would be to roll a fair dice that would have an equal probability outcome of 1-6.
Normal (Gaussian) Distribution
The most common distribution. Few would recognize this by a ‘bell curve’. Most values are around the mean value making the distribution arrangement symmetric.
Central limit theorem suggests that sum of several independent random variables is normally distributed
The area under the distribution curve is equal to 1 (all the probabilities must sum up to 1)
A parameter Mew drives the distribution center (mean). It corresponds to the maximum height of the graph. A parameter Sigma corresponds to the range of variation (variance or standard deviation).
68–95–99.7 rule (empirical rule) – approximate percentage of the data covered by ranges defined by 1, 2, and 3 standard deviations from the mean
Exponential Distribution
It is where a few outcomes are most likely with a rapid decrease in probability to all other outcomes. An example of it would be a car battery life in months.
A parameter Beta deals with scale that defines the mean and standard deviation of the distribution. A parameter Lambda deals with rate of change in the distribution
Probability Distribution Choices
I came across an awesome representation of the probability distribution choices. It works as a cheat sheet to understand the provided data.
Though above is just an introduction, believe it should be good enough to start, correlate and understand some basics of machine learning algorithms. There would be more to it while working on algorithms and problems while analyzing data to predict trends, etc.
While working on a machine learning problem, Matplotlib is the most popular python library used for visualization that helps in representing & analyzing the data and work through insights.
Generally, it’s difficult to interpret much about data, just by looking at it. But, a presentation of the data in any visual form, helps a great deal to peek into it. It becomes easy to deduce correlations, identify patterns & parameters of importance.
In data science world, data visualization plays an important role around data pre-processing stage. It helps in picking appropriate features and apply appropriate machine learning algorithm. Later, it helps in representing the data in a meaningful way.
Data Insights via various plots
If needed, we will use these dataset for plot examples and discussions. Based on the need, following are the common plots that are used:
Line Chart | ax.plot(x,y)
It helps in representing series of data points against a given range of defined parameter. Real benefit is to plot multiple line charts in a single plot to compare and track changes.
Points next to each other are related that helps to identify repeated or a defined pattern
With the above, we have couple of quick assessments: Q: How a particular stock fared over last year? A: Stocks were roughly rising till Feb 2020 and then took a dip in April and then back up since then.
Q: How the three stocks behaved during the same period? A: Stock price of ADBE was more sensitive and AAPL being least sensitive to the change during the same period.
Histogram | ax.hist(data, n_bins)
It helps in showing distributions of variables where it plots quantitative data with range of the data grouped into intervals.
We can use Log scale if the data range is across several orders of magnitude.
import numpy as np
import matplotlib.pyplot as plt
mean = [0, 0]
cov = [[2,4], [5, 9]]
xn, yn = np.random.multivariate_normal(
mean, cov, 100).T
plt.hist(xn,bins=25,label="Distribution on x-axis");
plt.xlabel('x')
plt.ylabel('frequency')
plt.grid(True)
plt.legend()
Real world example
We will work with dataset of Indian Census data downloaded from here.
With the above, couple of quick assessments about population in states of India: Q: What’s the general population distribution of states in India? A: More than 50% of states have population less than 2 crores (20 million)
Q: How many states are having population more than 10 crores (100 million)? A: Only 3 states have that high a population.
Bar Chart | ax.bar(x_pos, heights)
It helps in comparing two or more variables by displaying values associated with categorical data.
Most commonly used plot in Media sharing data around surveys displaying every data sample.
With the above, couple of quick assessments about population in states of India: – Uttar Pradesh has the highest total population and Lakshadeep has lowest – Relative popluation across states with Uttar Pradesh almost double the second most populated state
Pie Chart | ax.pie(sizes, labels=[labels])
It helps in showing the percentage (or proportional) distribution of categories at a certain point of time. Usually, it works well if it’s limited to single digit categories.
A circular statistical graphic where the arc length of each slice is proportional to the quantity it represents.
import numpy as np
import matplotlib.pyplot as plt
# Slices will be ordered n plotted counter-clockwise
labels = ['Audi','BMW','LandRover','Tesla','Ferrari']
sizes = [90, 70, 35, 20, 25]
fig, ax = plt.subplots()
ax.pie(sizes,labels=labels, autopct='%1.1f%%')
ax.set_title('Car Sales')
plt.show()
Real world example
We will work with dataset of Alcohol Consumption downloaded from here.
