idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
|---|---|---|---|---|---|---|
12,301 | What is the difference between PCA and asymptotic PCA? | It's not about Math. only. To understand why it's different, you need to know the Finance theory background of factor models. For a factor model $R = BF + \epsilon$. It is like a multivariate regression, R has dimension the number of stocks.
F, the number of factors.
B is a matrix of loadings.
$\epsilon$ (epsilon) has... | What is the difference between PCA and asymptotic PCA? | It's not about Math. only. To understand why it's different, you need to know the Finance theory background of factor models. For a factor model $R = BF + \epsilon$. It is like a multivariate regressi | What is the difference between PCA and asymptotic PCA?
It's not about Math. only. To understand why it's different, you need to know the Finance theory background of factor models. For a factor model $R = BF + \epsilon$. It is like a multivariate regression, R has dimension the number of stocks.
F, the number of facto... | What is the difference between PCA and asymptotic PCA?
It's not about Math. only. To understand why it's different, you need to know the Finance theory background of factor models. For a factor model $R = BF + \epsilon$. It is like a multivariate regressi |
12,302 | Does LASSO suffer from the same problems stepwise regression does? | The probability interpretation of frequentist expressions of likelihood, p-values etcetera, for a LASSO model, and stepwise regression, are not correct.
Those expressions overestimate the probability. E.g. a 95% confidence interval for some parameter is supposed to say that you have a 95% probability that the method w... | Does LASSO suffer from the same problems stepwise regression does? | The probability interpretation of frequentist expressions of likelihood, p-values etcetera, for a LASSO model, and stepwise regression, are not correct.
Those expressions overestimate the probability | Does LASSO suffer from the same problems stepwise regression does?
The probability interpretation of frequentist expressions of likelihood, p-values etcetera, for a LASSO model, and stepwise regression, are not correct.
Those expressions overestimate the probability. E.g. a 95% confidence interval for some parameter i... | Does LASSO suffer from the same problems stepwise regression does?
The probability interpretation of frequentist expressions of likelihood, p-values etcetera, for a LASSO model, and stepwise regression, are not correct.
Those expressions overestimate the probability |
12,303 | Does LASSO suffer from the same problems stepwise regression does? | I have a new talk that addresses this. Bottom line: lasso has a low
probability of selecting the "correct" variables. The slides are at
http://fharrell.com/talk/stratos19
– Frank Harrell
Related to "Bottom line: lasso has a low probability of selecting the
"correct" variables": there's a section on the same to... | Does LASSO suffer from the same problems stepwise regression does? | I have a new talk that addresses this. Bottom line: lasso has a low
probability of selecting the "correct" variables. The slides are at
http://fharrell.com/talk/stratos19
– Frank Harrell
Relate | Does LASSO suffer from the same problems stepwise regression does?
I have a new talk that addresses this. Bottom line: lasso has a low
probability of selecting the "correct" variables. The slides are at
http://fharrell.com/talk/stratos19
– Frank Harrell
Related to "Bottom line: lasso has a low probability of sel... | Does LASSO suffer from the same problems stepwise regression does?
I have a new talk that addresses this. Bottom line: lasso has a low
probability of selecting the "correct" variables. The slides are at
http://fharrell.com/talk/stratos19
– Frank Harrell
Relate |
12,304 | Inverting the Fourier Transform for a Fisher distribution | There is no closed-form density for a convolution of F-statistics, so trying to invert the characteristic function analytically is not likely to lead to anything useful.
In mathematical statistics, the tilted Edgeworth expansion (also known as the saddlepoint approximation) is a famous and often used technique for appr... | Inverting the Fourier Transform for a Fisher distribution | There is no closed-form density for a convolution of F-statistics, so trying to invert the characteristic function analytically is not likely to lead to anything useful.
In mathematical statistics, th | Inverting the Fourier Transform for a Fisher distribution
There is no closed-form density for a convolution of F-statistics, so trying to invert the characteristic function analytically is not likely to lead to anything useful.
In mathematical statistics, the tilted Edgeworth expansion (also known as the saddlepoint ap... | Inverting the Fourier Transform for a Fisher distribution
There is no closed-form density for a convolution of F-statistics, so trying to invert the characteristic function analytically is not likely to lead to anything useful.
In mathematical statistics, th |
12,305 | Prediction interval based on cross-validation (CV) | After reading over this question again, I can give you the following bound:
Assume the samples are drawn iid, the distribution is fixed, and the loss is bounded by $B$, then with probability at least $1 - \delta$,
$$
\mathbb{E}[\mathcal{E}(h)] \leq \hat{\mathcal{E}}(h) + B\sqrt{\frac{\log \frac{1}{\delta}}{2m}}
$$
wher... | Prediction interval based on cross-validation (CV) | After reading over this question again, I can give you the following bound:
Assume the samples are drawn iid, the distribution is fixed, and the loss is bounded by $B$, then with probability at least | Prediction interval based on cross-validation (CV)
After reading over this question again, I can give you the following bound:
Assume the samples are drawn iid, the distribution is fixed, and the loss is bounded by $B$, then with probability at least $1 - \delta$,
$$
\mathbb{E}[\mathcal{E}(h)] \leq \hat{\mathcal{E}}(h)... | Prediction interval based on cross-validation (CV)
After reading over this question again, I can give you the following bound:
Assume the samples are drawn iid, the distribution is fixed, and the loss is bounded by $B$, then with probability at least |
12,306 | How to test equality of variances with circular data | 1) The Watson-Williams test is appropriate here.
2) It is parametric, and assumes a Von-Mises distribution. The second assumption is that each group has a common concentration parameter. I do not recall how robust the test is to violations of that assumption.
3) I have been using an implementation of the Watson test in... | How to test equality of variances with circular data | 1) The Watson-Williams test is appropriate here.
2) It is parametric, and assumes a Von-Mises distribution. The second assumption is that each group has a common concentration parameter. I do not reca | How to test equality of variances with circular data
1) The Watson-Williams test is appropriate here.
2) It is parametric, and assumes a Von-Mises distribution. The second assumption is that each group has a common concentration parameter. I do not recall how robust the test is to violations of that assumption.
3) I ha... | How to test equality of variances with circular data
1) The Watson-Williams test is appropriate here.
2) It is parametric, and assumes a Von-Mises distribution. The second assumption is that each group has a common concentration parameter. I do not reca |
12,307 | How to test equality of variances with circular data | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
Regarding your third question, I wrote a function in M... | How to test equality of variances with circular data | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
| How to test equality of variances with circular data
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
R... | How to test equality of variances with circular data
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
|
12,308 | Shrunken $r$ vs unbiased $r$: estimators of $\rho$ | Regarding the bias in the correlation: When sample sizes are small enough for bias to have any practical significance (e.g., the n < 30 you suggested), then bias is likely to be the least of your worries, because inaccuracy is terrible.
Regarding the bias of R2 in multiple regression, there are many different adjustmen... | Shrunken $r$ vs unbiased $r$: estimators of $\rho$ | Regarding the bias in the correlation: When sample sizes are small enough for bias to have any practical significance (e.g., the n < 30 you suggested), then bias is likely to be the least of your worr | Shrunken $r$ vs unbiased $r$: estimators of $\rho$
Regarding the bias in the correlation: When sample sizes are small enough for bias to have any practical significance (e.g., the n < 30 you suggested), then bias is likely to be the least of your worries, because inaccuracy is terrible.
Regarding the bias of R2 in mult... | Shrunken $r$ vs unbiased $r$: estimators of $\rho$
Regarding the bias in the correlation: When sample sizes are small enough for bias to have any practical significance (e.g., the n < 30 you suggested), then bias is likely to be the least of your worr |
12,309 | Shrunken $r$ vs unbiased $r$: estimators of $\rho$ | I think the answer is in the context of simple regression and multiple regression. In simple regression with one IV and one DV, the R sq is not positively biased, and in-fact may be negatively biased given r is negatively biased. But in multiple regression with several IV's which may be correlated themselves, R sq may ... | Shrunken $r$ vs unbiased $r$: estimators of $\rho$ | I think the answer is in the context of simple regression and multiple regression. In simple regression with one IV and one DV, the R sq is not positively biased, and in-fact may be negatively biased | Shrunken $r$ vs unbiased $r$: estimators of $\rho$
I think the answer is in the context of simple regression and multiple regression. In simple regression with one IV and one DV, the R sq is not positively biased, and in-fact may be negatively biased given r is negatively biased. But in multiple regression with several... | Shrunken $r$ vs unbiased $r$: estimators of $\rho$
I think the answer is in the context of simple regression and multiple regression. In simple regression with one IV and one DV, the R sq is not positively biased, and in-fact may be negatively biased |
12,310 | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5? | Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ than $\log(0.5+\epsilon)$ is. So when you average them you get something less than $\log(0.5)$.
Similarly, if you take a te... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na | Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ tha | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?
Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ than ... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na
Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ tha |
12,311 | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5? | This is another illustration of Jensen's inequality
$$\mathbb E[\log X] < \log \mathbb E[X]$$
(since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property that the expectation of the transform is not the transform of the expectation when the transform is not linear (plus a few exo... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na | This is another illustration of Jensen's inequality
$$\mathbb E[\log X] < \log \mathbb E[X]$$
(since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property that t | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?
This is another illustration of Jensen's inequality
$$\mathbb E[\log X] < \log \mathbb E[X]$$
(since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property that the... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na
This is another illustration of Jensen's inequality
$$\mathbb E[\log X] < \log \mathbb E[X]$$
(since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property that t |
12,312 | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5? | It is worthwhile to note that if $X \sim \operatorname{Uniform}(0,1)$, then $-\log X \sim \operatorname{Exponential}(\lambda = 1)$, so that $\operatorname{E}[\log X] = -1$. Explicitly, $$f_X(x) = \mathbb 1(0 < x < 1) = \begin{cases} 1, & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}$$ implies $$Y = g(X) = -\log X$$ h... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na | It is worthwhile to note that if $X \sim \operatorname{Uniform}(0,1)$, then $-\log X \sim \operatorname{Exponential}(\lambda = 1)$, so that $\operatorname{E}[\log X] = -1$. Explicitly, $$f_X(x) = \ma | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?
It is worthwhile to note that if $X \sim \operatorname{Uniform}(0,1)$, then $-\log X \sim \operatorname{Exponential}(\lambda = 1)$, so that $\operatorname{E}[\log X] = -1$. Explicitly, $$f_X(x) = \math... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na
It is worthwhile to note that if $X \sim \operatorname{Uniform}(0,1)$, then $-\log X \sim \operatorname{Exponential}(\lambda = 1)$, so that $\operatorname{E}[\log X] = -1$. Explicitly, $$f_X(x) = \ma |
12,313 | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5? | Note that the mean of a transformed uniform variable is just the mean value of the function doing the transformation over the domain (since we are expecting each value to be selected equally). This is simply,
$$
\frac{1}{b-a}\int_a^b{t(x)}dx = \int_0^1{t(x)}dx
$$
For example (in R):
$$
\int_0^1{log(x)}dx = (1\cdot log(... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na | Note that the mean of a transformed uniform variable is just the mean value of the function doing the transformation over the domain (since we are expecting each value to be selected equally). This is | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?
