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What is LDA: Linear Discriminant Analysis for Machine Learning

Linear Discriminant Analysis or LDA is a dimensionality reduction technique. It is used as a pre-processing step in Machine Learning and applications of pattern classification. The goal of LDA is to project the features in higher dimensional space onto a lower-dimensional space in order to avoid the curse of dimensionality and also reduce resources and dimensional costs.The original technique was developed in the year 1936 by Ronald A. Fisher and was named Linear Discriminant or Fisher's Discriminant Analysis. The original Linear Discriminant was described as a two-class technique. The multi-class version was later generalized by C.R Rao as Multiple Discriminant Analysis. They are all simply referred to as the Linear Discriminant Analysis.LDA is a supervised classification technique that is considered a part of crafting competitive machine learning models. This category of dimensionality reduction is used in areas like image recognition and predictive analysis in marketing.What is Dimensionality Reduction?The techniques of dimensionality reduction are important in applications of Machine Learning, Data Mining, Bioinformatics, and Information Retrieval. The main agenda is to remove the redundant and dependent features by changing the dataset onto a lower-dimensional space.In simple terms, they reduce the dimensions (i.e. variables) in a particular dataset while retaining most of the data.Multi-dimensional data comprises multiple features having a correlation with one another. You can plot multi-dimensional data in just 2 or 3 dimensions with dimensionality reduction. It allows the data to be presented in an explicit manner which can be easily understood by a layman.What are the limitations of Logistic Regression?Logistic Regression is a simple and powerful linear classification algorithm. However, it has some disadvantages which have led to alternate classification algorithms like LDA. Some of the limitations of Logistic Regression are as follows:Two-class problems – Logistic Regression is traditionally used for two-class and binary classification problems. Though it can be extrapolated and used in multi-class classification, this is rarely performed. On the other hand, Linear Discriminant Analysis is considered a better choice whenever multi-class classification is required and in the case of binary classifications, both logistic regression and LDA are applied.Unstable with Well-Separated classes – Logistic Regression can lack stability when the classes are well-separated. This is where LDA comes in.Unstable with few examples – If there are few examples from which the parameters are to be estimated, logistic regression becomes unstable. However, Linear Discriminant Analysis is a better option because it tends to be stable even in such cases.How to have a practical approach to an LDA model?Consider a situation where you have plotted the relationship between two variables where each color represents a different class. One is shown with a red color and the other with blue.If you are willing to reduce the number of dimensions to 1, you can just project everything to the x-axis as shown below: This approach neglects any helpful information provided by the second feature. However, you can use LDA to plot it. The advantage of LDA is that it uses information from both the features to create a new axis which in turn minimizes the variance and maximizes the class distance of the two variables.How does LDA work?LDA focuses primarily on projecting the features in higher dimension space to lower dimensions. You can achieve this in three steps:Firstly, you need to calculate the separability between classes which is the distance between the mean of different classes. This is called the between-class variance.Secondly, calculate the distance between the mean and sample of each class. It is also called the within-class variance.Finally, construct the lower-dimensional space which maximizes the between-class variance and minimizes the within-class variance. P is considered as the lower-dimensional space projection, also called Fisher’s criterion.How are LDA models represented?The representation of LDA is pretty straight-forward. The model consists of the statistical properties of your data that has been calculated for each class. The same properties are calculated over the multivariate Gaussian in the case of multiple variables. The multivariates are means and covariate matrix.Predictions are made by providing the statistical properties into the LDA equation. The properties are estimated from your data. Finally, the model values are saved to file to create the LDA model.How do LDA models learn?The assumptions made by an LDA model about your data:Each variable in the data is shaped in the form of a bell curve when plotted,i.e. Gaussian.The values of each variable vary around the mean by the same amount on the average,i.e. each attribute has the same variance.The LDA model is able to estimate the mean and variance from your data for each class with the help of these assumptions.The mean value of each input for each of the classes can be calculated by dividing the sum of values by the total number of values:Mean =Sum(x)/Nkwhere Mean = mean value of x for class           N = number of           k = number of           Sum(x) = sum of values of each input x.The variance is computed across all the classes as the average of the square of the difference of each value from the mean:Σ²=Sum((x - M)²)/(N - k)where  Σ² = Variance across all inputs x.            N = number of instances.            k = number of classes.            Sum((x - M)²) = Sum of values of all (x - M)².            M = mean for input x.How does an LDA model make predictions?LDA models use Bayes’ Theorem to estimate probabilities. They make predictions based upon the probability that a new input dataset belongs to each class. The class which has the highest probability is considered the output class and then the LDA makes a prediction.  The prediction is made simply by the use of Bayes’ Theorem which estimates the probability of the output class given the input. They also make use of the probability of each class and the probability of the data belonging to each class:P(Y=x|X=x)  = [(Plk * fk(x))] / [(sum(PlI * fl(x))]Where x = input.            k = output class.            Plk = Nk/n or base probability of each class observed in the training data. It is also called prior probability in Bayes’ Theorem.            fk(x) = estimated probability of x belonging to class k.The f(x) is plotted using a Gaussian Distribution function and then it is plugged into the equation above and the result we get is the equation as follows:Dk(x) = x∗(mean/Σ²) – (mean²/(2*Σ²)) + ln(PIk)The Dk(x) is called the discriminant function for class k given input x, mean,  Σ² and Plk are all estimated from the data and the class is calculated as having the largest value, will be considered in the output classification.  How to prepare data from LDA?Some suggestions you should keep in mind while preparing your data to build your LDA model:LDA is mainly used in classification problems where you have a categorical output variable. It allows both binary classification and multi-class classification.The standard LDA model makes use of the Gaussian Distribution of the input variables. You should check the univariate distributions of each attribute and transform them into a more Gaussian-looking distribution. For example, for the exponential distribution, use log and root function and for skewed distributions use BoxCox.Outliers can skew the primitive statistics used to separate classes in LDA, so it is preferable to remove them.Since LDA assumes that each input variable has the same variance, it is always better to standardize your data before using an LDA model. Keep the mean to be 0 and the standard deviation to be 1.How to implement an LDA model from scratch?You can implement a Linear Discriminant Analysis model from scratch using Python. Let’s start by importing the libraries that are required for the model:from sklearn.datasets import load_wine import pandas as pd import numpy as np np.set_printoptions(precision=4) from matplotlib import pyplot as plt import seaborn as sns sns.set() from sklearn.preprocessing import LabelEncoder from sklearn.tree import DecisionTreeClassifier from sklearn.model_selection import train_test_split from sklearn.metrics import confusion_matrixSince we will work with the wine dataset, you can obtain it from the UCI machine learning repository. The scikit-learn library in Python provides a wrapper function for downloading it:wine_info = load_wine() X = pd.DataFrame(wine_info.data, columns=wine_info.feature_names) y = pd.Categorical.from_codes(wine_info.target, wine_info.target_names)The wine dataset comprises of 178 rows of 13 columns each:X.shape(178, 13)The attributes of the wine dataset comprise of various characteristics such as alcohol content of the wine, magnesium content, color intensity, hue and many more:X.head()The wine dataset contains three different kinds of wine:wine_info.target_names array(['class_0', 'class_1', 'class_2'], dtype='<U7')Now we create a DataFrame which will contain both the features and the content of the dataset:df = X.join(pd.Series(y, name='class'))We can divide the process of Linear Discriminant Analysis into 5 steps as follows:Step 1 - Computing the within-class and between-class scatter matrices.Step 2 - Computing the eigenvectors and their corresponding eigenvalues for the scatter matrices.Step 3 - Sorting the eigenvalues and selecting the top k.Step 4 - Creating a new matrix that will contain the eigenvectors mapped to the k eigenvalues.Step 5 - Obtaining new features by taking the dot product of the data and the matrix from Step 4.Within-class scatter matrixTo calculate the within-class scatter matrix, you can use the following mathematical expression:where, c = total number of distinct classes andwhere, x = a sample (i.e. a row).            