Introduction to Method of Least Squares
The method of least squares is a statistical technique used to find the best-fitting curve or line for a set of data points. It does this by minimizing the sum of the squares of the offsets (residuals) of the points from the curve.
The method of least squares is used for
- solution of equations, and
- curve fitting
The principles of least squares consist of minimizing the sum of squares of deviations, errors, or residuals.
Mathematical Functions/ Models
Many types of mathematical functions (or models) can be used to model the response, i.e. a function of one or more independent variables. It can be classified into two categories, deterministic and probabilistic models. For example, $Y$ and $X$ are related according to the relation
$$Y=\beta_o + \beta_1 X,$$
where $\beta_o$ and $\beta_1$ are unknown parameter. $Y$ is a response variable and $X$ is an independent/auxiliary variable (regressor). The model above is called the deterministic model because it does not allow for any error in predicting $Y$ as a function of $X$.
Probabilistic and Deterministic Models
Suppose that we collect a sample of $n$ values of $Y$ corresponding to $n$ different settings for the independent random variable $X$ and the graph of the data is as shown below.
In the figure above it is clear that $E(Y)$ may increase as a function of $X$ but the deterministic model is far from an adequate description of reality.
Repeating the experiment when say $X=20$, we would find $Y$ fluctuates about a random error, which leads us to the probabilistic model (that is the model is not deterministic or not an exact representation between two variables). Further, if the mode is used to predict $Y$ when $X=20$, the prediction would be subjected to some known error. This of course leads us to use the statistical method predicting $Y$ for a given value of $X$ is an inferential process and we need to find if the error of prediction is to be valued in real life. In contrast to the deterministic model, the probabilistic model is
$$E(Y)=\beta_o + \beta_1 X + \varepsilon,$$
where $\varepsilon$ is a random variable having the specified distribution, with zero mean. One may think having the deterministic component with error $\varepsilon$.
The probabilistic model accounts for the random behaviour of $Y$ exhibited in the figure and provides a more accurate description of reality than the deterministic model.
The properties of error of prediction of $Y$ can be divided for many probabilistic models. If the deterministic model can be used to predict with negligible error, for all practical purposes, we use them, if not, we seek a probabilistic model which will not be a correct/exact characterization of nature but enable us to assess the reality of our nature.
Estimation of Linear Model: Least Squares Method
For the estimation of the parameters of a linear model, we consider fitting a line.
$$E(Y) = \beta_o + \beta_1 X, \qquad (where\,\, X\,\,\, is \,\,\, fixed).$$
For a set of points ($x_i, y_i$), we consider the real situation
$$Y=\beta_o+\beta_1X+\varepsilon, \qquad with\,\,\, E(\varepsilon)=0$$
where $\varepsilon$ posses specific probability distribution with zero mean and $\beta_o$ and $\beta_1$ are unknown parameters.
Minimizing the Vertical Distances of Data Points
Now if $\hat{\beta}_o$ and $\hat{\beta}_1$ are the estimates of $\beta_o$ and $\beta_1$, respectively then $\hat{Y}=\hat{\beta}_o+\hat{\beta}_1X$ is an estimate of $E(Y)$.
Suppose we have a set of $n$ data sets (points, $x_i, y_i$) and we want to minimize the sum of squares of the vertical distances of the data points from the fitted line $\hat{y}_i = \hat{\beta}_o + \hat{\beta}_1x_i; \,\,\, i=1,2,\cdots, n$. The $\hat{y}_i = \hat{\beta}_o + \hat{\beta}_1x_i$ is the predicted value of $i$th $Y$ when $X=x_i$. The deviation of observed values of $Y$ from $\hat{Y}$ line (sometimes called errors) is $y_i – \hat{y}_i$ and the sum of squares of deviations to be minimized is (vertical distance: $y_i – \hat{y}_i$).
\begin{align*}
SSE &= \sum\limits_{i=1}^n (y_i-\hat{y}_i)^2\\
&= \sum\limits_{i=1}^n (y_i – \hat{\beta}_o – \hat{\beta}_1x_i)^2
\end{align*}
The quantity SSE is called the sum of squares of errors. If SSE possesses minimum, it will occur for values of $\beta_o$ and $\beta_1$ that satisfied the equation $\frac{\partial SSE}{\partial \beta_o}=0$ and $\frac{\partial SSE}{\partial \beta_1}=0$.
