How do you find the error variance?

There are a variety of ways to find the error variance in a regression model . One common approach is to use a bootstrap. A bootstrap is an algorithm that samples data from a distribution and Estimates the mean , standard deviation, and Error variances for different subsamples. The bootstrap is especially useful when testing hypotheses about the relationship between dependent variables and independent variables.
How do you find the error variance? : Count the number of observations that were used to generate the standard error of the mean This number is the sample size Multiply the square of the standard error (calculated previously) by the sample size (calculated previously) The result is the variance of the sample
What is error variance in reliability? : Reliability = “True” Variance / Observed Variance. In Rasch terms, “True” valiance is the “adjusted” variance (observed variance adjusted for measurement error). Error Variance is a mean-square error (derived from the model) inflated by misfit to the model encountered in the data.
What is error variance Anova? : ANOVA tests make use of the concept of within-group variation, also known as error group or error variance. It speaks of variations brought on by distinctions among various groups (or levels). Or, to put it another way, not all of the values within each group (e. g. equivalent in (means).

Read Detail Answer On What is error variance Anova?

ANOVA > Within-Group Variation

Formula for within-group variation.

Within-group variation (sometimes called error group or errorvariance) is a term used in ANOVA tests. It refers to variations caused by differences within individual groups (or levels). In other words, not all the values within each group (e.g. means) are the same. These are differences not caused by the independentvariable.

Each sample is looked at on its own. In other words, no interactions between samples are considered. For example, let’s say you had four groups, representing drugs A B C D, with each group composed of 20 people in each group and you’re measuring people’s cholesterol levels. For within-group variation, you’ll look at variances in cholesterol levels for people in group A, withoutconsidering groups B,C, and D. Then you would look at cholesterol levels for people in group B, without considering groups A,C, and D. And so on.

ANOVA Output

Within groups variance shown on Excel ANOVA output.

ANOVA output indicates within-group variation as SS(W)or, which stands for Sum of Squares Within Groups or SSW: Sum of Squares Within It is inextricably linked to the variance difference between groups (Sum of Squares between), which is brought on by how groups communicate with one another. This is due to the fact that the main goal of an ANOVA is to compare the ratio of within-group variance to variance between groups. The F statistic, which forms the basis of the ANOVA test, is actually determined by dividing the variance within and between groups.

If the variance caused by interactions between different samples is muchgreater than the variance found inside values in a single group, that indicates the means aren’t equal.

Degrees of freedom for Within-group variaton equals the sum of the individual degrees of freedom for each sample in the test.

References

Kotz, S ; et al , eds (2006),Encyclopedia of Statistical Sciences, Wiley Everitt, B S ; Skrondal, A (2010), The Cambridge Dictionary of Statistics, Cambridge University Press Vogt, W P (2005) Dictionary of Statistics and Methodology: A Nontechnical Guide for the SocialSciences SAGE

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What causes high error variance? : The variance is a mistake caused by sensitivity to slight variations in the training set. An algorithm that is overfit to the random noise in the training data can have high variance.
Read Detail Answer On What causes high error variance?

Function and noisy data.

spread=5

spread=1

spread=0.1

Radial basis functions are utilized to approximate a function (red; blue). In each graph, several trials are displayed. A training set of a few noisy data points is offered for each trial (top). For a wide spread (image 2), the bias is high; however, the variance between different trials is low. The RBFs are unable to accurately approximate the function (especially the central dip). The blue curves more closely resemble the red as spread decreases (images 3 and 4). The variance between trials, however, rises in relation to the noise in various trials. Depending on where the data points were located, the approximations for x=0 in the lower image vary greatly.

