- What Does Low Variance Mean Deviation
- What Does Low Variance Mean Math
- What Does Low Variance Mean Statistics
Aug 25, 2020 There are normal distributions where the variance is small compared to the mean (e.g., the heights of 30 year old men) and others where the mean is zero and the variance is significant. And there are long tailed distributions where the mean is finite and the variance is 'infinite'. In artificial neural networks, the variance increases and the bias decreases as the number of hidden units increase, although this classical assumption has been the subject of recent debate. 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).
Any supervised learning model is the result of optimizing the errors due to model complexity and the training error(prediction error on examples during training).
Example:
Ridge Regression or Regularized Linear Regression cost function(with parameters ) is given by
is the mean squared error of the prediction made(by the model with parameters ) on training samples. is given by
where is the actual label and is the prediction given by
From all the above equations,
OR
i.e. values determines the model complexity and training error is determined from the prediction made on training samples.
Other models:
In a model like K-NN classification, lower the value K, more complex the model is or in a Polynomial Regression, higher the degree of polynomial, more complex the model is. These are the examples of model complexity.
Training Error: Craps bets explained.
Prediction error on training examples(called Training Error above) exists due to the way data has been collected for training. Suppose we have to build a model to predict the political party which can win the election. We collect the data such that we chose only those people who have access to internet. Thus, we introduced bias in the data by not selecting samples from entire population. Hence, in a way, training error is introduced due to data and model complexity is due to the model obviously.
As we can see from above explanation and figure 1, there are two kinds of errors in supervised learning optimization problem. One might argue why do we need to optimize model complexity as complex model can do better prediction. A valid argument but not good for generalisation. Read here more on overfitting.
Error due to model complexity is also called the variance error. Error introduced due to some biases in the data is called bias error. Note that this may not be intentional but the way data was collected or features were extracted. Now that you're convinced variance error is related to model complexity, it also means that reducing variance error will lead to less irregular decision boundary.
Understanding Bias and Variance Error:
Imagine if model building task could be repeated for different training datasets(D_1, …, D_n), i.e. we train a new model for different dataset every time(shown in figure 2). Let us fix a test data point Test_1 whose actual prediction value is P. If we evaluate the model prediction on this point only, the predictions(P_1, …, P_n) will be different(varying) due to randomness in the model generation process. That is, every time we train a model(M_i) on a different dataset(D_i), we get a different model and hence different prediction(P_i) for the same data point(Test_1).
Bias Error is due to the difference between mean of these predictions and the correct value.
Variance Error is nothing but the variance in these predictions, i.e. how varied are these predictions.
where
This is the intuition behind bias and variance error. Till now, we have shown how does bias and variance error gets introduced. Next we need to understand why is there a tradeoff between bias and variance error?
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Understanding the bias variance tradeoff:
Figure 3 shows the plot of predictions P_1, … ,P_n(dark blue dots) against the correct value,P(dark red), at the centre. Note that the predictions P_1, …, P_n might be concentrated(Low Variance Error) but may not be close to the correct value(High Bias Error). Also they may be close to the correct value(Low Bias Error) but may not be concentrated(High Variance Error) as shown in the figure. This should be clear from the above equations of bias error and variance error. For bias error to be low, all the predictions or the mean of predictions should be close to the actual one(near the centre). For variance error to be low, all the predictions should lie close to the mean of predictions. In other words, variance error will be low if all the predictions lie close enough to each other. Space wars io. No deposit bonus codes usa.
Image Source: Understanding the Bias Variance tradeoff
So ideally we want low bias and low variance, i.e. not only the predictions should be close to the centre but also should very close to each other(concentrated). In reality low bias and low variance is possible only if there is an infinite amount of data for training which is impractical. Hence both low bias and low variance are difficult to achieve and there is a trade off. This is called bias-variance tradeoff.
Note that there are other explanation to bias-variance tradeoff and error. This is just another way to look at bias-variance error and not the only way.
Today we're going to introduce some terms that are important to machine learning:
- Variance
- r2 score
- Mean square error
We illustrate these concepts using scikit-learn.
(This article is part of our scikit-learn Guide. Use the right-hand menu to navigate.)
Why these terms are important
You need to understand these metrics in order to determine whether regression models are accurate or misleading. Following a flawed model is a bad idea, so it is important that you can quantify how accurate your model is. Understanding that is not so simple.
These first metrics are just a few of them. Other concepts, like bias and overtraining models, also yield misleading results and incorrect predictions.
(Learn more in Bias and Variance in Machine Learning.)
What Does Low Variance Mean Deviation
To provide examples, let's use the code from our last blog post, and add additional logic. We'll also introduce some randomness in the dependent variable (y) so that there is some error in our predictions. (Recall that, in the last blog post we made the independent y and dependent variables x perfectly correlate to illustrate the basics of how to do linear regression with scikit-learn.)
What is variance?
In terms of linear regression, variance is a measure of how far observed values differ from the average of predicted values, i.e., their difference from the predicted value mean. The goal is to have a value that is low. What low means is quantified by the r2 score (explained below).
In the code below, this is np.var(err), where err is an array of the differences between observed and predicted values and np.var() is the numpy array variance function.
What is r2 score?
The r2 score varies between 0 and 100%. It is closely related to the MSE (see below), but not the same. Wikipedia defines r2 as
' …the proportion of the variance in the dependent variable that is predictable from the independent variable(s).'
Another definition is '(total variance explained by model) / total variance.' So if it is 100%, the two variables are perfectly correlated, i.e., with no variance at all. A low value would show a low level of correlation, meaning a regression model that is not valid, but not in all cases.
