This article by the authors, Luis Pedro Coelho and Willi Richert, of the book, Building Machine Learning Systems with Python – Second Edition, focuses on the topic of classification.
(For more resources related to this topic, see here.)
You have probably already used this form of machine learning as a consumer, even if you were not aware of it. If you have any modern e-mail system, it will likely have the ability to automatically detect spam. That is, the system will analyze all incoming e-mails and mark them as either spam or not-spam. Often, you, the end user, will be able to manually tag e-mails as spam or not, in order to improve its spam detection ability. This is a form of machine learning where the system is taking examples of two types of messages: spam and ham (the typical term for “non spam e-mails”) and using these examples to automatically classify incoming e-mails.
The general method of classification is to use a set of examples of each class to learn rules that can be applied to new examples. This is one of the most important machine learning modes and is the topic of this article.
Working with text such as e-mails requires a specific set of techniques and skills. For the moment, we will work with a smaller, easier-to-handle dataset. The example question for this article is, “Can a machine distinguish between flower species based on images?” We will use two datasets where measurements of flower morphology are recorded along with the species for several specimens.
We will explore these small datasets using a few simple algorithms. At first, we will write classification code ourselves in order to understand the concepts, but we will quickly switch to using scikit-learn whenever possible. The goal is to first understand the basic principles of classification and then progress to using a state-of-the-art implementation.
The Iris dataset is a classic dataset from the 1930s; it is one of the first modern examples of statistical classification.
The dataset is a collection of morphological measurements of several Iris flowers. These measurements will enable us to distinguish multiple species of the flowers. Today, species are identified by their DNA fingerprints, but in the 1930s, DNA’s role in genetics had not yet been discovered.
The following four attributes of each plant were measured:
In general, we will call the individual numeric measurements we use to describe our data features. These features can be directly measured or computed from intermediate data.
This dataset has four features. Additionally, for each plant, the species was recorded. The problem we want to solve is, “Given these examples, if we see a new flower out in the field, could we make a good prediction about its species from its measurements?”
This is the supervised learning or classification problem: given labeled examples, can we design a rule to be later applied to other examples? A more familiar example to modern readers who are not botanists is spam filtering, where the user can mark e-mails as spam, and systems use these as well as the non-spam e-mails to determine whether a new, incoming message is spam or not.
For the moment, the Iris dataset serves our purposes well. It is small (150 examples, four features each) and can be easily visualized and manipulated.
Datasets will grow to thousands of features. With only four in our starting example, we can easily plot all two-dimensional projections on a single page. We will build intuitions on this small example, which can then be extended to large datasets with many more features. Visualizations are excellent at the initial exploratory phase of the analysis as they allow you to learn the general features of your problem as well as catch problems that occurred with data collection early.
Each subplot in the following plot shows all points projected into two of the dimensions. The outlying group (triangles) are the Iris Setosa plants, while Iris Versicolor plants are in the center (circle) and Iris Virginica are plotted with x marks. We can see that there are two large groups: one is of Iris Setosa and another is a mixture of Iris Versicolor and Iris Virginica.
In the following code snippet, we present the code to load the data and generate the plot:
>>> from matplotlib import pyplot as plt >>> import numpy as np >>> # We load the data with load_iris from sklearn >>> from sklearn.datasets import load_iris >>> data = load_iris() >>> # load_iris returns an object with several fields >>> features = data.data >>> feature_names = data.feature_names >>> target = data.target >>> target_names = data.target_names >>> for t in range(3): ... if t == 0: ... c = 'r' ... marker = '>' ... elif t == 1: ... c = 'g' ... marker = 'o' ... elif t == 2: ... c = 'b' ... marker = 'x' ... plt.scatter(features[target == t,0], ... features[target == t,1], ... marker=marker, ... c=c)
If the goal is to separate the three types of flowers, we can immediately make a few suggestions just by looking at the data. For example, petal length seems to be able to separate Iris Setosa from the other two flower species on its own. We can write a little bit of code to discover where the cut-off is:
>>> # We use NumPy fancy indexing to get an array of strings:
>>> labels = target_names[target]
>>> # The petal length is the feature at position 2
>>> plength = features[:, 2]
>>> # Build an array of booleans:
>>> is_setosa = (labels == 'setosa')
>>> # This is the important step:
>>> max_setosa =plength[is_setosa].max()
>>> min_non_setosa = plength[~is_setosa].min()
>>> print('Maximum of setosa: {0}.'.format(max_setosa))
Maximum of setosa: 1.9.