n = total number of samples within a given class.Now we create a vector with the mean values of each feature:feature_means1 = pd.DataFrame(columns=wine_info.target_names) for c, rows in df.groupby('class'): feature_means1[c] = rows.mean() feature_means1The mean vectors (mi ) are now plugged into the above equations to obtain the within-class scatter matrix:withinclass_scatter_matrix = np.zeros((13,13)) for c, rows in df.groupby('class'): rows = rows.drop(['class'], axis=1) s = np.zeros((13,13)) for index, row in rows.iterrows(): x, mc = row.values.reshape(13,1), feature_means1[c].values.reshape(13,1) s += (x - mc).dot((x - mc).T) withinclass_scatter_matrix += sBetween-class scatter matrixWe can calculate the between-class scatter matrix using the following mathematical expression:where,andfeature_means2 = df.mean() betweenclass_scatter_matrix = np.zeros((13,13)) for c in feature_means1:        n = len(df.loc[df['class'] == c].index)    mc, m = feature_means1[c].values.reshape(13,1), feature_means2.values.reshape(13,1) betweenclass_scatter_matrix += n * (mc - m).dot((mc - m).T)Now we will solve the generalized eigenvalue problem to obtain the linear discriminants for:eigen_values, eigen_vectors = np.linalg.eig(np.linalg.inv(withinclass_scatter_matrix).dot(betweenclass_scatter_matrix))We will sort the eigenvalues from the highest to the lowest since the eigenvalues with the highest values carry the most information about the distribution of data is done. Next, we will first k eigenvectors. Finally, we will place the eigenvalues in a temporary array to make sure the eigenvalues map to the same eigenvectors after the sorting is done:eigen_pairs = [(np.abs(eigen_values[i]), eigen_vectors[:,i]) for i in range(len(eigen_values))] eigen_pairs = sorted(eigen_pairs, key=lambda x: x[0], reverse=True) for pair in eigen_pairs: print(pair[0])237.46123198302251 46.98285938758684 1.4317197551638386e-14 1.2141209883217706e-14 1.2141209883217706e-14 8.279823065850476e-15 7.105427357601002e-15 6.0293733655173466e-15 6.0293733655173466e-15 4.737608877108813e-15 4.737608877108813e-15 2.4737196789039026e-15 9.84629525010022e-16Now we will transform the values into percentage since it is difficult to understand how much of the variance is explained by each component.sum_of_eigen_values = sum(eigen_values) print('Explained Variance') for i, pair in enumerate(eigen_pairs):    print('Eigenvector {}: {}'.format(i, (pair[0]/sum_of_eigen_values).real))Explained Variance Eigenvector 0: 0.8348256799387275 Eigenvector 1: 0.1651743200612724 Eigenvector 2: 5.033396012077518e-17 Eigenvector 3: 4.268399397827047e-17 Eigenvector 4: 4.268399397827047e-17 Eigenvector 5: 2.9108789097898625e-17 Eigenvector 6: 2.498004906118145e-17 Eigenvector 7: 2.119704204950956e-17 Eigenvector 8: 2.119704204950956e-17 Eigenvector 9: 1.665567688286435e-17 Eigenvector 10: 1.665567688286435e-17 Eigenvector 11: 8.696681541121664e-18 Eigenvector 12: 3.4615924706522496e-18First, we will create a new matrix W using the first two eigenvectors:W_matrix = np.hstack((eigen_pairs[0][1].reshape(13,1), eigen_pairs[1][1].reshape(13,1))).realNext, we will save the dot product of X and W into a new matrix Y:Y = X∗Wwhere, X = n x d matrix with n sample and d dimensions.            Y = n x k matrix with n sample and k dimensions.In simple terms, Y is the new matrix or the new feature space.X_lda = np.array(X.dot(W_matrix))Our next work is to encode every class a member in order to incorporate the class labels into our plot. This is done because matplotlib cannot handle categorical variables directly.Finally, we plot the data as a function of the two LDA components using different color for each class:plt.xlabel('LDA1') plt.ylabel('LDA2') plt.scatter( X_lda[:,0], X_lda[:,1], c=y, cmap='rainbow', alpha=0.7, edgecolors='b' )<matplotlib.collections.PathCollection at 0x7fd08a20e908>How to implement LDA using scikit-learn?For implementing LDA using scikit-learn, let’s work with the same wine dataset. You can also obtain it from the  UCI machine learning repository. You can use the predefined class LinearDiscriminant Analysis made available to us by the scikit-learn library to implement LDA rather than implementing from scratch every time:from sklearn.discriminant_analysis import LinearDiscriminantAnalysis lda_model = LinearDiscriminantAnalysis() X_lda = lda_model.fit_transform(X, y)To obtain the variance corresponding to each component, you can access the following property:lda.explained_variance_ratio_array([0.6875, 0.3125])Again, we will plot the two LDA components just like we did before:plt.xlabel('LDA1') plt.ylabel('LDA2') plt.scatter( X_lda[:,0], X_lda[:,1],    c=y, cmap='rainbow',    alpha=0.7, edgecolors='b' )<matplotlib.collections.PathCollection at 0x7fd089f60358>Linear Discriminant Analysis vs PCABelow are the differences between LDA and PCA:PCA ignores class labels and focuses on finding the principal components that maximizes the variance in a given data. Thus it is an unsupervised algorithm. On the other hand, LDA is a supervised algorithm that intends to find the linear discriminants that represents those axes which maximize separation between different classes.LDA performs better multi-class classification tasks than PCA. However, PCA performs better when the sample size is comparatively small. An example would be comparisons between classification accuracies that are used in image classification.Both LDA and PCA are used in case of dimensionality reduction. PCA is first followed by LDA.Let us create and fit an instance of the PCA class:from sklearn.decomposition import PCA pca_class = PCA(n_components=2) X_pca = pca.fit_transform(X, y)Again, to view the values in percentage for a better understanding, we will access the explained_variance_ratio_ property:pca.explained_variance_ratio_array([0.9981, 0.0017])Clearly, PCA selected the components which will be able to retain the most information and ignores the ones which maximize the separation between classes.plt.xlabel('PCA1') plt.ylabel('PCA2') plt.scatter( X_pca[:,0], X_pca[:,1],    c=y, cmap='rainbow',    alpha=0.7, edgecolors='bNow to create a classification model using the LDA components as features, we will divide the data into training datasets and testing datasets:X_train, X_test, y_train, y_test = train_test_split(X_lda, y, random_state=1)The next thing we will do is create a Decision Tree. Then, we will predict the category of each sample test and create a confusion matrix to evaluate the LDA model’s performance:data = DecisionTreeClassifier() data.fit(X_train, y_train) y_pred = data.predict(X_test) confusion_matrix(y_test, y_pred)array([[18,  0,  0],  [ 0, 17,  0],  [ 0,  0, 10]])So it is clear that the Decision Tree Classifier has correctly classified everything in the test dataset.What are the extensions to LDA?LDA is considered to be a very simple and effective method, especially for classification techniques. Since it is simple and well understood, so it has a lot of extensions and variations:Quadratic Discriminant Analysis(QDA) – When there are multiple input variables, each of the class uses its own estimate of variance and covariance.Flexible Discriminant Analysis(FDA) – This technique is performed when a non-linear combination of inputs is used as splines.Regularized Discriminant Analysis(RDA) – It moderates the influence of various variables in LDA by regularizing the estimate of the covariance.Real-Life Applications of LDASome of the practical applications of LDA are listed below:Face Recognition – LDA is used in face recognition to reduce the number of attributes to a more manageable number before the actual classification. The dimensions that are generated are a linear combination of pixels that forms a template. These are called Fisher’s faces.Medical – You can use LDA to classify the patient disease as mild, moderate or severe. The classification is done upon the various parameters of the patient and his medical trajectory. Customer Identification – You can obtain the features of customers by performing a simple question and answer survey. LDA helps in identifying and selecting which describes the properties of a group of customers who are most likely to buy a particular item in a shopping mall. SummaryLet us take a look at the topics we have covered in this article: Dimensionality Reduction and need for LDA Working of an LDA model Representation, Learning, Prediction and preparing data in LDA Implementation of an LDA model Implementation of LDA using scikit-learn LDA vs PCA Extensions and Applications of LDA The Linear Discriminant Analysis in Python is a very simple and well-understood approach of classification in machine learning. Though there are other dimensionality reduction techniques like Logistic Regression or PCA, but LDA is preferred in many special classification cases. If you want to be an expert in machine learning, knowledge of Linear Discriminant Analysis would lead you to that position effortlessly. Enrol in our  Data Science and Machine Learning Courses for more lucrative career options in this landscape and become a certified Data Scientist.

