6 Advanced Statistical Concepts in Data Science

Nagesh Singh Chauhan

Contents Outline

6 Advanced Statistical Concepts in Data Science

Oct 16, 2021 17 minutes read

The article contains some of the most commonly used advanced statistical concepts along with their Python implementation.

In my previous articles Beginners Guide to Statistics in Data Science and The Inferential Statistics Data Scientists Should Know we have talked about almost all the basics(Descriptive and Inferential) of statistics which are commonly used in understanding and working with any data science case study. In this article, lets go a little beyond and talk about some advance concepts which are not part of the buzz.


Concept #1 - Q-Q(quantile-quantile) Plots


Before understanding QQ plots first understand what is a Quantile?

A quantile defines a particular part of a data set, i.e. a quantile determines how many values in a distribution are above or below a certain limit. Special quantiles are the quartile (quarter), the quintile (fifth), and percentiles (hundredth).



An example:

If we divide a distribution into four equal portions, we will speak of four quartiles. The first quartile includes all values that are smaller than a quarter of all values. In a graphical representation, it corresponds to 25% of the total area of distribution. The two lower quartiles comprise 50% of all distribution values. The interquartile range between the first and third quartile equals the range in which 50% of all values lie that are distributed around the mean. 


In Statistics, A Q-Q(quantile-quantile) plot is a scatterplot created by plotting two sets of quantiles against one another. If both sets of quantiles came from the same distribution, we should see the points forming a line that’s roughly straight(y=x).







For example, the median is a quantile where 50% of the data fall below that point and 50% lie above it. The purpose of Q Q plots is to find out if two sets of data come from the same distribution. A 45-degree angle is plotted on the Q Q plot; if the two data sets come from a common distribution, the points will fall on that reference line.


It’s very important for you to know whether the distribution is normal or not so as to apply various statistical measures on the data and interpret it in much more human-understandable visualization and their Q-Q plot comes into the picture. The most fundamental question answered by the Q-Q plot is if the curve is Normally Distributed or not.


Normally distributed, but why?


The Q-Q plots are used to find the type of distribution for a random variable whether it is a Gaussian Distribution, Uniform Distribution, Exponential Distribution, or even Pareto Distribution, etc. 

You can tell the type of distribution using the power of the Q-Q plot just by looking at the plot. In general, we are talking about Normal distributions only because we have a very beautiful concept of the 68–95–99.7 rule which perfectly fits into the normal distribution So we know how much of the data lies in the range of the first standard deviation, second standard deviation and third standard deviation from the mean. So knowing if a distribution is Normal opens up new doors for us to experiment with 




 Types of Q-Q plots. Source 


Skewed Q-Q plots


Q-Q plots can find skewness(measure of asymmetry) of the distribution. 

If the bottom end of the Q-Q plot deviates from the straight line but the upper end is not, then the distribution is Left skewed(Negatively skewed).


Now if upper end of the Q-Q plot deviates from the staright line and the lower is not, then the distribution is Right skewed(Positively skewed).



Tailed Q-Q plots


Q-Q plots can find Kurtosis(measure of tailedness) of the distribution.

The distribution with the fat tail will have both the ends of the Q-Q plot to deviate from the straight line and its centre follows the line, where as a thin tailed distribution will term Q-Q plot with very less or negligible deviation at the ends thus making it a perfect fit for normal distribution.



Q-Q plots in Python(Source)


Suppose we have the following dataset of 100 values:


import numpy as np

#create dataset with 100 values that follow a normal distribution
np.random.seed(0)
data = np.random.normal(0,1, 1000)

#view first 10 values
data[:10] 

array([ 1.76405235,  0.40015721,  0.97873798,  2.2408932 ,  1.86755799,
       -0.97727788,  0.95008842, -0.15135721, -0.10321885,  0.4105985 ])


To create a Q-Q plot for this dataset, we can use the qqplot() function from the statsmodels library:

import statsmodels.api as sm
import matplotlib.pyplot as plt

#create Q-Q plot with 45-degree line added to plot
fig = sm.qqplot(data, line='45')
plt.show()





In a Q-Q plot, the x-axis displays the theoretical quantiles. This means it doesn’t show your actual data, but instead, it represents where your data would be if it were normally distributed.

The y-axis displays your actual data. This means that if the data values fall along a roughly straight line at a 45-degree angle, then the data is normally distributed.

We can see in our Q-Q plot above that the data values tend to closely follow the 45-degree, which means the data is likely normally distributed. This shouldn’t be surprising since we generated the 100 data values by using the numpy.random.normal() function.


Consider instead if we generated a dataset of 100 uniformly distributed values and created a Q-Q plot for that dataset:


#create dataset of 100 uniformally distributed values
data = np.random.uniform(0,1, 1000)

#generate Q-Q plot for the dataset
fig = sm.qqplot(data, line='45')
plt.show()





The data values clearly do not follow the red 45-degree line, which is an indication that they do not follow a normal distribution.



Concept #2- Chebyshev's Inequality



In probability, Chebyshev’s Inequality, also known as “Bienayme-Chebyshev” Inequality guarantees that, for a wide class of probability distributions, only a definite fraction of values will be found within a specific distance from the mean of a distribution.








