# 1 How is matrix multiplication defined?

Validate that $$(A_{k,n}\times B_{n,m})^t = (B^t_{m,n})\times (A^t_{n,k})$$, both using math notation (first principles) and using R functions for some example matrices.

# 2 Scalar vs. Matrix Multiplication

Demonstrate the differences between the scalar multiplication ($$*$$) and matrix multiplication ($$\%*\%$$).

# 3 Matrix Equations

Write a simple matrix solver ($$b = Ax$$, i.e.,$$x= A^{-1}b$$) and validate its accuracy using the R command solve(A,b). Solve this equation.

\begin{align*} 2a - b + 2c &= 5\\ -a - 2b + c &= 3\\ a + b - c &= 2 \end{align*}.

# 4 Least Square Estimation

Use the SOCR Knee Pain dataset, extract the RB = Right-Back locations $$(x,y)$$, and fit in a linear model for vertical location ($$y$$) in terms of the horizontal location ($$x$$). Display the linear model on top of the scatter plot of the paired data.

# 5 Matrix manipulation

Create a matrix $$A$$ with elements seq(1, 15, length = 6) and argument nrow = 3, add a row to this matrix; add two columns to A to obtain a new matrix $$C_{4,4}$$. Then generate a diagonal matrix $$D$$ with $$dim = 4$$ and elements rnorm(4,0,1). Apply element wise addition, subtraction, multiplication and division to the matrices $$C$$ and $$D$$. Apply matrix multiplication to $$D$$ and $$C$$. Obtain the inverse of $$C$$ and compare the result to ginv().

# 6 Matrix Transposition

Validate the multiplicative transpose formula ($$(A_{k,n}\cdot B_{n,m})^{T} = (B_{n,m})^{T}\cdot (A_{k,n})^{T}$$), both using math notation, as well as using R calls, e.g., you can try A = matrix(1:6,nrow=3), B = matrix(2:7, nrow = 2).

# 7 Sample Statistics

Use the SOCR Data Iris Sepal Petal Classes to compute sample mean and variance of each variables. Then calculate the sample covariance - use both math notation and R built-in functions.

# 8 Eigenvalues and Eigenvectors

Generate a random matrix with A = matrix(rnorm(9,0,1),nrow = 3), compute the eigenvalues and eigenvectors for $$A$$. Then try to solve this equation $$det(A-\lambda I) = 0$$, where $$\lambda$$ is a vector of length $$3$$. Compare $$\lambda$$ and the eigenvalues you computed above.

Example of manual and automated calculations of eigen-spectra (eigen-values and eigen-vectors)

 # A <- matrix(rnorm(9,0,1),nrow = 3); A # define a random design matrix, may generate complex solutions A <- matrix(c(0,1/4,1/4,3/4,0,1/4,1/4,3/4,1/2),3,3,byrow=T); A eigen_spectrum <- eigen(A); eigen_spectrum # compute the eigen-spectrum (eigen-values, $$l$$, and eigen-vectors, $$v$$), $A v = l v$$. B <- A-eigen(A)$$valuesdiag(3); B # compute B = (A - eigen_value I) det(A-eigen(A)$valuesdiag(3)) # verify that the det(A-eigen(A)$valuesdiag(3)) is not trivial ($$0$$) A%%eigen(A)$$vector - eigen(A)$$valuediag(3) # validate that$ A v = l v$$. all.equal(A, eigen(A)\vector %*% diag(eigen(A)\values) %*% solve(eigen(A)$$vector)) # compare A = vlinv(v) all.equal(diag(3), A%%eigen(A)$$vector - eigen(A)$$values * eigen(A)\$vector) # The last line compares I == AV - lambda*v, mind the $$*$$ and $$%*%$$ scalar and matrix operators

# 9 Regression Forecasting using Numerical Data

Use the Quality of Life data (Case06_QoL_Symptom_ChronicIllness) to fit several different Multiple Linear Regression models predicting clinically relevant outcomes, e.g., Chronic Disease Score.

• Summarize and visualize the data using summary, str, pairs.panels, ggplot, and plot_ly().
• Report paired correlations for numeric data and try to visualize these (e.g., heatmap, pairs plot, etc.)
• Examine potential dependencies of the predictors and the dependent response variables
• Fit a Multiple Linear Regression model, report the results, and explain the summary, residuals, effect-size coefficients, and the coefficient of determination, $$R^2$$
• Draw model diagnostic plots, at least QQ plot, residuals plot and leverage plot (half norm plot)
• Predict outcomes for new data
• Try to improve the model performance using the step function based on AIC and BIC.
• Fit a regression tree model and compare it with OLS model.
• Try to use M5P to improve the model.
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