Taking the partial derivatives of SSE with respect to $\hat{\beta}_o$ and $\hat{\beta}_1$ and setting them equal to zero, gives us
\begin{align*}
\frac{\partial SSE}{\partial \beta_o} &= \sum\limits_{i=1}^n (y_i – \hat{\beta}_o – \hat{\beta}_1 x_i)^2\\
&= -2 \sum\limits_{i=1}^n (y_i – \hat{\beta}_o – \hat{\beta}_1 x_i) =0\\
&= \sum\limits_{i=1}^n y_i – n\hat{\beta}_o – \hat{\beta}_1 \sum\limits_{i=1}^n x_i =0\\
\Rightarrow \overline{y} &= \hat{\beta}_o + \beta_1\overline{x} \tag*{eq (1)}
\end{align*}
and
\begin{align*}
\frac{\partial SSE}{\partial \beta_1} &= -2 \sum\limits_{i=1}^n (y_i – \hat{\beta}_o – \hat{\beta}_1 x_i)x_i =0\\
&= \sum\limits_{i=1}^n (y_i – \hat{\beta}_o – \hat{\beta}_1 x_i)x_i=0\\
\Rightarrow \sum\limits_{i=1}^n x_iy_i &= \hat{\beta}_o \sum\limits_{i=1}^n x_i – \hat{\beta}_1 \sum\limits_{i=1}^n x_i^2\tag*{eq (2)}
\end{align*}
The equation $\frac{\partial SSE}{\hat{\beta}_o}=0$ and $\frac{\partial SSE}{\partial \hat{\beta}_1}=0$ are called the least squares for estimating the parameters of a straight line. On solving the least squares equation, we have from equation (1),
$$\hat{\beta}_o = \overline{Y} – \hat{\beta}_1 \overline{X}$$
Putting $\hat{\beta}_o$ in equation (2)
\begin{align*}
\sum\limits_{i=1}^n x_i y_i &= (\overline{Y} – \hat{\beta}\overline{X}) \sum\limits_{i=1}^n x_i + \hat{\beta}_1 \sum\limits_{i=1}^n x_i^2\\
&= n\overline{X}\,\overline{Y} – n \hat{\beta}_1 \overline{X}^2 + \hat{\beta}_1 \sum\limits_{i=1}^n x_i^2\\
&= n\overline{X}\,\overline{Y} + (\sum\limits_{i=1}^n x_i^2 – n\overline{X}^2)\\
\Rightarrow \hat{\beta}_1 &= \frac{\sum\limits_{i=1}^n x_iy_i – n\overline{X}\,\overline{Y} }{\sum\limits_{i=1}^n x_i^2 – n\overline{X}^2} = \frac{\sum\limits_{i=1}^n (x_i-\overline{X})(y_i-\overline{Y})}{\sum\limits_{i=1}^n(x_i-\overline{X})^2}
\end{align*}
Applications of Least Squares Method
The method of least squares is a powerful statistical technique. It provides a systematic way to find the best-fitting curve or line for a set of data points. It enables us to model relationships between variables, make predictions, and gain insights from data. The method of least squares is widely used in various fields, such as:
- Regression Analysis: To model the relationship between variables and make predictions.
- Curve Fitting: To find the best-fitting curve for a set of data points.
- Data Analysis: To analyze trends and patterns in data.
- Machine Learning: As a foundation for many machine learning algorithms.
Frequently Asked Questions about Least Squares Method
- What is the method of Least Squares?
- Write down the applications of the Least Squares method.
- How vertical distance of the data points from the regression line is minimized?
- What is the principle of the Method of Least Squares?
- What is meant by probabilistic and deterministic models?
- Give an example of deterministic and probabilistic models.
- What is the mathematical model?
- What is the statistical model?
- What is curve fitting?
- State and prove the Least Squares Method?
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