least squares approximation

(noun)

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

Related Terms

  • curve fitting
  • linear regression
  • outlier

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+0.3025y = 0.554x+0.3025y=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 xxx .
    • In other words, find a linear combination of the columns of AAA that is as close as possible in a least squares sense to the data.
    • Let's call this approximate solution xls\mathbf{x_{ls}}x​ls​​ .
    • If we look again at the normal equations and assume for the moment that the matrix ATAA^TAA​T​​A is invertible, then the least squares solution is:
    • Now AAAAxlsA \mathbf{x_{ls}}Ax​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: ∑x2\sum x^2∑x​2​​
    • The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions.
    • If MSbetweenMS_{\text{between}}MS​between​​ and MSwithinMS_{\text{within}}MS​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 FFF-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 SSerrSS_\text{err}SS​err​​ is the residual sum of squares and SStotSS_\text{tot}SS​tot​​ is the total sum of squares.
    • In many (but not all) instances where r2r^2r​2​​ is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing SSerrSS_\text{err}SS​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.
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