Chebyshev’s inequality is similar to The Empirical rule(68-95-99.7); however, the latter rule only applies to normal distributions. Chebyshev’s inequality is broader; it can be applied to any distribution so long as the distribution includes a defined variance and mean.


So Chebyshev’s inequality says that at least (1-1/k^2) of data from a sample must fall within K standard deviations from the mean (or equivalently, no more than 1/k^2 of the distribution’s values can be more than k standard deviations away from the mean).

Where K --> Positive real number


If the data is not normally distributed then different amounts of data could be in one standard deviation. Chebyshev’s inequality provides a way to know what fraction of data falls within K standard deviations from the mean for any data distribution.

Also read: 22 Statistics Questions to Prepare for Data Science Interviews








Chebyshev’s inequality is of great value because it can be applied to any probability distribution in which the mean and variance are provided.


Let us consider an example, Assume 1,000 contestants show up for a job interview, but there are only 70 positions available. In order to select the finest 70 contestants amongst the total contestants, the proprietor gives tests to judge their potential. The mean score on the test is 60, with a standard deviation of 6. If an applicant scores an 84, can they presume that they are getting the job?





The results show that about 63 people scored above a 60, so with 70 positions available, a contestant who scores an 84 can be assured they got the job.


Chebyshev's Inequality in Python(Source)

 
Create a population of 1,000,000 values, I use a gamma distribution(also works with other distributions) with shape = 2 and scale = 2.

import numpy as np
import random
import matplotlib.pyplot as plt

#create a population with a gamma distribution
shape, scale = 2., 2.  #mean=4, std=2*sqrt(2)
mu = shape*scale #mean and standard deviation
sigma = scale*np.sqrt(shape)

s = np.random.gamma(shape, scale, 1000000)


Now sample 10,000 values from the population.

#sample 10000 values
rs = random.choices(s, k=10000)


Count the sample that has a distance from the expected value larger than k standard deviation and use the count to calculate the probabilities. I want to depict a trend of probabilities when k is increasing, so I use a range of k from 0.1 to 3.

#set k
ks = [0.1,0.5,1.0,1.5,2.0,2.5,3.0]

#probability list
probs = [] #for each k

for k in ks: 
    #start count
    c = 0
    for i in rs:
        # count if far from mean in k standard deviation
        if abs(i - mu) > k * sigma :
            c += 1
    probs.append(c/10000)


Plot the results:

plot = plt.figure(figsize=(20,10))
#plot each probability
plt.xlabel('K')
plt.ylabel('probability')
plt.plot(ks,probs, marker='o')
plot.show()
#print each probability
print("Probability of a sample far from mean more than k standard deviation:")
for i, prob in enumerate(probs):
    print("k:" + str(ks[i]) + ", probability: " \
          + str(prob)[0:5] + \
          " | in theory, probability should less than: " \
          + str(1/ks[i]**2)[0:5])










From the above plot and result, we can see that as the k increases, the probability is decreasing, and the probability of each k follows the inequality. Moreover, only the case that k is larger than 1 is useful. If k is less than 1, the right side of the inequality is larger than 1 which is not useful because the probability cannot be larger than 1.



Concept #3- Log-Normal Distribution



In probability theory, a Log-normal distribution also known as Galton's distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.


Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y i.e, X = exp(Y), has a log-normal distribution. 


Skewed distributions with low mean and high variance and all positive values fit under this type of distribution. A random variable that is log-normally distributed takes only positive real values. 


The general formula for the probability density function of the lognormal distribution is:






The location and scale parameters are equivalent to the mean and standard deviation of the logarithm of the random variable.


The shape of Lognormal distribution is defined by 3 parameters:

  1. σis the shape parameter, (and is the standard deviation of the log of the distribution)

  2. θ or μ is the location parameter (and is the mean of the distribution)

  3. m is the scale parameter (and is also the median of the distribution)

The location and scale parameters are equivalent to the mean and standard deviation of the logarithm of the random variable as explained above.


If x = θ, then f(x) = 0. The case where θ = 0 and m = 1 is called the standard lognormal distribution. The case where θ equals zero is called the 2-parameter lognormal distribution.


The following graph illustrates the effect of the location(μ) and scale(σ) parameter on the probability density function of the lognormal distribution: 








Log-Normal Distribution in Python(Source)

Let us consider an example to generate random numbers from a log-normal distribution with μ=1 and σ=0.5 using scipy.stats.lognorm function.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import lognorm

np.random.seed(42)

data = lognorm.rvs(s=0.5, loc=1, scale=1000, size=1000)

plt.figure(figsize=(10,6))
ax = plt.subplot(111)
plt.title('Generate wrandom numbers from a Log-normal distribution')
ax.hist(data, bins=np.logspace(0,5,200), density=True)
ax.set_xscale("log")

shape,loc,scale = lognorm.fit(data)

x = np.logspace(0, 5, 200)
pdf = lognorm.pdf(x, shape, loc, scale)

ax.plot(x, pdf, 'y')
plt.show()







Concept #4- Power Law distribution



In statistics, a Power Law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. 



For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.



A power law distribution has the form Y = k Xα, 

where:

  • X and Y are variables of interest,

  • α is the law’s exponent,

  • k is a constant.