Bias and variance as function of model complexity

In statistics andmachine learning, the bias–variance tradeoff is the property of a model that the variance of the parameter estimated across samples can be reduced by increasing thebias in the estimated parameters. The bias–variance dilemma or bias–variance problem is the conflict in trying to simultaneously minimize these two sources oferror that prevent supervised learning algorithms from generalizing beyond their trainingset:[1][2]

  • The bias error is an error from erroneous assumptions in the learningalgorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting).
  • The variance is an error from sensitivity to small fluctuations in the training set. High variance may result from an algorithm modeling the randomnoise in the training data (overfitting).
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The bias–variance decomposition is a way of analyzing a learning algorithm’s expectedgeneralization error with respect to a particular problem as a sum of three terms, the bias, variance, and a quantity called the irreducible error, resulting from noise in the problem itself.

Motivation[edit]

  • bias low, variance low

  • bias high,variance low

  • bias low,variance high

  • bias high,variance high

A major challenge in supervised learning is the biasvariance tradeoff. To achieve the best results, one should pick a model that accurately captures the regularities in its training data and generalizes well to new data. Unfortunately, it is usually not possible to do both at once. Although high-variance learning techniques may be able to accurately represent their training set, they run the risk of overfitting to noisy or inaccurate training data. On the other hand, algorithms with high bias typically result in simpler models that may miss crucial regularities (i. e. in the data (underfit).

It is an often madefallacy[3][4] to assume that complex models must have high variance;High variance models are ‘complex’ in some sense, but the reverse needs not be true[clarification needed]. In addition, one has to be careful how to define complexity: In particular, the number ofparameters used to describe the model is a poor measure of complexity. This is illustrated by an example adapted from:[5] The model has only two parameters() but it can interpolate any number of points by oscillating with a high enough frequency, resulting in both a high bias and high variance.

An analogy can be made to the relationship between accuracy and precision. Accuracy is adescription of bias and can intuitively be improved by selecting from only local information. Consequently, a sample will appear accurate (i.e. have low bias) under the aforementioned selection conditions, but may result in underfitting. In other words, testdata may not agree as closely with training data, which would indicate imprecision and therefore inflated variance. A graphical example would be a straight line fit to data exhibiting quadratic behavior overall. Precision is a description of variance and generally can only be improved by selecting information from a comparatively larger space. The option to select many data points over a broad sample space is the ideal condition for any analysis. However, intrinsic constraints (whetherphysical, theoretical, computational, etc.) will always play a limiting role. The limiting case where only a finite number of data points are selected over a broad sample space may result in improved precision and lower variance overall, but may also result in an overreliance on the training data (overfitting). This means that test data would also not agree as closely with the training data, but in this case the reason is due to inaccuracy or high bias. To borrow from the previous example, thegraphical representation would appear as a high-order polynomial fit to the same data exhibiting quadratic behavior. Note that error in each case is measured the same way, but the reason ascribed to the error is different depending on the balance between bias and variance. To mitigate how much information is used from neighboring observations, a model can be smoothed via explicitregularization, such as shrinkage.

Bias–variance decomposition of mean squared error[edit]

Suppose that we have a training set consisting of a set of points and real values associated with each point . We assume that there is a function with noise, where the noise, , has zero mean and variance.

We want to find a function , that approximates the true function as well as possible, by means of some learning algorithm based on a training dataset (sample) . We make “as well as possible” precise by measuringthe mean squared error between and : we want to be minimal, both for and for points outside of our sample. Of course, we cannot hope to do so perfectly, since the contain noise ; this means we must be prepared to accept an irreducible error in any function we come up with.

Finding an that generalizes to points outside of the training set can be done with any of the countless algorithms used for supervised learning. It turns out that whichever function we select, wecan decompose its expected error on an unseen sample as follows:[6]: 34[7]: 223

where

and

The expectation ranges over different choices of the training set, all sampled from the same joint distribution which can for example be done viabootstrapping. The three terms represent:

Since all three terms are non-negative, the irreducible error forms a lower bound on the expected error on unseen samples.[6]: 34

The more complex the model is, the more data points it will capture, and the lower the bias will be. However, complexity will make the model “move” more to capture the data points, and hence its variance will be larger.