Note that the mean of a transformed uniform variable is just the mean value of the function doing the transformation over the domain (since we are expecting each value to be selected equally). This is s... | Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the na
Note that the mean of a transformed uniform variable is just the mean value of the function doing the transformation over the domain (since we are expecting each value to be selected equally). This is |
12,314 | When A and B are positively related variables, can they have opposite effect on their outcome variable C? | The other answers are truly marvelous - they give real life examples.
I want to explain why this can happen despite our intuition to the contrary.
See this geometrically!
Correlation is the cosine of the angle between the (centered) vectors.
Essentially, you are asking whether it is possible that
$A$ makes an acute an... | When A and B are positively related variables, can they have opposite effect on their outcome variab | The other answers are truly marvelous - they give real life examples.
I want to explain why this can happen despite our intuition to the contrary.
See this geometrically!
Correlation is the cosine of | When A and B are positively related variables, can they have opposite effect on their outcome variable C?
The other answers are truly marvelous - they give real life examples.
I want to explain why this can happen despite our intuition to the contrary.
See this geometrically!
Correlation is the cosine of the angle betw... | When A and B are positively related variables, can they have opposite effect on their outcome variab
The other answers are truly marvelous - they give real life examples.
I want to explain why this can happen despite our intuition to the contrary.
See this geometrically!
Correlation is the cosine of |
12,315 | When A and B are positively related variables, can they have opposite effect on their outcome variable C? | Yes, two co-occuring conditions can have opposite effects.
For example:
Making outrageous statements (A) is positively related to being entertaining (B).
Making outrageous statements (A) has a negative effect on winning elections (C).
Being entertaining (B) has a positive effect on winning elections (C). | When A and B are positively related variables, can they have opposite effect on their outcome variab | Yes, two co-occuring conditions can have opposite effects.
For example:
Making outrageous statements (A) is positively related to being entertaining (B).
Making outrageous statements (A) has a negati | When A and B are positively related variables, can they have opposite effect on their outcome variable C?
Yes, two co-occuring conditions can have opposite effects.
For example:
Making outrageous statements (A) is positively related to being entertaining (B).
Making outrageous statements (A) has a negative effect on w... | When A and B are positively related variables, can they have opposite effect on their outcome variab
Yes, two co-occuring conditions can have opposite effects.
For example:
Making outrageous statements (A) is positively related to being entertaining (B).
Making outrageous statements (A) has a negati |
12,316 | When A and B are positively related variables, can they have opposite effect on their outcome variable C? | I've heard this car analogy which applies well to the question:
Driving uphill (A) is positively related to the driver stepping on the gas (B)
Driving uphill (A) has a negative effect on vehicle speed (C)
Stepping on the gas (B) has a positive effect on vehicle speed (C)
The key here is the driver's intention to main... | When A and B are positively related variables, can they have opposite effect on their outcome variab | I've heard this car analogy which applies well to the question:
Driving uphill (A) is positively related to the driver stepping on the gas (B)
Driving uphill (A) has a negative effect on vehicle spee | When A and B are positively related variables, can they have opposite effect on their outcome variable C?
I've heard this car analogy which applies well to the question:
Driving uphill (A) is positively related to the driver stepping on the gas (B)
Driving uphill (A) has a negative effect on vehicle speed (C)
Stepping... | When A and B are positively related variables, can they have opposite effect on their outcome variab
I've heard this car analogy which applies well to the question:
Driving uphill (A) is positively related to the driver stepping on the gas (B)
Driving uphill (A) has a negative effect on vehicle spee |
12,317 | When A and B are positively related variables, can they have opposite effect on their outcome variable C? | Yes, this is trivial to demonstrate with a simulation:
Simulate 2 variables, A and B that are positively correlated:
> require(MASS)
> set.seed(1)
> Sigma <- matrix(c(10,3,3,2),2,2)
> dt <- data.frame(mvrnorm(n = 1000, rep(0, 2), Sigma))
> names(dt) <- c("A","B")
> cor(dt)
A B
A 1.0000000 0.6707593
B... | When A and B are positively related variables, can they have opposite effect on their outcome variab | Yes, this is trivial to demonstrate with a simulation:
Simulate 2 variables, A and B that are positively correlated:
> require(MASS)
> set.seed(1)
> Sigma <- matrix(c(10,3,3,2),2,2)
> dt <- data.frame | When A and B are positively related variables, can they have opposite effect on their outcome variable C?
Yes, this is trivial to demonstrate with a simulation:
Simulate 2 variables, A and B that are positively correlated:
> require(MASS)
> set.seed(1)
> Sigma <- matrix(c(10,3,3,2),2,2)
> dt <- data.frame(mvrnorm(n = 1... | When A and B are positively related variables, can they have opposite effect on their outcome variab
Yes, this is trivial to demonstrate with a simulation:
Simulate 2 variables, A and B that are positively correlated:
> require(MASS)
> set.seed(1)
> Sigma <- matrix(c(10,3,3,2),2,2)
> dt <- data.frame |
12,318 | When A and B are positively related variables, can they have opposite effect on their outcome variable C? | $$
C = mB + n (A-proj_B(A))
$$
then
$$
\left<C,A\right> = m\left<B,A\right> + n\left<A,A\right> -n \left<B,A\right>
$$
Then covariance between C and A could be negative in two conditions:
$n>m,\ \left<A,A\right> < \left<B,A\right> (n-m)/n $
$n<-m,\ \left<A,A\right> > \left<B,A\right> (n-m)/n$ | When A and B are positively related variables, can they have opposite effect on their outcome variab | $$
C = mB + n (A-proj_B(A))
$$
then
$$
\left<C,A\right> = m\left<B,A\right> + n\left<A,A\right> -n \left<B,A\right>
$$
Then covariance between C and A could be negative in two conditions:
$n>m,\ \l | When A and B are positively related variables, can they have opposite effect on their outcome variable C?
$$
C = mB + n (A-proj_B(A))
$$
then
$$
\left<C,A\right> = m\left<B,A\right> + n\left<A,A\right> -n \left<B,A\right>
$$
Then covariance between C and A could be negative in two conditions:
$n>m,\ \left<A,A\right>... | When A and B are positively related variables, can they have opposite effect on their outcome variab
$$
C = mB + n (A-proj_B(A))
$$
then
$$
\left<C,A\right> = m\left<B,A\right> + n\left<A,A\right> -n \left<B,A\right>
$$
Then covariance between C and A could be negative in two conditions:
$n>m,\ \l |
12,319 | Does machine learning really need data-efficient algorithms? | You are not entirely wrong, often it will be a lot easier to collect more/better data to improve an algorithm than to squeeze minor improvements out of the algorithm.
However, in practice, there are many settings, where it is difficult to get really large dataset.
Sure, it's easy to get really large datasets when you u... | Does machine learning really need data-efficient algorithms? | You are not entirely wrong, often it will be a lot easier to collect more/better data to improve an algorithm than to squeeze minor improvements out of the algorithm.
However, in practice, there are m | Does machine learning really need data-efficient algorithms?
You are not entirely wrong, often it will be a lot easier to collect more/better data to improve an algorithm than to squeeze minor improvements out of the algorithm.
However, in practice, there are many settings, where it is difficult to get really large dat... | Does machine learning really need data-efficient algorithms?
You are not entirely wrong, often it will be a lot easier to collect more/better data to improve an algorithm than to squeeze minor improvements out of the algorithm.
However, in practice, there are m |
12,320 | Does machine learning really need data-efficient algorithms? | I work in retail forecasting. When you need to forecast tomorrow's demand for product X at store Y, you only have a limited amount of data available: possibly only the last two years' worth of sales of this particular product at this particular store, or potentially sales of all products at all stores, if you use a cro... | Does machine learning really need data-efficient algorithms? | I work in retail forecasting. When you need to forecast tomorrow's demand for product X at store Y, you only have a limited amount of data available: possibly only the last two years' worth of sales o | Does machine learning really need data-efficient algorithms?
I work in retail forecasting. When you need to forecast tomorrow's demand for product X at store Y, you only have a limited amount of data available: possibly only the last two years' worth of sales of this particular product at this particular store, or pote... | Does machine learning really need data-efficient algorithms?
I work in retail forecasting. When you need to forecast tomorrow's demand for product X at store Y, you only have a limited amount of data available: possibly only the last two years' worth of sales o |
12,321 | Does machine learning really need data-efficient algorithms? | While it is true that nowadays it is fairly easy to gather large piles of data, this doesn't mean that it is good data. The large datasets are usually gathered by scraping the resources freely available on Internet, for example, for textual data those may be Reddit post, news articles, Wikipedia entries, for images tho... | Does machine learning really need data-efficient algorithms? | While it is true that nowadays it is fairly easy to gather large piles of data, this doesn't mean that it is good data. The large datasets are usually gathered by scraping the resources freely availab | Does machine learning really need data-efficient algorithms?
While it is true that nowadays it is fairly easy to gather large piles of data, this doesn't mean that it is good data. The large datasets are usually gathered by scraping the resources freely available on Internet, for example, for textual data those may be ... | Does machine learning really need data-efficient algorithms?
While it is true that nowadays it is fairly easy to gather large piles of data, this doesn't mean that it is good data. The large datasets are usually gathered by scraping the resources freely availab |
12,322 | Does machine learning really need data-efficient algorithms? | I was once asked to build a model that puts archeological artifacts into classes according to their manufactoring process. A big problem: for some classes, there were only four samples. And many artifacts are broken, so even for the samples we had, not all measurements were known (like their total length).
Yes, "smal... | Does machine learning really need data-efficient algorithms? | I was once asked to build a model that puts archeological artifacts into classes according to their manufactoring process. A big problem: for some classes, there were only four samples. And many art | Does machine learning really need data-efficient algorithms?
I was once asked to build a model that puts archeological artifacts into classes according to their manufactoring process. A big problem: for some classes, there were only four samples. And many artifacts are broken, so even for the samples we had, not all ... | Does machine learning really need data-efficient algorithms?
I was once asked to build a model that puts archeological artifacts into classes according to their manufactoring process. A big problem: for some classes, there were only four samples. And many art |
12,323 | Does machine learning really need data-efficient algorithms? | Here are a couple thoughts to add to what has been posted so far.
You might be interested in taking a look at the famous machine learning paper, Domingos, P. (2012). "A Few Useful Things to Know about Machine Learning". Communications of the ACM (pdf). It should contain some food for thought. Specifically, here are t... | Does machine learning really need data-efficient algorithms? | Here are a couple thoughts to add to what has been posted so far.
You might be interested in taking a look at the famous machine learning paper, Domingos, P. (2012). "A Few Useful Things to Know about | Does machine learning really need data-efficient algorithms?
Here are a couple thoughts to add to what has been posted so far.
You might be interested in taking a look at the famous machine learning paper, Domingos, P. (2012). "A Few Useful Things to Know about Machine Learning". Communications of the ACM (pdf). It sh... | Does machine learning really need data-efficient algorithms?