Power-law distribution is just one of many probability distributions, but it is considered a valuable tool to assess uncertainty issues that normal distribution cannot handle when they occur at a certain probability.



Many processes have been found to follow power laws over substantial ranges of values. From the distribution in incomes, size of meteoroids, earthquake magnitudes, the spectral density of weight matrices in deep neural networks, word usage, number of neighbors in various networks, etc. (Note: The power law here is a continuous distribution. The last two examples are discrete, but on a large scale can be modeled as if continuous).

Also read: Statistical Measures of Central Tendency



Power-law distribution in Python(Source) 


Let us plot the Pareto distribution which is one form of a power-law probability distribution. Pareto distribution is sometimes known as the Pareto Principle or ‘80–20’ rule, as the rule states that 80% of society’s wealth is held by 20% of its population. Pareto distribution is not a law of nature, but an observation. It is useful in many real-world problems. It is a skewed heavily tailed distribution.



import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import pareto

x_m = 1 #scale
alpha = [1, 2, 3] #list of values of shape parameters
plt.figure(figsize=(10,6))
samples = np.linspace(start=0, stop=5, num=1000)
for a in alpha:
    output = np.array([pareto.pdf(x=samples, b=a, loc=0, scale=x_m)])
    plt.plot(samples, output.T, label='alpha {0}' .format(a))

plt.xlabel('samples', fontsize=15)
plt.ylabel('PDF', fontsize=15)
plt.title('Probability Density function', fontsize=15)
plt.legend(loc='best')
plt.show()






Concept #5- Box cox transformation




The Box-Cox transformation transforms our data so that it closely resembles a normal distribution.


The one-parameter Box-Cox transformations are defined as In many statistical techniques, we assume that the errors are normally distributed. This assumption allows us to construct confidence intervals and conduct hypothesis tests. By transforming your target variable, we can (hopefully) normalize our errors (if they are not already normal).


Additionally, transforming our variables can improve the predictive power of our models because transformations can cut away white noise.






Original distribution(Left) and near-normal distribution after applying Box cox transformation. Source  


At the core of the Box-Cox transformation is an exponent, lambda (λ), which varies from -5 to 5. All values of λ are considered and the optimal value for your data is selected; The “optimal value” is the one that results in the best approximation of a normal distribution curve.

 

The one-parameter Box-Cox transformations are defined as:



and the two-parameter Box-Cox transformations as:




Moreover, the one-parameter Box-Cox transformation holds for y > 0, i.e. only for positive values and two-parameter Box-Cox transformation for y > -λ, i.e. negative values. 



The parameter λ is estimated using the profile likelihood function and using goodness-of-fit tests.


If we talk about some drawbacks of Box-cox transformation, then if interpretation is what you want to do, then Box-cox is not recommended. Because if λ is some non-zero number, then the transformed target variable may be more difficult to interpret than if we simply applied a log transform.



A second stumbling block is that the Box-Cox transformation usually gives the median of the forecast distribution when we revert the transformed data to its original scale. Occasionally, we want the mean and not the median.


Box-Cox transformation in Python(Source)


SciPy’s stats package provides a function called boxcox for performing box-cox power transformation that takes in original non-normal data as input and returns fitted data along with the lambda value that was used to fit the non-normal distribution to normal distribution.

#load necessary packages
import numpy as np 
from scipy.stats import boxcox 
import seaborn as sns 

#make this example reproducible
np.random.seed(0)

#generate dataset
data = np.random.exponential(size=1000)

fig, ax = plt.subplots(1, 2)
#plot the distribution of data values
sns.distplot(data, hist=False, kde=True, 
             kde_kws = {'shade': True, 'linewidth': 2}, 
            label = "Non-Normal", color ="red", ax = ax[0]) 

#perform Box-Cox transformation on original data
transformed_data, best_lambda = boxcox(data) 

sns.distplot(transformed_data, hist = False, kde = True,
            kde_kws = {'shade': True, 'linewidth': 2}, 
            label = "Normal", color ="red", ax = ax[1])
  
#adding legends to the subplots
plt.legend(loc = "upper right")

#rescaling the subplots
fig.set_figheight(5)
fig.set_figwidth(10)
#display optimal lambda value
print(f"Lambda value used for Transformation: {best_lambda}")






Concept #6- Poisson distribution



In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.


In very simple terms, A Poisson distribution can be used to estimate how likely it is that something will happen "X" number of times. 

Some examples of Poisson processes are customers calling a help center, radioactive decay in atoms, visitors to a website, photons arriving at a space telescope, and movements in a stock price. Poisson processes are usually associated with time, but they do not have to be. 



The Formula for the Poisson Distribution Is:




Where:

  • e is Euler's number (e = 2.71828...)

  • k is the number of occurrences

  • k! is the factorial of k

  • λ is equal to the expected value of kwhen that is also equal to its variance



Lambda(λ) can be thought of as the expected number of events in the interval. As we change the rate parameter, λ, we change the probability of seeing different numbers of events in one interval. The below graph is the probability mass function of the Poisson distribution showing the probability of a number of events occurring in an interval with different rate parameters.





   Probability Mass function for Poisson Distribution with varying rate parameters.Source 




The Poisson distribution is also commonly used to model financial count data where the tally is small and is often zero. For one example, in finance, it can be used to model the number of trades that a typical investor will make in a given day, which can be 0 (often), or 1, or 2, etc.