With the above, we can have a quick assessment that alcohol consumption is distributed overall. This view helps if we have less number of slices (categories).
Scatter plot | ax.scatter(x_points, y_points)
It helps representing paired numerical data either to compare how one variable is affected by another or to see how multiple dependent variables value is spread for each value of independent variable.
Sometimes the data points in a scatter plot form distinct groups and are called as clusters.
import numpy as np
import matplotlib.pyplot as plt
# random but focused cluster data
x1 = np.random.randn(100) + 8
y1 = np.random.randn(100) + 8
x2 = np.random.randn(100) + 3
y2 = np.random.randn(100) + 3
x = np.append(x1,x2)
y = np.append(y1,y2)
plt.scatter(x,y, label="xy distribution")
plt.legend()
Real world example
We will work with dataset of Alcohol Consumption downloaded from here.
import pandas as pd
import matplotlib.pyplot as plt
drinksdf = pd.read_csv('data-files/drinks.csv',
skiprows=1,
names = ['country', 'beer', 'spirit',
'wine', 'alcohol', 'continent'])
drinksdf['total'] = drinksdf['beer']
+ drinksdf['spirit']
+ drinksdf['wine']
+ drinksdf['alcohol']
# drinksdf.corr() tells beer and alcochol
# are highly corelated
fig = plt.figure()
# Compare beet and alcohol consumption
# Use color to show a third variable.
# Can also use size (s) to show a third variable.
scat = plt.scatter(drinksdf['beer'],
drinksdf['alcohol'],
c=drinksdf['total'],
cmap=plt.cm.rainbow)
# colorbar to explain the color scheme
fig.colorbar(scat, label='Total drinks')
plt.xlabel('Beer')
plt.ylabel('Alcohol')
plt.title('Comparing beer and alcohol consumption')
plt.grid(True)
plt.show()
With the above, we can have a quick assessment that beer and alcohol consumption have strong positive correlation which would suggest a large overlap of people who drink beer and alcohol.
2. We will work with dataset of Mall Customers downloaded from here.
With the above, we can have a quick assessment that there are five clusters there and thus five segments or types of customers one can make plan for.
Box Plot | ax.boxplot([data list])
A statistical plot that helps in comparing distributions of variables because the center, spread and range are immediately visible. It only shows the summary statistics like mean, median and interquartile range.
Easy to identify if data is symmetrical, how tightly it is grouped, and if and how data is skewed
import numpy as np
import matplotlib.pyplot as plt
# some random data
data1 = np.random.normal(0, 2, 100)
data2 = np.random.normal(0, 4, 100)
data3 = np.random.normal(0, 3, 100)
data4 = np.random.normal(0, 5, 100)
data = list([data1, data2, data3, data4])
fig, ax = plt.subplots()
bx = ax.boxplot(data, patch_artist=True)
ax.set_title('Box Plot Sample')
ax.set_ylabel('Spread')
xticklabels=['category A',
'category B',
'category B',
'category D']
colors = ['pink','lightblue','lightgreen','yellow']
for patch, color in zip(bx['boxes'], colors):
patch.set_facecolor(color)
ax.set_xticklabels(xticklabels)
ax.yaxis.grid(True)
plt.show()
Real world example
We will work with dataset of Tips downloaded from here.
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
tipsdf = pd.read_csv('data-files/tips.csv')
sns.boxplot(x="time", y="tip",
hue='sex', data=tipsdf,
order=["Dinner", "Lunch"],
palette='coolwarm')
With the above, we can have a quick couple of assessments: – male gender gives more tip compared to females – tips during dinner time can vary a lot (more) by males mean tip
Violen Plot | ax.violinplot([data list])
A statistical plot that helps in comparing distributions of variables because the center, spread and range are immediately visible. It shows the full distribution of data.
A quick way to compare distributions across multiple variables
import numpy as np
import matplotlib.pyplot as plt
data = [np.random.normal(0, std, size=100)
for std in range(2, 6)]
fig, ax = plt.subplots()
bx = ax.violinplot(data)
ax.set_title('Violin Plot Sample')
ax.set_ylabel('Spread')
xticklabels=['category A',
'category B',
'category B',
'category D']
ax.set_xticks([1,2,3,4])
ax.set_xticklabels(xticklabels)
ax.yaxis.grid(True)
plt.show()
Real world example
We will work with dataset of Tips downloaded from here.