Derivation[edit]

The derivation of the bias–variance decomposition for squared error proceeds asfollows.[8][9] For notational convenience, we abbreviate , and we drop the subscript on our expectation operators. First, recall that, by definition, for any random variable, we have

Rearranging, we get:

Since is deterministic, i.e. independent of ,

Thus, given and (because is noise), implies

Also, since

Thus, since and are independent, we can write

Finally, MSE loss function (or negative log-likelihood) is obtained by taking the expectationvalue over :

Approaches[edit]

Dimensionality reduction and featureselection can decrease variance by simplifying models. Similarly, a larger training set tends to decrease variance. Adding features (predictors) tends to decrease bias, at the expense of introducing additional variance. Learning algorithms typically have some tunable parameters that control bias and variance; for example,

  • linear andGeneralized linear models can be regularized to decrease their variance at the cost of increasing their bias.[10]
  • Inartificial neural networks, the variance increases and the bias decreases as the number of hidden units increase,[11] although this classical assumption has been the subject of recentdebate.[4] Like in GLMs, regularization is typically applied.
  • In k-nearest neighbor models, a high value of k leads to high bias and low variance (see below).
  • Ininstance-based learning, regularization can be achieved varying the mixture of prototypes and exemplars.[12]
  • Indecision trees, the depth of the tree determines the variance. Decision trees are commonly pruned to control variance.[6]: 307
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One way of resolving the trade-off is to usemixture models and ensemblelearning.[13][14] For example, boosting combines many “weak” (high bias) models in an ensemble that haslower bias than the individual models, while bagging combines “strong” learners in a way that reduces their variance.

Model validation methods such ascross-validation (statistics) can be used to tune models so as to optimize the trade-off.

k-nearestneighbors[edit]

In the case of k-nearest neighbors regression, when the expectation is taken over the possible labeling of a fixedtraining set, a closed-form expression exists that relates the bias–variance decomposition to the parameter k:[7]: 37, 223

where are thek nearest neighbors of x in the training set. The bias (first term) is a monotone rising function of k, while the variance (second term) drops off as k is increased. In fact, under “reasonable assumptions” the bias of the first-nearest neighbor (1-NN) estimator vanishes entirely as the size of the training set approachesinfinity.[11]

Applications[edit]

Inregression[edit]

The bias–variance decomposition forms the conceptual basis for regression regularization methods such asLasso and ridge regression. Regularization methods introduce bias into the regression solution that can reduce variance considerably relative to the ordinary least squares (OLS)solution. Although the OLS solution provides non-biased regression estimates, the lower variance solutions produced by regularization techniques provide superior MSE performance.

In classification[edit]

The biasvariance decomposition was initially developed for least-squares regression. It is possible to locate a similar decomposition for the classification scenario with a loss of 0–1 (misclassification rate). As an alternative, the classification problem can be expressed as a probabilistic classification, in which case the expected squared error of the predicted probabilities with respect to the true probabilities can be decomposed as before. [17]

It has been argued that as training data increases, the variance of learned models will tend to decrease, and hence that as training data quantity increases, error is minimized by methods that learn modelswith lesser bias, and that conversely, for smaller training data quantities it is ever more important to minimize variance.[18]

In reinforcement learning[edit]

Even though the bias–variance decomposition does not directly apply in reinforcement learning, a similar tradeoff can also characterize generalization. When an agent has limited information on its environment, the suboptimality of an RL algorithm can be decomposed into the sum of two terms: a term related to anasymptotic bias and a term due to overfitting. The asymptotic bias is directly related to the learning algorithm (independently of the quantity of data) while the overfitting term comes from the fact that the amount of data is limited.[19]

In humanlearning[edit]

Although frequently discussed in relation to machine learning, the bias-variance conundrum has been studied in relation to human cognition, most notably by Gerd Gigerenzer and colleagues in relation to learned heuristics. They have argued that the human brain uses high-bias/low variance heuristics to solve the problem when faced with the typically sparse, poorly characterized training sets provided by experience (see references below). This is a reflection of the fact that a zero-bias approach is not generally applicable to novel situations and also unreasonably assumes exact knowledge of the real state of the world. The resulting heuristics are fairly straightforward but result in better inferences across a wider range of scenarios. [20].