As another example, this model can be used to predict the number of "shocks" to the market that will occur in a given time period, say over a decade.



Poisson distribution in Python


from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns

lam_list = [1, 4, 9] #list of Lambda values  

plt.figure(figsize=(10,6))
samples = np.linspace(start=0, stop=5, num=1000)

for lam in lam_list:
    sns.distplot(random.poisson(lam=lam, size=10), hist=False, label='lambda {0}'.format(lam))

plt.xlabel('Poisson Distribution', fontsize=15)
plt.ylabel('Frequency', fontsize=15)
plt.legend(loc='best')
plt.show()









As λ becomes bigger, the graph looks more like a normal distribution.


I hope you have enjoyed reading this article, If you have any questions or suggestions, please leave a comment.

Also read: False Positives vs. False Negatives


Feel free to connect me on LinkedIn for any query.


Thanks for reading!!!


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But if the null hypothesis is true, then, in reality, the drug does not combat the disease at all. The drug is falsely claimed to have a positive effect on a disease.Type II Error:- equivalent to False Negatives(FN).The other kind of error that occurs when we accept a false null hypothesis. This sort of error is called a type II error and is also referred to as an error of the second kind.If we think back again to the scenario in which we are testing a drug, what would a type II error look like? A type II error would occur if we accepted that the drug hs no effect on disease, but in reality, it did.A sample python implementation of the Confusion matrix.import warnings import pandas as pd from sklearn import model_selection from sklearn.linear_model import LogisticRegression from sklearn.metrics import confusion_matrix import matplotlib.pyplot as plt %matplotlib inline   #ignore warnings warnings.filterwarnings('ignore') # Load digits dataset url = "http://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data" df = pd.read_csv(url) # df = df.values X = df.iloc[:,0:4] y = df.iloc[:,4] #test size test_size = 0.33 #generate the same set of random numbers seed = 7 #Split data into train and test set. X_train, X_test, y_train, y_test = model_selection.train_test_split(X, y, test_size=test_size, random_state=seed) #Train Model model = LogisticRegression() model.fit(X_train, y_train) pred = model.predict(X_test)  #Construct the Confusion Matrix labels = ['Iris-setosa', 'Iris-versicolor', 'Iris-virginica'] cm = confusion_matrix(y_test, pred, labels) print(cm) fig = plt.figure() ax = fig.add_subplot(111) cax = ax.matshow(cm) plt.title('Confusion matrix') fig.colorbar(cax) ax.set_xticklabels([''] + labels) ax.set_yticklabels([''] + labels) plt.xlabel('Predicted Values') plt.ylabel('Actual Values') plt.show()Confusion matrix with 3 class labels.The diagonal elements represent the number of points for which the predicted label is equal to the true label, while anything off the diagonal was mislabeled by the classifier. Therefore, the higher the diagonal values of the confusion matrix the better, indicating many correct predictions.In our case, the classifier predicted all the 13 setosa and 18 virginica plants in the test data perfectly. However, it incorrectly classified 4 of the versicolor plants as virginica.There is also a list of rates that are often computed from a confusion matrix for a binary classifier:1. AccuracyOverall, how often is the classifier correct?Accuracy = (TP+TN)/totalWhen our classes are roughly equal in size, we can use accuracy, which will give us correctly classified values.Accuracy is a common evaluation metric for classification problems. It’s the number of correct predictions made as a ratio of all predictions made.Misclassification Rate(Error Rate): Overall, how often is it wrong. Since accuracy is the percent we correctly classified (success rate), it follows that our error rate (the percentage we got wrong) can be calculated as follows:Misclassification Rate = (FP+FN)/totalWe use the sklearn module to compute the accuracy of a classification task, as shown below.#import modules import warnings import pandas as pd import numpy as np from sklearn import model_selection from sklearn.linear_model import LogisticRegression from sklearn import datasets from sklearn.metrics import accuracy_score #ignore warnings warnings.filterwarnings('ignore') # Load digits dataset iris = datasets.load_iris() # # Create feature matrix X = iris.data # Create target vector y = iris.target #test size test_size = 0.33 #generate the same set of random numbers seed = 7 #cross-validation settings kfold = model_selection.KFold(n_splits=10, random_state=seed) #Model instance model = LogisticRegression() #Evaluate model performance scoring = 'accuracy' results = model_selection.cross_val_score(model, X, y, cv=kfold, scoring=scoring) print('Accuracy -val set: %.2f%% (%.2f)' % (results.mean()*100, results.std()))  #split data X_train, X_test, y_train, y_test = model_selection.train_test_split(X, y, test_size=test_size, random_state=seed) #fit model model.fit(X_train, y_train) #accuracy on test set result = model.score(X_test, y_test) print("Accuracy - test set: %.2f%%" % (result*100.0))The classification accuracy is 88% on the validation set.2. PrecisionWhen it predicts yes, how often is it correct?Precision=TP/predicted yesWhen we have a class imbalance, accuracy can become an unreliable metric for measuring our performance. For instance, if we had a 99/1 split between two classes, A and B, where the rare event, B, is our positive class, we could build a model that was 99% accurate by just saying everything belonged to class A. Clearly, we shouldn’t bother building a model if it doesn’t do anything to identify class B; thus, we need different metrics