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
tipsdf = pd.read_csv('data-files/tips.csv')
sns.violinplot(x="day", y="tip",
split="True", data=tipsdf)
With the above, we can have a quick assessment that the tips on Saturday has more relaxed distribution whereas Friday has much narrow distribution in comparison.
2. We will work with dataset of Indian Census data downloaded from here.
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
populationdf = pd.read_csv(
"./data-files/census-population.csv")
mask1 = populationdf['Level']=='DISTRICT'
mask2 = populationdf['TRU']!='Total'
statesdf = populationdf[mask1 & mask2]
maskUP = statesdf['State']==9
maskM = statesdf['State']==27
data = statesdf.loc[maskUP | maskM]
sns.violinplot( x='State', y='P_06',
inner='quartile', hue='TRU',
palette={'Rural':'green','Urban':'blue'},
scale='count', split=True,
data=data, size=6)
plt.title('In districts of UP and Maharashtra')
plt.show()
With the above, we can have couple of quick assessments: – Uttar Pradesh has high volume and distribution of rural child population. – Maharashtra has almost equal spread of rural and urban child population
Heatmap
It helps in representing a 2-D matrix form of data using variation of color for different values. Variation of color maybe hue or intensity.
Generally used to visualize correlation matrix which in turn helps in features (variables) selection.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
# create 2D array
array_2d = np.random.rand(4, 6)
sns.heatmap(array_2d, annot=True)
Real world example
We will work with dataset of Alcohol Consumption downloaded from here.
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
drinksdf = pd.read_csv('data-files/drinks.csv',
skiprows=1,
names = ['country', 'beer', 'spirit',
'wine', 'alcohol', 'continent'])
sns.heatmap(drinksdf.corr(),annot=True,cmap='YlGnBu')
With the above, we can have a quick couple of assessments: – there is a strong correlation between beer and alcohol and thus a strong overlap there. – wine and spirit are almost not correlated and thus it would be rare to have a place where wine and spirit consumption equally high. One would be preferred over other.
If we notice, upper and lower halves along the diagonal are same. Correlation of A is to B is same as B is to A. Further, A correlation with A will always be 1. Such case, we can make a small tweak to make it more presentable and avoid any correlation confusion.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
drinksdf = pd.read_csv(
'data-files/drinks.csv',
skiprows=1,
names = ['country', 'beer', 'spirit',
'wine', 'alcohol', 'continent'])
# correlation and masks
drinks_cr = drinksdf.corr()
drinks_mask = np.triu(drinks_cr)
# remove the last ones on both axes
drinks_cr = drinks_cr.iloc[1:,:-1]
drinks_mask = drinks_mask[1:, :-1]
sns.heatmap(drinks_cr,
mask=drinks_mask,
annot=True,
cmap='coolwarm')
It is the same correlation data but just the needed one is represented.
Data Image
It helps in displaying data as an image, i.e. on a 2D regular raster.
Images are internally just arrays. Any 2D numpy array can be displayed as an image.
import pandas as pd
import matplotlib.pyplot as plt
M,N = 25,30
data = np.random.random((M,N))
plt.imshow(data)
Real world example
Let’s read an image and then try to display it back to see how it looks
It read the image as an array of matrix and then drew it as plot that turned to be same as the image. Since, images are like any other plots, we can plot other objects (like annotations) on top of it.
Generally, it is used in comparing multiple variables (in pairs) against each other. With multiple plots stacked against each other in the same figure, it helps in quick assessment for correlation and distribution for a pair.
Parameters are: number of rows, number of columns, the index of the subplot
(Index are counted row wise starting with 1)
The widths of the different subplots may be different with use of GridSpec.
import numpy as np
import matplotlib.pyplot as plt
import math
# data setup
x = np.arange(1, 100, 5)
y1 = x**2
y2 = 2*x+4
y3 = [ math.sqrt(i) for i in x]
y4 = [ math.log(j) for j in x]
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2)
ax1.plot(x, y1)
ax1.set_title('f(x) = quadratic')
ax1.grid()
ax2.plot(x, y2)
ax2.set_title('f(x) = linear')
ax2.grid()
ax3.plot(x, y3)
ax3.set_title('f(x) = sqareroot')
ax3.grid()
ax4.plot(x, y4)
ax4.set_title('f(x) = log')
ax4.grid()
fig.tight_layout()
plt.show()
We can stack up m x n view of the variables and have a quick look on how they are correlated. With the above, we can quickly assess that second graph parameters are linearly correlated.