Geman etal.[11] argue that the bias–variance dilemma implies that abilities such as generic object recognition cannot be learned from scratch, but require a certain degree of “hard wiring” that is later tuned by experience. This is because model-freeapproaches to inference require impractically large training sets if they are to avoid high variance.

See also[edit]

  • Accuracy andprecision
  • Bias of an estimator
  • Double descent
  • Gauss–Markov theorem
  • Hyperparameter optimization
  • Law of total variance
  • Minimum-variance unbiased estimator
  • Model selection
  • Regression model validation
  • Supervised learning

References[edit]

  • ^ Kohavi, Ron; Wolpert, David H. (1996). “Bias Plus VarianceDecomposition for Zero-One Loss Functions”. ICML. 96.
  • ^ Luxburg, Ulrike V.; Schölkopf, B. (2011). “Statistical learning theory: Models, concepts, and results”. Handbook of the History of Logic. 10: Section2.4.
  • ^ Neal, Brady (2019). “On the Bias-Variance Tradeoff: Textbooks Need an Update”. arXiv:1912.08286[cs.LG].
  • ^a b Neal, Brady; Mittal, Sarthak; Baratin, Aristide; Tantia, Vinayak; Scicluna, Matthew; Lacoste-Julien, Simon; Mitliagkas, Ioannis (2018). “A Modern Take on theBias-Variance Tradeoff in Neural Networks”. arXiv:1810.08591[cs.LG].
  • ^ Vapnik, Vladimir (2000). The nature of statistical learning theory. New York: Springer-Verlag. ISBN 978-1-4757-3264-1.
  • ^ a b cJames, Gareth; Witten, Daniela; Hastie, Trevor; Tibshirani, Robert (2013).An Introduction to Statistical Learning.Springer.
  • ^a b Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome H. (2009).The Elements of Statistical Learning. Archived from the original on 2015-01-26. Retrieved2014-08-20.
  • ^ Vijayakumar, Sethu (2007). “The Bias–Variance Tradeoff” (PDF).University of Edinburgh. Retrieved 19 August2014.
  • ^ Shakhnarovich, Greg (2011). “Notes on derivation of bias-variance decomposition in linear regression” (PDF). Archived fromthe original (PDF) on 21 August 2014. Retrieved 20 August2014.
  • ^ Belsley, David (1991). Conditioning diagnostics : collinearity and weak data in regression. New York (NY): Wiley. ISBN 978-0471528890.
  • ^a b cGeman, Stuart; Bienenstock, Élie; Doursat, René (1992). “Neural networks and the bias/variance dilemma” (PDF). Neural Computation. 4: 1–58.doi:10.1162/neco.1992.4.1.1.
  • ^ Gagliardi, Francesco (May 2011). “Instance-based classifiers applied to medical databases: diagnosis and knowledge extraction”. Artificial Intelligence in Medicine. 52 (3): 123–139. doi:10.1016/j.artmed.2011.04.002. PMID 21621400.
  • ^ Ting, Jo-Anne; Vijaykumar, Sethu; Schaal, Stefan (2011). “Locally Weighted Regression for Control”. In Sammut, Claude; Webb, Geoffrey I. (eds.). Encyclopedia of MachineLearning (PDF). Springer. p. 615.Bibcode:2010eoml.book…..S.
  • ^ Fortmann-Roe, Scott (2012). “Understanding the Bias–VarianceTradeoff”.
  • ^ Domingos, Pedro (2000). A unified bias-variance decomposition (PDF).ICML.
  • ^ Valentini, Giorgio; Dietterich, Thomas G. (2004). “Bias–variance analysis of support vector machines for the development ofSVM-based ensemble methods” (PDF). Journal of Machine Learning Research. 5:725–775.
  • ^ Manning, Christopher D.; Raghavan, Prabhakar; Schütze, Hinrich (2008). Introduction to Information Retrieval. Cambridge University Press. pp. 308–314.
  • ^ Brain, Damian; Webb, Geoffrey (2002). The Need for Low Bias Algorithms in Classification Learning From Large Data Sets (PDF).Proceedings of the Sixth European Conference on Principles of Data Mining and Knowledge Discovery (PKDD2002).
  • ^ Francois-Lavet, Vincent; Rabusseau, Guillaume; Pineau, Joelle; Ernst, Damien; Fonteneau, Raphael (2019). “On Overfitting and Asymptotic Bias in Batch Reinforcement Learning with Partial Observability”. Journal of AIResearch. 65: 1–30.doi:10.1613/jair.1.11478.
  • ^ Gigerenzer, Gerd; Brighton, Henry (2009). “Homo Heuristicus: Why Biased Minds Make Better Inferences”. Topics in Cognitive Science. 1 (1): 107–143.doi:10.1111/j.1756-8765.2008.01006.x.hdl:11858/00-001M-0000-0024-F678-0. PMID 25164802.
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    External links[edit]

