Multivariate Statistics - Introduction and Core Concepts (GEOG 316 / PLAN 351)

Course Foundations and Collaborative Expectations

  • Course Metadata:

    • Course Title: Multivariate Statistics
    • Course Codes: GEOG 316 / PLAN 351
    • Instructor: Derek T. Robinson, University of Waterloo School of Environment, Department of Geography & Environmental Management.
    • Date: May 13, 2026.

  • Instructor Expectations for Students:

    • Maintaining honesty and academic integrity.
    • Providing the best possible effort.
    • Starting assignments early to allow for proper troubleshooting.

  • Student Expectations from the Instructor:

    • A "no judgement" environment.
    • A genuine desire to help students succeed.
    • Providing useful and constructive feedback.
    • Effort from the instructor that matches the effort put in by the student.

Defining Multivariate Statistics

  • Etymological Breakdown:

    • Variate: Refers to an outcome variable of interest.
    • Multi: Means more than one.

  • Technical Definition:

    • Multivariate statistics involves the observation and analysis of more than one variable simultaneously.
    • It typically involves datasets with more than one independent variable (predictors).
    • It can also involve having more than one dependent variable or multiple outcomes/categories for a single dependent variable.

Multivariate Data and Dataset Structure

  • Data Representation: Multivariate data is typically viewed as a matrix where:

    • Each row represents an individual observation (e.g., a single tree).
    • Each column represents an attribute or feature of that observation (e.g., height, species, distance to road).

  • Sample Dataset Attributes (Forestry Research Example):

    • file_name: Laser scan file identifier (e.g., 208_GS0001.las).
    • Insitu_DBH: Diameter at Breast Height measured on the ground.
    • Height_m: Tree height measured in meters (mm).
    • Width_m: Crown width measured in meters (mm).
    • Point_density: Density of LIDAR points.
    • Total_points: Total LIDAR points for the specimen.
    • Forest_type: Classified environment (e.g., Natural).
    • Site: Location name (e.g., Hullet).
    • Species (common name): e.g., Green Ash, Sugar Maple, American Basswood, Eastern Hophornbeam, American Elm.
    • Species (scientific name): Fraxinus pennsylvanica, Acer saccharum, Tilia americana, Ostrya virginiana, Ulmus americana.
    • CrownHealth: Categorical health assessment score (values ranging from 00 to 44).

The Multivariate Necessity: Beyond Simple Univariate Models

  • The Limitation of One Variable: If two trees have the same height, it does not mean they have the same biomass or DBH. Differences occur due to species, age, and location.

  • Model Comparison and Performance (Predicting DBH):

    • Model 1: dbh=f(height)dbh = f(\text{height}) with R2=0.72R^2 = 0.72.
    • Model 2: dbh=f(height, width)dbh = f(\text{height, width}) with R2=0.84R^2 = 0.84.
    • Model 3: dbh=f(height, width, species, isStreetTree)dbh = f(\text{height, width, species, isStreetTree}).
    • Conclusion: Model 1 is not "wrong," but it is incomplete and simple. Model 3 is likely to perform better because it captures more of the variability in the system.

  • Flexibility of Roles: Variables are not inherently responses or predictors; their role depends on the specific question.

    • Example Model 4: height=f(dbh,species)\text{height} = f(dbh, \text{species}).
    • Example Model 5: species=f(height, width, dbh, isStreetTree)\text{species} = f(\text{height, width, dbh, isStreetTree}).

Omitted Variable Bias

  • Definition: This occurs when a statistical model excludes one or more relevant and influential variables.
  • Consequences:
    • It causes the included independent variables to receive more or less statistical importance than they actually deserve.
    • The model cannot isolate the "signal" or the true effect of a predictor.
    • The model attempts to compensate for the missing data and ultimately produces incorrect conclusions.

Feature Space and Similarity

  • 1D vs. 3D Feature Space:

    • In a 1D space (e.g., just Height), we are limited to fitting a single line or using simple thresholds.
    • In a multivariate space (3D or higher), we interrogate a "richer feature space" where differences between data points become more visible.