Data Representation
Plot Anatomy
Below picture will help with plots terminology and representation:
Credit: matplotlib.org
Figure above is the base space where the entire plot happens. Most of the parameters can be customized for better representation. For specific details, look here.
Plot annotations
It helps in highlighting few key findings or indicators on a plot. For advanced annotations, look here.
import numpy as np
import matplotlib.pyplot as plt
# A simple parabolic data
x = np.arange(-4, 4, 0.02)
y = x**2
# Setup plot with data
fig, ax = plt.subplots()
ax.plot(x, y)
# Setup axes
ax.set_xlim(-4,4)
ax.set_ylim(-1,8)
# Visual titles
ax.set_title('Annotation Sample')
ax.set_xlabel('X-values')
ax.set_ylabel('Parabolic values')
# Annotation
# 1. Highlighting specific data on the x,y data
ax.annotate('local minima of \n the parabola',
xy=(0, 0),
xycoords='data',
xytext=(2, 3),
arrowprops=
dict(facecolor='red', shrink=0.04),
horizontalalignment='left',
verticalalignment='top')
# 2. Highlighting specific data on the x/y axis
bbox_yproperties = dict(
boxstyle="round,pad=0.4", fc="w", ec="k", lw=2)
ax.annotate('Covers 70% of y-plot range',
xy=(0, 0.7),
xycoords='axes fraction',
xytext=(0.2, 0.7),
bbox=bbox_yproperties,
arrowprops=
dict(facecolor='green', shrink=0.04),
horizontalalignment='left',
verticalalignment='center')
bbox_xproperties = dict(
boxstyle="round,pad=0.4", fc="w", ec="k", lw=2)
ax.annotate('Covers 40% of x-plot range',
xy=(0.3, 0),
xycoords='axes fraction',
xytext=(0.1, 0.4),
bbox=bbox_xproperties,
arrowprops=
dict(facecolor='blue', shrink=0.04),
horizontalalignment='left',
verticalalignment='center')
plt.show()
Plot style | plt.style.use('style')
It helps in customizing representation of a plot, like color, fonts, line thickness, etc. Default styles get applied if the customization is not defined. Apart from adhoc customization, we can also choose one of the already defined template styles and apply them.
# To know all existing styles with package
for style in plt.style.available:
print(style)
# To use a defined style for plot
plt.style.use('seaborn')
# OR
with plt.style.context('Solarize_Light2'):
plt.plot(np.sin(np.linspace(0, 2 * np.pi)), 'r-o')
plt.show()
Saving plots | ax.savefig()
It helps in saving figure with plot as an image file of defined parameters. Parameters details are here. It will save the image file to the current directory by default.
It helps in filling missing data with some reasonable data as many statistical or machine learning packages do not work with data containing null values.
Data interpolation can be defined to use pre-defined functions such as linear, quadratic or cubic
import pandas as pd
import matplotlib.pyplot as plt
df = pd.DataFrame(np.random.randn(20,1))
df = df.where(df<0.5)
fig, (ax1, ax2) = plt.subplots(1, 2)
ax1.plot(df)
ax1.set_title('f(x) = data missing')
ax1.grid()
ax2.plot(df.interpolate())
ax2.set_title('f(x) = data interpolated')
ax2.grid()
fig.tight_layout()
plt.show()
With the above, we see all the missing data replaced with some probably interpolation supported by dataframe based on valid previous and next data.
Animation
At times, it helps in presenting the data as an animation. On a high level, it would need data to be plugged in a loop with delta changes translating into a moving view.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib import animation
fig = plt.figure()
def f(x, y):
return np.sin(x) + np.cos(y)
x = np.linspace(0, 2 * np.pi, 80)
y = np.linspace(0, 2 * np.pi, 70).reshape(-1, 1)
im = plt.imshow(f(x, y), animated=True)
def updatefig(*args):
global x, y
x += np.pi / 5.
y += np.pi / 10.
im.set_array(f(x, y))
return im,
ani = animation.FuncAnimation(
fig, updatefig, interval=100, blit=True)
plt.show()
3-D Plotting
If needed, we can also have an interactive 3-D plot though it might be slow with large datasets.
This is to get started with pandas and try few concrete examples. pandas is a Python based library that helps in reading, transforming, cleaning and analyzing data. It is built on the NumPy package.
pandas is a fast, powerful, flexible and easy to use open source data analysis and manipulation tool, built on top of the Python programming language.
https://pandas.pydata.org
Key data structure in pandas is called DataFrame – it helps to work with tabular data translated as rows of observations and columns of features.