What is LDA: Linear Discriminant Analysis for Machine Learning

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What is LDA: Linear Discriminant Analysis for Machine Learning

Linear Discriminant Analysis or LDA is a dimensionality reduction technique. It is used as a pre-processing step in Machine Learning and applications of pattern classification. The goal of LDA is to project the features in higher dimensional space onto a lower-dimensional space in order to avoid the curse of dimensionality and also reduce resources and dimensional costs.

The original technique was developed in the year 1936 by Ronald A. Fisher and was named Linear Discriminant or Fisher's Discriminant Analysis. The original Linear Discriminant was described as a two-class technique. The multi-class version was later generalized by C.R Rao as Multiple Discriminant Analysis. They are all simply referred to as the Linear Discriminant Analysis.

LDA is a supervised classification technique that is considered a part of crafting competitive machine learning models. This category of dimensionality reduction is used in areas like image recognition and predictive analysis in marketing.

What is Dimensionality Reduction?

The techniques of dimensionality reduction are important in applications of Machine Learning, Data Mining, Bioinformatics, and Information Retrieval. The main agenda is to remove the redundant and dependent features by changing the dataset onto a lower-dimensional space.

In simple terms, they reduce the dimensions (i.e. variables) in a particular dataset while retaining most of the data.

Multi-dimensional data comprises multiple features having a correlation with one another. You can plot multi-dimensional data in just 2 or 3 dimensions with dimensionality reduction. It allows the data to be presented in an explicit manner which can be easily understood by a layman.

What are the limitations of Logistic Regression?

Logistic Regression is a simple and powerful linear classification algorithm. However, it has some disadvantages which have led to alternate classification algorithms like LDA. Some of the limitations of Logistic Regression are as follows:

  • Two-class problems – Logistic Regression is traditionally used for two-class and binary classification problems. Though it can be extrapolated and used in multi-class classification, this is rarely performed. On the other hand, Linear Discriminant Analysis is considered a better choice whenever multi-class classification is required and in the case of binary classifications, both logistic regression and LDA are applied.
  • Unstable with Well-Separated classes – Logistic Regression can lack stability when the classes are well-separated. This is where LDA comes in.
  • Unstable with few examples – If there are few examples from which the parameters are to be estimated, logistic regression becomes unstable. However, Linear Discriminant Analysis is a better option because it tends to be stable even in such cases.

How to have a practical approach to an LDA model?

Consider a situation where you have plotted the relationship between two variables where each color represents a different class. One is shown with a red color and the other with blue.

How to have a practical approach to an LDA model?

If you are willing to reduce the number of dimensions to 1, you can just project everything to the x-axis as shown below: 

How to have a practical approach to an LDA model?

How to have a practical approach to an LDA model?

This approach neglects any helpful information provided by the second feature. However, you can use LDA to plot it. The advantage of LDA is that it uses information from both the features to create a new axis which in turn minimizes the variance and maximizes the class distance of the two variables.

How to have a practical approach to an LDA model?

How to have a practical approach to an LDA model?

How does LDA work?

LDA focuses primarily on projecting the features in higher dimension space to lower dimensions. You can achieve this in three steps:

  • Firstly, you need to calculate the separability between classes which is the distance between the mean of different classes. This is called the between-class variance.

How does LDA work?

  • Secondly, calculate the distance between the mean and sample of each class. It is also called the within-class variance.

How does LDA work?

  • Finally, construct the lower-dimensional space which maximizes the between-class variance and minimizes the within-class variance. P is considered as the lower-dimensional space projection, also called Fisher’s criterion.

How does LDA work?

How are LDA models represented?

The representation of LDA is pretty straight-forward. The model consists of the statistical properties of your data that has been calculated for each class. The same properties are calculated over the multivariate Gaussian in the case of multiple variables. The multivariates are means and covariate matrix.

Predictions are made by providing the statistical properties into the LDA equation. The properties are estimated from your data. Finally, the model values are saved to file to create the LDA model.

How do LDA models learn?

The assumptions made by an LDA model about your data:

  • Each variable in the data is shaped in the form of a bell curve when plotted,i.e. Gaussian.
  • The values of each variable vary around the mean by the same amount on the average,i.e. each attribute has the same variance.

The LDA model is able to estimate the mean and variance from your data for each class with the help of these assumptions.

The mean value of each input for each of the classes can be calculated by dividing the sum of values by the total number of values:

Mean =Sum(x)/Nk

where Mean = mean value of x for class
           N = number of
           k = number of
           Sum(x) = sum of values of each input x.

The variance is computed across all the classes as the average of the square of the difference of each value from the mean:

Σ²=Sum((x - M)²)/(N - k)

where  Σ² = Variance across all inputs x.
            N = number of instances.
            k = number of classes.
            Sum((x - M)²) = Sum of values of all (x - M)².
            M = mean for input x.

How does an LDA model make predictions?

LDA models use Bayes’ Theorem to estimate probabilities. They make predictions based upon the probability that a new input dataset belongs to each class. The class which has the highest probability is considered the output class and then the LDA makes a prediction.  

The prediction is made simply by the use of Bayes’ Theorem which estimates the probability of the output class given the input. They also make use of the probability of each class and the probability of the data belonging to each class:

P(Y=x|X=x)  = [(Plk * fk(x))] / [(sum(PlI * fl(x))]

Where x = input.
            k = output class.
            Plk = Nk/n or base probability of each class observed in the training data. It is also called prior probability in Bayes’ Theorem.
            fk(x) = estimated probability of x belonging to class k.

The f(x) is plotted using a Gaussian Distribution function and then it is plugged into the equation above and the result we get is the equation as follows:

Dk(x) = x∗(mean/Σ²) – (mean²/(2*Σ²)) + ln(PIk)

The Dk(x) is called the discriminant function for class k given input x, mean,  Σ² and Plk are all estimated from the data and the class is calculated as having the largest value, will be considered in the output classification.  

How to prepare data from LDA?

Some suggestions you should keep in mind while preparing your data to build your LDA model:

  • LDA is mainly used in classification problems where you have a categorical output variable. It allows both binary classification and multi-class classification.
  • The standard LDA model makes use of the Gaussian Distribution of the input variables. You should check the univariate distributions of each attribute and transform them into a more Gaussian-looking distribution. For example, for the exponential distribution, use log and root function and for skewed distributions use BoxCox.
  • Outliers can skew the primitive statistics used to separate classes in LDA, so it is preferable to remove them.
  • Since LDA assumes that each input variable has the same variance, it is always better to standardize your data before using an LDA model. Keep the mean to be 0 and the standard deviation to be 1.

How to implement an LDA model from scratch?

You can implement a Linear Discriminant Analysis model from scratch using Python. Let’s start by importing the libraries that are required for the model:

from sklearn.datasets import load_wine
import pandas as pd
import numpy as np
np.set_printoptions(precision=4)
from matplotlib import pyplot as plt
import seaborn as sns
sns.set()
from sklearn.preprocessing import LabelEncoder
from sklearn.tree import DecisionTreeClassifier
from sklearn.model_selection import train_test_split
from sklearn.metrics import confusion_matrix

Since we will work with the wine dataset, you can obtain it from the UCI machine learning repository. The scikit-learn library in Python provides a wrapper function for downloading it:

wine_info = load_wine()
X = pd.DataFrame(wine_info.data, columns=wine_info.feature_names)
y = pd.Categorical.from_codes(wine_info.target, wine_info.target_names)

The wine dataset comprises of 178 rows of 13 columns each:

X.shape
(178, 13)

The attributes of the wine dataset comprise of various characteristics such as alcohol content of the wine, magnesium content, color intensity, hue and many more:

X.head()

How to implement an LDA model from scratch?

The wine dataset contains three different kinds of wine:

wine_info.target_names 
array(['class_0', 'class_1', 'class_2'], dtype='<U7')

Now we create a DataFrame which will contain both the features and the content of the dataset:

df = X.join(pd.Series(y, name='class'))

We can divide the process of Linear Discriminant Analysis into 5 steps as follows:

Step 1 - Computing the within-class and between-class scatter matrices.
Step 2 - Computing the eigenvectors and their corresponding eigenvalues for the scatter matrices.
Step 3 - Sorting the eigenvalues and selecting the top k.
Step 4 - Creating a new matrix that will contain the eigenvectors mapped to the k eigenvalues.
Step 5 - Obtaining new features by taking the dot product of the data and the matrix from Step 4.

Within-class scatter matrix

To calculate the within-class scatter matrix, you can use the following mathematical expression:

Within-class scatter matrix

where, c = total number of distinct classes and

Within-class scatter matrix

Within-class scatter matrix

where, x = a sample (i.e. a row).
            n = total number of samples within a given class.

