least squares approximation

(noun)

An attempt to minimize the sums of the squared distance between the predicted point and the actual point.

Related Terms

Examples of least squares approximation in the following topics:

  • Fitting a Curve

    • The simplest and perhaps most common linear regression model is the ordinary least squares approximation.
    • This approximation attempts to minimize the sums of the squared distance between the line and every point.  
    • Example:  Write the least squares fit line and then graph the line that best fits the data 
    • The line found by the least squares approximation, $y = 0.554x+0.3025$.
    • Model a set of data points as a line using the least squares approximation

  • Introduction to Least Squares

    • where we have used $x _{\rm ls}$ to denote the least squares value of $x$ .
    • In other words, find a linear combination of the columns of $A$ that is as close as possible in a least squares sense to the data.
    • Let's call this approximate solution $\mathbf{x_{ls}}$ .
    • If we look again at the normal equations and assume for the moment that the matrix $A^TA$ is invertible, then the least squares solution is:
    • Now $A$$A \mathbf{x_{ls}}$ applied to the least squares solution is the approximation to the data from within the column space.

  • Estimating the Target Parameter: Point Estimation

    • Another popular estimation approach is the linear least squares method.
    • Mathematically, linear least squares is the problem of approximately solving an over-determined system of linear equations, where the best approximation is defined as that which minimizes the sum of squared differences between the data values and their corresponding modeled values.
    • The approach is called "linear" least squares since the assumed function is linear in the parameters to be estimated.
    • One basic form of such a model is an ordinary least squares model.
    • Contrast why MLE and linear least squares are popular methods for estimating parameters

  • Homework

    • Calculate the least squares line.
    • Calculate the least squares line.
    • Calculate the least squares line.
    • Calculate the least squares line.
    • Calculate the least squares line.

  • Least-Squares Regression

    • The criteria for determining the least squares regression line is that the sum of the squared errors is made as small as possible.
    • The criteria for the best fit line is that the sum of squared errors (SSE) is made as small as possible.
    • Therefore, this best fit line is called the least squares regression line.
    • Ordinary Least Squares (OLS) regression (or simply "regression") is a useful tool for examining the relationship between two or more interval/ratio variables assuming there is a linear relationship between said variables.
    • This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear approximation.

  • Unequal Sample Sizes

    • Although the sample sizes were approximately equal, the "Acquaintance Typical" condition had the most subjects.
    • SPSS calls them estimated marginal means, whereas SAS and SAS JMP call them least squares means.
    • When all confounded sums of squares are apportioned to sources of variation, the sums of squares are called Type I sums of squares.
    • As you can see, with Type I sums of squares, the sum of all sums of squares is the total sum of squares.
    • First, let's consider the case in which the differences in sample sizes arise because in the sampling of intact groups, the sample cell sizes reflect the population cell sizes (at least approximately).

  • Chi Square Distribution

    • Define the Chi Square distribution in terms of squared normal deviates
    • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
    • Therefore, Chi Square with one degree of freedom, written as χ2(1), is simply the distribution of a single normal deviate squared.
    • A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) being six or higher is 0.050.
    • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

  • Mean Squares and the F-Ratio

    • To find a "sum of squares" is to add together squared quantities which, in some cases, may be weighted.
    • Sum of squares of all values from every group combined: $\sum x^2$
    • The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions.
    • If $MS_{\text{between}}$ and $MS_{\text{within}}$ estimate the same value (following the belief that Ho is true), then the F-ratio should be approximately equal to one.
    • Demonstrate how sums of squares and mean squares produce the $F$-ratio and the implications that changes in mean squares have on it.

  • Coefficient of Determination

    • the regression sum of squares (also called the explained sum of squares); and
    • the sum of squares of residuals, also called the residual sum of squares.
    • where $SS_\text{err}$ is the residual sum of squares and $SS_\text{tot}$ is the total sum of squares.
    • In many (but not all) instances where $r^2$ is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing $SS_\text{err}$.
    • Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be explained by the variation in the grades on the third exam.

  • Conditions for the least squares line

    • The variability of points around the least squares line remains roughly constant.
    • Should we have concerns about applying least squares regression to the Elmhurst data in Figure 7.12?
    • Least squares regression can be applied to these data.