sum of squared errors

Examples of sum of squared errors in the following topics:

• Partitioning the Sums of Squares

  • Partition sum of squares Y into sum of squares predicted and sum of squares error
  • Define r2 in terms of sum of squares explained and sum of squares Y
  • The last column contains the squares of these errors of prediction.
  • SSY can be partitioned into two parts: the sum of squares predicted (SSY') and the sum of squares error (SSE).
  • The sum of squares error is the sum of the squared errors of prediction.

• Standard Error of the Estimate

  • Compute the standard error of the estimate based on errors of prediction
  • Recall that the regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error).
  • The numerator is the sum of squared differences between the actual scores and the predicted scores.
  • The last column shows that the sum of the squared errors of prediction is 2.791.
  • The reason N-2 is used rather than N-1 is that two parameters (the slope and the intercept) were estimated in order to estimate the sum of squares.

• 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.
  • This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear approximation.
  • It is considered optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated.
  • Under these conditions, the method of OLS provides minimum-variance, mean-unbiased estimation when the errors have finite variances.

• Proportion of Variance Explained

  • The sum of squares total (377.189) represents the variation when "Smile Condition" is ignored and the sum of squares error (377.189 - 27.544 = 349.654) is the variation left over when "Smile Condition" is accounted for.
  • where MSE is the mean square error and k is the number of conditions.
  • In one-factor designs, the sum of squares total is the sum of squares condition plus the sum of squares error.
  • The value of η2 for an effect is simply the sum of squares for this effect divided by the sum of squares total.
  • In the section "Partitioning the Sums of Squares" in the Regression chapter, we saw that the sum of squares for Y (the criterion variable) can be partitioned into the sum of squares explained and the sum of squares error.

• Introduction to Multiple Regression

  • Explain why the sum of squares explained in a multiple regression model is usually less than the sum of the sums of squares in simple regression
  • Just as in the case of simple linear regression, the sum of squares for the criterion (UGPA in this example) can be partitioned into the sum of squares predicted and the sum of squares error.
  • Table 3 shows the partitioning of the sums of squares into the sum of squares uniquely explained by each predictor variable, the sum of squares confounded between the two predictor variables, and the sum of squares error.
  • SSQT is the sum of squares total (the sum of squared deviations of the criterion variable from its mean), and
  • This sum of squares is 9.75.

• Expected Value and Standard Error

  • The standard error is the standard deviation of the sampling distribution of a statistic.
  • The new expected value of the sum of the numbers can be calculated by the number of draws multiplied by the expected value of the box: $25\cdot 2.2 = 55$.
  • The standard error of the sum can be calculated by the square root of number of draws multiplied by the standard deviation of the box: $\sqrt{25} \cdot \text{SD of box} = 5\cdot 1.17 = 5.8$.
  • This means that if this experiment were to be repeated many times, we could expect the sum of 25 numbers chosen to be within 5.8 of the expected value of 55, either higher or lower.
  • Solve for the standard error of a sum and the expected value of a random variable

• Unequal Sample Sizes

  • That is, if you add up the sums of squares for Diet, Exercise, D x E, and Error, you get 902.625.
  • When confounded sums of squares are not apportioned to any source of variation, the sums of squares are called Type III sums of squares.
  • 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.
  • Therefore, the Type II sums of squares are equal to the Type III sums of squares.

• Mean Squares and the F-Ratio

  • Most $F$-tests arise by considering a decomposition of the variability in a collection of data in terms of sums of squares.
  • Most $F$-tests arise by considering a decomposition of the variability in a collection of data in terms of sums of squares.
  • 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: $\sum x^2$
  • Total sum of squares: $\displaystyle \sum { { x }^{ 2 }- } \frac { { \left( \sum { x } \right) }^{ 2 } }{ n }$

• Computing R.M.S. Error

  • Root-mean-square error serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power.
  • RMS error is the square root of mean squared error (MSE), which is a risk function corresponding to the expected value of the squared error loss or quadratic loss.
  • MSE measures the average of the squares of the "errors. " The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator and its bias.
  • Like the variance, MSE has the same units of measurement as the square of the quantity being estimated.
  • RMS error is simply the square root of the resulting MSE quantity.

• Fitting a Curve

  • This approximation attempts to minimize the sums of the squared distance between the line and every point.  
  • Calculate the numerator:  The product of the $x$ and $y$-coordinates minus one-eighth the product of the sum of the $x$-coordinates and the sum of the $y$-coordinates.
  • Calculate the denominator:  The sum of the squares of the $x$-coordinates minus one-eighth the sum of the $x$-coordinates squared.
  • Indeed, trying to fit linear models to data that is quadratic, cubic, or anything non-linear, or data with many outliers or errors can result in bad approximations.
  • Model a set of data points as a line using the least squares approximation