Here are a couple thoughts to add to what has been posted so far.
You might be interested in taking a look at the famous machine learning paper, Domingos, P. (2012). "A Few Useful Things to Know about |
12,324 | Does machine learning really need data-efficient algorithms? | There's some ambiguity in saying a data set is large. To improve predictive performance of an algorithm, you need more observations. You need to increase your sample size ($n$) and not the number of things you measured/observed within an experimental unit.
These can be hard to come by depending on the field of research... | Does machine learning really need data-efficient algorithms? | There's some ambiguity in saying a data set is large. To improve predictive performance of an algorithm, you need more observations. You need to increase your sample size ($n$) and not the number of t | Does machine learning really need data-efficient algorithms?
There's some ambiguity in saying a data set is large. To improve predictive performance of an algorithm, you need more observations. You need to increase your sample size ($n$) and not the number of things you measured/observed within an experimental unit.
Th... | Does machine learning really need data-efficient algorithms?
There's some ambiguity in saying a data set is large. To improve predictive performance of an algorithm, you need more observations. You need to increase your sample size ($n$) and not the number of t |
12,325 | Does machine learning really need data-efficient algorithms? | Would an ML algorithm that is, say, 100x more data-efficient, while being 1000x slower, be useful?
You have almost answered your own question.
There are multiple factors at play here:
The cost of gathering a data point
The cost of training a model with an additional data point
The cost of making the model learn more ... | Does machine learning really need data-efficient algorithms? | Would an ML algorithm that is, say, 100x more data-efficient, while being 1000x slower, be useful?
You have almost answered your own question.
There are multiple factors at play here:
The cost of ga | Does machine learning really need data-efficient algorithms?
Would an ML algorithm that is, say, 100x more data-efficient, while being 1000x slower, be useful?
You have almost answered your own question.
There are multiple factors at play here:
The cost of gathering a data point
The cost of training a model with an a... | Does machine learning really need data-efficient algorithms?
Would an ML algorithm that is, say, 100x more data-efficient, while being 1000x slower, be useful?
You have almost answered your own question.
There are multiple factors at play here:
The cost of ga |
12,326 | Does machine learning really need data-efficient algorithms? | People who work on data-efficient algorithms often bring up robotics for "motivation". But even for robotics, large datasets can be collected, as is done in this data-collection factory at Google:
What if I want to (for example) use reinforcement learning on a task involving underwater robotics to classify arctic ocea... | Does machine learning really need data-efficient algorithms? | People who work on data-efficient algorithms often bring up robotics for "motivation". But even for robotics, large datasets can be collected, as is done in this data-collection factory at Google:
Wh | Does machine learning really need data-efficient algorithms?
People who work on data-efficient algorithms often bring up robotics for "motivation". But even for robotics, large datasets can be collected, as is done in this data-collection factory at Google:
What if I want to (for example) use reinforcement learning on... | Does machine learning really need data-efficient algorithms?
People who work on data-efficient algorithms often bring up robotics for "motivation". But even for robotics, large datasets can be collected, as is done in this data-collection factory at Google:
Wh |
12,327 | Does machine learning really need data-efficient algorithms? | To all the other answers I'd add that in Deep Learning, Neural Architecture Search benefits immensely from data efficiency. Think about it: each data point is a trained network.
If your NAS setup requires $N$ data points (networks), and each network requires $D$ samples to be trained, that's $ND$ forward- and backpropa... | Does machine learning really need data-efficient algorithms? | To all the other answers I'd add that in Deep Learning, Neural Architecture Search benefits immensely from data efficiency. Think about it: each data point is a trained network.
If your NAS setup requ | Does machine learning really need data-efficient algorithms?
To all the other answers I'd add that in Deep Learning, Neural Architecture Search benefits immensely from data efficiency. Think about it: each data point is a trained network.
If your NAS setup requires $N$ data points (networks), and each network requires ... | Does machine learning really need data-efficient algorithms?
To all the other answers I'd add that in Deep Learning, Neural Architecture Search benefits immensely from data efficiency. Think about it: each data point is a trained network.
If your NAS setup requ |
12,328 | Does machine learning really need data-efficient algorithms? | To generalize a bit what @FransrRodenburg says about sample size:
many data sets have structure from various influencing factors. In particular, there are certain situation that lead to what is called nested factors in statistics (clustered data sets, hierarchical data sets, 1 : n relationships between influencing fact... | Does machine learning really need data-efficient algorithms? | To generalize a bit what @FransrRodenburg says about sample size:
many data sets have structure from various influencing factors. In particular, there are certain situation that lead to what is called | Does machine learning really need data-efficient algorithms?
To generalize a bit what @FransrRodenburg says about sample size:
many data sets have structure from various influencing factors. In particular, there are certain situation that lead to what is called nested factors in statistics (clustered data sets, hierarc... | Does machine learning really need data-efficient algorithms?
To generalize a bit what @FransrRodenburg says about sample size:
many data sets have structure from various influencing factors. In particular, there are certain situation that lead to what is called |
12,329 | Does machine learning really need data-efficient algorithms? | Random thoughts, although does not fully answer the question.
There seems to be a wastage of information in training new models on new data even if it is plentiful. Using an analogy with another general purpose technology, fitting new models is not totally dissimilar to reinventing the wheel. Bayesian and transfer lear... | Does machine learning really need data-efficient algorithms? | Random thoughts, although does not fully answer the question.
There seems to be a wastage of information in training new models on new data even if it is plentiful. Using an analogy with another gener | Does machine learning really need data-efficient algorithms?
Random thoughts, although does not fully answer the question.
There seems to be a wastage of information in training new models on new data even if it is plentiful. Using an analogy with another general purpose technology, fitting new models is not totally di... | Does machine learning really need data-efficient algorithms?
Random thoughts, although does not fully answer the question.
There seems to be a wastage of information in training new models on new data even if it is plentiful. Using an analogy with another gener |
12,330 | How to write a linear model formula with 100 variables in R | Try this
df<-data.frame(y=rnorm(10),x1=rnorm(10),x2=rnorm(10))
lm(y~.,df) | How to write a linear model formula with 100 variables in R | Try this
df<-data.frame(y=rnorm(10),x1=rnorm(10),x2=rnorm(10))
lm(y~.,df) | How to write a linear model formula with 100 variables in R
Try this
df<-data.frame(y=rnorm(10),x1=rnorm(10),x2=rnorm(10))
lm(y~.,df) | How to write a linear model formula with 100 variables in R
Try this
df<-data.frame(y=rnorm(10),x1=rnorm(10),x2=rnorm(10))
lm(y~.,df) |
12,331 | How to write a linear model formula with 100 variables in R | Great answers!
I would add that by default, calling formula on a data.frame creates an additive formula to regress the first column onto the others.
So in the case of the answer of @danas.zuokas you can even do
lm(df)
which is interpreted correctly. | How to write a linear model formula with 100 variables in R | Great answers!
I would add that by default, calling formula on a data.frame creates an additive formula to regress the first column onto the others.
So in the case of the answer of @danas.zuokas you c | How to write a linear model formula with 100 variables in R
Great answers!
I would add that by default, calling formula on a data.frame creates an additive formula to regress the first column onto the others.
So in the case of the answer of @danas.zuokas you can even do
lm(df)
which is interpreted correctly. | How to write a linear model formula with 100 variables in R
Great answers!
I would add that by default, calling formula on a data.frame creates an additive formula to regress the first column onto the others.
So in the case of the answer of @danas.zuokas you c |
12,332 | How to write a linear model formula with 100 variables in R | If each row is an observation and each column is a predictor so that $Y$ is an $n$-length vector and $X$ is an $n \times p$ matrix ($p=100$ in this case), then you can do this with
Z = as.data.frame(cbind(Y,X))
lm(Y ~ .,data=Z)
If there are other columns you did not want to include as predictors, you would have to rem... | How to write a linear model formula with 100 variables in R | If each row is an observation and each column is a predictor so that $Y$ is an $n$-length vector and $X$ is an $n \times p$ matrix ($p=100$ in this case), then you can do this with
Z = as.data.frame(c | How to write a linear model formula with 100 variables in R
If each row is an observation and each column is a predictor so that $Y$ is an $n$-length vector and $X$ is an $n \times p$ matrix ($p=100$ in this case), then you can do this with
Z = as.data.frame(cbind(Y,X))
lm(Y ~ .,data=Z)
If there are other columns you ... | How to write a linear model formula with 100 variables in R
If each row is an observation and each column is a predictor so that $Y$ is an $n$-length vector and $X$ is an $n \times p$ matrix ($p=100$ in this case), then you can do this with
Z = as.data.frame(c |
12,333 | How to write a linear model formula with 100 variables in R | You can also use a combination of the formula and paste functions.
Setup data: Let's imagine we have a data.frame that contains the predictor variables x1 to x100 and our dependent variable y, but that there is also a nuisance variable asdfasdf. Also the predictor variables are arranged in an order such that they are n... | How to write a linear model formula with 100 variables in R | You can also use a combination of the formula and paste functions.
Setup data: Let's imagine we have a data.frame that contains the predictor variables x1 to x100 and our dependent variable y, but tha | How to write a linear model formula with 100 variables in R
You can also use a combination of the formula and paste functions.
Setup data: Let's imagine we have a data.frame that contains the predictor variables x1 to x100 and our dependent variable y, but that there is also a nuisance variable asdfasdf. Also the predi... | How to write a linear model formula with 100 variables in R
You can also use a combination of the formula and paste functions.
Setup data: Let's imagine we have a data.frame that contains the predictor variables x1 to x100 and our dependent variable y, but tha |
12,334 | What is the name of this plot that has rows with two connected dots? | Some call it a (horizontal) lollipop plot with two groups.
Here is how to make this plot in Python using matplotlib and seaborn (only used for the style), adapted from https://python-graph-gallery.com/184-lollipop-plot-with-2-groups/ and as requested by the OP in the comments.
import numpy as np
import pandas as pd
imp... | What is the name of this plot that has rows with two connected dots? | Some call it a (horizontal) lollipop plot with two groups.
Here is how to make this plot in Python using matplotlib and seaborn (only used for the style), adapted from https://python-graph-gallery.com | What is the name of this plot that has rows with two connected dots?
Some call it a (horizontal) lollipop plot with two groups.
Here is how to make this plot in Python using matplotlib and seaborn (only used for the style), adapted from https://python-graph-gallery.com/184-lollipop-plot-with-2-groups/ and as requested ... | What is the name of this plot that has rows with two connected dots?
Some call it a (horizontal) lollipop plot with two groups.
Here is how to make this plot in Python using matplotlib and seaborn (only used for the style), adapted from https://python-graph-gallery.com |
12,335 | What is the name of this plot that has rows with two connected dots? | That's a dot plot. It is sometimes called a "Cleveland dot plot" because there is a variant of a histogram made with dots that people sometimes call a dot plot as well. This particular version plots two dots per country (for the two years) and draws a thicker line between them. The countries are sorted by the latter... | What is the name of this plot that has rows with two connected dots? | That's a dot plot. It is sometimes called a "Cleveland dot plot" because there is a variant of a histogram made with dots that people sometimes call a dot plot as well. This particular version plots | What is the name of this plot that has rows with two connected dots?
