Glossary

6.1 Least squares and maximum likelihood estimation

2 min read•july 25, 2024

Mathematical biology uses estimation techniques to understand complex biological systems. and are two key methods for fitting models to data and inferring parameters.

These techniques have different strengths and applications in biology. Least squares is simpler and works well for linear models, while maximum likelihood is more flexible and can handle various probability distributions.

Estimation Techniques in Mathematical Biology

Principles of least squares estimation

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  • Least squares estimation minimizes sum of squared differences between observed and predicted values
  • Objective function: S=∑i=1n(yi−f(xi,β))2S = \sum_{i=1}^n (y_i - f(x_i, \beta))^2 where yiy_i are observed values, f(xi,β)f(x_i, \beta) are predicted values, and β\beta are model parameters
  • Applied in linear regression, nonlinear curve fitting, and model calibration (population growth models)
  • Assumes normally distributed errors and (constant error variance)
  • Computationally efficient and provides unbiased estimates under certain conditions
  • Sensitive to outliers and may not be optimal for non-Gaussian error distributions (skewed data)

Application of maximum likelihood estimation

  • Maximum likelihood estimation (MLE) finds parameter values maximizing likelihood of observing given data
  • : L(θ∣x)=P(x∣θ)L(\theta|x) = P(x|\theta) where θ\theta are model parameters and xx is observed data
  • MLE process:
    1. Define data probability distribution
    2. Construct likelihood function
    3. Take logarithm of likelihood function
    4. Find maximum by setting derivatives to zero
  • Used in population genetics (allele frequency estimation), phylogenetic tree reconstruction, and epidemiological models (disease transmission rates)
  • Asymptotically efficient and consistent estimator
  • Computationally intensive for complex models and requires knowledge of underlying probability distribution

Least squares vs maximum likelihood

  • Both estimate model parameters for linear and nonlinear models
  • Least squares assumes normally distributed errors, MLE accommodates various distributions
  • MLE more efficient for large samples, least squares more robust for small samples
  • Least squares simpler to implement, MLE more computationally demanding
  • MLE flexible in handling different data types, least squares primarily for continuous data
  • Least squares provides easily interpretable fit measures (), MLE offers likelihood-based model comparison (, BIC)

Implementation of estimation algorithms

  • Programming languages: Python, R, MATLAB
  • Libraries: NumPy, SciPy (numerical computations), Statsmodels (statistical modeling), Scikit-learn (machine learning)
  • Least squares algorithms:
    • Ordinary least squares (OLS) for linear models
    • Nonlinear least squares (NLS) for nonlinear models
    • Iterative methods (Gauss-Newton, Levenberg-Marquardt)
  • MLE algorithms:
    • Newton-Raphson method
    • Expectation-Maximization (EM) algorithm
    • Gradient descent and variants
  • techniques: Conjugate gradient method, Quasi-Newton methods (BFGS)
  • Model diagnostics: Residual analysis, goodness-of-fit tests, cross-validation techniques (k-fold)

Key Terms to Review (18)

