Error In Prediction
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In simple linear regression, we predict scores on one variable from the scores on a second variable. The variable we are predicting is called the criterion variable and is referred to as Y. The variable we are basing our predictions on is called the predictor variable and is referred to as X. When there is standard deviation prediction only one predictor variable, the prediction method is called simple regression. In simple linear regression, the topic
Error Dictionary
of this section, the predictions of Y when plotted as a function of X form a straight line. The example data in Table 1 are plotted
Error Theory
in Figure 1. You can see that there is a positive relationship between X and Y. If you were going to predict Y from X, the higher the value of X, the higher your prediction of Y. Table 1. Example data. X Y
Error Experiment
1.00 1.00 2.00 2.00 3.00 1.30 4.00 3.75 5.00 2.25 Figure 1. A scatter plot of the example data. Linear regression consists of finding the best-fitting straight line through the points. The best-fitting line is called a regression line. The black diagonal line in Figure 2 is the regression line and consists of the predicted score on Y for each possible value of X. The vertical lines from the points to the regression line represent the errors of prediction. As you can see, the red point error in prediction format of predictions is invalid is very near the regression line; its error of prediction is small. By contrast, the yellow point is much higher than the regression line and therefore its error of prediction is large. Figure 2. A scatter plot of the example data. The black line consists of the predictions, the points are the actual data, and the vertical lines between the points and the black line represent errors of prediction. The error of prediction for a point is the value of the point minus the predicted value (the value on the line). Table 2 shows the predicted values (Y') and the errors of prediction (Y-Y'). For example, the first point has a Y of 1.00 and a predicted Y (called Y') of 1.21. Therefore, its error of prediction is -0.21. Table 2. Example data. X Y Y' Y-Y' (Y-Y')2 1.00 1.00 1.210 -0.210 0.044 2.00 2.00 1.635 0.365 0.133 3.00 1.30 2.060 -0.760 0.578 4.00 3.75 2.485 1.265 1.600 5.00 2.25 2.910 -0.660 0.436 You may have noticed that we did not specify what is meant by "best-fitting line." By far, the most commonly-used criterion for the best-fitting line is the line that minimizes the sum of the squared errors of prediction. That is the criterion that was used to find the line in Figure 2. The last column in Table 2 shows the squared errors of prediction. The sum of the squared errors of prediction shown in Table 2 is lower than it would be for any other regression line. The formula for a regression
measure its prediction error is of key importance. Often, however, techniques of measuring error are used that give grossly misleading results. This can lead to the phenomenon of over-fitting where a model may fit the training data very well, but will do a error in prediction definition poor job of predicting results for new data not used in model training. Here error in prediction format of labels is invalid is an overview of methods to accurately measure model prediction error. Measuring Error When building prediction models, the primary goal should be prediction error regression to make a model that most accurately predicts the desired target value for new data. The measure of model error that is used should be one that achieves this goal. In practice, however, many modelers instead report http://onlinestatbook.com/2/regression/intro.html a measure of model error that is based not on the error for new data but instead on the error the very same data that was used to train the model. The use of this incorrect error measure can lead to the selection of an inferior and inaccurate model. Naturally, any model is highly optimized for the data it was trained on. The expected error the model exhibits on new data will always http://scott.fortmann-roe.com/docs/MeasuringError.html be higher than that it exhibits on the training data. As example, we could go out and sample 100 people and create a regression model to predict an individual's happiness based on their wealth. We can record the squared error for how well our model does on this training set of a hundred people. If we then sampled a different 100 people from the population and applied our model to this new group of people, the squared error will almost always be higher in this second case. It is helpful to illustrate this fact with an equation. We can develop a relationship between how well a model predicts on new data (its true prediction error and the thing we really care about) and how well it predicts on the training data (which is what many modelers in fact measure). $$ True\ Prediction\ Error = Training\ Error + Training\ Optimism $$ Here, Training Optimism is basically a measure of how much worse our model does on new data compared to the training data. The more optimistic we are, the better our training error will be compared to what the true error is and the worse our training error will be as an approximation of the true error. The Danger of Overfitting In general, we would like to
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About http://stats.stackexchange.com/questions/49821/estimate-error-of-prediction-from-r-square Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a https://www.researchgate.net/post/What_is_standard_error_of_prediction_from_linear_regression_with_known_SE_for_y-values question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can error in ask a question Anybody can answer The best answers are voted up and rise to the top Estimate error of prediction from R-square up vote 2 down vote favorite What I have: a linear model $y=a_0+a_1x$ with given parameter estimates, the number of values used for fitting the model, the Pearson R² value. I need to estimate errors of prediction. I don't see a way to calculate error in prediction it, but is there a way to at least get a rough estimate? regression error r-squared pearson share|improve this question edited Feb 13 '13 at 9:31 asked Feb 12 '13 at 12:58 Roland 2,5691227 Are you interested in the theoretical aspects or in the practical aspects of doing so with some software? If so, what software do you use? –Erik Feb 12 '13 at 13:29 I use R, but I am hopeful, that I would be able to implement a solution myself. Hence, I am mainly interested in a theoretical solution, but would be also happy with R code. –Roland Feb 12 '13 at 15:04 If that's all you have, the problem is hopeless, because $R^2$ is invariant under changes of scale in the predictand. What other information is available to you? –whuber♦ Feb 12 '13 at 17:49 @whuber That's what I thought and told the phd student. Unfortunately this really is all information, which has been published for this (empirical) model. You don't find much statistics in papers from soil science ... –Roland Feb 12 '13 at 18:21 1 It depends on what journals you read :-). T
error of prediction from linear regression, with known SE for y-values? There are 32 pairs of dependent and independent variables: labelled (yi, xi), where 1<=i<=32. The SE of yi was calculated earlier by GLM, but was NOT calculated from the regression of y on x. What is the formula for the SE of prediction of each yi, given R²y, x, the deviation of yi from the regression on xi, and the corrected sum of squares of x? Topics Statistical Testing × 442 Questions 65 Followers Follow Linear Regression × 362 Questions 367 Followers Follow Standard Error × 119 Questions 11 Followers Follow Mar 10, 2016·Modified Mar 10, 2016 by the commenter. Share Facebook Twitter LinkedIn Google+ 0 / 0 All Answers (3) James R Knaub · N/A Anthony - I did not really follow your explanation, so I'll just try to answer the question as written: "What is standard error of prediction from linear regression, with known SE for y-values?" That looks to me as if you probably meant you had the standard deviation of y, called sigma. Otherwise, if you had a standard error of say a mean, that is a function of sample size. But you did say something about a sample size in your explanation, so I am not certain. But that would still require knowledge of sigma. The estimated standard error of a prediction error is based on a sigma, but not of the population of y, but instead on the residuals, or for weighted least squares (WLS) regression, the estimated sigma is the estimated standard deviation of the random factors of the estimated residuals. So my thought is that you have confused sigma for the y-value population with sigma for the residuals of a regression, which help you find the standard errors of the prediction errors for y given x. For more on the case of WLS regression, especially for regression through the origin for simple regression cases, see https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys The graphic for the special case of the classical ratio estimator in the following might interest you: https://www.researchgate.net/publication/263265199_CRE_Prediction_%27Bounds%27_and_Graphs_Example_for_Section_4_of_Properties_of_WLS_article I hope this is of some help and that I have not