Mathematics for Data Science and Machine Learning using R

Explore foundational mathematics for data science with R, covering linear algebra, calculus, and vector spaces. Engage with brief lectures, lab exercises, and solution videos to reinforce learning and ask questions.

Explore the four-pane R studio interface, learn core commands for data frames (assignment, dollar sign, subsetting), and master tidyverse basics (filter, select, mutate, group_by, summarize) plus basic plotting.

Load the mpd data set, view six observations in two ways, compute city mpg for compact cars, and plot highway against city for 2008 models in blue.

Demonstrates loading the tidyverse, creating and mutating data frames, computing means by class, and plotting highway versus city for 2008 models.

Explore scalars, vectors, and matrices as the building blocks of machine learning mathematics, defining scalars, vectors, matrices, and their dimensions, and introducing M_ij notation and vectors in space.

Practice scalar, vector, and matrix concepts by building a 3x4 matrix with 1–12, identifying a_{2,4}, and creating and plotting vectors V1, V2, and V3.

Explore vectors, matrices, and plotting in R: create vectors with c, colon, sequence, and rep; form matrices with matrix, rbind, and cbind; plot 2D and 3D with plot and plot3D.

Create a 3x4 matrix A in R filled row-wise with 1-12, verify A[2,4] equals 8, and plot vectors V1, V2 in 2D and V3 in 3D to illustrate vector operations.

Explore scalar multiplication, vector addition and subtraction with graphical interpretations in two and three space, and learn how dilation, dimension matching, and operation order shape these vector operations in R.

Explore vector operations in R: plotting vectors in 2D and 3D, scalar multiplication, vector addition and subtraction, and visualizing vector sums with custom plotting functions.

Practice vector operations by multiplying V1 by -3, adding V2 to V1, and computing V2 plus twice V1, then plot V1, V2, and their sum on a coordinate grid.

Demonstrate scalar multiplication and addition for v1 and v2, compute sums, and visualize in 2D and 3D using vec plot toD and vec add plot toD, including v3 and v4.

Explore how to perform scalar multiplication, addition, subtraction, and multiplication of matrices, with focus on inner dimensions and non-commutativity.

Explore scalar multiplication, addition, subtraction, and true matrix multiplication in R. Identify dimension requirements, errors from nonconforming arrays, and the difference between element-wise and %*% matrix multiplication.

Construct a 3x4 matrix A and a 4x2 matrix B in R; they cannot be added, create a compatible C, perform addition and A times B by hand, and verify.

Explore matrix operations in R by showing why a 3x4 matrix a and a 4x2 matrix b cannot be added, then form a 3x4 c and compute a %*% b.

Explore transposes, inverses, and the identity matrix, showing how transposes swap rows and columns and how inverses exist only for certain square matrices.

Learn to transpose matrices in R with t, including vector cases. Use solve for inverses of square matrices and verify that a times its inverse yields the identity.

Practice transposing a square matrix and finding its inverse by hand and in R, then multiply the matrix by its inverse to verify the identity.

Explore the lab on 3x3 matrices by computing the transpose, testing whether the inverse exists with the solve function, and confirming A times its inverse is the identity.

Learn how linear regression models a dependent variable from one or more independent variables, using intercepts, coefficients, and an error term. Includes deterministic and stochastic examples.

Explore linear regression in R using lm to model total cost from hotdogs and fries in a baseball example, then assess model error with fitted vs actual values and RMSE.

Practice linear regression in R using the 2018 baseball dataset to model runs as a function of home run and age 9, then plot, estimate, and evaluate predictions with MSE.

Load and clean the baseball data, fit a linear regression model predicting runs from home runs and hits per nine innings, and evaluate predictions with RMSE.

Translate the linear regression model into matrices, defining Y, X, beta, and epsilon with an intercept term, and compute beta as (X^T X)^{-1} X^T Y, illustrated with a three-observation example.

Demonstrates building a matrix representation of linear regression in R, forming the design matrix with an intercept, and computing beta via the inverse cross-product to predict deterministic and stochastic data.

Develop and apply a matrix-based linear regression model to predict win percentage from run differential and ERA using the baseball 2018 dataset in R, including data visualization and prediction evaluation.

Represent linear regression with Y, X, and beta matrices and estimate coefficients via (X'X)^{-1} X'Y. Interpret the intercept, run differential, and ERA weights and compare actual vs predicted win percentages.

Explore calculus foundations for data science with functions and tangent lines. See how slopes describe rate of change and how linear regression uses matrix representations.

Define and plot functions in R, then estimate the slope of a tangent line using secret lines and animation, with examples like x^2 and sqrt(x).

Practice calculating tangent slopes for x^3 and x^4 at x = -1, 0, 2, compare patterns, and determine slope signs for x^5, ln x, e^x, and 1/x without plotting.

Explore how tangent line slopes indicate function growth by computing at x = -1, 0, and 2 for x^3 and x^4, with plotting and patterns.

Learn how derivatives map inputs to tangent slopes and rates of change, from single-variable examples to partial derivatives like f_x and f_y in f(x,y)=x^2+y^2.