>>> print('Minimum of others: {0}.'.format(min_non_setosa))
Minimum of others: 3.0. Therefore, we can build a simple model: if the petal length is smaller than 2, then this is an Iris Setosa flower; otherwise it is either Iris Virginica or Iris Versicolor. This is our first model and it works very well in that it separates Iris Setosa flowers from the other two species without making any mistakes. In this case, we did not actually do any machine learning. Instead, we looked at the data ourselves, looking for a separation between the classes. Machine learning happens when we write code to look for this separation automatically.
The problem of recognizing Iris Setosa apart from the other two species was very easy. However, we cannot immediately see what the best threshold is for distinguishing Iris Virginica from Iris Versicolor. We can even see that we will never achieve perfect separation with these features. We could, however, look for the best possible separation, the separation that makes the fewest mistakes. For this, we will perform a little computation.
We first select only the non-Setosa features and labels:
>>> # ~ is the boolean negation operator >>> features = features[~is_setosa] >>> labels = labels[~is_setosa] >>> # Build a new target variable, is_virginica >>> is_virginica = (labels == 'virginica')
Here we are heavily using NumPy operations on arrays. The is_setosa array is a Boolean array and we use it to select a subset of the other two arrays, features and labels. Finally, we build a new boolean array, virginica, by using an equality comparison on labels.
Now, we run a loop over all possible features and thresholds to see which one results in better accuracy. Accuracy is simply the fraction of examples that the model classifies correctly.
>>> # Initialize best_acc to impossibly low value >>> best_acc = -1.0 >>> for fi in range(features.shape[1]): ... # We are going to test all possible thresholds ... thresh = features[:,fi] ... for t in thresh: ... # Get the vector for feature `fi` ... feature_i = features[:, fi] ... # apply threshold `t` ... pred = (feature_i > t) ... acc = (pred == is_virginica).mean() ... rev_acc = (pred == ~is_virginica).mean() ... if rev_acc > acc: ... reverse = True ... acc = rev_acc ... else: ... reverse = False ... ... if acc > best_acc: ... best_acc = acc ... best_fi = fi ... best_t = t ... best_reverse = reverse
We need to test two types of thresholds for each feature and value: we test a greater than threshold and the reverse comparison. This is why we need the rev_acc variable in the preceding code; it holds the accuracy of reversing the comparison.
The last few lines select the best model. First, we compare the predictions, pred, with the actual labels, is_virginica. The little trick of computing the mean of the comparisons gives us the fraction of correct results, the accuracy. At the end of the for loop, all the possible thresholds for all the possible features have been tested, and the variables best_fi, best_t, and best_reverse hold our model. This is all the information we need to be able to classify a new, unknown object, that is, to assign a class to it. The following code implements exactly this method:
def is_virginica_test(fi, t, reverse, example): "Apply threshold model to a new example" test = example[fi] > t if reverse: test = not test return test
What does this model look like? If we run the code on the whole data, the model that is identified as the best makes decisions by splitting on the petal width. One way to gain intuition about how this works is to visualize the decision boundary. That is, we can see which feature values will result in one decision versus the other and exactly where the boundary is. In the following screenshot, we see two regions: one is white and the other is shaded in grey. Any datapoint that falls on the white region will be classified as Iris Virginica, while any point that falls on the shaded side will be classified as Iris Versicolor.
In a threshold model, the decision boundary will always be a line that is parallel to one of the axes. The plot in the preceding screenshot shows the decision boundary and the two regions where points are classified as either white or grey. It also shows (as a dashed line) an alternative threshold, which will achieve exactly the same accuracy. Our method chose the first threshold it saw, but that was an arbitrary choice.
The model discussed in the previous section is a simple model; it achieves 94 percent accuracy of the whole data. However, this evaluation may be overly optimistic. We used the data to define what the threshold will be, and then we used the same data to evaluate the model. Of course, the model will perform better than anything else we tried on this dataset. The reasoning is circular.