That's a dot plot. It is sometimes called a "Cleveland dot plot" because there is a variant of a histogram made with dots that people sometimes call a dot plot as well. This particular version plots two dots per country (for the two years) and draws... | What is the name of this plot that has rows with two connected dots?
That's a dot plot. It is sometimes called a "Cleveland dot plot" because there is a variant of a histogram made with dots that people sometimes call a dot plot as well. This particular version plots |
12,336 | What is the name of this plot that has rows with two connected dots? | The answer by @gung is correct in identifying the chart type and providing a link to how to implement in Excel, as requested by the OP. But for others wanting to know how to do this in R/tidyverse/ggplot, below is complete code:
library(dplyr) # for data manipulation
library(tidyr) # for reshaping the data frame
li... | What is the name of this plot that has rows with two connected dots? | The answer by @gung is correct in identifying the chart type and providing a link to how to implement in Excel, as requested by the OP. But for others wanting to know how to do this in R/tidyverse/ggp | What is the name of this plot that has rows with two connected dots?
The answer by @gung is correct in identifying the chart type and providing a link to how to implement in Excel, as requested by the OP. But for others wanting to know how to do this in R/tidyverse/ggplot, below is complete code:
library(dplyr) # for... | What is the name of this plot that has rows with two connected dots?
The answer by @gung is correct in identifying the chart type and providing a link to how to implement in Excel, as requested by the OP. But for others wanting to know how to do this in R/tidyverse/ggp |
12,337 | How to generate random integers between 1 and 4 that have a specific mean? | I agree with X'ian that the problem is under-specified. However, there is an elegant, scalable, efficient, effective, and versatile solution worth considering.
Because the product of the sample mean and sample size equals the sample sum, the problem concerns generating a random sample of $n$ values in the set $\{1,2,\... | How to generate random integers between 1 and 4 that have a specific mean? | I agree with X'ian that the problem is under-specified. However, there is an elegant, scalable, efficient, effective, and versatile solution worth considering.
Because the product of the sample mean | How to generate random integers between 1 and 4 that have a specific mean?
I agree with X'ian that the problem is under-specified. However, there is an elegant, scalable, efficient, effective, and versatile solution worth considering.
Because the product of the sample mean and sample size equals the sample sum, the pr... | How to generate random integers between 1 and 4 that have a specific mean?
I agree with X'ian that the problem is under-specified. However, there is an elegant, scalable, efficient, effective, and versatile solution worth considering.
Because the product of the sample mean |
12,338 | How to generate random integers between 1 and 4 that have a specific mean? | The question is under-specified in that the constraints on the frequencies
\begin{align}n_1+2n_2+3n_3+4n_4&=100M\\n_1+n_2+n_3+n_4&=100\end{align}
do not determine a distribution: "random" is not associated with a particular distribution, unless the OP means "uniform". For instance, if there exists one solution $(n_1^0,... | How to generate random integers between 1 and 4 that have a specific mean? | The question is under-specified in that the constraints on the frequencies
\begin{align}n_1+2n_2+3n_3+4n_4&=100M\\n_1+n_2+n_3+n_4&=100\end{align}
do not determine a distribution: "random" is not assoc | How to generate random integers between 1 and 4 that have a specific mean?
The question is under-specified in that the constraints on the frequencies
\begin{align}n_1+2n_2+3n_3+4n_4&=100M\\n_1+n_2+n_3+n_4&=100\end{align}
do not determine a distribution: "random" is not associated with a particular distribution, unless ... | How to generate random integers between 1 and 4 that have a specific mean?
The question is under-specified in that the constraints on the frequencies
\begin{align}n_1+2n_2+3n_3+4n_4&=100M\\n_1+n_2+n_3+n_4&=100\end{align}
do not determine a distribution: "random" is not assoc |
12,339 | How to generate random integers between 1 and 4 that have a specific mean? | I want to ... uh ... "attenuate" @whuber's amazing answer, which @TomZinger says is too difficult to follow. By that I mean I want to re-describe it in terms that I think Tom Zinger will understand, because it's clearly the best answer here. And as Tom gradually uses the method and finds that he needs, say, to know the... | How to generate random integers between 1 and 4 that have a specific mean? | I want to ... uh ... "attenuate" @whuber's amazing answer, which @TomZinger says is too difficult to follow. By that I mean I want to re-describe it in terms that I think Tom Zinger will understand, b | How to generate random integers between 1 and 4 that have a specific mean?
I want to ... uh ... "attenuate" @whuber's amazing answer, which @TomZinger says is too difficult to follow. By that I mean I want to re-describe it in terms that I think Tom Zinger will understand, because it's clearly the best answer here. And... | How to generate random integers between 1 and 4 that have a specific mean?
I want to ... uh ... "attenuate" @whuber's amazing answer, which @TomZinger says is too difficult to follow. By that I mean I want to re-describe it in terms that I think Tom Zinger will understand, b |
12,340 | How to generate random integers between 1 and 4 that have a specific mean? | You can use sample() and select specific probabilities for each integer. If you sum the product of the probabilities and the integers, you get the expected value of the distribution. So, if you have a mean value in mind, say $k$, you can solve the following equation:
$$k = 1\times P(1) + 2\times P(2) + 3\times P(3) + 4... | How to generate random integers between 1 and 4 that have a specific mean? | You can use sample() and select specific probabilities for each integer. If you sum the product of the probabilities and the integers, you get the expected value of the distribution. So, if you have a | How to generate random integers between 1 and 4 that have a specific mean?
You can use sample() and select specific probabilities for each integer. If you sum the product of the probabilities and the integers, you get the expected value of the distribution. So, if you have a mean value in mind, say $k$, you can solve t... | How to generate random integers between 1 and 4 that have a specific mean?
You can use sample() and select specific probabilities for each integer. If you sum the product of the probabilities and the integers, you get the expected value of the distribution. So, if you have a |
12,341 | How to generate random integers between 1 and 4 that have a specific mean? | Here is a simple algorithm: Create $n-1$ random integers in the range $[1,4]$ and calculate the $n^{th}$ integer for the mean to be equal to the specified value. If that number is smaller than $1$ or larger than $4$, then one by one distribute the surplus/lacking onto other integers, e.g. if the integer is $5$, we have... | How to generate random integers between 1 and 4 that have a specific mean? | Here is a simple algorithm: Create $n-1$ random integers in the range $[1,4]$ and calculate the $n^{th}$ integer for the mean to be equal to the specified value. If that number is smaller than $1$ or | How to generate random integers between 1 and 4 that have a specific mean?
Here is a simple algorithm: Create $n-1$ random integers in the range $[1,4]$ and calculate the $n^{th}$ integer for the mean to be equal to the specified value. If that number is smaller than $1$ or larger than $4$, then one by one distribute t... | How to generate random integers between 1 and 4 that have a specific mean?
Here is a simple algorithm: Create $n-1$ random integers in the range $[1,4]$ and calculate the $n^{th}$ integer for the mean to be equal to the specified value. If that number is smaller than $1$ or |
12,342 | How to generate random integers between 1 and 4 that have a specific mean? | As a supplement to whuber's answer, I've written a script in Python which goes through each step of the sampling scheme. Note that this is meant for illustrative purposes and is not necessarily performant.
Example output:
n=10, s=20, k=4
Starting grid
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
X X X X... | How to generate random integers between 1 and 4 that have a specific mean? | As a supplement to whuber's answer, I've written a script in Python which goes through each step of the sampling scheme. Note that this is meant for illustrative purposes and is not necessarily perfor | How to generate random integers between 1 and 4 that have a specific mean?
As a supplement to whuber's answer, I've written a script in Python which goes through each step of the sampling scheme. Note that this is meant for illustrative purposes and is not necessarily performant.
Example output:
n=10, s=20, k=4
Starti... | How to generate random integers between 1 and 4 that have a specific mean?
As a supplement to whuber's answer, I've written a script in Python which goes through each step of the sampling scheme. Note that this is meant for illustrative purposes and is not necessarily perfor |
12,343 | How to generate random integers between 1 and 4 that have a specific mean? | I've turned whuber's answer into an r function. I hope it helps someone.
n is how many integers you want;
t is the mean you want; and
k is the upper limit you want for your returned values
whubernator<-function(n=NULL, t=NULL, kMax=5){
z = tabulate(sample.int(kMax*(n), (n)*(t),replace =F) %% (n)+1, (n))
return(... | How to generate random integers between 1 and 4 that have a specific mean? | I've turned whuber's answer into an r function. I hope it helps someone.
n is how many integers you want;
t is the mean you want; and
k is the upper limit you want for your returned values
whubern | How to generate random integers between 1 and 4 that have a specific mean?
I've turned whuber's answer into an r function. I hope it helps someone.
n is how many integers you want;
t is the mean you want; and
k is the upper limit you want for your returned values
whubernator<-function(n=NULL, t=NULL, kMax=5){
z =... | How to generate random integers between 1 and 4 that have a specific mean?
I've turned whuber's answer into an r function. I hope it helps someone.
n is how many integers you want;
t is the mean you want; and
k is the upper limit you want for your returned values
whubern |
12,344 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity? | The other answers already here do a great job of explaining why Gaussian RVs don't converge to anything as the variance increases without bound, but I want to point out a seemingly-uniform property that such a collection of Gaussians does satisfy that I think might be enough for someone to guess that they are becoming ... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to | The other answers already here do a great job of explaining why Gaussian RVs don't converge to anything as the variance increases without bound, but I want to point out a seemingly-uniform property th | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity?
The other answers already here do a great job of explaining why Gaussian RVs don't converge to anything as the variance increases without bound, but I want to point out a seemingly-uniform property that such a... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to
The other answers already here do a great job of explaining why Gaussian RVs don't converge to anything as the variance increases without bound, but I want to point out a seemingly-uniform property th |
12,345 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity? | A common mistake in probability is to think that a distribution is uniform because it looks visually flat when all its values are near zero. This is because we tend to see that $f(x)=0.001 \approx 0.000001=f(y)$ and yet $f(x)/f(y)=0.001/0.000001=1000$, i.e. a small interval around $x$ is 1000 times more likely than a s... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to | A common mistake in probability is to think that a distribution is uniform because it looks visually flat when all its values are near zero. This is because we tend to see that $f(x)=0.001 \approx 0.0 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity?
A common mistake in probability is to think that a distribution is uniform because it looks visually flat when all its values are near zero. This is because we tend to see that $f(x)=0.001 \approx 0.000001=f(y... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to
A common mistake in probability is to think that a distribution is uniform because it looks visually flat when all its values are near zero. This is because we tend to see that $f(x)=0.001 \approx 0.0 |
12,346 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity? | Your statement the pdf starts looking like a uniform distribution with bounds given by $[−2σ,2σ]$ is not correct if you adjust $\sigma$ to match the wider standard deviation.
