A data scientist is modeling a manufacturing process outcome (Y) based on a sensor reading (X). The sensor's specifications indicate that its output is the natural logarithm of the underlying physical pressure (P), such that X = log(P). Previous domain research has established a strong linear relationship between the process outcome and the pressure itself, meaning Y is linearly proportional to P. When plotting the collected data, the relationship between Y and the sensor reading X is non-linear, rising with a visibly accelerating slope. To use a standard linear regression model, the data scientist must first transform the input feature.
Which of the following transformations should be applied to feature X to establish a linear relationship with Y?
The correct answer is the exponential transformation. The scenario states that the process outcome (Y) is linearly proportional to the physical pressure (P), which can be written as Y ≈ aP + b. It is also known that the sensor reading X is the natural logarithm of the pressure, so X = log(P). To express Y as a function of X, we first need to solve for P in terms of X, which gives P = eX (the exponential function). By substituting this into the linear relationship, we get Y ≈ a(eX) + b. This shows that Y has a linear relationship with the new transformed feature, e^X. Therefore, applying an exponential transformation to X is the correct step to linearize the relationship for a linear regression model.
The logarithmic transformation is incorrect. Applying a log transform would mean modeling Y against log(X), which is equivalent to modeling Y against log(log(P)). This would not result in a linear relationship.
The Box-Cox transformation is a family of power transformations used to find an optimal transformation, typically to stabilize variance or correct for non-normality. While it might identify a transformation similar to exponential, the known theoretical relationship from the sensor's specifications makes a direct exponential transformation more precise and appropriate than this empirical method.
The square root transformation is a type of power transformation but is not suitable for this specific non-linear relationship. It would not linearize the relationship between Y and X as defined in the scenario.
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