Reading the code below, we do this calculation in three steps to make it easier to understand. g is the sum of the differences between the observed values and the predicted ones. (ytest[i] – preds[i]) **2. y is each observed value y[i] minus the average of observed values np.mean(ytest). And then the results are printed thus:
Our goal here is to explain. We can of course let scikit-learn to this with the r2_score() method:
What is mean square error (MSE)?
Mean square error (MSE) is the average of the square of the errors. The larger the number the larger the error. Error in this case means the difference between the observed values y1, y2, y3, … and the predicted ones pred(y1), pred(y2), pred(y3), … We square each difference (pred(yn) – yn)) ** 2 so that negative and positive values do not cancel each other out.
The complete code
What Does Low Variance Mean Math
So here is the complete code:
Results in:
You can see by looking at the data np.array([[[1],[2],[3]], [[2.01],[4.03],[6.04]]]) that every dependent variable is roughly twice the independent variable. That is confirmed as the calculated coefficient reg.coef_ is 2.015.
Understanding the bias variance tradeoff:
Figure 3 shows the plot of predictions P_1, … ,P_n(dark blue dots) against the correct value,P(dark red), at the centre. Note that the predictions P_1, …, P_n might be concentrated(Low Variance Error) but may not be close to the correct value(High Bias Error). Also they may be close to the correct value(Low Bias Error) but may not be concentrated(High Variance Error) as shown in the figure. This should be clear from the above equations of bias error and variance error. For bias error to be low, all the predictions or the mean of predictions should be close to the actual one(near the centre). For variance error to be low, all the predictions should lie close to the mean of predictions. In other words, variance error will be low if all the predictions lie close enough to each other. Space wars io. No deposit bonus codes usa.
Image Source: Understanding the Bias Variance tradeoff
So ideally we want low bias and low variance, i.e. not only the predictions should be close to the centre but also should very close to each other(concentrated). In reality low bias and low variance is possible only if there is an infinite amount of data for training which is impractical. Hence both low bias and low variance are difficult to achieve and there is a trade off. This is called bias-variance tradeoff.
Note that there are other explanation to bias-variance tradeoff and error. This is just another way to look at bias-variance error and not the only way.
Today we're going to introduce some terms that are important to machine learning:
- Variance
- r2 score
- Mean square error
We illustrate these concepts using scikit-learn.
(This article is part of our scikit-learn Guide. Use the right-hand menu to navigate.)
Why these terms are important
You need to understand these metrics in order to determine whether regression models are accurate or misleading. Following a flawed model is a bad idea, so it is important that you can quantify how accurate your model is. Understanding that is not so simple.
These first metrics are just a few of them. Other concepts, like bias and overtraining models, also yield misleading results and incorrect predictions.
(Learn more in Bias and Variance in Machine Learning.)
What Does Low Variance Mean Deviation
To provide examples, let's use the code from our last blog post, and add additional logic. We'll also introduce some randomness in the dependent variable (y) so that there is some error in our predictions. (Recall that, in the last blog post we made the independent y and dependent variables x perfectly correlate to illustrate the basics of how to do linear regression with scikit-learn.)
What is variance?
In terms of linear regression, variance is a measure of how far observed values differ from the average of predicted values, i.e., their difference from the predicted value mean. The goal is to have a value that is low. What low means is quantified by the r2 score (explained below).
In the code below, this is np.var(err), where err is an array of the differences between observed and predicted values and np.var() is the numpy array variance function.
What is r2 score?
The r2 score varies between 0 and 100%. It is closely related to the MSE (see below), but not the same. Wikipedia defines r2 as
' …the proportion of the variance in the dependent variable that is predictable from the independent variable(s).'
Another definition is '(total variance explained by model) / total variance.' So if it is 100%, the two variables are perfectly correlated, i.e., with no variance at all. A low value would show a low level of correlation, meaning a regression model that is not valid, but not in all cases.
Reading the code below, we do this calculation in three steps to make it easier to understand. g is the sum of the differences between the observed values and the predicted ones. (ytest[i] – preds[i]) **2. y is each observed value y[i] minus the average of observed values np.mean(ytest). And then the results are printed thus:
Our goal here is to explain. We can of course let scikit-learn to this with the r2_score() method:
What is mean square error (MSE)?
Mean square error (MSE) is the average of the square of the errors. The larger the number the larger the error. Error in this case means the difference between the observed values y1, y2, y3, … and the predicted ones pred(y1), pred(y2), pred(y3), … We square each difference (pred(yn) – yn)) ** 2 so that negative and positive values do not cancel each other out.
The complete code
What Does Low Variance Mean Math
So here is the complete code:
Results in:
You can see by looking at the data np.array([[[1],[2],[3]], [[2.01],[4.03],[6.04]]]) that every dependent variable is roughly twice the independent variable. That is confirmed as the calculated coefficient reg.coef_ is 2.015.
What Does Low Variance Mean Statistics
There is no correct value for MSE. Simply put, the lower the value the better and 0 means the model is perfect. Since there is no correct answer, the MSE's basic value is in selecting one prediction model over another.
Similarly, there is also no correct answer as to what R2 Scrivener 2 7 download free. should be. 100% means perfect correlation. Yet, there are models with a low R2 that are still good models.
Our take away message here is that you cannot look at these metrics in isolation in sizing up your model. You have to look at other metrics as well, plus understand the underlying math. We will get into all of this in subsequent blog posts.
Additional Resources
Extending R-squared beyond ordinary least-squares linear regression from pcdjohnson
