Why minimize quadratic errors? Repair immediately

July 15, 2020 by Logan Robertson


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It is worth reading these corrective recommendations when you find out why you minimize the square error on your computer. Minimizing the smallest quadratic loss is equivalent to minimizing the variance! This explains why the slightest loss of square works for a lot of problems. The main noise due to CLT is very often Gaussian, and minimizing the squared error is correct!



In statistics, the mean square error (MSE) or standard deviation (MSD) of an estimate (a method for estimating an unobservable value) measures the mean of the squared errors, i.e. the standard deviation between the estimated values ​​and the current value. MSE is a risk function that corresponds to the expected value of the squared error loss. The fact that ESM is almost always strictly positive (and not null) is due to the random nature or the fact that the evaluator does not take into account information that can provide a more accurate estimate. [1]

why minimize squared error

MSE is an indicator of the quality of the evaluator - it is not always negative, and values ​​close to zero are better.

How does regression minimize squared errors?

The line assigns each height the expected weight. We subtract the actual weights from the predicted weights and add these errors. There will be lines that work well, while others will not. Regression finds a line that minimizes quadratic errors.

MSE is the second point (around the source) of the error, taking into account both the variance of the estimate (how far the estimates are distributed from the data sample to another) and its bias (how far from the average) the estimated value comes from the truth). For an undistorted assessment, MSE is the variance of the assessment. Like variance, MSE has the same units as the square of the calculated sum. Similar to standard deviation, square root from the MSE gives the standard error or standard error (RMSE or RMSD), which has the same units as the calculated value. For an undistorted estimate, the RMSE is the square root of the variance called the standard error.

Definition And Basic Properties [edit]

MSE evaluates the quality of the population predictor (that is, a function that maps arbitrary input data to a sample of random values) or an estimate (that is, say a mathematical function that maps a sample of data to a parameter estimate) where the data comes from). The definition of MSE differs depending on whether you are describing a predictor or evaluator.

Predictor [edit]

If the vector n {\ displaystyle n} Predictions are created from a sample of n data points for all variables and O {\ displaystyle Y} is the vector of the observed values ​​of the predicted variables using O ^ {\ displaystyle {\ hat {Y}}} Since these are the predicted values ​​(for example, based on the least squares adjustment), the MSE in the predictor sample is calculated as follows:

Should mean squared error be high or low?

There is no correct value for MSE. In other words, the lower the value, the better, and 0 means the model is perfect. Since there is no right answer, the fundamental value of MSE is to prefer one forecasting model to another.

In other words, MSE is the mean ( 1 n I = 1 n ) {\ displaystyle \ left ({\ frac {1} {n}} \ sum _ {i = 1} ^ {n} \ right)} error squares ( O I - O I ^ ) 2 {\ displaystyle (Y_ {i} - {\ hat {Y_ {i}}} ^ {2}} . This is an easily calculated quantity for a particular sample (and therefore depends on the sample).

MSE can also be calculated for q data points that were not used in the evaluation of the model, either because they were stored for this purpose, or because these data were obtained recently. In this process, known as cross-validation, MSEs are often referred to as root-mean-square errorazanias and calculate

Estimator [edit]

Why is error squared?

The root-mean-square error shows how close the regression line is to a certain number of points. To do this, the distances from the points to the regression line (these distances are “errors”) are taken and squared. A square is necessary to eliminate negative signs. There are also big differences in weight.

MSE evaluator θ ^ {\ displaystyle {\ hat {\ theta}}} regarding the unknown parameter θ {\ displaystyle \ theta} is defined as

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This definition depends on an unknown parameter, but the MSE is a priori a property of the evaluator. MSE may be a function of unknown parameters. In this case, any MSE evaluator based on estimates of these parameters will be a data function and therefore a random variable. If the evaluator has θ ^ {\ displaystyle {\ hat {\ theta}}} is displayed as a sample of statistics and is used to estimate certain parameters of a population. The wait then refers to the sample distribution of sample statistics.

MSE can be written as the sum of the variance of the estimate and the squared bias of the estimate, which is a useful method for calculating MSE and implies that in the case of distorted estimates, the MSE and variance are equivalent. [2]

Regression [edit]

In regression analysis, plotting is a more natural way to show the general trend of all data. The average distance from each point to the predicted regression model can be calculated and displayed as the standard error. Squaring is necessary to reduce the complexity of negative symptoms. To minimize this, the model can be more accurate, which means that the model is close enough to the actual data. An example of linear regression using This method is the least squares method. This is a method for assessing the ability of a linear regression model to model a two-dimensional data set [3] , but the limitation is related to the known distribution of data.

The term rms error is sometimes used to denote an undistorted estimate of the variance of the error: the remaining sum of squares divided by the number of degrees of freedom. This definition for a known calculated quantity differs from the above definition for a calculated MSE predictor in that a different denominator is used. The denominator is the sample size, which is reduced by the number of model parameters (n-p) estimated from the same data.



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