What we really want to do is estimate the ability of the model to generalize to new instances. We should measure its performance in instances that the algorithm has not seen at training. Therefore, we are going to do a more rigorous evaluation and use held-out data. For this, we are going to break up the data into two groups: on one group, we’ll train the model, and on the other, we’ll test the one we held out of training. The full code, which is an adaptation of the code presented earlier, is available on the online support repository. Its output is as follows:
Training accuracy was 96.0%. Testing accuracy was 90.0% (N = 50).
The result on the training data (which is a subset of the whole data) is apparently even better than before. However, what is important to note is that the result in the testing data is lower than that of the training error. While this may surprise an inexperienced machine learner, it is expected that testing accuracy will be lower than the training accuracy. To see why, look back at the plot that showed the decision boundary. Consider what would have happened if some of the examples close to the boundary were not there or that one of them between the two lines was missing. It is easy to imagine that the boundary will then move a little bit to the right or to the left so as to put them on the wrong side of the border.
The accuracy on the training data, the training accuracy, is almost always an overly optimistic estimate of how well your algorithm is doing. We should always measure and report the testing accuracy, which is the accuracy on a collection of examples that were not used for training.
These concepts will become more and more important as the models become more complex. In this example, the difference between the accuracy measured on training data and on testing data is not very large. When using a complex model, it is possible to get 100 percent accuracy in training and do no better than random guessing on testing!
One possible problem with what we did previously, which was to hold out data from training, is that we only used half the data for training. Perhaps it would have been better to use more training data. On the other hand, if we then leave too little data for testing, the error estimation is performed on a very small number of examples. Ideally, we would like to use all of the data for training and all of the data for testing as well, which is impossible.
We can achieve a good approximation of this impossible ideal by a method called cross-validation. One simple form of cross-validation is leave-one-out cross-validation. We will take an example out of the training data, learn a model without this example, and then test whether the model classifies this example correctly. This process is then repeated for all the elements in the dataset.
The following code implements exactly this type of cross-validation:
>>> correct = 0.0
>>> for ei in range(len(features)):
# select all but the one at position `ei`:
training = np.ones(len(features), bool)
training[ei] = False
testing = ~training
model = fit_model(features[training], is_virginica[training])
predictions = predict(model, features[testing])
correct += np.sum(predictions == is_virginica[testing])
>>> acc = correct/float(len(features))
>>> print('Accuracy: {0:.1%}'.format(acc))
Accuracy: 87.0% At the end of this loop, we will have tested a series of models on all the examples and have obtained a final average result. When using cross-validation, there is no circularity problem because each example was tested on a model which was built without taking that datapoint into account. Therefore, the cross-validated estimate is a reliable estimate of how well the models would generalize to new data.
The major problem with leave-one-out cross-validation is that we are now forced to perform many times more work. In fact, you must learn a whole new model for each and every example and this cost will increase as our dataset grows.
We can get most of the benefits of leave-one-out at a fraction of the cost by using x-fold cross-validation, where x stands for a small number. For example, to perform five-fold cross-validation, we break up the data into five groups, so-called five folds.
Then you learn five models: each time you will leave one fold out of the training data. The resulting code will be similar to the code given earlier in this section, but we leave 20 percent of the data out instead of just one element. We test each of these models on the left-out fold and average the results.
The preceding figure illustrates this process for five blocks: the dataset is split into five pieces. For each fold, you hold out one of the blocks for testing and train on the other four. You can use any number of folds you wish. There is a trade-off between computational efficiency (the more folds, the more computation is necessary) and accurate results (the more folds, the closer you are to using the whole of the data for training). Five folds is often a good compromise. This corresponds to training with 80 percent of your data, which should already be close to what you will get from using all the data. If you have little data, you can even consider using 10 or 20 folds. In the extreme case, if you have as many folds as datapoints, you are simply performing leave-one-out cross-validation. On the other hand, if computation time is an issue and you have more data, 2 or 3 folds may be the more appropriate choice.
When generating the folds, you need to be careful to keep them balanced. For example, if all of the examples in one fold come from the same class, then the results will not be representative. We will not go into the details of how to do this, because the machine learning package scikit-learn will handle them for you.
We have now generated several models instead of just one. So, “What final model do we return and use for new data?” The simplest solution is now to train a single overall model on all your training data. The cross-validation loop gives you an estimate of how well this model should generalize.
A cross-validation schedule allows you to use all your data to estimate whether your methods are doing well. At the end of the cross-validation loop, you can then use all your data to train a final model.