Consider this chart of two normal densities centred on zero. The red curve corresponds to a standard deviation of $1$ and the blue curve to a sta... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to | Your statement the pdf starts looking like a uniform distribution with bounds given by $[−2σ,2σ]$ is not correct if you adjust $\sigma$ to match the wider standard deviation.
Consider this chart of tw | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity?
Your statement the pdf starts looking like a uniform distribution with bounds given by $[−2σ,2σ]$ is not correct if you adjust $\sigma$ to match the wider standard deviation.
Consider this chart of two normal ... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to
Your statement the pdf starts looking like a uniform distribution with bounds given by $[−2σ,2σ]$ is not correct if you adjust $\sigma$ to match the wider standard deviation.
Consider this chart of tw |
12,347 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity? | The limit of normal distributions leads to another nice property that reflects a uniform distribution, which is that conditional probabilities for any two bounded sets converge in the limit to the conditional probability that applies for the uniform distribution. I will show this below.
To facilitate our analysis, we... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to | The limit of normal distributions leads to another nice property that reflects a uniform distribution, which is that conditional probabilities for any two bounded sets converge in the limit to the con | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity?
The limit of normal distributions leads to another nice property that reflects a uniform distribution, which is that conditional probabilities for any two bounded sets converge in the limit to the conditional ... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to
The limit of normal distributions leads to another nice property that reflects a uniform distribution, which is that conditional probabilities for any two bounded sets converge in the limit to the con |
12,348 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity? | Your question is fundamentally flawed. The standard normal distribution is scaled so that $\sigma = 1$. So for some other Gaussian distribution ($\mu = 0, \sigma = \sigma^*$) then the curve between bounds $[-2\sigma^*, 2\sigma^*]$ has the same shape as the standard normal distribution. The only difference is the scali... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to | Your question is fundamentally flawed. The standard normal distribution is scaled so that $\sigma = 1$. So for some other Gaussian distribution ($\mu = 0, \sigma = \sigma^*$) then the curve between b | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity?
Your question is fundamentally flawed. The standard normal distribution is scaled so that $\sigma = 1$. So for some other Gaussian distribution ($\mu = 0, \sigma = \sigma^*$) then the curve between bounds $[-... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to
Your question is fundamentally flawed. The standard normal distribution is scaled so that $\sigma = 1$. So for some other Gaussian distribution ($\mu = 0, \sigma = \sigma^*$) then the curve between b |
12,349 | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity? | Here is an alternative view of the problem that shows that of $X_n\sim N(\mu;n\sigma)$, where $\mu$ and $\sigma>0$ are fixed and $\{x\}=x-\lfloor x\rfloor$ is the fractional part function, then $\{X_n\}$ converges weakly to a random variable $U$ uniformly distributed over $[0,1]$, in other words, $$X_n\mod 1\stackrel{n... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to | Here is an alternative view of the problem that shows that of $X_n\sim N(\mu;n\sigma)$, where $\mu$ and $\sigma>0$ are fixed and $\{x\}=x-\lfloor x\rfloor$ is the fractional part function, then $\{X_n | Does the normal distribution converge to a uniform distribution when the standard deviation grows to infinity?
Here is an alternative view of the problem that shows that of $X_n\sim N(\mu;n\sigma)$, where $\mu$ and $\sigma>0$ are fixed and $\{x\}=x-\lfloor x\rfloor$ is the fractional part function, then $\{X_n\}$ conve... | Does the normal distribution converge to a uniform distribution when the standard deviation grows to
Here is an alternative view of the problem that shows that of $X_n\sim N(\mu;n\sigma)$, where $\mu$ and $\sigma>0$ are fixed and $\{x\}=x-\lfloor x\rfloor$ is the fractional part function, then $\{X_n |
12,350 | How do we know that the probability of rolling 1 and 2 is 1/18? | Imagine that you threw your fair six-sided die and you got ⚀. The
result was so fascinating that you called your friend Dave and told
him about it. Since he was curious what he'd get when
throwing his fair six-sided die, he threw it and got ⚁.
A standard die has six sides. If you are not cheating then it lands ... | How do we know that the probability of rolling 1 and 2 is 1/18? | Imagine that you threw your fair six-sided die and you got ⚀. The
result was so fascinating that you called your friend Dave and told
him about it. Since he was curious what he'd get when
throw | How do we know that the probability of rolling 1 and 2 is 1/18?
Imagine that you threw your fair six-sided die and you got ⚀. The
result was so fascinating that you called your friend Dave and told
him about it. Since he was curious what he'd get when
throwing his fair six-sided die, he threw it and got ⚁.
A st... | How do we know that the probability of rolling 1 and 2 is 1/18?
Imagine that you threw your fair six-sided die and you got ⚀. The
result was so fascinating that you called your friend Dave and told
him about it. Since he was curious what he'd get when
throw |
12,351 | How do we know that the probability of rolling 1 and 2 is 1/18? | I think you are overlooking the fact that it does not matter whether "we" can distinguish the dice or not, but rather it matters that the dice are unique and distinct, and act on their own accord.
So if in the closed box scenario, you open the box and see a 1 and a 2, you don't know whether it is $(1,2)$ or $(2,1)$, be... | How do we know that the probability of rolling 1 and 2 is 1/18? | I think you are overlooking the fact that it does not matter whether "we" can distinguish the dice or not, but rather it matters that the dice are unique and distinct, and act on their own accord.
So | How do we know that the probability of rolling 1 and 2 is 1/18?
I think you are overlooking the fact that it does not matter whether "we" can distinguish the dice or not, but rather it matters that the dice are unique and distinct, and act on their own accord.
So if in the closed box scenario, you open the box and see ... | How do we know that the probability of rolling 1 and 2 is 1/18?
I think you are overlooking the fact that it does not matter whether "we" can distinguish the dice or not, but rather it matters that the dice are unique and distinct, and act on their own accord.
So |
12,352 | How do we know that the probability of rolling 1 and 2 is 1/18? | Lets imagine that the first scenario involves rolling one red die and one blue die, while the second involves you rolling a pair of white dice.
In the first case, can write down every possible outcome as (red die, blue die), which gives you this table (reproduced from your question):
\begin{array} {|c|c|c|c|c|c|c|}
\hl... | How do we know that the probability of rolling 1 and 2 is 1/18? | Lets imagine that the first scenario involves rolling one red die and one blue die, while the second involves you rolling a pair of white dice.
In the first case, can write down every possible outcome | How do we know that the probability of rolling 1 and 2 is 1/18?
Lets imagine that the first scenario involves rolling one red die and one blue die, while the second involves you rolling a pair of white dice.
In the first case, can write down every possible outcome as (red die, blue die), which gives you this table (rep... | How do we know that the probability of rolling 1 and 2 is 1/18?
Lets imagine that the first scenario involves rolling one red die and one blue die, while the second involves you rolling a pair of white dice.
In the first case, can write down every possible outcome |
12,353 | How do we know that the probability of rolling 1 and 2 is 1/18? | The key idea is that if you list the 36 possible outcomes of two distinguishable dice, you are listing equally probable outcomes. This is not obvious, or axiomatic; it's true only if your dice are fair and not somehow connected. If you list the outcomes of indistinguishable dice, they are not equally probable, because ... | How do we know that the probability of rolling 1 and 2 is 1/18? | The key idea is that if you list the 36 possible outcomes of two distinguishable dice, you are listing equally probable outcomes. This is not obvious, or axiomatic; it's true only if your dice are fai | How do we know that the probability of rolling 1 and 2 is 1/18?
The key idea is that if you list the 36 possible outcomes of two distinguishable dice, you are listing equally probable outcomes. This is not obvious, or axiomatic; it's true only if your dice are fair and not somehow connected. If you list the outcomes of... | How do we know that the probability of rolling 1 and 2 is 1/18?
The key idea is that if you list the 36 possible outcomes of two distinguishable dice, you are listing equally probable outcomes. This is not obvious, or axiomatic; it's true only if your dice are fai |
12,354 | How do we know that the probability of rolling 1 and 2 is 1/18? | If you translate this into terms of coins - say, flipping two indistinguishable pennies - it becomes a question of only three outcomes: 2 heads, 2 tails, 1 of each, and the problem is easier to spot. The same logic applies, and we see that it's more likely to get 1 of each than to get 2 heads or 2 tails.
That's the sl... | How do we know that the probability of rolling 1 and 2 is 1/18? | If you translate this into terms of coins - say, flipping two indistinguishable pennies - it becomes a question of only three outcomes: 2 heads, 2 tails, 1 of each, and the problem is easier to spot. | How do we know that the probability of rolling 1 and 2 is 1/18?
If you translate this into terms of coins - say, flipping two indistinguishable pennies - it becomes a question of only three outcomes: 2 heads, 2 tails, 1 of each, and the problem is easier to spot. The same logic applies, and we see that it's more likely... | How do we know that the probability of rolling 1 and 2 is 1/18?
If you translate this into terms of coins - say, flipping two indistinguishable pennies - it becomes a question of only three outcomes: 2 heads, 2 tails, 1 of each, and the problem is easier to spot. |
12,355 | How do we know that the probability of rolling 1 and 2 is 1/18? | Let's start by stating the assumption: indistinguishable dice only roll 21 possible outcomes, while distinguishable dice roll 36 possible outcomes.
To test the difference, get a pair of identical white dice. Coat one in a UV-absorbent material like sunscreen, which is invisible to the naked eye. The dice still appear i... | How do we know that the probability of rolling 1 and 2 is 1/18? | Let's start by stating the assumption: indistinguishable dice only roll 21 possible outcomes, while distinguishable dice roll 36 possible outcomes.
To test the difference, get a pair of identical whit | How do we know that the probability of rolling 1 and 2 is 1/18?
Let's start by stating the assumption: indistinguishable dice only roll 21 possible outcomes, while distinguishable dice roll 36 possible outcomes.
To test the difference, get a pair of identical white dice. Coat one in a UV-absorbent material like sunscre... | How do we know that the probability of rolling 1 and 2 is 1/18?
Let's start by stating the assumption: indistinguishable dice only roll 21 possible outcomes, while distinguishable dice roll 36 possible outcomes.
To test the difference, get a pair of identical whit |
12,356 | How do we know that the probability of rolling 1 and 2 is 1/18? | We can deduce that your second table does not represent the scenario accurately.
You have eliminated all the cells below and left of the diagonal, on the supposed basis that (1, 2) and (2, 1) are congruent and therefore redundant outcomes.
Instead suppose that you roll one die twice in a row. Is it valid to count 1-th... | How do we know that the probability of rolling 1 and 2 is 1/18? | We can deduce that your second table does not represent the scenario accurately.
You have eliminated all the cells below and left of the diagonal, on the supposed basis that (1, 2) and (2, 1) are cong | How do we know that the probability of rolling 1 and 2 is 1/18?
We can deduce that your second table does not represent the scenario accurately.
You have eliminated all the cells below and left of the diagonal, on the supposed basis that (1, 2) and (2, 1) are congruent and therefore redundant outcomes.
Instead suppose ... | How do we know that the probability of rolling 1 and 2 is 1/18?