Now we create a vector with the mean values of each feature:

feature_means1 = pd.DataFrame(columns=wine_info.target_names)
for c, rows in df.groupby('class'):
feature_means1[c] = rows.mean()
feature_means1

Within-class scatter matrix

The mean vectors (mi ) are now plugged into the above equations to obtain the within-class scatter matrix:

withinclass_scatter_matrix = np.zeros((13,13))
for c, rows in df.groupby('class'):
rows = rows.drop(['class'], axis=1)

s = np.zeros((13,13))
for index, row in rows.iterrows():
x, mc = row.values.reshape(13,1),
feature_means1[c].values.reshape(13,1)

s += (x - mc).dot((x - mc).T)

withinclass_scatter_matrix += s

Between-class scatter matrix

We can calculate the between-class scatter matrix using the following mathematical expression:

Between-class scatter matrix

where,

Between-class scatter matrix

and

Between-class scatter matrix

feature_means2 = df.mean()
betweenclass_scatter_matrix = np.zeros((13,13))
for c in feature_means1:    
   n = len(df.loc[df['class'] == c].index)
   mc, m = feature_means1[c].values.reshape(13,1), 
feature_means2.values.reshape(13,1)
betweenclass_scatter_matrix += n * (mc - m).dot((mc - m).T)

Now we will solve the generalized eigenvalue problem to obtain the linear discriminants for:

eigen_values, eigen_vectors = 
np.linalg.eig(np.linalg.inv(withinclass_scatter_matrix).dot(betweenclass_scatter_matrix))

We will sort the eigenvalues from the highest to the lowest since the eigenvalues with the highest values carry the most information about the distribution of data is done. Next, we will first k eigenvectors. Finally, we will place the eigenvalues in a temporary array to make sure the eigenvalues map to the same eigenvectors after the sorting is done:

eigen_pairs = [(np.abs(eigen_values[i]), eigen_vectors[:,i]) for i in range(len(eigen_values))]
eigen_pairs = sorted(eigen_pairs, key=lambda x: x[0], reverse=True)
for pair in eigen_pairs:
print(pair[0])
237.46123198302251
46.98285938758684
1.4317197551638386e-14
1.2141209883217706e-14
1.2141209883217706e-14
8.279823065850476e-15
7.105427357601002e-15
6.0293733655173466e-15
6.0293733655173466e-15
4.737608877108813e-15
4.737608877108813e-15
2.4737196789039026e-15
9.84629525010022e-16

Now we will transform the values into percentage since it is difficult to understand how much of the variance is explained by each component.

sum_of_eigen_values = sum(eigen_values)
print('Explained Variance')
for i, pair in enumerate(eigen_pairs):
   print('Eigenvector {}: {}'.format(i, (pair[0]/sum_of_eigen_values).real))
Explained Variance
Eigenvector 0: 0.8348256799387275
Eigenvector 1: 0.1651743200612724
Eigenvector 2: 5.033396012077518e-17
Eigenvector 3: 4.268399397827047e-17
Eigenvector 4: 4.268399397827047e-17
Eigenvector 5: 2.9108789097898625e-17
Eigenvector 6: 2.498004906118145e-17
Eigenvector 7: 2.119704204950956e-17
Eigenvector 8: 2.119704204950956e-17
Eigenvector 9: 1.665567688286435e-17
Eigenvector 10: 1.665567688286435e-17
Eigenvector 11: 8.696681541121664e-18
Eigenvector 12: 3.4615924706522496e-18

First, we will create a new matrix W using the first two eigenvectors:

W_matrix = np.hstack((eigen_pairs[0][1].reshape(13,1), eigen_pairs[1][1].reshape(13,1))).real

Next, we will save the dot product of X and W into a new matrix Y:

Y = X∗W

where, X = n x d matrix with n sample and d dimensions.
            Y = n x k matrix with n sample and k dimensions.

In simple terms, Y is the new matrix or the new feature space.

X_lda = np.array(X.dot(W_matrix))

Our next work is to encode every class a member in order to incorporate the class labels into our plot. This is done because matplotlib cannot handle categorical variables directly.

Finally, we plot the data as a function of the two LDA components using different color for each class:

plt.xlabel('LDA1')
plt.ylabel('LDA2')
plt.scatter(
X_lda[:,0],
X_lda[:,1],
c=y,
cmap='rainbow',
alpha=0.7,
edgecolors='b'
)
<matplotlib.collections.PathCollection at 0x7fd08a20e908>

Between-class scatter matrix

How to implement LDA using scikit-learn?

For implementing LDA using scikit-learn, let’s work with the same wine dataset. You can also obtain it from the  UCI machine learning repository. 

You can use the predefined class LinearDiscriminant Analysis made available to us by the scikit-learn library to implement LDA rather than implementing from scratch every time:

from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
lda_model = LinearDiscriminantAnalysis()
X_lda = lda_model.fit_transform(X, y)

To obtain the variance corresponding to each component, you can access the following property:

lda.explained_variance_ratio_
array([0.6875, 0.3125])

Again, we will plot the two LDA components just like we did before:

plt.xlabel('LDA1')
plt.ylabel('LDA2')
plt.scatter(
X_lda[:,0],
X_lda[:,1],
   c=y,
cmap='rainbow',
   alpha=0.7,
edgecolors='b'
)
<matplotlib.collections.PathCollection at 0x7fd089f60358>

How to implement LDA using scikit-learn?

Linear Discriminant Analysis vs PCA

Below are the differences between LDA and PCA:

  • PCA ignores class labels and focuses on finding the principal components that maximizes the variance in a given data. Thus it is an unsupervised algorithm. On the other hand, LDA is a supervised algorithm that intends to find the linear discriminants that represents those axes which maximize separation between different classes.
  • LDA performs better multi-class classification tasks than PCA. However, PCA performs better when the sample size is comparatively small. An example would be comparisons between classification accuracies that are used in image classification.
  • Both LDA and PCA are used in case of dimensionality reduction. PCA is first followed by LDA.

Linear Discriminant Analysis vs PCA

Let us create and fit an instance of the PCA class:

from sklearn.decomposition import PCA
pca_class = PCA(n_components=2)
X_pca = pca.fit_transform(X, y)

Again, to view the values in percentage for a better understanding, we will access the explained_variance_ratio_ property:

pca.explained_variance_ratio_
array([0.9981, 0.0017])

Clearly, PCA selected the components which will be able to retain the most information and ignores the ones which maximize the separation between classes.

plt.xlabel('PCA1')
plt.ylabel('PCA2')
plt.scatter(
   X_pca[:,0],
   X_pca[:,1],
   c=y,
   cmap='rainbow',
   alpha=0.7,
   edgecolors='b

Linear Discriminant Analysis vs PCA

Now to create a classification model using the LDA components as features, we will divide the data into training datasets and testing datasets:

X_train, X_test, y_train, y_test = train_test_split(X_lda, y, random_state=1)

The next thing we will do is create a Decision Tree. Then, we will predict the category of each sample test and create a confusion matrix to evaluate the LDA model’s performance:

data = DecisionTreeClassifier()
data.fit(X_train, y_train)
y_pred = data.predict(X_test)
confusion_matrix(y_test, y_pred)
array([[18,  0,  0], 
       [ 0, 17,  0], 
       [ 0,  0, 10]])

So it is clear that the Decision Tree Classifier has correctly classified everything in the test dataset.

What are the extensions to LDA?

LDA is considered to be a very simple and effective method, especially for classification techniques. Since it is simple and well understood, so it has a lot of extensions and variations:

  • Quadratic Discriminant Analysis(QDA) – When there are multiple input variables, each of the class uses its own estimate of variance and covariance.
  • Flexible Discriminant Analysis(FDA) – This technique is performed when a non-linear combination of inputs is used as splines.
  • Regularized Discriminant Analysis(RDA) – It moderates the influence of various variables in LDA by regularizing the estimate of the covariance.

Real-Life Applications of LDA

Some of the practical applications of LDA are listed below:

  • Face Recognition – LDA is used in face recognition to reduce the number of attributes to a more manageable number before the actual classification. The dimensions that are generated are a linear combination of pixels that forms a template. These are called Fisher’s faces.
  • Medical – You can use LDA to classify the patient disease as mild, moderate or severe. The classification is done upon the various parameters of the patient and his medical trajectory. 
  • Customer Identification – You can obtain the features of customers by performing a simple question and answer survey. LDA helps in identifying and selecting which describes the properties of a group of customers who are most likely to buy a particular item in a shopping mall. 

Summary

Let us take a look at the topics we have covered in this article: 

  • Dimensionality Reduction and need for LDA 
  • Working of an LDA model 
  • Representation, Learning, Prediction and preparing data in LDA 
  • Implementation of an LDA model 
  • Implementation of LDA using scikit-learn 
  • LDA vs PCA 
  • Extensions and Applications of LDA 

The Linear Discriminant Analysis in Python is a very simple and well-understood approach of classification in machine learning. Though there are other dimensionality reduction techniques like Logistic Regression or PCA, but LDA is preferred in many special classification cases. If you want to be an expert in machine learning, knowledge of Linear Discriminant Analysis would lead you to that position effortlessly. Enrol in our  Data Science and Machine Learning Courses for more lucrative career options in this landscape and become a certified Data Scientist.