Although it was not properly recognized when machine learning was starting out as a field, nowadays, it is seen as a very bad sign to even discuss the training accuracy of a classification system. This is because the results can be very misleading and even just presenting them marks you as a newbie in machine learning. We always want to measure and compare either the error on a held-out dataset or the error estimated using a cross-validation scheme.
In the previous section, we used a very simple model: a threshold on a single feature. Are there other types of systems? Yes, of course! Many others.
To think of the problem at a higher abstraction level, “What makes up a classification model?” We can break it up into three parts:
We can play around with these three aspects of classifiers and get different systems. A simple threshold is one of the simplest models available in machine learning libraries and only works well when the problem is very simple, such as with the Iris dataset. In the next section, we will tackle a more difficult classification task that requires a more complex structure.
In our case, we optimized the threshold to minimize the number of errors. Alternatively, we might have different loss functions. It might be that one type of error is much costlier than the other. In a medical setting, false negatives and false positives are not equivalent. A false negative (when the result of a test comes back negative, but that is false) might lead to the patient not receiving treatment for a serious disease. A false positive (when the test comes back positive even though the patient does not actually have that disease) might lead to additional tests to confirm or unnecessary treatment (which can still have costs, including side effects from the treatment, but are often less serious than missing a diagnostic). Therefore, depending on the exact setting, different trade-offs can make sense. At one extreme, if the disease is fatal and the treatment is cheap with very few negative side-effects, then you want to minimize false negatives as much as you can.
What the gain/cost function should be is always dependent on the exact problem you are working on. When we present a general-purpose algorithm, we often focus on minimizing the number of mistakes, achieving the highest accuracy. However, if some mistakes are costlier than others, it might be better to accept a lower overall accuracy to minimize the overall costs.
We will now look at a slightly more complex dataset. This will motivate the introduction of a new classification algorithm and a few other ideas.
We now look at another agricultural dataset, which is still small, but already too large to plot exhaustively on a page as we did with Iris. This dataset consists of measurements of wheat seeds. There are seven features that are present, which are as follows:
There are three classes, corresponding to three wheat varieties: Canadian, Koma, and Rosa. As earlier, the goal is to be able to classify the species based on these morphological measurements. Unlike the Iris dataset, which was collected in the 1930s, this is a very recent dataset and its features were automatically computed from digital images.
This is how image pattern recognition can be implemented: you can take images, in digital form, compute a few relevant features from them, and use a generic classification system. For the moment, we will work with the features that are given to us.
UCI Machine Learning Dataset RepositoryThe University of California at Irvine (UCI) maintains an online repository of machine learning datasets (at the time of writing, they list 233 datasets). Both the Iris and the Seeds dataset used in this article were taken from there.
The repository is available online at http://archive.ics.uci.edu/ml/.
One interesting aspect of these features is that the compactness feature is not actually a new measurement, but a function of the previous two features, area and perimeter. It is often very useful to derive new combined features. Trying to create new features is generally called feature engineering. It is sometimes seen as less glamorous than algorithms, but it often matters more for performance (a simple algorithm on well-chosen features will perform better than a fancy algorithm on not-so-good features).
In this case, the original researchers computed the compactness, which is a typical feature for shapes. It is also sometimes called roundness. This feature will have the same value for two kernels, one of which is twice as big as the other one, but with the same shape. However, it will have different values for kernels that are very round (when the feature is close to one) when compared to kernels that are elongated (when the feature is closer to zero).
The goals of a good feature are to simultaneously vary with what matters (the desired output) and be invariant with what does not. For example, compactness does not vary with size, but varies with the shape. In practice, it might be hard to achieve both objectives perfectly, but we want to approximate this ideal.
You will need to use background knowledge to design good features. Fortunately, for many problem domains, there is already a vast literature of possible features and feature-types that you can build upon. For images, all of the previously mentioned features are typical and computer vision libraries will compute them for you. In text-based problems too, there are standard solutions that you can mix and match. When possible, you should use your knowledge of the problem to design a specific feature or to select which ones from the literature are more applicable to the data at hand.
Even before you have data, you must decide which data is worthwhile to collect. Then, you hand all your features to the machine to evaluate and compute the best classifier.
A natural question is whether we can select good features automatically. This problem is known as feature selection. There are many methods that have been proposed for this problem, but in practice very simple ideas work best. For the small problems we are currently exploring, it does not make sense to use feature selection, but if you had thousands of features, then throwing out most of them might make the rest of the process much faster.
