a) In class we derived a least-squares (LS) estimate for the slope of a linear model. Suppose instead that we believe our data can be represented well by a quadratic model (a parabola), y = ax². Assuming that we have training data pairs (x, y), i=1,K, N, derive a general expression for the LS estimate à of the model parameter a. b) Use your expression from part (a) to compute the LS estimate à from the following noisy training data (N=5). (The blank columns are there so you'll have space to do calculations.) i 1 -2 5 21 1 2345 -0.5 1 5 2 3 3 0 4 1 Sketch the estimated model function y = âx², and on the same graph plot the noisy data points (x, y),i=1,K ,5. c) State the definition of the k nearest-neighbor (k-NN) model for regression.
a) In class we derived a least-squares (LS) estimate for the slope of a linear model. Suppose instead that we believe our data can be represented well by a quadratic model (a parabola), y = ax². Assuming that we have training data pairs (x, y), i=1,K, N, derive a general expression for the LS estimate à of the model parameter a. b) Use your expression from part (a) to compute the LS estimate à from the following noisy training data (N=5). (The blank columns are there so you'll have space to do calculations.) i 1 -2 5 21 1 2345 -0.5 1 5 2 3 3 0 4 1 Sketch the estimated model function y = âx², and on the same graph plot the noisy data points (x, y),i=1,K ,5. c) State the definition of the k nearest-neighbor (k-NN) model for regression.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Regression/curve fitting
![a) In class we derived a least-squares (LS) estimate for the slope of a linear model. Suppose instead that we believe our data can be represented well by a quadratic model (a parabola), \( y = ax^2 \). Assuming that we have training data pairs \((x_i, y_i)\), \( i = 1, K, N \), derive a general expression for the LS estimate \(\hat{a}\) of the model parameter \( a \).
b) Use your expression from part (a) to compute the LS estimate \(\hat{a}\) from the following noisy training data (\(N = 5\)). (The blank columns are there so you’ll have space to do calculations.)
\[
\begin{array}{|c|c|c|}
\hline
i & x_i & y_i \\
\hline
1 & -2 & 5 \\
2 & 1 & 1 \\
3 & 0 & -0.5 \\
4 & 1 & 1 \\
5 & 2 & 3 \\
\hline
\end{array}
\]
Sketch the estimated model function \( y = \hat{a}x^2 \), and on the same graph plot the noisy data points \((x_i, y_i), i = 1, K, 5\).
c) State the definition of the k nearest-neighbor (k-NN) model for regression.
d) Use this definition to compute the 2-NN (two nearest-neighbor) model of the function \( y = f(x) \) for the training data in part (b), and sketch the resulting function.
e) Which model do you expect to generalize better to data outside the training set, the LS model or the 2-NN model? What is the word used to describe careful fitting to training data, which results in failure to generalize to test data?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3831194d-a4dd-43eb-950f-f7d241d3bcca%2Ff2a3c7e5-e9ee-4e85-8f42-5d1ad8a94f3e%2F2wm1xsy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a) In class we derived a least-squares (LS) estimate for the slope of a linear model. Suppose instead that we believe our data can be represented well by a quadratic model (a parabola), \( y = ax^2 \). Assuming that we have training data pairs \((x_i, y_i)\), \( i = 1, K, N \), derive a general expression for the LS estimate \(\hat{a}\) of the model parameter \( a \).
b) Use your expression from part (a) to compute the LS estimate \(\hat{a}\) from the following noisy training data (\(N = 5\)). (The blank columns are there so you’ll have space to do calculations.)
\[
\begin{array}{|c|c|c|}
\hline
i & x_i & y_i \\
\hline
1 & -2 & 5 \\
2 & 1 & 1 \\
3 & 0 & -0.5 \\
4 & 1 & 1 \\
5 & 2 & 3 \\
\hline
\end{array}
\]
Sketch the estimated model function \( y = \hat{a}x^2 \), and on the same graph plot the noisy data points \((x_i, y_i), i = 1, K, 5\).
c) State the definition of the k nearest-neighbor (k-NN) model for regression.
d) Use this definition to compute the 2-NN (two nearest-neighbor) model of the function \( y = f(x) \) for the training data in part (b), and sketch the resulting function.
e) Which model do you expect to generalize better to data outside the training set, the LS model or the 2-NN model? What is the word used to describe careful fitting to training data, which results in failure to generalize to test data?
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