Priyankur

Priyankur Sarkar

Data Science Enthusiast

Priyankur Sarkar loves to play with data and get insightful results out of it, then turn those data insights and results in business growth. He is an electronics engineer with a versatile experience as an individual contributor and leading teams, and has actively worked towards building Machine Learning capabilities for organizations.

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Types of Probability Distributions Every Data Science Expert Should know

Data Science has become one of the most popular interdisciplinary fields. It uses scientific approaches, methods, algorithms, and operations to obtain facts and insights from unstructured, semi-structured, and structured datasets. Organizations use these collected facts and insights for efficient production, business growth, and to predict user requirements. Probability distribution plays a significant role in performing data analysis equipping a dataset for training a model. In this article, you will learn about the types of Probability Distribution, random variables, types of discrete distributions, and continuous distribution.  What is Probability Distribution? A Probability Distribution is a statistical method that determines all the probable values and possibilities that a random variable can deliver from a particular range. This range of values will have a lower bound and an upper bound, which we call the minimum and the maximum possible values.  Various factors on which plotting of a value depends are standard deviation, mean (or average), skewness, and kurtosis. All of these play a significant role in Data science as well. We can use probability distribution in physics, engineering, finance, data analysis, machine learning, etc. Significance of Probability distributions in Data Science In a way, most of the data science and machine learning operations are dependent on several assumptions about the probability of your data. Probability distribution allows a skilled data analyst to recognize and comprehend patterns from large data sets; that is, otherwise, entirely random variables and values. Thus, it makes probability distribution a toolkit based on which we can summarize a large data set. The density function and distribution techniques can also help in plotting data, thus supporting data analysts to visualize data and extract meaning. General Properties of Probability Distributions Probability distribution determines the likelihood of any outcome. The mathematical expression takes a specific value of x and shows the possibility of a random variable with p(x). Some general properties of the probability distribution are – The total of all probabilities for any possible value becomes equal to 1. In a probability distribution, the possibility of finding any specific value or a range of values must lie between 0 and 1. Probability distributions tell us the dispersal of the values from the random variable. Consequently, the type of variable also helps determine the type of probability distribution.Common Data Types Before jumping directly into explaining the different probability distributions, let us first understand the different types of probability distributions or the main categories of the probability distribution. Data analysts and data engineers have to deal with a broad spectrum of data, such as text, numerical, image, audio, voice, and many more. Each of these have a specific means to be represented and analyzed. Data in a probability distribution can either be discrete or continuous. Numerical data especially takes one of the two forms. Discrete data: They take specific values where the outcome of the data remains fixed. Like, for example, the consequence of rolling two dice or the number of overs in a T-20 match. In the first case, the result lies between 2 and 12. In the second case, the event will be less than 20. Different types of discrete distributions that use discrete data are: Binomial Distribution Hypergeometric Distribution Geometric Distribution Poisson Distribution Negative Binomial Distribution Multinomial Distribution  Continuous data: It can obtain any value irrespective of bound or limit. Example: weight, height, any trigonometric value, age, etc. Different types of continuous distributions that use continuous data are: Beta distribution Cauchy distribution Exponential distribution Gamma distribution Logistic distribution Weibull distribution Types of Probability Distribution explained Here are some of the popular types of Probability distributions used by data science professionals. (Try all the code using Jupyter Notebook) Normal Distribution: It is also known as Gaussian distribution. It is one of the simplest types of continuous distribution. This probability distribution is symmetrical around its mean value. It also shows that data at close proximity of the mean is frequently occurring, compared to data that is away from it. Here, mean = 0, variance = finite valueHere, you can see 0 at the center is the Normal Distribution for different mean and variance values. Here is a code example showing the use of Normal Distribution: from scipy.stats import norm  import matplotlib.pyplot as mpl  import numpy as np  def normalDist() -> None:      fig, ax = mpl.subplots(1, 1)      mean, var, skew, kurt = norm.stats(moments = 'mvsk')      x = np.linspace(norm.ppf(0.01),  norm.ppf(0.99), 100)      ax.plot(x, norm.pdf(x),          'r-', lw = 5, alpha = 0.6, label = 'norm pdf')      ax.plot(x, norm.cdf(x),          'b-', lw = 5, alpha = 0.6, label = 'norm cdf')      vals = norm.ppf([0.001, 0.5, 0.999])      np.allclose([0.001, 0.5, 0.999], norm.cdf(vals))      r = norm.rvs(size = 1000)      ax.hist(r, normed = True, histtype = 'stepfilled', alpha = 0.2)      ax.legend(loc = 'best', frameon = False)      mpl.show()  normalDist() Output: Bernoulli Distribution: It is the simplest type of probability distribution. It is a particular case of Binomial distribution, where n=1. It means a binomial distribution takes 'n' number of trials, where n > 1 whereas, the Bernoulli distribution takes only a single trial.   Probability Mass Function of a Bernoulli’s Distribution is:  where p = probability of success and q = probability of failureHere is a code example showing the use of Bernoulli Distribution: from scipy.stats import bernoulli  import seaborn as sb    def bernoulliDist():      data_bern = bernoulli.rvs(size=1200, p = 0.7)      ax = sb.distplot(          data_bern,           kde = True,           color = 'g',           hist_kws = {'alpha' : 1},          kde_kws = {'color': 'y', 'lw': 3, 'label': 'KDE'})      ax.set(xlabel = 'Bernouli Values', ylabel = 'Frequency Distribution')  bernoulliDist() Output:Continuous Uniform Distribution: In this type of continuous distribution, all outcomes are equally possible; each variable gets the same probability of hit as a consequence. This symmetric probabilistic distribution has random variables at an equal interval, with the probability of 1/(b-a). Here is a code example showing the use of Uniform Distribution: from numpy import random  import matplotlib.pyplot as mpl  import seaborn as sb  def uniformDist():      sb.distplot(random.uniform(size = 1200), hist = True)      mpl.show()  uniformDist() Output: Log-Normal Distribution: A Log-Normal distribution is another type of continuous distribution of logarithmic values that form a normal distribution. We can transform a log-normal distribution into a normal distribution. Here is a code example showing the use of Log-Normal Distribution import matplotlib.pyplot as mpl  def lognormalDist():      muu, sig = 3, 1      s = np.random.lognormal(muu, sig, 1000)      cnt, bins, ignored = mpl.hist(s, 80, normed = True, align ='mid', color = 'y')      x = np.linspace(min(bins), max(bins), 10000)      