Describe the process of feature scaling and provide a scenario where standardization would be preferred over normalization and vice-versa.

Feature scaling is a preprocessing step applied to independent variables or features of data, primarily used in machine learning, to transform the range of data. This is essential because many machine learning algorithms, especially those using distance-based calculations such as K-Nearest Neighbors or gradient descent-based algorithms such as linear regression and neural networks, are sensitive to the scale of the input features. When features have vastly different scales (e.g., one ranging from 0 to 1 and another from 100 to 1000), the larger-scale features might dominate or skew the learning process if not scaled to the same range. Feature scaling techniques bring all features to a similar scale, ensuring that each feature contributes proportionally and equally during the learning process.

There are two common types of feature scaling methods: Normalization and Standardization.

Normalization, also known as min-max scaling, transforms data to fit within a specified range, often between 0 and 1. The formula for normalization is typically: x_scaled = (x - x_min) / (x_max - x_min), where x is the original feature value, x_min is the minimum value of the feature in the dataset, and x_max is the maximum value of the feature in the dataset. Normalization is particularly useful when you know the upper and lower bounds of your data, or when you need to represent features as percentages or proportions. For example, consider a dataset containing information about housing prices and their corresponding sizes in square footage. If the prices range from $100,000 to $1,000,000, and the sizes range from 500 to 5,000 square feet, normalization would scale both of these feature to the same 0 to 1 range. This can ensure that both the price and the size of a house have equal importance when training a predictive model. Another use case might be image processing where pixel values are commonly normalized to a 0 to 1 range which makes it simpler for image processing algortihms to be applied to them. Normalization maintains the original distribution of the data, though the range is changed, making it effective when working with data that doesn't follow a Gaussian (normal) distribution. However, this does not mitigate the effect of outliers.

Standardization, also known as Z-score normalization, transforms the data to have a mean of 0 and a standard deviation of 1. The formula for standardization is: x_scaled = (x - mean) / standard deviation, where x is the original feature value, mean is the average of the feature in the dataset, and standard deviation is the standard deviation of the feature in the dataset. Standardization is generally favored when the data is approximately Gaussian or normal or when the algorithm assumes normally distributed data, as it does not compress the data to a limited range. Consider for example a situation where a model will be trained on height and weight data. Because these variables are measured in very different units (height in centimeters, weight in kilograms), a model would prioritize weight over height because weights generally have a higher range of numbers than heights. Applying standardization to the dataset would ensure that both features will be given equal importance. This technique is also less sensitive to outliers compared to normalization as outliers don't drastically alter the mean or standard deviation. Furthermore, many machine learning algorithms such as logistic regression and support vector machines, work more reliably with standardized data.

In summary, while both methods bring features to a similar scale, they achieve this in different ways. Normalization scales features to a specific range, often 0 to 1, and is helpful when the range of the data is known or when the data is not normally distributed. On the other hand, Standardization is more useful for algorithms that assume normally distributed data or that are sensitive to the data scale and is also less sensitive to outliers in the data. Choosing between normalization and standardization often depends on the data and the specific algorithm.