We can deduce that your second table does not represent the scenario accurately.
You have eliminated all the cells below and left of the diagonal, on the supposed basis that (1, 2) and (2, 1) are cong |
12,357 | How do we know that the probability of rolling 1 and 2 is 1/18? | If we just observe "Somebody gives me a box. I open the box. There is a $1$ and a $2$", without further information, we don't know anything about the probability.
If we know that the two dice are fair and that they have been rolled, then the probability is 1/18 as all other answer have explained. The fact we don't know... | How do we know that the probability of rolling 1 and 2 is 1/18? | If we just observe "Somebody gives me a box. I open the box. There is a $1$ and a $2$", without further information, we don't know anything about the probability.
If we know that the two dice are fair | How do we know that the probability of rolling 1 and 2 is 1/18?
If we just observe "Somebody gives me a box. I open the box. There is a $1$ and a $2$", without further information, we don't know anything about the probability.
If we know that the two dice are fair and that they have been rolled, then the probability is... | How do we know that the probability of rolling 1 and 2 is 1/18?
If we just observe "Somebody gives me a box. I open the box. There is a $1$ and a $2$", without further information, we don't know anything about the probability.
If we know that the two dice are fair |
12,358 | How do we know that the probability of rolling 1 and 2 is 1/18? | The probability of event A and B is calculated by multiplying both probabilities.
The probability of rolling a 1 when there are six possible options is 1/6. The probability of rolling a 2 when there are six possible options is 1/6.
1/6 * 1/6 = 1/36.
However, the event is not contingent on time (in other words, it is no... | How do we know that the probability of rolling 1 and 2 is 1/18? | The probability of event A and B is calculated by multiplying both probabilities.
The probability of rolling a 1 when there are six possible options is 1/6. The probability of rolling a 2 when there a | How do we know that the probability of rolling 1 and 2 is 1/18?
The probability of event A and B is calculated by multiplying both probabilities.
The probability of rolling a 1 when there are six possible options is 1/6. The probability of rolling a 2 when there are six possible options is 1/6.
1/6 * 1/6 = 1/36.
Howeve... | How do we know that the probability of rolling 1 and 2 is 1/18?
The probability of event A and B is calculated by multiplying both probabilities.
The probability of rolling a 1 when there are six possible options is 1/6. The probability of rolling a 2 when there a |
12,359 | How do we know that the probability of rolling 1 and 2 is 1/18? | The naive definition of probability is the ratio of favourable outcomes to total outcomes, as you worked it out for your example to be 2/36 = 1/18. The naive definition is applicable when the following two conditions are met:
All outcomes are equally likely
Sample space is finite.
We meet the first requirement (by sy... | How do we know that the probability of rolling 1 and 2 is 1/18? | The naive definition of probability is the ratio of favourable outcomes to total outcomes, as you worked it out for your example to be 2/36 = 1/18. The naive definition is applicable when the followin | How do we know that the probability of rolling 1 and 2 is 1/18?
The naive definition of probability is the ratio of favourable outcomes to total outcomes, as you worked it out for your example to be 2/36 = 1/18. The naive definition is applicable when the following two conditions are met:
All outcomes are equally like... | How do we know that the probability of rolling 1 and 2 is 1/18?
The naive definition of probability is the ratio of favourable outcomes to total outcomes, as you worked it out for your example to be 2/36 = 1/18. The naive definition is applicable when the followin |
12,360 | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | Let the CDF $F$ equal $1-1/n$ at the integers $n=1,2,\ldots,$ piecewise constant everywhere else, and subject to all criteria to be a CDF. The expectation is
$$\int_{0}^\infty (1-F(x))\mathrm{d}x = 1/2 + 1/3 + 1/4 + \cdots$$
which diverges. In this sense the first moment (and therefore all higher moments) is infinite... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | Let the CDF $F$ equal $1-1/n$ at the integers $n=1,2,\ldots,$ piecewise constant everywhere else, and subject to all criteria to be a CDF. The expectation is
$$\int_{0}^\infty (1-F(x))\mathrm{d}x = 1 | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
Let the CDF $F$ equal $1-1/n$ at the integers $n=1,2,\ldots,$ piecewise constant everywhere else, and subject to all criteria to be a CDF. The expectation is
$$\int_{0}^\infty (1-F(x))\mathrm{d}x = 1/2 + 1/3 + 1/4 + \cdo... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
Let the CDF $F$ equal $1-1/n$ at the integers $n=1,2,\ldots,$ piecewise constant everywhere else, and subject to all criteria to be a CDF. The expectation is
$$\int_{0}^\infty (1-F(x))\mathrm{d}x = 1 |
12,361 | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | Here's a famous example: Let $X$ take value $2^k$ with probability $2^{-k}$, for each integer $k\ge1$. Then $X$ takes values in (a subset of) the positive integers; the total mass is $\sum_{k=1}^\infty 2^{-k}=1$, but its expectation is
$$E(X) = \sum_{k=1}^\infty 2^k P(X=2^k) = \sum_{k=1}^\infty 1 = \infty.
$$
This rand... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | Here's a famous example: Let $X$ take value $2^k$ with probability $2^{-k}$, for each integer $k\ge1$. Then $X$ takes values in (a subset of) the positive integers; the total mass is $\sum_{k=1}^\inft | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
Here's a famous example: Let $X$ take value $2^k$ with probability $2^{-k}$, for each integer $k\ge1$. Then $X$ takes values in (a subset of) the positive integers; the total mass is $\sum_{k=1}^\infty 2^{-k}=1$, but its ... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
Here's a famous example: Let $X$ take value $2^k$ with probability $2^{-k}$, for each integer $k\ge1$. Then $X$ takes values in (a subset of) the positive integers; the total mass is $\sum_{k=1}^\inft |
12,362 | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | The zeta distribution is a fairly well-known discrete distribution on the positive integers that doesn't have finite mean (for $1<\theta\leq 2$) .
$P(X=x|\theta)={{\frac {1}{\zeta (\theta)}}x^{-\theta}}\,,\: x=1,2,...,\:\theta>1$
where the normalizing constant involves $\zeta(\cdot)$, the Riemann zeta function
(edit: T... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | The zeta distribution is a fairly well-known discrete distribution on the positive integers that doesn't have finite mean (for $1<\theta\leq 2$) .
$P(X=x|\theta)={{\frac {1}{\zeta (\theta)}}x^{-\theta | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
The zeta distribution is a fairly well-known discrete distribution on the positive integers that doesn't have finite mean (for $1<\theta\leq 2$) .
$P(X=x|\theta)={{\frac {1}{\zeta (\theta)}}x^{-\theta}}\,,\: x=1,2,...,\:\... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
The zeta distribution is a fairly well-known discrete distribution on the positive integers that doesn't have finite mean (for $1<\theta\leq 2$) .
$P(X=x|\theta)={{\frac {1}{\zeta (\theta)}}x^{-\theta |
12,363 | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | some discretized version of the Cauchy distribution
Yes, if you take $p(n)$ as being the average value of the Cauchy distribution in the interval around $n$, then clearly its zeroth moment is the same as that of the Cauchy distribution, and its first moment asymptotically approaches the first moment of the Cauchy dist... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist? | some discretized version of the Cauchy distribution
Yes, if you take $p(n)$ as being the average value of the Cauchy distribution in the interval around $n$, then clearly its zeroth moment is the sam | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
some discretized version of the Cauchy distribution
Yes, if you take $p(n)$ as being the average value of the Cauchy distribution in the interval around $n$, then clearly its zeroth moment is the same as that of the Cauc... | Example of a non-negative discrete distribution where the mean (or another moment) does not exist?
some discretized version of the Cauchy distribution
Yes, if you take $p(n)$ as being the average value of the Cauchy distribution in the interval around $n$, then clearly its zeroth moment is the sam |
12,364 | How I can convert distance (Euclidean) to similarity score | If $d(p_1,p_2)$ represents the euclidean distance from point $p_1$ to point $p_2$,
$$\frac{1}{1 + d(p_1, p_2)}$$
is commonly used. | How I can convert distance (Euclidean) to similarity score | If $d(p_1,p_2)$ represents the euclidean distance from point $p_1$ to point $p_2$,
$$\frac{1}{1 + d(p_1, p_2)}$$
is commonly used. | How I can convert distance (Euclidean) to similarity score
If $d(p_1,p_2)$ represents the euclidean distance from point $p_1$ to point $p_2$,
$$\frac{1}{1 + d(p_1, p_2)}$$
is commonly used. | How I can convert distance (Euclidean) to similarity score
If $d(p_1,p_2)$ represents the euclidean distance from point $p_1$ to point $p_2$,
$$\frac{1}{1 + d(p_1, p_2)}$$
is commonly used. |
12,365 | How I can convert distance (Euclidean) to similarity score | You could also use: $\frac{1}{e^{dist}}$ where dist is your desired distance function. | How I can convert distance (Euclidean) to similarity score | You could also use: $\frac{1}{e^{dist}}$ where dist is your desired distance function. | How I can convert distance (Euclidean) to similarity score
You could also use: $\frac{1}{e^{dist}}$ where dist is your desired distance function. | How I can convert distance (Euclidean) to similarity score
You could also use: $\frac{1}{e^{dist}}$ where dist is your desired distance function. |
12,366 | How I can convert distance (Euclidean) to similarity score | It sounds like you want something akin to cosine similarity, which is itself a similarity score in the unit interval. In fact, a direct relationship between Euclidean distance and cosine similarity exists!
Observe that
$$
||x-x^\prime||^2=(x-x^\prime)^T(x-x^\prime)=||x||+||x^\prime||-2||x-x^\prime||.
$$
While cosine si... | How I can convert distance (Euclidean) to similarity score | It sounds like you want something akin to cosine similarity, which is itself a similarity score in the unit interval. In fact, a direct relationship between Euclidean distance and cosine similarity ex | How I can convert distance (Euclidean) to similarity score
It sounds like you want something akin to cosine similarity, which is itself a similarity score in the unit interval. In fact, a direct relationship between Euclidean distance and cosine similarity exists!
Observe that
$$
||x-x^\prime||^2=(x-x^\prime)^T(x-x^\pr... | How I can convert distance (Euclidean) to similarity score
It sounds like you want something akin to cosine similarity, which is itself a similarity score in the unit interval. In fact, a direct relationship between Euclidean distance and cosine similarity ex |
12,367 | How I can convert distance (Euclidean) to similarity score | How about a Gaussian kernel ?
$K(x, x') = \exp\left( -\frac{\| x - x' \|^2}{2\sigma^2} \right)$
The distance $\|x - x'\|$ is used in the exponent. The kernel value is in the range $[0, 1]$. There is one tuning parameter $\sigma$. Basically if $\sigma$ is high, $K(x, x')$ will be close to 1 for any $x, x'$. If $\sigma$... | How I can convert distance (Euclidean) to similarity score | How about a Gaussian kernel ?
$K(x, x') = \exp\left( -\frac{\| x - x' \|^2}{2\sigma^2} \right)$
The distance $\|x - x'\|$ is used in the exponent. The kernel value is in the range $[0, 1]$. There is | How I can convert distance (Euclidean) to similarity score
How about a Gaussian kernel ?
