The Method of Least Squares Introduction to Statistics

Let’s lock this line in place, and attach springs between the data points and the line. One of the first applications of the method of least squares was to settle a controversy involving Earth’s shape. The English mathematician Isaac Newton asserted in the Principia (1687) that Earth has an oblate (grapefruit) shape due to its spin—causing the equatorial diameter to exceed the polar diameter by about 1 part in 230.

  1. But for any specific observation, the actual value of Y can deviate from the predicted value.
  2. Quickly generate a regression scatter graph online, using the least squares method to generate a line of best fit.
  3. Having calculated the b of our model, we can go ahead and calculate the a.

We add some rules so we have our inputs and table to the left and our graph to the right. For WLS, the ordinary objective function above is replaced for a weighted average of residuals. These values can be used for a statistical criterion as to the goodness of fit. When unit weights are used, the numbers should be divided by the variance of an observation. Some of the data points are further from the mean line, so these springs are stretched more than others.

Least Square Method Definition

An early demonstration of the strength of Gauss’s method came when it was used to predict the future location of the newly discovered asteroid Ceres. On 1 January 1801, the Italian astronomer Giuseppe Piazzi discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the Sun. Based on these data, astronomers desired to determine the location of Ceres after it emerged from behind the Sun without solving Kepler’s complicated nonlinear equations of planetary motion. The only predictions that successfully allowed Hungarian astronomer Franz Xaver von Zach to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis. In statistics, the lower error means better explanatory power of the regression model.

So, when we square each of those errors and add them all up, the total is as small as possible. There isn’t much to be said about the code here since it’s all the theory that we’ve been through earlier. We loop through the values to get sums, averages, and all the other values we need to obtain the coefficient (a) and the slope (b). Before we jump into the formula and code, let’s define the data we’re going to use. For example, say we have a list of how many topics future engineers here at freeCodeCamp can solve if they invest 1, 2, or 3 hours continuously.

Cost Function

The red points in the above plot represent the data points for the sample data available. Independent variables are plotted as x-coordinates and dependent ones are plotted as y-coordinates. The equation of the line of best fit obtained from the least squares method is plotted as the red line in the graph.

The measurements seemed to support Newton’s theory, but the relatively large error estimates for the measurements left too much uncertainty for a definitive conclusion—although this was not immediately recognized. In fact, while Newton was essentially right, later observations showed that his prediction for excess wave software for water treatment plant design equatorial diameter was about 30 percent too large. The method of least squares actually defines the solution for the minimization of the sum of squares of deviations or the errors in the result of each equation. Find the formula for sum of squares of errors, which help to find the variation in observed data.

So, we try to get an equation of a line that fits best to the given data points with the help of the Least Square Method. The resulting fitted model can be used to summarize the data, to predict unobserved values from the same system, and to understand the mechanisms that may underlie the system. The least-squares method is a crucial statistical method that is practised to find a regression line or a best-fit line for the given pattern. The method of least squares is generously used in evaluation and regression.

If the conditions of the Gauss–Markov theorem apply, the arithmetic mean is optimal, whatever the distribution of errors of the measurements might be. But for any specific observation, the actual value of Y can deviate from the predicted value. The deviations between the actual and predicted values are called errors, or residuals. The Least Squares Model for a set of data (x1, y1), (x2, y2), (x3, y3), …, (xn, yn) passes through the point (xa, ya) where xa is the average of the xi‘s and ya is the average of the yi‘s. The below example explains how to find the equation of a straight line or a least square line using the least square method.

What Is the Least Squares Method?

The difference \(b-A\hat x\) is the vertical distance of the graph from the data points, as indicated in the above picture. The best-fit linear function minimizes the sum of these vertical distances. Traders and analysts have a number of tools available to help make predictions about the future performance of the markets and economy. The least squares method is a form of regression analysis that is used by many technical analysts to identify trading opportunities and market trends. It uses two variables that are plotted on a graph to show how they’re related.

Adding functionality

In this subsection we give an application of the method of least squares to data modeling. The least-squares method is a very beneficial method of curve fitting. Use the least square method to determine the equation of line of best fit for the data. Solving these two normal equations we can get the required trend line equation. Suppose when we have to determine the equation of line of best fit for the given data, then we first use the following formula. In order to find the best-fit line, we try to solve the above equations in the unknowns M and B.

This method of fitting equations which approximates the curves to given raw data is the least squares. In practice, the vertical offsets from a line (polynomial, surface, hyperplane, etc.) are almost always minimized instead of the perpendicular offsets. In addition, the fitting technique can be easily generalized from a best-fit line to a best-fit polynomial when sums of vertical distances are used.

A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets (“the residuals”) of the points from the curve. The sum of the squares of the offsets is used instead of the offset absolute values because this allows the residuals to be treated as a continuous differentiable quantity. However, because squares of the offsets are used, outlying points can have a disproportionate effect on the fit, a property which may or may not be desirable depending on the problem at hand. Consider the case of an investor considering whether to invest in a gold mining company.

Another thing you might note is that the formula for the slope \(b\) is just fine providing you have statistical software to make the calculations. But, what would you do if you were stranded on a desert island, and were in need of finding the least squares regression line for the relationship between the depth of the tide and the time of day? You might also appreciate understanding the relationship between the slope \(b\) and the sample correlation coefficient \(r\). It is quite obvious that the fitting of curves for a particular data set are not always unique. Thus, it is required to find a curve having a minimal deviation from all the measured data points. This is known as the best-fitting curve and is found by using the least-squares method.

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