Explore derivatives in data science with R by plotting a function and its derivative, discovering the power rule. Learn to compute partial derivatives for two-variable functions.

Practice deriving and plotting derivatives for sqrt(x) and f(x)=x, verify f' with the power rule, and then study partial derivatives of x^3+y^3 with plotting of functions and partials.

Lab 2 demonstrates deriving derivatives for sqrt(x) and x, plotting functions with their derivatives to show tangent slopes, and practicing partial derivatives of x^3+y^3.

Explore optimization of single-variable functions by using derivatives to locate local and global extrema, identify critical points where f'(x)=0, and apply the second derivative test to distinguish maxima and minima.

Solve for zero of the derivative to locate the critical point near 18.75 and maximum sales, then use the second derivative test and uniroot in R to verify.

Investigate optimization of a single-variable function on the interval [-1.5, 1.5] using derivatives. Estimate extreme values from the graph and verify maxima or minima with the second derivative test.

Analyze f(x)=x^4-2x^2+3 on [-1.5,1.5], identify critical points at -1, 0, 1, and confirm minima 2 at -1 and 1 and maximum 3 at 0 using f' and f''.

Explore optimization using derivatives for two-variable functions, finding critical points where partial derivatives vanish, and applying the second derivative test to identify a local minimum at (1,3) with value four.

Explore two-variable optimization in r by solving partial derivatives equal to zero and applying the two-variable second derivative test. Verify a local minimum at (1,3) for f(x,y)=x^2+y^2-2x-6y+14 using D>0.

Explore two-variable optimization of f(x,y)=x^2+xy+y^2+y by computing first and second partial derivatives, locating critical point (1/3, -2/3), and confirming a local minimum with the second derivative test using R.

Explore a two-variable function f(x,y) by plotting in three-space, derive all first and second partials, and verify a local minimum at (1/3, -2/3) using the second derivative test.

Tie calculus to linear regression through matrix algebra, showing how minimizing the sum of squared errors yields the beta estimates via the normal equation (X^T X)^{-1} X^T Y.

Learn to view and interpret model error in linear regression by plotting actuals versus predicted values and residuals, and computing SSE and RMSE (via MSE).

Explore linear regression with matrix calculus in r: define beta via X^T Y, verify derivatives, fit lm with weight and hp, and interpret residuals and mse for mpg.

Explore linear regression with weight and HP, derive beta, validate derivatives, fit the model, and evaluate residuals and model choice to discuss overfitting and predictive reliability.

Explore orthogonal vectors and linear independence in n dimensional data spaces, applying them to improve linear regression with orthogonal features that provide distinct information beyond other inputs.

Test orthogonality to assess linear independence via vector products, then simulate regression where x3 = 2x1+3x2 and y = 4x3. Show that 8x1+12x2 matches 4x3, highlighting preprocessing to detect relations.

Develop intuition for orthogonal two- and three-dimensional vectors and linear independence, then use baseball 2018 data to express a variable as a linear combination and compare simple and multiple regression.

the lecture shows constructing orthogonal vector pairs in two and three dimensions, explains diff as a linear combination of runs and runs against, and analyzes baseball data models.

Explore eigenvectors and eigenvalues and how the covariance matrix reveals directions and magnitudes of variation, enabling dimensionality reduction in data sets.

Compute the covariance matrix with cov and interpret its diagonal and off-diagonal entries. Apply eigenvalues and eigenvectors to identify main directions of variation and orthogonality in data.

Compute the covariance matrix for the baseball 2018 dataset, interpret its entries, and determine how many eigenvectors and eigenvalues are needed to explain nearly all data variation.

Compute the covariance matrix for a three-variable baseball data set and explain diagonal and off-diagonal entries. Use eigenvectors and eigenvalues to show two components capture almost all variation.

Explore gradient descent, a popular optimization method, using the gradient and partial derivatives to update a two-variable function toward its minimum with learning rate alpha, plus a practical R example.

Learn to run gradient descent in R on a two-variable function, compute partial derivatives, apply a 0.01 learning rate for 500 iterations, and visualize convergence to minimum at (2, -3).

Plot the function f(x,y) and apply gradient descent from (0,0) and (5,5), comparing convergence and iterations, then visualize descent paths to identify starting points with fewer, more, or equal iterations.

Explore gradient descent on a two-variable function f(x,y) to locate its minimum, using starting points (0,0) and (5,5). Analyze convergence, gradient calculations, and plotting histories to compare paths.

Learn how gradient descent minimizes linear regression by reducing the sum of squared errors through updating beta0 and beta1 with a learning rate.

Demonstrate gradient descent for linear regression by generating data, fitting with lm, and illustrating convergence of beta0 and beta1 to the intercept and the coefficient on x.

Use gradient descent to fit simple regression models predicting win percent from individual inputs—ops, homerun, and strikeouts—on baseball 2018 data set, and compare convergence to find the best predictor.

Use gradient descent to fit a linear regression predicting win percentage from baseball data, comparing diff, ops, homerun, and strikeout, and explain convergence and why diff is the best predictor.