that will discourage this behavior. For this, we use precision and recall instead of accuracy.3. Recall or SensitivityWhen it’s actually yes, how often does it predict yes?True Positive Rate = TP/actual yesRecall gives us the true positive rate (TPR), which is the ratio of true positives to everything positive.In the case of the 99/1 split between classes A and B, the model that classifies everything as A would have a recall of 0% for the positive class, B (precision would be undefined — 0/0). Precision and recall provide a better way of evaluating model performance in the face of a class imbalance. They will correctly tell us that the model has little value for our use case.Just like accuracy, both precision and recall are easy to compute and understand but require thresholds. Besides, precision and recall only consider half of the confusion matrix:4. F1 ScoreThe F1 score is the harmonic mean of the precision and recall, where an F1 score reaches its best value at 1 (perfect precision and recall) and worst at 0.Why harmonic mean? Since the harmonic mean of a list of numbers skews strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.An F1 score punishes extreme values more. Ideally, an F1 Score could be an effective evaluation metric in the following classification scenarios:When FP and FN are equally costly — meaning they miss on true positives or find false positives — both impact the model almost the same way, as in our cancer detection classification exampleAdding more data doesn’t effectively change the outcome effectivelyTN is high (like with flood predictions, cancer predictions, etc.)A sample python implementation of the F1 score.import warnings import pandas from sklearn import model_selection from sklearn.linear_model import LogisticRegression from sklearn.metrics import log_loss from sklearn.metrics import precision_recall_fscore_support as score, precision_score, recall_score, f1_score  warnings.filterwarnings('ignore')  url = "https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv" dataframe = pandas.read_csv(url) dat = dataframe.values X = dat[:,:-1] y = dat[:,-1] test_size = 0.33 seed = 7  model = LogisticRegression() #split data X_train, X_test, y_train, y_test = model_selection.train_test_split(X, y, test_size=test_size, random_state=seed) model.fit(X_train, y_train) precision = precision_score(y_test, pred) print('Precision: %f' % precision) # recall: tp / (tp + fn) recall = recall_score(y_test, pred) print('Recall: %f' % recall) # f1: tp / (tp + fp + fn) f1 = f1_score(y_test, pred) print('F1 score: %f' % f1)5. SpecificityWhen it’s no, how often does it predict no?True Negative Rate=TN/actual noIt is the true negative rate or the proportion of true negatives to everything that should have been classified as negative.Note that, together, specificity and sensitivity consider the full confusion matrix:6. Receiver Operating Characteristics (ROC) CurveMeasuring the area under the ROC curve is also a very useful method for evaluating a model. By plotting the true positive rate (sensitivity) versus the false-positive rate (1 — specificity), we get the Receiver Operating Characteristic (ROC) curve. This curve allows us to visualize the trade-off between the true positive rate and the false positive rate.The following are examples of good ROC curves. The dashed line would be random guessing (no predictive value) and is used as a baseline; anything below that is considered worse than guessing. We want to be toward the top-left corner:A sample python implementation of the ROC curves.#Classification Area under curve import warnings import pandas from sklearn import model_selection from sklearn.linear_model import LogisticRegression from sklearn.metrics import roc_auc_score, roc_curve  warnings.filterwarnings('ignore')  url = "https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv" dataframe = pandas.read_csv(url) dat = dataframe.values X = dat[:,:-1] y = dat[:,-1] seed = 7 #split data X_train, X_test, y_train, y_test = model_selection.train_test_split(X, y, test_size=test_size, random_state=seed) model.fit(X_train, y_train)  # predict probabilities probs = model.predict_proba(X_test) # keep probabilities for the positive outcome only probs = probs[:, 1]  auc = roc_auc_score(y_test, probs) print('AUC - Test Set: %.2f%%' % (auc*100))  # calculate roc curve fpr, tpr, thresholds = roc_curve(y_test, probs) # plot no skill plt.plot([0, 1], [0, 1], linestyle='--') # plot the roc curve for the model plt.plot(fpr, tpr, marker='.') plt.xlabel('False positive rate') plt.ylabel('Sensitivity/ Recall') # show the plot plt.show()In the example above, the AUC is relatively close to 1 and greater than 0.5. A perfect classifier will have the ROC curve go along the Y-axis and then along the X-axisLog LossLog Loss is the most important classification metric based on probabilities.As the predicted probability of the true class gets closer to zero, the loss increases exponentially:It measures the performance of a classification model where the prediction input is a probability value between 0 and 1. Log loss increases as the predicted probability diverge from the actual label. The goal of any machine learning model is to minimize this value. As such, smaller log loss is better, with a perfect model having a log loss of 0.A sample python implementation of the Log Loss.