Learning Curves

Sigmoidal Functions for Modeling Learning Curves

Sigmoidal functions, such as the logistic function, are often used to model learning curves due to their characteristic S-shaped curve. This shape captures the three main phases of learning:

• Initial Slow Improvement: At the beginning, the rate of improvement is slow as individuals or teams are just starting to learn and adapt.

• Rapid Improvement: Once the basics are understood, the rate of improvement accelerates significantly as efficiency and proficiency increase.

• Plateau: Eventually, the rate of improvement slows down again as the maximum potential or efficiency is approached.

The general form of a sigmoidal function (logistic function) is:

\[ L(t) = \frac{L_{\max}}{1 + e^{-k(t - t_0)}} \]

where:

• L(t) is the learning outcome (e.g., efficiency, proficiency) at time

• L_max is the maximum learning outcome (the plateau).

• k is the growth rate, determining how quickly the learning improves.

• t_o is the time at which the learning outcome is at its midpoint (the point of inflection).

Applications in Project Management

• Task Estimation: Using learning curves to estimate the time required for future tasks based on the performance of similar past tasks.

• Resource Allocation: Allocating resources efficiently by understanding the learning curve and optimizing the workforce deployment over time.

• Training Programs: Designing training programs to maximize the rate of learning and minimize the time to reach optimal performance levels.

• Performance Monitoring: Continuously monitoring performance and adjusting strategies to ensure that the learning curve progresses as expected.

Example

First, load the package:

library(PRA)

Then, set up a data frame of time and completion percentage data:

data <- data.frame(time = 1:10, completion = c(5, 15, 40, 60, 70, 75, 80, 85, 90, 95))

Fit a logistic model to the data:

fit <- fit_sigmoidal(data, "time", "completion", "logistic")

Use the model to predict future completion times:

predictions <- predict_sigmoidal(fit, seq(min(data$time), max(data$time),
  length.out = 100), "logistic")

Plot the results:

p <- ggplot2::ggplot(data, ggplot2::aes_string(x = "time", y = "completion")) +
  ggplot2::geom_point() +
  ggplot2::geom_line(data = predictions, ggplot2::aes(x = x, y = pred), color = "red") +
  ggplot2::labs(title = "Fitted Logistic Model", x = "time", y = "completion %") + ggplot2::theme_minimal()
p

Summary

In summary, learning curves provide valuable insights into how performance improves with experience, and sigmoidal functions offer a mathematical model to describe and predict these improvements in project management.