A certain experiment produces the data (1,1.8),(2,2.7),...

A certain experiment produces the data $(1,1.8),(2,2.7),$ $(3,3.4),(4,3.8),(5,3.9) .$ Describe the model that produces a least-squares fit of these points by a function of the form $y=\beta_{1} x+\beta_{2} x^{2}$ Such a function might arise, for example, as the revenue from the sale of $x$ units of a product, when the amount offered for sale affects the price to be set for the product. a. Give the design matrix, the observation vector, and the unknown parameter vector. b. [M] Find the associated least-squares curve for the data.

A certain experiment produces the data $(1,1.8),(2,2.7),$ $(3,3.4),(4,3.8),(5,3.9) .$ Describe the model that produces a least-squares fit of these points by a function of the form
$y=\beta_{1} x+\beta_{2} x^{2}$
Such a function might arise, for example, as the revenue from the sale of $x$ units of a product, when the amount offered for sale affects the price to be set for the product.
a. Give the design matrix, the observation vector, and the unknown parameter vector.
b. [M] Find the associated least-squares curve for the data.

Key Concepts

Design Matrix

In a regression problem, the design matrix is the matrix that organizes the independent variable data in a structured way so that each row corresponds to an observation and each column to a predictor (or term) in the model. For instance, when fitting a model like y = ??x + ??x², the design matrix contains columns representing the values of x and x² for each observation, allowing the regression algorithm to compute the corresponding contributions to the predicted response.

Observation Vector

The observation vector (or response vector) is a column vector that contains the observed values of the dependent variable for each data point in the experiment. It represents the outcomes that the regression model seeks to predict based on the independent variables present in the design matrix.

Parameter Vector

The parameter vector contains the unknown coefficients that need to be estimated by the regression model. In the context of the model y = ??x + ??x², this vector comprises the polynomial coefficients associated with x and x², which are determined by minimizing the discrepancy between the predicted and observed responses.

Least Squares Estimation

Least squares estimation is a method used to find the parameter values that minimize the sum of squared differences between the observed values and the values predicted by the model. This technique is foundational in statistical modeling because it provides a systematic way to derive the best-fitting curve through the given data points by optimizing the parameters.

Normal Equations

Normal equations arise when the derivative of the sum of squared errors is taken with respect to each parameter in the model, set to zero to find the minimum error. These equations form a linear system that can be solved to obtain the least squares estimates, serving as the backbone of the analytical solution to polynomial regression problems.

Transcript

00:01 Okay, so a question in a certain experiment produces a data as given.

00:04 I'm going to describe a model that produces a least square best feed for a model of equation as y equals to beta 1x plus beta 2 x square.

00:15 We need to find design matrix, observation vector and a known parameter vector and also the associated equation with that.

00:23 Okay, as the model given is y equals to beta 1x plus beta 2 x square.

00:32 Okay, so for every point for y 1, let's say for x1, y 1 pair will have y1 is equal to beta 1 x1 plus beta 2 x2 square.

00:45 Similarly, y2 will be beta 2x2 plus beta 1 x2 plus beta 2 square.

00:54 Will be beta 1 x3 plus beta 2 x3 square and it will keep on going okay so this can be model in a matrix as right so we'll write this in left hand side because this is our input and this is our output okay so we can write this as x1 x2 square sorry x1 square x2 x2 square x3 x3 square x4 x4 square and we can go till x n right x n square and here this will be multiplied with beta so that is beta 1 and beta 2 okay so this is our left hand side right hand side is directly y1, y2, y3, y4, yn.

01:54 Okay, so this is our model.

01:57 Now, so part one is done.

02:00 We form the model.

02:01 We can substitute the values here, the given values, that is, so this can be written as, therefore we have x1, x1 square, so one, one square, so that is one only, then two, 2, 2, 4, let me write directly the square values, then 3, 3, square, 4, 4, square, then we have 5, 5, square, right? so, oops, we don't have this one, right? we have 4 and 3, 8.

02:37 Okay, so this is our model.

02:39 Then beta 1, beta 2, we need to find.

02:42 And again, y1, y2 is given to us, which is 1 .8, 2.

02:48 7, 3 .4, 3 .8 and 3 .9.

02:56 Okay.

02:57 So from here, this is our a solution.

03:01 Now b is asking for some naming or nomenclature.

03:05 So this is called as this is a design matrix.

03:13 Okay.

03:14 This is an unknown parameter vector and this is observation vector.

03:32 Okay...