#Classification LogLoss import warnings import pandas from sklearn import model_selection from sklearn.linear_model import LogisticRegression from sklearn.metrics import log_loss  warnings.filterwarnings('ignore') url = "https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv" dataframe = pandas.read_csv(url) dat = dataframe.values X = dat[:,:-1] y = dat[:,-1] seed = 7 #split data X_train, X_test, y_train, y_test = model_selection.train_test_split(X, y, test_size=test_size, random_state=seed) model.fit(X_train, y_train) #predict and compute logloss pred = model.predict(X_test) accuracy = log_loss(y_test, pred) print("Logloss: %.2f" % (accuracy))Logloss: 8.02 Jaccard IndexJaccard Index is one of the simplest ways to calculate and find out the accuracy of a classification ML model. Let’s understand it with an example. Suppose we have a labeled test set, with labels as –y = [0,0,0,0,0,1,1,1,1,1]And our model has predicted the labels as –y1 = [1,1,0,0,0,1,1,1,1,1]The above Venn diagram shows us the labels of the test set and the labels of the predictions, and their intersection and union.Jaccard Index or Jaccard similarity coefficient is a statistic used in understanding the similarities between sample sets. The measurement emphasizes the similarity between finite sample sets and is formally defined as the size of the intersection divided by the size of the union of the two labeled sets, with formula as –Jaccard Index or Intersection over Union(IoU)So, for our example, we can see that the intersection of the two sets is equal to 8 (since eight values are predicted correctly) and the union is 10 + 10–8 = 12. So, the Jaccard index gives us the accuracy as –So, the accuracy of our model, according to Jaccard Index, becomes 0.66, or 66%.Higher the Jaccard index higher the accuracy of the classifier.A sample python implementation of the Jaccard index.import numpy as np def compute_jaccard_similarity_score(x, y):     intersection_cardinality = len(set(x).intersection(set(y)))     union_cardinality = len(set(x).union(set(y)))     return intersection_cardinality / float(union_cardinality)  score = compute_jaccard_similarity_score(np.array([0, 1, 2, 5, 6]), np.array([0, 2, 3, 5, 7, 9])) print "Jaccard Similarity Score : %s" %score passJaccard Similarity Score : 0.375Kolomogorov Smirnov chartK-S or Kolmogorov-Smirnov chart measures the performance of classification models. More accurately, K-S is a measure of the degree of separation between positive and negative distributions.The cumulative frequency for the observed and hypothesized distributions is plotted against the ordered frequencies. The vertical double arrow indicates the maximal vertical difference.The K-S is 100 if the scores partition the population into two separate groups in which one group contains all the positives and the other all the negatives. On the other hand, If the model cannot differentiate between positives and negatives, then it is as if the model selects cases randomly from the population. The K-S would be 0.In most classification models the K-S will fall between 0 and 100, and that the higher the value the better the model is at separating the positive from negative cases.The K-S may also be used to test whether two underlying one-dimensional probability distributions differ. It is a very efficient way to determine if two samples are significantly different from each other.A sample python implementation of the Kolmogorov-Smirnov.from scipy.stats import kstest import random   # N = int(input("Enter number of random numbers: ")) N = 10   actual =[] print("Enter outcomes: ")   for i in range(N):     # x = float(input("Outcomes of class "+str(i + 1)+": "))     actual.append(random.random())   print(actual) x = kstest(actual, "norm")    print(x)The Null hypothesis used here assumes that the numbers follow the normal distribution. It returns statistics and p-value. If the p-value is < alpha, we reject the Null hypothesis.Alpha is defined as the probability of rejecting the null hypothesis given the null hypothesis(H0) is true. For most of the practical applications, alpha is chosen as 0.05.Gain and Lift ChartGain or Lift is a measure of the effectiveness of a classification model calculated as the ratio between the results obtained with and without the model. Gain and lift charts are visual aids for evaluating the performance of classification models. However, in contrast to the confusion matrix that evaluates models on the whole population gain or lift chart evaluates model performance in a portion of the population.The higher the lift (i.e. the further up it is from the baseline), the better the model.The following gains chart, run on a validation set, shows that with 50% of the data, the model contains 90% of targets, Adding more data adds a negligible increase in the percentage of targets included in the model.Gain/lift chartLift charts are often shown as a cumulative lift chart, which is also known as a gains chart. Therefore, gains charts are sometimes (perhaps confusingly) called “lift charts”, but they are more accurately cumulative lift charts.It is one of their most common uses is in marketing, to decide if a prospective client is worth calling.Gini CoefficientThe Gini coefficient or Gini Index is a popular metric for imbalanced class values. The coefficient ranges from 0 to 1 where 0 represents perfect equality and 1 represents perfect inequality. Here, if the value of an index is higher, then the data will be more dispersed.Gini coefficient can be computed from the area under the ROC curve using the following formula:Gini Coefficient = (2 * ROC_curve) — 1ConclusionUnderstanding how well a machine learning model is going to perform on unseen data is the ultimate purpose behind working with these evaluation metrics. Metrics like accuracy, precision, recall are good ways to evaluate classification models for balanced datasets, but if the data is imbalanced and there’s a class disparity, then other methods like ROC/AUC, Gini coefficient perform better in evaluating the model performance.Well, this concludes this article. I hope you guys have enjoyed reading it, feel free to share your comments/thoughts/feedback in the comment section.Thanks for reading !!!