calc = (np.exp( -(np.log(x) - muu) **2 / (2 * sig**2))             / (x * sig * np.sqrt(2 * np.pi)))      mpl.plot(x, calc, linewidth = 2.5, color = 'g')      mpl.axis('tight')      mpl.show()  lognormalDist() Output: Pareto Distribution: It is one of the most critical types of continuous distribution. The Pareto Distribution is a skewed statistical distribution that uses power-law to describe quality control, scientific, social, geophysical, actuarial, and many other types of observable phenomena. The distribution shows slow or heavy-decaying tails in the plot, where much of the data reside at its extreme end. Here is a code example showing the use of Pareto Distribution – import numpy as np  from matplotlib import pyplot as plt  from scipy.stats import pareto  def paretoDist():      xm = 1.5        alp = [2, 4, 6]       x = np.linspace(0, 4, 800)      output = np.array([pareto.pdf(x, scale = xm, b = a) for a in alp])      plt.plot(x, output.T)      plt.show()  paretoDist() Output:Exponential Distribution: It is a type of continuous distribution that determines the time elapsed between events (in a Poisson process). Let’s suppose, that you have the Poisson distribution model that holds the number of events happening in a given period. We can model the time between each birth using an exponential distribution.Here is a code example showing the use of Pareto Distribution – from numpy import random  import matplotlib.pyplot as mpl  import seaborn as sb  def expDist():      sb.distplot(random.exponential(size = 1200), hist = True)      mpl.show()   expDist()Output:Types of the Discrete probability distribution – There are various types of Discrete Probability Distribution a Data science aspirant should know about. Some of them are – Binomial Distribution: It is one of the popular discrete distributions that determine the probability of x success in the 'n' trial. We can use Binomial distribution in situations where we want to extract the probability of SUCCESS or FAILURE from an experiment or survey which went through multiple repetitions. A Binomial distribution holds a fixed number of trials. Also, a binomial event should be independent, and the probability of obtaining failure or success should remain the same. Here is a code example showing the use of Binomial Distribution – from numpy import random  import matplotlib.pyplot as mpl  import seaborn as sb    def binomialDist():      sb.distplot(random.normal(loc = 50, scale = 6, size = 1200), hist = False, label = 'normal')      sb.distplot(random.binomial(n = 100, p = 0.6, size = 1200), hist = False, label = 'binomial')      plt.show()    binomialDist() Output:Geometric Distribution: The geometric probability distribution is one of the crucial types of continuous distributions that determine the probability of any event having likelihood ‘p’ and will happen (occur) after 'n' number of Bernoulli trials. Here 'n' is a discrete random variable. In this distribution, the experiment goes on until we encounter either a success or a failure. The experiment does not depend on the number of trials. Here is a code example showing the use of Geometric Distribution – import matplotlib.pyplot as mpl  def probability_to_occur_at(attempt, probability):      return (1-p)**(attempt - 1) * probability  p = 0.3  attempt = 4  attempts_to_show = range(21)[1:]  print('Possibility that this event will occur on the 7th try: ', probability_to_occur_at(attempt, p))  mpl.xlabel('Number of Trials')  mpl.ylabel('Probability of the Event')  barlist = mpl.bar(attempts_to_show, height=[probability_to_occur_at(x, p) for x in attempts_to_show], tick_label=attempts_to_show)  barlist[attempt].set_color('g')  mpl.show() Output:Poisson Distribution: Poisson distribution is one of the popular types of discrete distribution that shows how many times an event has the possibility of occurrence in a specific set of time. We can obtain this by limiting the Bernoulli distribution from 0 to infinity. Data analysts often use the Poisson distributions to comprehend independent events occurring at a steady rate in a given time interval. Here is a code example showing the use of Poisson Distribution from scipy.stats import poisson  import seaborn as sb  import numpy as np  import matplotlib.pyplot as mpl  def poissonDist():       mpl.figure(figsize = (10, 10))      data_binom = poisson.rvs(mu = 3, size = 5000)      ax = sb.distplot(data_binom, kde=True, color = 'g',                       bins=np.arange(data_binom.min(), data_binom.max() + 1),                       kde_kws={'color': 'y', 'lw': 4, 'label': 'KDE'})      ax.set(xlabel = 'Poisson Distribution', ylabel='Data Frequency')      mpl.show()      poissonDist() Output:Multinomial Distribution: A multinomial distribution is another popular type of discrete probability distribution that calculates the outcome of an event having two or more variables. The term multi means more than one. The Binomial distribution is a particular type of multinomial distribution with two possible outcomes - true/false or heads/tails. Here is a code example showing the use of Multinomial Distribution – import numpy as np  import matplotlib.pyplot as mpl  np.random.seed(99)   n = 12                      pvalue = [0.3, 0.46, 0.22]     s = []  p = []     for size in np.logspace(2, 3):      outcomes = np.random.multinomial(n, pvalue, size=int(size))        prob = sum((outcomes[:,0] == 7) & (outcomes[:,1] == 2) & (outcomes[:,2] == 3))/len(outcomes)      p.append(prob)      s.append(int(size))  fig1 = mpl.figure()  mpl.plot(s, p, 'o-')  mpl.plot(s, [0.0248]*len(s), '--r')  mpl.grid()  mpl.xlim(xmin = 0)  mpl.xlabel('Number of Events')  mpl.ylabel('Function p(X = K)') Output:Negative Binomial Distribution: It is also a type of discrete probability distribution for random variables having negative binomial events. It is also known as the Pascal distribution, where the random variable tells us the number of repeated trials produced during a specific number of experiments.  Here is a code example showing the use of Negative Binomial Distribution – import matplotlib.pyplot as mpl   import numpy as np   from scipy.stats import nbinom    x = np.linspace(0, 6, 70)   gr, kr = 0.3, 0.7        g = nbinom.ppf(x, gr, kr)   s = nbinom.pmf(x, gr, kr)   mpl.plot(x, g, "*", x, s, "r--") Output: Apart from these mentioned distribution types, various other types of probability distributions exist that data science professionals can use to extract reliable datasets. In the next topic, we will understand some interconnections & relationships between various types of probability distributions. Relationship between various Probability distributions – It is surprising to see that different types of probability distributions are interconnected. In the chart shown below, the dashed line is for limited connections between two families of distribution, whereas the solid lines show the exact relationship between them in terms of transformation, variable, type, etc. Conclusion  Probability distributions are prevalent among data analysts and data science professionals because of their wide usage. Today, companies and enterprises hire data science professionals in many sectors, namely, computer science, health, insurance, engineering, and even social science, where probability distributions appear as fundamental tools for application. It is essential for Data analysts and data scientists. to know the core of statistics. Probability Distributions perform a requisite role in analyzing data and cooking a dataset to train the algorithms efficiently. If you want to learn more about data science - particularly probability distributions and their uses, check out KnowledgeHut's comprehensive Data science course. 
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Role of Unstructured Data in Data Science