For use with this dataset, we will introduce a new classifier: the nearest neighbor classifier. The nearest neighbor classifier is very simple. When classifying a new element, it looks at the training data for the object that is closest to it, its nearest neighbor. Then, it returns its label as the answer. Notice that this model performs perfectly on its training data! For each point, its closest neighbor is itself, and so its label matches perfectly (unless two examples with different labels have exactly the same feature values, which will indicate that the features you are using are not very descriptive). Therefore, it is essential to test the classification using a cross-validation protocol.
The nearest neighbor method can be generalized to look not at a single neighbor, but to multiple ones and take a vote amongst the neighbors. This makes the method more robust to outliers or mislabeled data.
We have been using handwritten classification code, but Python is a very appropriate language for machine learning because of its excellent libraries. In particular, scikit-learn has become the standard library for many machine learning tasks, including classification. We are going to use its implementation of nearest neighbor classification in this section.
The scikit-learn classification API is organized around classifier objects. These objects have the following two essential methods:
Here is how we could use its implementation of k-nearest neighbors for our data. We start by importing the KneighborsClassifier object from the sklearn.neighbors submodule:
>>> from sklearn.neighbors import KNeighborsClassifier
The scikit-learn module is imported as sklearn (sometimes you will also find that scikit-learn is referred to using this short name instead of the full name). All of the sklearn functionality is in submodules, such as sklearn.neighbors.
We can now instantiate a classifier object. In the constructor, we specify the number of neighbors to consider, as follows:
>>> classifier = KNeighborsClassifier(n_neighbors=1)
If we do not specify the number of neighbors, it defaults to 5, which is often a good choice for classification.
We will want to use cross-validation (of course) to look at our data. The scikit-learn module also makes this easy:
>>> from sklearn.cross_validation import KFold
>>> kf = KFold(len(features), n_folds=5, shuffle=True)
>>> # `means` will be a list of mean accuracies (one entry per fold)
>>> means = []
>>> for training,testing in kf:
... # We fit a model for this fold, then apply it to the
... # testing data with `predict`:
... classifier.fit(features[training], labels[training])
... prediction = classifier.predict(features[testing])
...
... # np.mean on an array of booleans returns fraction
... # of correct decisions for this fold:
... curmean = np.mean(prediction == labels[testing])
... means.append(curmean)
>>> print("Mean accuracy: {:.1%}".format(np.mean(means)))
Mean accuracy: 90.5% Using five folds for cross-validation, for this dataset, with this algorithm, we obtain 90.5 percent accuracy. As we discussed in the earlier section, the cross-validation accuracy is lower than the training accuracy, but this is a more credible estimate of the performance of the model.
We will now examine the decision boundary. In order to plot these on paper, we will simplify and look at only two dimensions. Take a look at the following plot:
Canadian examples are shown as diamonds, Koma seeds as circles, and Rosa seeds as triangles. Their respective areas are shown as white, black, and grey. You might be wondering why the regions are so horizontal, almost weirdly so. The problem is that the x axis (area) ranges from 10 to 22, while the y axis (compactness) ranges from 0.75 to 1.0. This means that a small change in x is actually much larger than a small change in y. So, when we compute the distance between points, we are, for the most part, only taking the x axis into account. This is also a good example of why it is a good idea to visualize our data and look for red flags or surprises.
If you studied physics (and you remember your lessons), you might have already noticed that we had been summing up lengths, areas, and dimensionless quantities, mixing up our units (which is something you never want to do in a physical system). We need to normalize all of the features to a common scale. There are many solutions to this problem; a simple one is to normalize to z-scores. The z-score of a value is how far away from the mean it is, in units of standard deviation. It comes down to this operation:
In this formula, f is the old feature value, f’ is the normalized feature value, μ is the mean of the feature, and σ is the standard deviation. Both μ and σ are estimated from training data. Independent of what the original values were, after z-scoring, a value of zero corresponds to the training mean, positive values are above the mean, and negative values are below it.
The scikit-learn module makes it very easy to use this normalization as a preprocessing step. We are going to use a pipeline of transformations: the first element will do the transformation and the second element will do the classification. We start by importing both the pipeline and the feature scaling classes as follows:
>>> from sklearn.pipeline import Pipeline
>>> from sklearn.preprocessing import StandardScaler
Now, we can combine them.
