Linked Questions

5
votes
0answers
4k views

Can I calculate prediction intervals in scikit-learn for a linear model without bootstrapping? [duplicate]

I'd like to produce 95% prediction intervals along with predictions from my model. I'm using a moderately large dataset and making thousands of predictions, so I was wondering if there was some way ...
0
votes
0answers
168 views

Variance of a predicted $y$ when $x_0$ is given? [duplicate]

I cannot figure out why the variance of a prediction of $y$, when given $x_0$ is $$\sigma^2\bigg(1+\frac 1 n +\frac{(x_0-\bar x)^2}{Sxx}\bigg)$$ Isn't this the variance of a prediction error? ...
0
votes
0answers
41 views

How to obtain the equation of a predictive interval around the regression line in R? [duplicate]

I have a some data set on which I fit a linear regression model using the lm function in R. I can also visually obtain the prediction Interval around that regression line using predict. My question is ...
0
votes
0answers
16 views

How to prove the Standard error of fitted value given the standard error of each coefficients [duplicate]

I know those Equations but I cannot show the standard error of fitted value, please help me out, much appreciated. $$ SE(\hat{\beta_1})=\hat{\sigma}\sqrt{\frac{1}{(n-1)s_x^2}}\\ SE(\hat{\beta_0})=\...
61
votes
6answers
59k views

What is the difference between estimation and prediction?

For example, I have historical loss data and I am calculating extreme quantiles (Value-at-Risk or Probable Maximum Loss). The results obtained is for estimating the loss or predicting them? Where can ...
39
votes
2answers
33k views

Understanding shape and calculation of confidence bands in linear regression

I am trying to understand the origin of the curved shaped of confidence bands associated with an OLS linear regression and how it relates to the confidence intervals of the regression parameters (...
21
votes
3answers
9k views

Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)

Let's take the following example: set.seed(342) x1 <- runif(100) x2 <- runif(100) y <- x1+x2 + 2*x1*x2 + rnorm(100) fit <- lm(y~x1*x2) This creates a ...
14
votes
1answer
15k views

Why does the standard error of the intercept increase the further $\bar x$ is from 0?

The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^...
14
votes
2answers
4k views

Prediction and Tolerance Intervals

I have a couple of questions for prediction and tolerance intervals. Let's agree on the definition of the tolerance intervals first: We are given a confidence level, say 90%, the percentage of the ...
9
votes
2answers
16k views

SE of fit versus SE of prediction

I would like to get the standard error on a prediction. Using R glm, I can get the SE of the fit for a specific prediction: ...
8
votes
2answers
5k views

Gaussian process prediction interval

How can the prediction interval of a Gaussian process be evaluated? I don't know how to estimate this interval though I can find a 95 % confidence interval for the mean line.
3
votes
2answers
12k views

Calculation of variance of prediction

Could anybody show me how @Rob Hyndman calculates the variance of $\hat{y}$ in the following link Obtaining a formula for prediction limits in a linear model : EDIT: Basically I don't understand how ...
8
votes
5answers
643 views

Confidence Interval on a random quantity?

Suppose $\vec{a}$ is an unknown $p$-vector, and one observes $\vec{b} \sim \mathcal{N}\left(\vec{a}, I\right)$. I would like to compute confidence intervals on the random quantity $\vec{b}^{\top} \...
0
votes
2answers
9k views

What is the best way to extrapolate when working with a linear regression model?

There's not much more to ask than what I've written in the title. Some of the values I want to predict are outside of the range used to build the regression model.
12
votes
2answers
1k views

Can we make probabilistic statements with prediction intervals?

I've read through the many excellent discussions on the site regarding interpretation of confidence intervals and prediction intervals, but one concept is still a bit puzzling: Consider the OLS ...

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