    • MLU-Explain: The Bias Variance Tradeoff — An interactive visualization of the bias-variance tradeoff in LOESS Regressionand K-Nearest Neighbors.

    Additional Question — How do you find the error variance?

    How do you reduce error variance?

    Making auxiliary variables constant will allow you to treat subjects equally, reducing error variance. subjects are compared based on key characteristics. Reduce some types of carryover by employing strategies like pre-training, practice sessions, or rest periods between treatments. use an inside-subjects design.

    Is a high variance good or bad?

    Low-variance stocks are typically better for conservative investors who are less risk-tolerant, whereas high-variance stocks are typically better for aggressive investors who are less risk-averse. The degree of risk associated with an investment is gauged by variance.

    What does high variance mean?

    It is a statistical measure of variability that shows how much a set of numbers deviates from the mean. A high variance indicates that the data collected has higher variability and is typically further away from the mean.

    What is high variance in machine learning?

    Simply put, variance refers to how much the ML function can vary its predictions from the model based on the available data. Models with many features and a high level of complexity are the source of variation. Low variance is a characteristic of models with high bias. Models with a low bias will have a high variance.

    What is high variance and high bias?

    A model with a high variance may accurately represent the data set, but it also runs the risk of being overfit to training data that is noisy or otherwise unrepresentative. A model with high bias, in contrast, may underfit the training data because it is a simpler model that ignores data regularities.

    What is variance error in machine learning?

    Variance Error Variance is the amount by which using different training sets will alter the target function estimate. A machine learning algorithm estimates the target function from the training data, so we should expect the algorithm to have some variance.

    Why is overfitting called high variance?

    A high variance model is more likely to be overly complex. Overfitting of the model results from this. Assume that the high variance model will have very high training accuracy (or very low training loss), but low testing accuracy (or low testing loss).

    What is the difference between bias and error?

    Simply put, bias is the discrepancy between the expected value of your estimate (denote as) and the actual value of what you are estimating. Error is the discrepancy between your estimate and the actual value of the thing you are projecting.

    What is the difference between bias and variance?

    Variance describes the degree of variation that would occur if different training sets were used, changing the estimate of the target function. The discrepancy between expected and actual values is referred to as bias. A random variable’s variation from the expected value is indicated by its variance.

    How do I stop overfitting?

    Cross-validation is a method for preventing overfitting in machine learning. Cross-validation is an effective safeguard against overfitting. Use more data when training. Even though it won’t always work, training with more data can help algorithms identify the signal more accurately. Take features away. early stopping Regularization. Ensembling.

    What is high bias and low variance?

    Predictions are generally accurate but consistent when there is a high bias and low variance. When a model learns poorly from the training dataset or only uses a small number of parameter values, this situation arises. Underfitting issues in the model result from it.

    Conclusion :

    When studying stocks, it is important to remember the Error variance. This measure of how often different prices vary by ±1% or more from their fundamental value. The greater the Error variance, the more likely a stock is to be wrong by 1%. A price can also have an Error variance if it varies by ±2% or more from its fundamental value. This means that a price could be significantly off its true value.

    Dannie Jarrod

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