Data has become the new game changer for businesses. Typically, data scientists categorize data into three broad divisions - structured, semi-structured, and unstructured data. In this article, you will get to know about unstructured data, sources of unstructured data, unstructured data vs. structured data, the use of structured and unstructured data in machine learning, and the difference between structured and unstructured data. Let us first understand what is unstructured data with examples. What is unstructured data? Unstructured data is a kind of data format where there is no organized form or type of data. Videos, texts, images, document files, audio materials, email contents and more are considered to be unstructured data. It is the most copious form of business data, and cannot be stored in a structured database or relational database. Some examples of unstructured data are the photos we post on social media platforms, the tagging we do, the multimedia files we upload, and the documents we share. Seagate predicts that the global data-sphere will expand to 163 zettabytes by 2025, where most of the data will be in the unstructured format. Characteristics of Unstructured DataUnstructured data cannot be organized in a predefined fashion, and is not a homogenous data model. This makes it difficult to manage. Apart from that, these are the other characteristics of unstructured data. You cannot store unstructured data in the form of rows and columns as we do in a database table. Unstructured data is heterogeneous in structure and does not have any specific data model. The creation of such data does not follow any semantics or habits. Due to the lack of any particular sequence or format, it is difficult to manage. Such data does not have an identifiable structure. Sources of Unstructured Data There are various sources of unstructured data. Some of them are: Content websites Social networking sites Online images Memos Reports and research papers Documents, spreadsheets, and presentations Audio mining, chatbots Surveys Feedback systems Advantages of Unstructured Data Unstructured data has become exceptionally easy to store because of MongoDB, Cassandra, or even using JSON. Modern NoSQL databases and software allows data engineers to collect and extract data from various sources. There are numerous benefits that enterprises and businesses can gain from unstructured data. These are: With the advent of unstructured data, we can store data that lacks a proper format or structure. There is no fixed schema or data structure for storing such data, which gives flexibility in storing data of different genres. Unstructured data is much more portable by nature. Unstructured data is scalable and flexible to store. Database systems like MongoDB, Cassandra, etc., can easily handle the heterogeneous properties of unstructured data. Different applications and platforms produce unstructured data that becomes useful in business intelligence, unstructured data analytics, and various other fields. Unstructured data analysis allows finding comprehensive data stories from data like email contents, website information, social media posts, mobile data, cache files and more. Unstructured data, along with data analytics, helps companies improve customer experience. Detection of the taste of consumers and their choices becomes easy because of unstructured data analysis. Disadvantages of Unstructured data Storing and managing unstructured data is difficult because there is no proper structure or schema. Data indexing is also a substantial challenge and hence becomes unclear due to its disorganized nature. Search results from an unstructured dataset are also not accurate because it does not have predefined attributes. Data security is also a challenge due to the heterogeneous form of data. Problems faced and solutions for storing unstructured data. Until recently, it was challenging to store, evaluate, and manage unstructured data. But with the advent of modern data analysis tools, algorithms, CAS (content addressable storage system), and big data technologies, storage and evaluation became easy. Let us first take a look at the various challenges used for storing unstructured data. Storing unstructured data requires a large amount of space. Indexing of unstructured data is a hectic task. Database operations such as deleting and updating become difficult because of the disorganized nature of the data. Storing and managing video, audio, image file, emails, social media data is also challenging. Unstructured data increases the storage cost. For solving such issues, there are some particular approaches. These are: CAS system helps in storing unstructured data efficiently. We can preserve unstructured data in XML format. Developers can store unstructured data in an RDBMS system supporting BLOB. We can convert unstructured data into flexible formats so that evaluating and storage becomes easy. Let us now understand the differences between unstructured data vs. structured data. Unstructured Data Vs. Structured Data In this section, we will understand the difference between structured and unstructured data with examples. STRUCTUREDUNSTRUCTUREDStructured data resides in an organized format in a typical database.Unstructured data cannot reside in an organized format, and hence we cannot store it in a typical database.We can store structured data in SQL database tables having rows and columns.Storing and managing unstructured data requires specialized databases, along with a variety of business intelligence and analytics applications.It is tough to scale a database schema.It is highly scalable.Structured data gets generated in colleges, universities, banks, companies where people have to deal with names, date of birth, salary, marks and so on.We generate or find unstructured data in social media platforms, emails, analyzed data for business intelligence, call centers, chatbots and so on.Queries in structured data allow complex joining.Unstructured data allows only textual queries.The schema of a structured dataset is less flexible and dependent.An unstructured dataset is flexible but does not have any particular schema.It has various concurrency techniques.It has no concurrency techniques.We can use SQL, MySQL, SQLite, Oracle DB, Teradata to store structured data.We can use NoSQL (Not Only SQL) to store unstructured data.Types of Unstructured Data Do you have any idea just how much of unstructured data we produce and from what sources? Unstructured data includes all those forms of data that we cannot actively manage in an RDBMS system that is a transactional system. We can store structured data in the form of records. But this is not the case with unstructured data. Before the advent of object-based storage, most of the unstructured data was stored in file-based systems. Here are some of the types of unstructured data. Rich media content: Entertainment files, surveillance data, multimedia email attachments, geospatial data, audio files (call center and other recorded audio), weather reports (graphical), etc., comes under this genre. Document data: Invoices, text-file records, email contents, productivity applications, etc., are included under this genre. Internet of Things (IoT) data: Ticker data, sensor data, data from other IoT devices come under this genre. Apart from all these, data from business intelligence and analysis, machine learning datasets, and artificial intelligence data training datasets are also a separate genre of unstructured data. Examples of Unstructured Data There are various sources from where we can obtain unstructured data. The prominent use of this data is in unstructured data analytics. Let us now understand what are some examples of unstructured data and their sources – Healthcare industries generate a massive volume of human as well as machine-generated unstructured data. Human-generated unstructured data could be in the form of patient-doctor or patient-nurse conversations, which are usually recorded in audio or text formats. Unstructured data generated by machines includes emergency video camera footage, surgical robots, data accumulated from medical imaging devices like endoscopes, laparoscopes and more.  Social Media is an intrinsic entity of our daily life. Billions of people come together to join channels, share different thoughts, and exchange information with their loved ones. They create and share such data over social media platforms in the form of images, video clips, audio messages, tagging people (this helps companies to map relations between two or more people), entertainment data, educational data, geolocations, texts, etc. Other spectra of data generated from social media platforms are behavior patterns, perceptions, influencers, trends, news, and events. Business and corporate documents generate a multitude of unstructured data such as emails, presentations, reports containing texts, images, presentation reports, video contents, feedback and much more. These documents help to create knowledge repositories within an organization to make better implicit operations. Live chat, video conferencing, web meeting, chatbot-customer messages, surveillance data are other prominent examples of unstructured data that companies can cultivate to get more insights into the details of a person. Some prominent examples of unstructured data used in enterprises and organizations are: Reports and documents, like Word files or PDF files Multimedia files, such as audio, images, designed texts, themes, and videos System logs Medical images Flat files Scanned documents (which are images that hold numbers and text – for example, OCR) Biometric data Unstructured Data Analytics Tools  You might be wondering what tools can come into use to gather and analyze information that does not have a predefined structure or model. Various tools and programming languages use structured and unstructured data for machine learning and data analysis. These are: Tableau MonkeyLearn Apache Spark SAS Python MS. Excel RapidMiner KNIME QlikView Python programming R programming Many cloud services (like Amazon AWS, Microsoft Azure, IBM Cloud, Google Cloud) also offer unstructured data analysis solutions bundled with their services. How to analyze unstructured data? In the past, the process of storage and analysis of unstructured data was not well defined. Enterprises used to carry out this kind of analysis manually. But with the advent of modern tools and programming languages, most of the unstructured data analysis methods became highly advanced. AI-powered tools use algorithms designed precisely to help to break down unstructured data for analysis. Unstructured data analytics tools, along with Natural language processing (NLP) and machine learning algorithms, help advanced software tools analyze and extract analytical data from the unstructured datasets. Before using these tools for analyzing unstructured data, you must properly go through a few steps and keep these points in mind. Set a clear goal for analyzing the data: It is essential to clear your intention about what insights you want to extract from your unstructured data. Knowing this will help you distinguish what type of data you are planning to accumulate. Collect relevant data: Unstructured data is available everywhere, whether it's a social media platform, online feedback or reviews, or a survey form. Depending on the previous point, that is your goal - you have to be precise about what data you want to collect in real-time. Also, keep in mind whether your collected details are relevant or not. Clean your data: Data cleaning or data cleansing is a significant process to detect corrupt or irrelevant data from the dataset, followed by modifying or deleting the coarse and sloppy data. This phase is also known as the data-preprocessing phase, where you have to reduce the noise, carry out data slicing for meaningful representation, and remove unnecessary data. Use Technology and tools: Once you perform the data cleaning, it is time to utilize unstructured data analysis tools to prepare and cultivate the insights from your data. Technologies used for unstructured data storage (NoSQL) can help in managing your flow of data. Other tools and programming libraries like Tableau, Matplotlib, Pandas, and Google Data Studio allows us to extract and visualize unstructured data. Data can be visualized and presented in the form of compelling graphs, plots, and charts. How to Extract information from Unstructured Data? With the growth in digitization during the information era, repetitious transactions in data cause data flooding. The exponential accretion in the speed of digital data creation has brought a whole new domain of understanding user interaction with the online world. According to Gartner, 80% of the data created by an organization or its application is unstructured. While extracting exact information through appropriate analysis of organized data is not yet possible, even obtaining a decent sense of this unstructured data is quite tough. Until now, there are no perfect tools to analyze unstructured data. But algorithms and tools designed using machine learning, Natural language processing, Deep learning, and Graph Analysis (a mathematical method for estimating graph structures) help us to get the upper hand in extracting information from unstructured data. Other neural network models like modern linguistic models follow unsupervised learning techniques to gain a good 'knowledge' about the unstructured dataset before going into a specific supervised learning step. AI-based algorithms and technologies are capable enough to extract keywords, locations, phone numbers, analyze image meaning (through digital image processing). We can then understand what to evaluate and identify information that is essential to your business. ConclusionUnstructured data is found abundantly from sources like documents, records, emails, social media posts, feedbacks, call-records, log-in session data, video, audio, and images. Manually analyzing unstructured data is very time-consuming and can be very boring at the same time. With the growth of data science and machine learning algorithms and models, it has become easy to gather and analyze insights from unstructured information.  According to some research, data analytics tools like MonkeyLearn Studio, Tableau, RapidMiner help analyze unstructured data 1200x faster than the manual approach. Analyzing such data will help you learn more about your customers as well as competitors. Text analysis software, along with machine learning models, will help you dig deep into such datasets and make you gain an in-depth understanding of the overall scenario with fine-grained analyses.
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Role of Unstructured Data in Data Science