AIC: Akaike Information Criterion (AIC) is a statistical measure used to compare the goodness of fit of different models while penalizing for the number of parameters to prevent overfitting. It balances model complexity with how well a model describes the data, making it a key tool in model selection. A lower AIC value indicates a better model relative to others being compared, which helps in identifying the most appropriate model for the data at hand.
Binomial Distribution: The binomial distribution is a statistical distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is widely used to describe scenarios where there are two possible outcomes, such as success or failure, and it helps in estimating probabilities based on known parameters. This distribution is key in various statistical methods, including least squares and maximum likelihood estimation, where it provides a framework for analyzing and interpreting data that follow this binary outcome pattern.
Carl Friedrich Gauss: Carl Friedrich Gauss was a renowned German mathematician and scientist who made significant contributions to various fields including number theory, statistics, and astronomy. He is particularly known for developing the method of least squares, which is essential for statistical data fitting, and his work laid the groundwork for maximum likelihood estimation, a fundamental concept in statistical inference.
Error minimization: Error minimization refers to the process of reducing the difference between observed values and predicted values in statistical models. This concept is central to techniques that aim to fit a model to data, ensuring that the predictions made by the model are as close as possible to the actual outcomes. By minimizing error, statisticians can improve the accuracy and reliability of their models, making it essential for robust analysis and decision-making.
Generalized Linear Model: A generalized linear model (GLM) is an extension of traditional linear regression that allows for response variables to have distributions other than a normal distribution, enabling the modeling of various types of data. GLMs unify the concepts of linear regression and statistical distributions through a link function that connects the mean of the distribution of the response variable to the linear predictors. This flexibility makes GLMs particularly useful for analyzing non-normally distributed data, such as binary outcomes or count data.
Homoscedasticity: Homoscedasticity refers to the property of a dataset where the variance of the errors is constant across all levels of the independent variable(s). This concept is crucial in statistical modeling because when data exhibits homoscedasticity, it implies that the model's predictions are reliable, leading to valid statistical inferences. If the variance changes, it could indicate model mis-specification or the presence of outliers, which can severely impact the effectiveness of techniques like least squares and maximum likelihood estimation.
Independence: Independence refers to the property of random variables where the occurrence or value of one variable does not affect the occurrence or value of another. In statistics, this concept is crucial as it influences how data can be analyzed and interpreted, particularly in methods such as least squares and maximum likelihood estimation where assumptions about independence can affect the reliability of estimates and the validity of inferential statistics.
Least squares: Least squares is a mathematical approach used to minimize the sum of the squares of the residuals, which are the differences between observed and predicted values. This technique is often utilized in regression analysis to estimate the parameters of a model, providing a best-fit line that represents the relationship between variables. Its effectiveness makes it a fundamental concept in statistics, particularly in contexts requiring data fitting and modeling.
Likelihood Function: The likelihood function is a fundamental concept in statistics that represents the probability of observing the given data under different parameter values of a statistical model. It plays a critical role in estimating parameters, particularly in methods like maximum likelihood estimation, where the goal is to find the parameter values that maximize this likelihood function. In Bayesian inference, the likelihood function is used to update prior beliefs about parameters in light of new data, working alongside prior distributions to derive posterior distributions.
Linear regression model: A linear regression model is a statistical method used to describe the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. This model helps in understanding how changes in the independent variables influence the dependent variable, making it crucial for predictions and inference. The approach relies on principles like least squares and maximum likelihood estimation to determine the best-fitting line through the data points.
Maximum Likelihood Estimation: Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a model by maximizing the likelihood function, ensuring that the observed data is most probable given the parameters. This technique connects various fields by providing a framework for model fitting, particularly in understanding population dynamics and validating models through comparative analysis of different parameter estimates.
Model Fitting: Model fitting refers to the process of adjusting a statistical model so that it accurately describes the relationship between variables in a given dataset. This involves finding the best parameters that minimize the difference between observed data and the values predicted by the model. A good fit can help make reliable predictions and inform decisions based on the data, employing techniques like least squares estimation or Bayesian inference methods.
Normal Distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, where the majority of the observations cluster around the central peak, and its tails extend infinitely in both directions. This distribution is fundamental in statistics and forms the basis for various estimation techniques.
Optimization: Optimization is the mathematical process of finding the best solution from a set of feasible options, often by maximizing or minimizing a particular function. This concept is central to various fields, including statistics and ecology, where it helps in making informed decisions based on data. In essence, optimization seeks to identify parameters that yield the most favorable outcomes, whether that's fitting a model to data or managing resources in conservation efforts.
Parameter Estimation: Parameter estimation is the process of using statistical methods to determine the values of parameters in a mathematical model that best fit a set of observed data. This concept is crucial in developing accurate models for biological systems, as it allows researchers to refine their predictions and enhance their understanding of complex biological phenomena. It connects directly to statistical methods like least squares and maximum likelihood estimation, which provide frameworks for quantifying uncertainty and optimizing model parameters based on empirical data.
R-squared: R-squared, also known as the coefficient of determination, measures the proportion of variance in the dependent variable that can be predicted from the independent variable(s). It indicates how well the data fit the statistical model, helping to evaluate its explanatory power and effectiveness. A higher r-squared value suggests a better fit, which is crucial when using methods like least squares and maximum likelihood estimation to determine the best model for the data.
Residuals: Residuals are the differences between observed values and the values predicted by a statistical model. They represent the portion of the data that is not explained by the model and play a crucial role in assessing how well the model fits the data. Analyzing residuals helps to identify patterns, check for violations of assumptions, and ultimately improve model accuracy.
Ronald A. Fisher: Ronald A. Fisher was a British statistician, geneticist, and biologist known for his pioneering contributions to the field of statistics and for founding the modern science of population genetics. He developed key statistical methods, particularly in the areas of least squares and maximum likelihood estimation, which are essential for analyzing data and estimating parameters in various scientific fields, including biology.
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