Data Science
DataSource AI Hosts KTM AG's Inaugural AI Challenge: "Code the Light Fantastic"

DataSource AI announces the launch of the KTM AG inaugural AI Challenge, an unprecedented 3-month online competition that aims to revolutionise two-wheeler innovation through artificial intelligence and deep learning. KTM AG is a global frontrunner in two-wheeler innovation, known for pushing the boundaries of what's possible in the world of motorcycles. With a rich history of groundbreaking engineering and a commitment to cutting-edge technology, KTM AG has set new standards in performance, design, and safety. As a global leader in two-wheeler innovation, KTM AG invites participants to embark on this groundbreaking innovation journey. At the core of this competition lies a challenge set to redefine the future of motorcycle lighting systems. Participants are tasked with developing an algorithm for a high-beam lighting system utilizing a pixel matrix. Participants can find detailed guidelines in the Datathon competition. The datathon unfolds in a 3-tiered cascade model: This Code Challenge by KTM AG promises not only substantial rewards but also an exciting opportunity to shape the future of two-wheeler technology, along with supporting the participants to upscale and test their knowledge in a global AI competition. The cumulative budget for this remarkable Code Challenge by KTM AG is a substantial €24,000, motivating participants with not only the opportunity to push the boundaries of two-wheeler technology but also significant rewards for those who rise to the occasion. With cumulative prizes, contestants have the chance to potentially take home a maximum reward of €10,800 in addition to contributing to cutting-edge advancements in the field.  We invite all aspiring innovators, data scientists, and AI enthusiasts to join us in this journey to "Code the Light Fantastic." For more information, rules, and registration details, please register hereAbout DataSource:  At DataSource AI, we are driven by a singular mission - to democratise the immense power of data science and AI/ML for businesses of all sizes and budgets. We facilitate AI competitions, for businesses of all sizes and budgets by harnessing our extensive data expert community that's collaborating over our intelligent AI algorithm crowdsourcing platform. Our community is at the heart of what we do. We've built a diverse and talented pool of data experts who are passionate about solving real-world problems. They collaborate, ideate, and innovate, driving forward the frontiers of data science.  

Data Science
Data-Driven Creativity: Enhancing Video Content through Data Science

