MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH2922 Regression Analysis回归分析课程的代写代考和辅导服务！
这是悉尼大学 回归分析课程的代写成功案例。
MATH2922课程简介
Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit is an advanced version of MATH2022, with more emphasis on the underlying concepts and on mathematical rigour. This unit investigates and explores properties of vector spaces, matrices and linear transformations, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract groups theory. The unit culminates in studying inner spaces, quadratic forms and normal forms of matrices together with their applications to problems both in mathematics and in the sciences and engineering.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. appreciate the basic concepts and problems of linear algebra and be able to apply linear algebra to solve problems in mathematics, science and engineering
- LO2. understand the definitions of fields and vector spaces and be able to perform calculations in real and complex vector spaces, both algebraically and geometrically
- LO3. determine if a system of equations is consistent and find its general solution
- LO4. compute the rank of a matrix and understand how the rank of a matrix relates to the solution set of a corresponding system of linear equations
- LO5. compute the eigenvalues, eigenvectors, minimal polynomials and normal forms for linear transformations
- LO6. use the definition and properties of linear transformations and matrices of linear transformations and change of basis, including kernel, range and isomorphism
- LO7. compute inner products and determine orthogonality on vector spaces, including Gram-Schmidt orthogonalisation
- LO8. identify self-adjoint transformations and apply the spectral theorem and orthogonal decomposition of inner product spaces, and the Jordan canonical form, to solving systems of ordinary differential equations
- LO9. calculate the exponential of a matrix and use it to solve a linear system of ordinary differential equations with constant coefficients
- LO10. identify special properties of a matrix, such as symmetric of Hermitian, positive definite, etc., and use this information to facilitate the calculation of matrix characteristics
- LO11. demonstrate accurate and efficient use of advanced algebraic techniques and the capacity for mathematical reasoning through analysing, proving and explaining concepts from advanced algebra
- LO12. apply problem-solving using advanced algebraic techniques applied to diverse situations in physics, engineering and other mathematical contexts
MATH2922 Regression Analysis HELP（EXAM HELP， ONLINE TUTOR）
The theoretical problems should be neatly numbered, written out, and solved. Do not turn in messy work.
In the simple regression model, $Y_i=\beta_0+\beta_1 x_i+\epsilon_i$, where $\epsilon_i \sim \mathcal{N}\left(0, \sigma^2\right)$.
(a) Write down the likelihood function for $\beta_0$ and $\beta_1$ for $Y_1, \ldots, Y_n$.
(b) Find the MLEs for $\beta_0$ and $\beta_1$. To do this, you need to assume two equations with two unknowns.
(c) Is the least-squares estimator (LSE) different than the MLE for either parameter? (You may use class notes for this – you do not need to re-derive the LSE.)
The efficiency ratio for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in $\mathrm{mg} / \mathrm{ft}^2$ ). The article “Statistical Process Control of a Phosphate Coating Line” (Wire J. Intl., May 1997: 78-81) provides data on tank temperature $(x)$ and efficiency ratio $(y)$, which can be found in the “HW5SteelData.txt” file.
(a) Plot a histogram of both temperature and efficiency ratio and comment on any interesting features.
(b) Is the value of the efficiency ratio completely and uniquely determined by tank temperature? Justify.
(c) Construct a scatter plot of the data Does it appear that efficiency ratio could be well predicted by the value of temperature?
(d) i. Determine the equation of the estimated regressions line, plot the regression line and plot the data.
ii. Plot a histogram of the standardized residuals, does the error appear to be normally distributed?
iii. Create a plot of the fitted values vs. the standardized residuals, does the homoscedastic assumption seem reasonable?
(e) Calculate a point estimate for true average efficiency ratio when tank temperature is $182^{\circ}$.
(f) Calculate the values of the residuals from the least squares line for the four observation for which the temperature is $182^{\circ}$. Why do they not all have the same sign?
(g) What proportion of the observed variation in efficiency ratio can be attributed to the simple linear regression relationship between the two variables?
Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article “Ion Beam-Assisted Etching of Aluminum with Chlorine” (J. of the Electrochem. Soc., 1985: 2010-2012) gives the accompanying data (read from a graph) on chlorine flow, ( $x$, in units of SCCM) through a nozzle used in the etching mechanism and etch rate $(y$, in $100 \mathrm{~A} / \mathrm{min})$. Fit a linear model predicting etch rate with chlorine flow.
(a) i. Construct a scatter plot. Does the simple linear regression model appear to be reasonable in this situation?
\begin{tabular}{c|ccccccccc}
$x$ & 1.5 & 1.5 & 2.0 & 2.5 & 2.5 & 3.0 & 3.5 & 3.5 & 4.0 \
\hline$y$ & 23.0 & 24.5 & 25.0 & 30.0 & 33.5 & 40.0 & 40.5 & 47.0 & 49.0
\end{tabular}
ii. Plot a histogram of the standardized residuals, does the error appear to be normally distributed?
iii. Create a plot of the fitted values vs. the standardized residuals, does the homoscedastic assumption seem reasonable?
iv. What proportion of observed variation in etch rate an be explained by the approximate linear relationship between the two variables?
(b) Does the simple linear regression model specify a useful relationship between chlorine flow and etch rate? Plot the regression line and the data.
(c) Estimate the true average change in etch rate associated with a 1-SCCM increase in flow rate using a $95 \%$ confidence interval $(\mathrm{CI})$ and interpret the interval.
(d) Calculate a $95 \% \mathrm{CI}$ for $\mu_{Y \mid X=3.0}$, the true average etch rate when flow $=3.0$. Has this average been precisely estimated?
(e) Calculate a $95 \%$ prediction interval (PI) for a single future observation on etch rate to be made when flow = 3.0. Is the predication likely to be accurate?
(f) Would the $95 \% \mathrm{CI}$ and PI when flow $=2.5$ be wider or narrower than the corresponding intervals of parts (d) and (e)? Answer without actually computing the intervals.
(g) Would you recommend calculating a 95\% PI for a flow of 6.0? Explain.
APPM 5570 only: Write a function titled ‘my.lm’ that accepts parameters $X$ and $Y$, and that performs simple linear regression on the two variables. Your output should look similar to that produced by the $\operatorname{lm}($ ) function in $\mathrm{R}$, and should include:
- The estimates for $\beta_0$ and $\beta_1$.
- The standard errors for $\hat{\beta}_0$ and $\hat{\beta}_1$.
- $R^2$, the correlation coefficient.
- The SSE, SSR, and SST.
Email me the function by the due date. Example test code that I will enter is:
$\mathrm{my} \cdot \operatorname{lm}(\mathrm{X}=$ predictorVar, $\mathrm{Y}=$ responseVar $)$
MY-ASSIGNMENTEXPERT™可以为您提供SYDNEY MATH2922 REGRESSION ANALYSIS回归分析课程的代写代考和辅导服务！