$K(x, x') = \exp\left( -\frac{\| x - x' \|^2}{2\sigma^2} \right)$
The distance $\|x - x'\|$ is used in the exponent. The kernel value is in the range $[0, 1]$. There is one tuning parameter $\sigma$. Basically if $\sigma$ is high,... | How I can convert distance (Euclidean) to similarity score
How about a Gaussian kernel ?
$K(x, x') = \exp\left( -\frac{\| x - x' \|^2}{2\sigma^2} \right)$
The distance $\|x - x'\|$ is used in the exponent. The kernel value is in the range $[0, 1]$. There is |
12,368 | How I can convert distance (Euclidean) to similarity score | If you are using a distance metric that is naturally between 0 and 1, like Hellinger distance. Then you can use 1 - distance to obtain similarity. | How I can convert distance (Euclidean) to similarity score | If you are using a distance metric that is naturally between 0 and 1, like Hellinger distance. Then you can use 1 - distance to obtain similarity. | How I can convert distance (Euclidean) to similarity score
If you are using a distance metric that is naturally between 0 and 1, like Hellinger distance. Then you can use 1 - distance to obtain similarity. | How I can convert distance (Euclidean) to similarity score
If you are using a distance metric that is naturally between 0 and 1, like Hellinger distance. Then you can use 1 - distance to obtain similarity. |
12,369 | ARIMA model interpretation | I think that you need to remember that ARIMA models are atheoretic models, so the usual approach to interpreting estimated regression coefficients does not really carry over to ARIMA modelling.
In order to interpret (or understand) estimated ARIMA models, one would do well to be cognizant of the different features disp... | ARIMA model interpretation | I think that you need to remember that ARIMA models are atheoretic models, so the usual approach to interpreting estimated regression coefficients does not really carry over to ARIMA modelling.
In ord | ARIMA model interpretation
I think that you need to remember that ARIMA models are atheoretic models, so the usual approach to interpreting estimated regression coefficients does not really carry over to ARIMA modelling.
In order to interpret (or understand) estimated ARIMA models, one would do well to be cognizant of ... | ARIMA model interpretation
I think that you need to remember that ARIMA models are atheoretic models, so the usual approach to interpreting estimated regression coefficients does not really carry over to ARIMA modelling.
In ord |
12,370 | ARIMA model interpretation | Note that due to Wold's decomposition theorem you can rewrite any stationary ARMA model as a $MA(\infty)$ model, i.e. :
$$\Delta Y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}$$
In this form there are no lagged variables, so any interpretation involving notion of a lagged variable is not very convincing. However looking at th... | ARIMA model interpretation | Note that due to Wold's decomposition theorem you can rewrite any stationary ARMA model as a $MA(\infty)$ model, i.e. :
$$\Delta Y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}$$
In this form there are no lag | ARIMA model interpretation
Note that due to Wold's decomposition theorem you can rewrite any stationary ARMA model as a $MA(\infty)$ model, i.e. :
$$\Delta Y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}$$
In this form there are no lagged variables, so any interpretation involving notion of a lagged variable is not very convin... | ARIMA model interpretation
Note that due to Wold's decomposition theorem you can rewrite any stationary ARMA model as a $MA(\infty)$ model, i.e. :
$$\Delta Y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}$$
In this form there are no lag |
12,371 | ARIMA model interpretation | I totally agree with the sentiment of the previous commentators. I would like to add that all ARIMA model can also be represented as a pure AR model. These weights are referred to as the Pi weights as compared to the pure MA form (Psi weights) . In this way you can view (interpret) an ARIMA model as an optimized weight... | ARIMA model interpretation | I totally agree with the sentiment of the previous commentators. I would like to add that all ARIMA model can also be represented as a pure AR model. These weights are referred to as the Pi weights as | ARIMA model interpretation
I totally agree with the sentiment of the previous commentators. I would like to add that all ARIMA model can also be represented as a pure AR model. These weights are referred to as the Pi weights as compared to the pure MA form (Psi weights) . In this way you can view (interpret) an ARIMA m... | ARIMA model interpretation
I totally agree with the sentiment of the previous commentators. I would like to add that all ARIMA model can also be represented as a pure AR model. These weights are referred to as the Pi weights as |
12,372 | Random forest is overfitting? | This is a common rookie error when using RF models (I'll put my hand up as a previous perpetrator). The forest that you build using the training set will in many cases fit the training data almost perfectly (as you are finding) when considered in totality. However, as the algorithm builds the forest it remembers the ou... | Random forest is overfitting? | This is a common rookie error when using RF models (I'll put my hand up as a previous perpetrator). The forest that you build using the training set will in many cases fit the training data almost per | Random forest is overfitting?
This is a common rookie error when using RF models (I'll put my hand up as a previous perpetrator). The forest that you build using the training set will in many cases fit the training data almost perfectly (as you are finding) when considered in totality. However, as the algorithm builds ... | Random forest is overfitting?
This is a common rookie error when using RF models (I'll put my hand up as a previous perpetrator). The forest that you build using the training set will in many cases fit the training data almost per |
12,373 | Random forest is overfitting? | I think the answer is the max_features parameter: int, string or None, optional (default=”auto”) parameter. basically for this problem you should set it to None , so that each tree is built with all the inputs, since clearly you can't build a proper classifier using only a fraction of the cards ( default "auto" is sel... | Random forest is overfitting? | I think the answer is the max_features parameter: int, string or None, optional (default=”auto”) parameter. basically for this problem you should set it to None , so that each tree is built with all | Random forest is overfitting?
I think the answer is the max_features parameter: int, string or None, optional (default=”auto”) parameter. basically for this problem you should set it to None , so that each tree is built with all the inputs, since clearly you can't build a proper classifier using only a fraction of the... | Random forest is overfitting?
I think the answer is the max_features parameter: int, string or None, optional (default=”auto”) parameter. basically for this problem you should set it to None , so that each tree is built with all |
12,374 | What is the name of this chart showing false and true positive rates and how is it generated? | The plot is ROC curve and the (False Positive Rate, True Positive Rate) points are calculated for different thresholds. Assuming you have an uniform utility function, the optimal threshold value is the one for the point closest to (0, 1). | What is the name of this chart showing false and true positive rates and how is it generated? | The plot is ROC curve and the (False Positive Rate, True Positive Rate) points are calculated for different thresholds. Assuming you have an uniform utility function, the optimal threshold value is th | What is the name of this chart showing false and true positive rates and how is it generated?
The plot is ROC curve and the (False Positive Rate, True Positive Rate) points are calculated for different thresholds. Assuming you have an uniform utility function, the optimal threshold value is the one for the point closes... | What is the name of this chart showing false and true positive rates and how is it generated?
The plot is ROC curve and the (False Positive Rate, True Positive Rate) points are calculated for different thresholds. Assuming you have an uniform utility function, the optimal threshold value is th |
12,375 | What is the name of this chart showing false and true positive rates and how is it generated? | To generate ROC curves (= Receiver Operating Characteristic curves):
Assume we have a probabilistic, binary classifier such as logistic regression. Before presenting the ROC curve, the concept of confusion matrix must be understood. When we make a binary prediction, there can be 4 types of errors:
We predict 0 while w... | What is the name of this chart showing false and true positive rates and how is it generated? | To generate ROC curves (= Receiver Operating Characteristic curves):
Assume we have a probabilistic, binary classifier such as logistic regression. Before presenting the ROC curve, the concept of conf | What is the name of this chart showing false and true positive rates and how is it generated?
To generate ROC curves (= Receiver Operating Characteristic curves):
Assume we have a probabilistic, binary classifier such as logistic regression. Before presenting the ROC curve, the concept of confusion matrix must be under... | What is the name of this chart showing false and true positive rates and how is it generated?
To generate ROC curves (= Receiver Operating Characteristic curves):
Assume we have a probabilistic, binary classifier such as logistic regression. Before presenting the ROC curve, the concept of conf |
12,376 | What is the name of this chart showing false and true positive rates and how is it generated? | Morten's answer correctly addresses the question in the title -- the figure is, indeed, a ROC curve. It's produced by plotting a sequence of false positive rates (FPR) against their corresponding true positive rates.
However, I'd like to reply to the question that you ask in the body of your post.
If a method is appl... | What is the name of this chart showing false and true positive rates and how is it generated? | Morten's answer correctly addresses the question in the title -- the figure is, indeed, a ROC curve. It's produced by plotting a sequence of false positive rates (FPR) against their corresponding true | What is the name of this chart showing false and true positive rates and how is it generated?
Morten's answer correctly addresses the question in the title -- the figure is, indeed, a ROC curve. It's produced by plotting a sequence of false positive rates (FPR) against their corresponding true positive rates.
However, ... | What is the name of this chart showing false and true positive rates and how is it generated?
Morten's answer correctly addresses the question in the title -- the figure is, indeed, a ROC curve. It's produced by plotting a sequence of false positive rates (FPR) against their corresponding true |
12,377 | What is the name of this chart showing false and true positive rates and how is it generated? | From Wikipedia:
The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battlefields and was soon introduced to psychology to account for perceptual detection of stimuli. ROC analysis since then has been used in medicine, radiology, biometrics, a... | What is the name of this chart showing false and true positive rates and how is it generated? | From Wikipedia:
The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battlefields and was soon introduced to psychology to | What is the name of this chart showing false and true positive rates and how is it generated?
From Wikipedia:
The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battlefields and was soon introduced to psychology to account for perceptual det... | What is the name of this chart showing false and true positive rates and how is it generated?