In the age of digital marketing and content creation, data-driven creativity is becoming an increasingly important concept. It's the fusion of artistic vision with the insights gleaned from data science to enhance the impact and effectiveness of video content. This 2500-word blog will explore how data science can be leveraged to elevate video content creation, ensuring that it not only engages but also resonates with the intended audience.Introduction to Data-Driven CreativityData-driven creativity marks a groundbreaking shift in video content creation, blending artistic vision with the insights provided by data science. This combination allows creators to break free from conventional creative limits, using data analytics to develop content that is both visually captivating and strategically significant. By delving into viewer behavior, preferences, and interactions, creators can refine their stories and visuals, achieving a deeper connection with their audience. This technique effectively transforms data into a guide for storytelling, steering content towards increased relevance and attractiveness. Consequently, video content becomes a more potent medium for engaging viewers and delivering impactful messages. Fundamentally, data-driven creativity is about converting data points into compelling stories and turning analytical insights into creative masterpieces, thereby redefining the standards of digital video content.Understanding the Role of Data in Video Content CreationExploring the Role of Data in Video Content Creation ventures into the rapidly growing realm of data-driven creativity, where data science emerges as a key instrument in enriching video content. In this realm, data transcends mere figures to become a narrative element, providing rich insights into what audiences prefer, how they behave, and emerging trends. Utilizing data, video creators can break free from conventional creative constraints, shaping their stories to more deeply connect with viewers. This process involves a detailed examination of viewer interactions, demographics, and feedback to hone storytelling skills, aiming to create videos that are not only watched but also emotionally impactful and memorable. Data-driven creativity is a fusion of art and science, where each view, reaction, and comment plays a role in directing the trajectory of video content, enhancing its relevance, engagement, and effect. This marks a transformative phase in content creation, where data equips creators to weave narratives that are not just creatively rich but also finely tuned to the dynamic preferences and interests of their audience.The Process of Gathering and Analyzing DataCollecting and analyzing data forms the foundation of data-driven creativity, especially in the realm of video content enhancement. This process involves the acquisition of key information, including audience demographics, interaction metrics, and performance measures, utilizing sophisticated tools and technologies. These range from social media analytics to advanced data mining applications designed to track a broad spectrum of viewer interactions. Once collected, this data undergoes thorough analysis to identify trends, preferences, and behaviors within the target audience. Such analysis equips content creators with insightful knowledge, allowing them to adjust their video content for greater appeal and connection with their audience. Leveraging these insights, creators can modify elements such as the tone, style, and themes of their content, revolutionizing storytelling methods and ensuring their content is both captivating and impactful. This integration of data science with creative storytelling heralds a transformative phase in video content production, where analytical findings significantly enhance artistic expression.Tailoring Content to Audience PreferencesAdapting content to audience preferences through data-driven creativity signifies a vital evolution in video content production. By incorporating data science, creators gain profound insights into audience behaviors, likes, and engagement patterns. This approach facilitates the creation of content that better resonates with viewers, ensuring everything from the plot to visual elements aligns with their interests. Utilizing analytics such as viewer habits and interaction rates, creators can pinpoint engaging aspects for better video content. Using a high-quality video editor tool is important to make the video look better. This knowledge allows precise adjustments, making the content not only captivating but also highly relevant. Ultimately, incorporating data in video content creation leads to more impactful and resonant viewer experiences, forging a deeper bond between the audience and the content.Enhancing Storytelling with Data InsightsUtilizing data insights to enhance storytelling is a groundbreaking method in video content production. Termed data-driven creativity, this technique blends the storytelling craft with data science accuracy. Content creators leverage analysis of viewer engagement, preferences, and behavior to fine-tune their narratives, ensuring a deeper connection with their audience. This integration results in not only engaging narratives but also ones that are in tune with audience interests and emerging trends. Insights from data grant a clearer understanding of what truly engages viewers, empowering creators to optimize their storytelling for the greatest effect. This modern approach reinvents traditional storytelling into an experience that's both more impactful and centered around the audience, with each creative decision being shaped and enriched by data.Using Data to Predict Future TrendsUtilizing data for future trends in data-driven creativity marks a revolutionary step in improving video content via data science. This technique focuses on analyzing viewer interactions, demographic information, and behavioral tendencies to predict future content direction. Using data enables creators to be proactive, crafting video content that resonates with emerging audience preferences and interests. Such a forward-thinking approach guarantees ongoing relevance in a dynamic digital world and fosters innovation and leadership among content creators. The blend of data analytics and artistic insight leads to the production of not just captivating but also pioneering videos, demonstrating the significant role of data in shaping the future of video content creation.Balancing Creativity and DataAchieving a harmonious blend of creativity and data in video content production is both subtle and potent. Data-driven creativity embodies the convergence of artistic flair and data analytics, providing an innovative method to boost video effectiveness. By weaving in data analysis, video creators unlock insights into what their audience prefers and how they behave, guiding their artistic choices. This integration results in content that is not only enthralling but also deeply meaningful to viewers. It is essential, however, to ensure that data serves as a guide, not a ruler, in the creative journey. This equilibrium keeps the content fresh and appealing while aligning it thoughtfully with data-driven knowledge. In essence, data-driven creativity in video content merges the narrative craft with analytical insights, culminating in videos that are both compelling and influential.Overcoming Challenges in Data-Driven CreativityOvercoming hurdles in data-driven creativity necessitates a nuanced integration of data science into the creation of video content. It involves striking a delicate balance between analytical methodologies and artistic expression, ensuring that data serves as an informative tool rather than a constraint on creativity. Accurate interpretation of data empowers content creators to avoid formulaic outputs, utilizing insights to enrich storytelling and enhance audience engagement. This intricate process demands a comprehensive understanding of the artistry of video creation and the scientific principles behind data analysis. Ethical considerations, including respecting audience privacy and obtaining data consent, are pivotal in this approach. Innovative strategies within data-driven creativity empower creators to produce content that forges deeper connections with viewers, setting new benchmarks in the digital landscape. Embracing these challenges is essential for unlocking the full potential of data-enhanced video content.Ethical Considerations in Data-Driven CreativityIn the domain of data-driven creativity, ethical considerations play a crucial role, especially when utilizing data science to enhance video content. While utilizing data insights can enhance creative processes, it is essential to address privacy concerns and ensure transparent, responsible data usage. Achieving the right equilibrium between creativity and ethical considerations becomes paramount as brands employ data to customize video content. Upholding user privacy and securing informed consent are fundamental principles in ethical data-driven creativity, fostering trust among audiences. Moreover, there is an obligation to avoid perpetuating biases and stereotypes in content creation, championing inclusivity and diversity. Ethical practices not only maintain brand integrity but also contribute to a positive and respectful digital environment for consumers.Tools and Resources for Data-Driven Video CreationExplore the potential of data-driven creativity using state-of-the-art tools and resources for crafting videos. In the current digital landscape, integrating data science and video content is transforming the landscape of creative processes. Immerse yourself in a domain where insights derived from data direct every facet of video production. These tools empower creators to customize content according to audience preferences, ensuring that each video is not only visually captivating but also strategically aligned. From scriptwriting informed by analytics to incorporating personalized visual elements, the utilization of data science takes video content to unprecedented levels. Delve into the crossroads of technology and creativity, where strategies driven by data redefine storytelling, captivating audiences in a personalized and meaningful manner.The Future of Data-Driven Creativity in Video ContentThe evolution of data-driven creativity in video content is set to transform our interaction with digital media. Through the incorporation of data science, creators gain valuable insights into viewer preferences, behavior, and trends. This collaboration enables a personalized and captivating viewing experience, heightening audience engagement. With the utilization of data-driven creativity, content producers can shape videos to suit the unique preferences of their target audience, resulting in more impactful storytelling and brand communication. As technology progresses, we anticipate a shift towards highly personalized content, driven by data insights, leading to innovative approaches in video production. This convergence of creativity and data science holds significant promise for the future development of video content within the digital landscape.ConclusionIn summary, the convergence of data-driven insights and creative components represents a transformative shift in the realm of video content creation. The fusion of Data Science and creativity provides content producers with the tools to precisely tailor videos to audience preferences, resulting in more impactful and engaging content. Leveraging the potential of data facilitates a deeper comprehension of viewer behavior, enabling targeted storytelling. Amidst the digital landscape, the symbiosis of data and creativity not only elevates video content but also fosters innovation and personalized experiences. Looking ahead, embracing Data-Driven Creativity becomes crucial for maintaining a leading edge in the continually evolving landscape of video content creation.

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