Data has become the new game changer for busines... Read More

What Is Statistical Analysis and Its Business Applications?

Statistics is a science concerned with collection, analysis, interpretation, and presentation of data. In Statistics, we generally want to study a population. You may consider a population as a collection of things, persons, or objects under experiment or study. It is usually not possible to gain access to all of the information from the entire population due to logistical reasons. So, when we want to study a population, we generally select a sample. In sampling, we select a portion (or subset) of the larger population and then study the portion (or the sample) to learn about the population. Data is the result of sampling from a population.Major ClassificationThere are two basic branches of Statistics – Descriptive and Inferential statistics. Let us understand the two branches in brief. Descriptive statistics Descriptive statistics involves organizing and summarizing the data for better and easier understanding. Unlike Inferential statistics, Descriptive statistics seeks to describe the data, however, it does not attempt to draw inferences from the sample to the whole population. We simply describe the data in a sample. It is not developed on the basis of probability unlike Inferential statistics. Descriptive statistics is further broken into two categories – Measure of Central Tendency and Measures of Variability. Inferential statisticsInferential statistics is the method of estimating the population parameter based on the sample information. It applies dimensions from sample groups in an experiment to contrast the conduct group and make overviews on the large population sample. Please note that the inferential statistics are effective and valuable only when examining each member of the group is difficult. Let us understand Descriptive and Inferential statistics with the help of an example. Task – Suppose, you need to calculate the score of the players who scored a century in a cricket tournament.  Solution: Using Descriptive statistics you can get the desired results.   Task – Now, you need the overall score of the players who scored a century in the cricket tournament.  Solution: Applying the knowledge of Inferential statistics will help you in getting your desired results.  Top Five Considerations for Statistical Data AnalysisData can be messy. Even a small blunder may cost you a fortune. Therefore, special care when working with statistical data is of utmost importance. Here are a few key takeaways you must consider to minimize errors and improve accuracy. Define the purpose and determine the location where the publication will take place.  Understand the assets to undertake the investigation. Understand the individual capability of appropriately managing and understanding the analysis.  Determine whether there is a need to repeat the process.  Know the expectation of the individuals evaluating reviewing, committee, and supervision. Statistics and ParametersDetermining the sample size requires understanding statistics and parameters. The two being very closely related are often confused and sometimes hard to distinguish.  StatisticsA statistic is merely a portion of a target sample. It refers to the measure of the values calculated from the population.  A parameter is a fixed and unknown numerical value used for describing the entire population. The most commonly used parameters are: Mean Median Mode Mean :  The mean is the average or the most common value in a data sample or a population. It is also referred to as the expected value. Formula: Sum of the total number of observations/the number of observations. Experimental data set: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20  Calculating mean:   (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20)/10  = 110/10   = 11 Median:  In statistics, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. It’s the mid-value obtained by arranging the data in increasing order or descending order. Formula:  Let n be the data set (increasing order) When data set is odd: Median = n+1/2th term Case-I: (n is odd)  Experimental data set = 1, 2, 3, 4, 5  Median (n = 5) = [(5 +1)/2]th term      = 6/2 term       = 3rd term   Therefore, the median is 3 When data set is even: Median = [n/2th + (n/2 + 1)th] /2 Case-II: (n is even)  Experimental data set = 1, 2, 3, 4, 5, 6   Median (n = 6) = [n/2th + (n/2 + 1)th]/2     = ( 6/2th + (6/2 +1)th]/2     = (3rd + 4th)/2      = (3 + 4)/2      = 7/2      = 3.5  Therefore, the median is 3.5 Mode: The mode is the value that appears most often in a set of data or a population. Experimental data set= 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4,4,5, 6  Mode = 3 (Since 3 is the most repeated element in the sequence.) Terms Used to Describe DataWhen working with data, you will need to search, inspect, and characterize them. To understand the data in a tech-savvy and straightforward way, we use a few statistical terms to denote them individually or in groups.  The most frequently used terms used to describe data include data point, quantitative variables, indicator, statistic, time-series data, variable, data aggregation, time series, dataset, and database. Let us define each one of them in brief: Data points: These are the numerical files formed and organized for interpretations. Quantitative variables: These variables present the information in digit form.  Indicator: An indicator explains the action of a community's social-economic surroundings.  Time-series data: The time-series defines the sequential data.  Data aggregation: A group of data points and data set. Database: A group of arranged information for examination and recovery.  Time-series: A set of measures of a variable documented over a specified time. Step-by-Step Statistical Analysis ProcessThe statistical analysis process involves five steps followed one after another. Step 1: Design the study and find the population of the study. Step 2: Collect data as samples. Step 3: Describe the data in the sample. Step 4: Make inferences with the help of samples and calculations Step 5: Take action Data distributionData distribution is an entry that displays entire imaginable readings of data. It shows how frequently a value occurs. Distributed data is always in ascending order, charts, and graphs enabling visibility of measurements and frequencies. The distribution function displaying the density of values of reading is known as the probability density function. Percentiles in data distributionA percentile is the reading in a distribution with a specified percentage of clarifications under it.  Let us understand percentiles with the help of an example.  Suppose you have scored 90th percentile on a math test. A basic interpretation is that merely 4-5% of the scores were higher than your scores. Right? The median is 50th percentile because the assumed 50% of the values are higher than the median. Dispersion Dispersion explains the magnitude of distribution readings anticipated for a specific variable and multiple unique statistics like range, variance, and standard deviation. For instance, high values of a data set are widely scattered while small values of data are firmly clustered. Histogram The histogram is a pictorial display that arranges a group of data facts into user detailed ranges. A histogram summarizes a data series into a simple interpreted graphic by obtaining many data facts and combining them into reasonable ranges. It contains a variety of results into columns on the x-axis. The y axis displays percentages of data for each column and is applied to picture data distributions. Bell Curve distribution Bell curve distribution is a pictorial representation of a probability distribution whose fundamental standard deviation obtained from the mean makes the bell, shaped curving. The peak point on the curve symbolizes the maximum likely occasion in a pattern of data. The other possible outcomes are symmetrically dispersed around the mean, making a descending sloping curve on both sides of the peak. The curve breadth is therefore known as the standard deviation. Hypothesis testingHypothesis testing is a process where experts experiment with a theory of a population parameter. It aims to evaluate the credibility of a hypothesis using sample data. The five steps involved in hypothesis testing are:  Identify the no outcome hypothesis.  (A worthless or a no-output hypothesis has no outcome, connection, or dissimilarities amongst many factors.) Identify the alternative hypothesis.  Establish the importance level of the hypothesis.  Estimate the experiment statistic and equivalent P-value. P-value explains the possibility of getting a sample statistic.  Sketch a conclusion to interpret into a report about the alternate hypothesis. Types of variablesA variable is any digit, amount, or feature that is countable or measurable. Simply put, it is a variable characteristic that varies. The six types of variables include the following: Dependent variableA dependent variable has values that vary according to the value of another variable known as the independent variable.  Independent variableAn independent variable on the other side is controllable by experts. Its reports are recorded and equated.  Intervening variableAn intervening variable explicates fundamental relations between variables. Moderator variableA moderator variable upsets the power of the connection between dependent and independent variables.  Control variableA control variable is anything restricted to a research study. The values are constant throughout the experiment. Extraneous variableExtraneous variable refers to the entire variables that are dependent but can upset experimental outcomes. Chi-square testChi-square test records the contrast of a model to actual experimental data. Data is unsystematic, underdone, equally limited, obtained from independent variables, and a sufficient sample. It relates the size of any inconsistencies among the expected outcomes and the actual outcomes, provided with the sample size and the number of variables in the connection. Types of FrequenciesFrequency refers to the number of repetitions of reading in an experiment in a given time. Three types of frequency distribution include the following: Grouped, ungrouped Cumulative, relative Relative cumulative frequency distribution. Features of FrequenciesThe calculation of central tendency and position (median, mean, and mode). The measure of dispersion (range, variance, and standard deviation). Degree of symmetry (skewness). Peakedness (kurtosis). Correlation MatrixThe correlation matrix is a table that shows the correlation coefficients of unique variables. It is a powerful tool that summarises datasets points and picture sequences in the provided data. A correlation matrix includes rows and columns that display variables. Additionally, the correlation matrix exploits in aggregation with other varieties of statistical analysis. Inferential StatisticsInferential statistics use random data samples for demonstration and to create inferences. They are measured when analysis of each individual of a whole group is not likely to happen. Applications of Inferential StatisticsInferential statistics in educational research is not likely to sample the entire population that has summaries. For instance, the aim of an investigation study may be to obtain whether a new method of learning mathematics develops mathematical accomplishment for all students in a class. Marketing organizations: Marketing organizations use inferential statistics to dispute a survey and request inquiries. It is because carrying out surveys for all the individuals about merchandise is not likely. Finance departments: Financial departments apply inferential statistics for expected financial plan and resources expenses, especially when there are several indefinite aspects. However, economists cannot estimate all that use possibility. Economic planning: In economic planning, there are potent methods like index figures, time series investigation, and estimation. Inferential statistics measures national income and its components. It gathers info about revenue, investment, saving, and spending to establish links among them. Key TakeawaysStatistical analysis is the gathering and explanation of data to expose sequences and tendencies.   Two divisions of statistical analysis are statistical and non-statistical analyses.  Descriptive and Inferential statistics are the two main categories of statistical analysis. Descriptive statistics describe data, whereas Inferential statistics equate dissimilarities between the sample groups.  Statistics aims to teach individuals how to use restricted samples to generate intellectual and precise results for a large group.   Mean, median, and mode are the statistical analysis parameters used to measure central tendency.   Conclusion Statistical analysis is the procedure of gathering and examining data to recognize sequences and trends. It uses random samples of data obtained from a population to demonstrate and create inferences on a group. Inferential statistics applies economic planning with potent methods like index figures, time series investigation, and estimation.  Statistical analysis finds its applications in all the major sectors – marketing, finance, economic, operations, and data mining. Statistical analysis aids marketing organizations in disputing a survey and requesting inquiries concerning their merchandise. 
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What Is Statistical Analysis and Its Business Appl...

Statistics is a science concerned with collection,... Read More