  • Similarity and Tobler’s First Law:

    • In feature space, groups/clusters emerge because similar observations are located near each other.
    • This reflects Tobler’s First Law: Nearby points are more similar than distant points.
    • Mathematically: similarity=f(distance)\text{similarity} = f(\text{distance}).

  • Distance Measurements:

    • Manhattan Distance (L1L^1): d1=∣x2−x1∣+∣y2−y1∣d_1 = |x_2 - x_1| + |y_2 - y_1|
    • Euclidean Distance (L2L^2): d2=sqrt((x2−x1)2+(y2−y1)2)d_2 = \text{sqrt}((x_2 - x_1)^2 + (y_2 - y_1)^2)

Multivariate Statistics vs. Machine Learning (ML)

  • Multivariate Statistics: Focuses on building the feature space to understand patterns and relationships. Emphasizes statistical inference, interpretability, and hypothesis testing.
  • Machine Learning: Focuses on identifying and learning the complex shapes and patterns within the data. Emphasizes maximizing prediction and accuracy.

Advanced Tree Height Prediction Models (Publication Reference)

  • Chapman-Richards: H_{ij} = \beta_0 (1 - e^{-\beta_1 D_{ij}})^{\beta_2} + \text{\epsilon}_{ij}
  • Exponential: H_{ij} = \text{exp}(\beta_0 + \beta_1 D_{ij}^{\beta_2}) + \text{\epsilon}_{ij}
  • OLS Polynomial: H_{ij} = \beta_0 + \beta_1 D_{ij} + \beta_2 D_{ij}^2 + \beta_3 D_{ij}^3 + \text{\epsilon}_{ij}
  • Random Intercept: H_{ij} = \beta_0 + \beta_1 D_{ij} + \beta_2 D_{ij}^2 + \beta_3 D_{ij}^3 + b_i + \text{\epsilon}_{ij}
  • Hierarchical: H_{ij} = \text{\alpha}_0 + \text{\alpha}_1(D_{ij} - \bar{D}_i) + \text{\alpha}_2(D_{ij} - \bar{D}_i)^2 + \text{\alpha}_3(D_{ij} - \bar{D}_i)^3 + b_i + \text{\epsilon}_{ij}
  • Variable Definitions:
    • HijH_{ij} and DijD_{ij}: Height and diameter of the jj-th tree in the ii-th plot.
    • Dˉi\bar{D}_i: Average tree diameter for plot ii.
    • β\beta and \text{\alpha}: Fixed parameters.
    • b_i \text{ \sim } N(0, \text{\sigma}_b^2).
    • \text{\epsilon} \text{ \sim } N(0, \text{\sigma}^2).

Course Personnel and Logistics

  • Instructional Team:

    • Derek Robinson: Degrees and Post Docs from Waterloo, Michigan, and Edinburgh. Research focus on Land-use change and the carbon cycle.
    • Lan Zhao (TA): MSc Simon Fraser, PhD Waterloo, NSERC CGRS-D recipient.
    • Jasmine Klassen (TA): BES and MSc Waterloo.
    • Non-official Support: Brandon Giura, Hayden Lam.

  • Learning Resources:

    • No single mandatory textbook, but multiple reference books are available via the library.
    • LEARN Platform: Used for all content, assignment submissions. (If LEARN is down, a 24-hour extension is granted).
    • Jupyter Notebooks: Used for programming and computational data science.

  • Communication Policy:

    • Subject lines must include the course shortcode (e.g., GEOG 316).
    • 24-hour response time (excluding weekends: Friday 5:00…pm5:00…pm to Monday 8:00…am8:00…am).
    • Mark-related questions must be handled in person.

  • Generative AI Policy:

    • Avoid using it for direct assignment answers.
    • Never copy the assignment question into an AI tool.
    • It should only be used as one of many sources for information/code.

Strategies for Success in GEOG 316 / PLAN 351

  • Operational Tips:
    • Read the entire assignment before starting.
    • Save frequently and use version control (e.g., descriptive fileName spelling).
    • Keep a log/list of actions performed (Google Docs or Notepad).
  • Engagement Strategy:
    • Attend all classes and start assignments early.
    • Utilize local software help, Google, and online resources before asking instructors.
    • Three Points of Contact: Text readings, Lectures/Slides, Assignments/Studying for finals.