>>> classifier = KNeighborsClassifier(n_neighbors=1)
>>> classifier = Pipeline([('norm', StandardScaler()), ('knn', classifier)]) The Pipeline constructor takes a list of pairs (str,clf). Each pair corresponds to a step in the pipeline: the first element is a string naming the step, while the second element is the object that performs the transformation. Advanced usage of the object uses these names to refer to different steps.
After normalization, every feature is in the same units (technically, every feature is now dimensionless; it has no units) and we can more confidently mix dimensions. In fact, if we now run our nearest neighbor classifier, we obtain 93 percent accuracy, estimated with the same five-fold cross-validation code shown previously!
Look at the decision space again in two dimensions:
The boundaries are now different and you can see that both dimensions make a difference for the outcome. In the full dataset, everything is happening on a seven-dimensional space, which is very hard to visualize, but the same principle applies; while a few dimensions are dominant in the original data, after normalization, they are all given the same importance.
The first classifier we used, the threshold classifier, was a simple binary classifier. Its result is either one class or the other, as a point is either above the threshold value or it is not. The second classifier we used, the nearest neighbor classifier, was a natural multiclass classifier, its output can be one of the several classes.
It is often simpler to define a simple binary method than the one that works on multiclass problems. However, we can reduce any multiclass problem to a series of binary decisions. This is what we did earlier in the Iris dataset, in a haphazard way: we observed that it was easy to separate one of the initial classes and focused on the other two, reducing the problem to two binary decisions:
Of course, we want to leave this sort of reasoning to the computer. As usual, there are several solutions to this multiclass reduction.
The simplest is to use a series of one versus the rest classifiers. For each possible label ℓ, we build a classifier of the type is this ℓ or something else? When applying the rule, exactly one of the classifiers will say yes and we will have our solution. Unfortunately, this does not always happen, so we have to decide how to deal with either multiple positive answers or no positive answers.
Alternatively, we can build a classification tree. Split the possible labels into two, and build a classifier that asks, “Should this example go in the left or the right bin?” We can perform this splitting recursively until we obtain a single label. The preceding diagram depicts the tree of reasoning for the Iris dataset. Each diamond is a single binary classifier. It is easy to imagine that we could make this tree larger and encompass more decisions. This means that any classifier that can be used for binary classification can also be adapted to handle any number of classes in a simple way.
There are many other possible ways of turning a binary method into a multiclass one. There is no single method that is clearly better in all cases. The scikit-learn module implements several of these methods in the sklearn.multiclass submodule.
Some classifiers are binary systems, while many real-life problems are naturally multiclass. Several simple protocols reduce a multiclass problem to a series of binary decisions and allow us to apply the binary models to our multiclass problem. This means methods that are apparently only for binary data can be applied to multiclass data with little extra effort.
Classification means generalizing from examples to build a model (that is, a rule that can automatically be applied to new, unclassified objects). It is one of the fundamental tools in machine.
In a sense, this was a very theoretical article, as we introduced generic concepts with simple examples. We went over a few operations with the Iris dataset. This is a small dataset. However, it has the advantage that we were able to plot it out and see what we were doing in detail. This is something that will be lost when we move on to problems with many dimensions and many thousands of examples. The intuitions we gained here will all still be valid.
You also learned that the training error is a misleading, over-optimistic estimate of how well the model does. We must, instead, evaluate it on testing data that has not been used for training. In order to not waste too many examples in testing, a cross-validation schedule can get us the best of both worlds (at the cost of more computation).
We also had a look at the problem of feature engineering. Features are not predefined for you, but choosing and designing features is an integral part of designing a machine learning pipeline. In fact, it is often the area where you can get the most improvements in accuracy, as better data beats fancier methods.
Further resources on this subject:
I remember deciding to pursue my first IT certification, the CompTIA A+. I had signed…
Key takeaways The transformer architecture has proved to be revolutionary in outperforming the classical RNN…
Once we learn how to deploy an Ubuntu server, how to manage users, and how…
Key-takeaways: Clean code isn’t just a nice thing to have or a luxury in software projects; it's a necessity. If we…
While developing a web application, or setting dynamic pages and meta tags we need to deal with…
Software architecture is one of the most discussed topics in the software industry today, and…