From Wikipedia:
The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battlefields and was soon introduced to psychology to |
12,378 | Continuous dependent variable with ordinal independent variable | @Scortchi's got you covered with this answer on Coding for an ordered covariate. I've repeated the recommendation on my answer to Effect of two demographic IVs on survey answers (Likert scale). Specifically, the recommendation is to use Gertheiss' (2013) ordPens package, and to refer to Gertheiss and Tutz (2009a) for t... | Continuous dependent variable with ordinal independent variable | @Scortchi's got you covered with this answer on Coding for an ordered covariate. I've repeated the recommendation on my answer to Effect of two demographic IVs on survey answers (Likert scale). Specif | Continuous dependent variable with ordinal independent variable
@Scortchi's got you covered with this answer on Coding for an ordered covariate. I've repeated the recommendation on my answer to Effect of two demographic IVs on survey answers (Likert scale). Specifically, the recommendation is to use Gertheiss' (2013) o... | Continuous dependent variable with ordinal independent variable
@Scortchi's got you covered with this answer on Coding for an ordered covariate. I've repeated the recommendation on my answer to Effect of two demographic IVs on survey answers (Likert scale). Specif |
12,379 | Continuous dependent variable with ordinal independent variable | When there are multiple predictors, and the predictor of interest is ordinal, it is often difficult to decide how to code the variable. Coding it as categorical loses the order information, while coding it as numerical imposes linearity on the effects of the ordered categories that may be far from their true effects. ... | Continuous dependent variable with ordinal independent variable | When there are multiple predictors, and the predictor of interest is ordinal, it is often difficult to decide how to code the variable. Coding it as categorical loses the order information, while cod | Continuous dependent variable with ordinal independent variable
When there are multiple predictors, and the predictor of interest is ordinal, it is often difficult to decide how to code the variable. Coding it as categorical loses the order information, while coding it as numerical imposes linearity on the effects of ... | Continuous dependent variable with ordinal independent variable
When there are multiple predictors, and the predictor of interest is ordinal, it is often difficult to decide how to code the variable. Coding it as categorical loses the order information, while cod |
12,380 | Continuous dependent variable with ordinal independent variable | Generally there is lot of literature on ordinal variables as the dependent and little on using them as predictors. In statistical practice they are usually either assumed to be continous or categorical. You can check whether a linear model with the predictor as a continous variable looks like a good fit, by checking th... | Continuous dependent variable with ordinal independent variable | Generally there is lot of literature on ordinal variables as the dependent and little on using them as predictors. In statistical practice they are usually either assumed to be continous or categorica | Continuous dependent variable with ordinal independent variable
Generally there is lot of literature on ordinal variables as the dependent and little on using them as predictors. In statistical practice they are usually either assumed to be continous or categorical. You can check whether a linear model with the predict... | Continuous dependent variable with ordinal independent variable
Generally there is lot of literature on ordinal variables as the dependent and little on using them as predictors. In statistical practice they are usually either assumed to be continous or categorica |
12,381 | What is the intuitive meaning of having a linear relationship between the logs of two variables? | You just need to take exponential of both sides of the equation and you will get a potential relation, that may make sense for some data.
$$\log(Y) = a\log(X) + b$$
$$\exp(\log(Y)) = \exp(a \log(X) + b)$$
$$Y = e^b\cdot X^a$$
And since $e^b$ is just a parameter that can take any positive value, this model is equivalent... | What is the intuitive meaning of having a linear relationship between the logs of two variables? | You just need to take exponential of both sides of the equation and you will get a potential relation, that may make sense for some data.
$$\log(Y) = a\log(X) + b$$
$$\exp(\log(Y)) = \exp(a \log(X) + | What is the intuitive meaning of having a linear relationship between the logs of two variables?
You just need to take exponential of both sides of the equation and you will get a potential relation, that may make sense for some data.
$$\log(Y) = a\log(X) + b$$
$$\exp(\log(Y)) = \exp(a \log(X) + b)$$
$$Y = e^b\cdot X^a... | What is the intuitive meaning of having a linear relationship between the logs of two variables?
You just need to take exponential of both sides of the equation and you will get a potential relation, that may make sense for some data.
$$\log(Y) = a\log(X) + b$$
$$\exp(\log(Y)) = \exp(a \log(X) + |
12,382 | What is the intuitive meaning of having a linear relationship between the logs of two variables? | You can take your model $\log(Y)=a\log(X)+b$ and calculate the total differential, you will end up with something like :
$$\frac{1}YdY=a\frac{1}XdX$$
which yields to
$$\frac{dY}{dX}\frac{X}{Y}=a$$
Hence one simple interpretation of the coefficient $a$ will be the percent change in $Y$ for a percent change in $X$.
Thi... | What is the intuitive meaning of having a linear relationship between the logs of two variables? | You can take your model $\log(Y)=a\log(X)+b$ and calculate the total differential, you will end up with something like :
$$\frac{1}YdY=a\frac{1}XdX$$
which yields to
$$\frac{dY}{dX}\frac{X}{Y}=a$$
He | What is the intuitive meaning of having a linear relationship between the logs of two variables?
You can take your model $\log(Y)=a\log(X)+b$ and calculate the total differential, you will end up with something like :
$$\frac{1}YdY=a\frac{1}XdX$$
which yields to
$$\frac{dY}{dX}\frac{X}{Y}=a$$
Hence one simple interpre... | What is the intuitive meaning of having a linear relationship between the logs of two variables?
You can take your model $\log(Y)=a\log(X)+b$ and calculate the total differential, you will end up with something like :
$$\frac{1}YdY=a\frac{1}XdX$$
which yields to
$$\frac{dY}{dX}\frac{X}{Y}=a$$
He |
12,383 | What is the intuitive meaning of having a linear relationship between the logs of two variables? | Intuitively $\log$ gives us the order of magnitude of a variable, so we can view the relationship as the orders of magnitudes of the two variables are linearly related. For example, increasing the predictor by one order of magnitude may be associated with an increase of three orders of magnitude of the response.
When ... | What is the intuitive meaning of having a linear relationship between the logs of two variables? | Intuitively $\log$ gives us the order of magnitude of a variable, so we can view the relationship as the orders of magnitudes of the two variables are linearly related. For example, increasing the pre | What is the intuitive meaning of having a linear relationship between the logs of two variables?
Intuitively $\log$ gives us the order of magnitude of a variable, so we can view the relationship as the orders of magnitudes of the two variables are linearly related. For example, increasing the predictor by one order of ... | What is the intuitive meaning of having a linear relationship between the logs of two variables?
Intuitively $\log$ gives us the order of magnitude of a variable, so we can view the relationship as the orders of magnitudes of the two variables are linearly related. For example, increasing the pre |
12,384 | What is the intuitive meaning of having a linear relationship between the logs of two variables? | Reconciling the answer by @Rscrill with actual discrete data, consider
$$\log(Y_t) = a\log(X_t) + b,\;\;\; \log(Y_{t-1}) = a\log(X_{t-1}) + b$$
$$\implies \log(Y_t) - \log(Y_{t-1}) = a\left[\log(X_t)-\log(X_{t-1})\right]$$
But
$$\log(Y_t) - \log(Y_{t-1}) = \log\left(\frac{Y_t}{Y_{t-1}}\right) \equiv \log\left(\frac{Y_... | What is the intuitive meaning of having a linear relationship between the logs of two variables? | Reconciling the answer by @Rscrill with actual discrete data, consider
$$\log(Y_t) = a\log(X_t) + b,\;\;\; \log(Y_{t-1}) = a\log(X_{t-1}) + b$$
$$\implies \log(Y_t) - \log(Y_{t-1}) = a\left[\log(X_t)- | What is the intuitive meaning of having a linear relationship between the logs of two variables?
Reconciling the answer by @Rscrill with actual discrete data, consider
$$\log(Y_t) = a\log(X_t) + b,\;\;\; \log(Y_{t-1}) = a\log(X_{t-1}) + b$$
$$\implies \log(Y_t) - \log(Y_{t-1}) = a\left[\log(X_t)-\log(X_{t-1})\right]$$
... | What is the intuitive meaning of having a linear relationship between the logs of two variables?
Reconciling the answer by @Rscrill with actual discrete data, consider
$$\log(Y_t) = a\log(X_t) + b,\;\;\; \log(Y_{t-1}) = a\log(X_{t-1}) + b$$
$$\implies \log(Y_t) - \log(Y_{t-1}) = a\left[\log(X_t)- |
12,385 | What is the intuitive meaning of having a linear relationship between the logs of two variables? | A linear relationship between the logs is equivalent to a power law dependence:
$$Y \sim X^\alpha$$
In physics such behavior means that the system is scale free or scale invariant. As an example, if $X$ is distance or time this means that the dependence on $X$ cannot be characterized by a characteristic length or time... | What is the intuitive meaning of having a linear relationship between the logs of two variables? | A linear relationship between the logs is equivalent to a power law dependence:
$$Y \sim X^\alpha$$
In physics such behavior means that the system is scale free or scale invariant. As an example, if | What is the intuitive meaning of having a linear relationship between the logs of two variables?
A linear relationship between the logs is equivalent to a power law dependence:
$$Y \sim X^\alpha$$
In physics such behavior means that the system is scale free or scale invariant. As an example, if $X$ is distance or time... | What is the intuitive meaning of having a linear relationship between the logs of two variables?
A linear relationship between the logs is equivalent to a power law dependence:
$$Y \sim X^\alpha$$
In physics such behavior means that the system is scale free or scale invariant. As an example, if |
12,386 | Simulating draws from a Uniform Distribution using draws from a Normal Distribution | In the spirit of using simple algebraic calculations which are unrelated to computation of the Normal distribution, I would lean towards the following. They are ordered as I thought of them (and therefore needed to get more and more creative), but I have saved the best--and most surprising--to last.
Reverse the Box-M... | Simulating draws from a Uniform Distribution using draws from a Normal Distribution | In the spirit of using simple algebraic calculations which are unrelated to computation of the Normal distribution, I would lean towards the following. They are ordered as I thought of them (and ther | Simulating draws from a Uniform Distribution using draws from a Normal Distribution
In the spirit of using simple algebraic calculations which are unrelated to computation of the Normal distribution, I would lean towards the following. They are ordered as I thought of them (and therefore needed to get more and more cr... | Simulating draws from a Uniform Distribution using draws from a Normal Distribution
In the spirit of using simple algebraic calculations which are unrelated to computation of the Normal distribution, I would lean towards the following. They are ordered as I thought of them (and ther |
12,387 | Simulating draws from a Uniform Distribution using draws from a Normal Distribution | You can use a trick very similar to what you mention. Let's say that $X \sim N(\mu, \sigma^2)$ is a normal random variable with known parameters. Then we know its distribution function, $\Phi_{\mu,\sigma^2}$, and $\Phi_{\mu,\sigma^2}(X)$ will be uniformly distributed on $(0,1)$. To prove this, note that for $d \in (0,1... | Simulating draws from a Uniform Distribution using draws from a Normal Distribution | You can use a trick very similar to what you mention. Let's say that $X \sim N(\mu, \sigma^2)$ is a normal random variable with known parameters. Then we know its distribution function, $\Phi_{\mu,\si | Simulating draws from a Uniform Distribution using draws from a Normal Distribution
You can use a trick very similar to what you mention. Let's say that $X \sim N(\mu, \sigma^2)$ is a normal random variable with known parameters. Then we know its distribution function, $\Phi_{\mu,\sigma^2}$, and $\Phi_{\mu,\sigma^2}(X)... | Simulating draws from a Uniform Distribution using draws from a Normal Distribution
You can use a trick very similar to what you mention. Let's say that $X \sim N(\mu, \sigma^2)$ is a normal random variable with known parameters. Then we know its distribution function, $\Phi_{\mu,\si |
12,388 | Simulating draws from a Uniform Distribution using draws from a Normal Distribution | Adding on to 5:
The trick of transforming random variables to bits works for any independent pair of absolutely continuous random variables X and Y, even if X and Y are dependent on each other or the two variables are not identically distributed, as long as the two variables are statistically indifferent (Montes Gutiér... | Simulating draws from a Uniform Distribution using draws from a Normal Distribution | Adding on to 5:
The trick of transforming random variables to bits works for any independent pair of absolutely continuous random variables X and Y, even if X and Y are dependent on each other or the | Simulating draws from a Uniform Distribution using draws from a Normal Distribution
Adding on to 5:
The trick of transforming random variables to bits works for any independent pair of absolutely continuous random variables X and Y, even if X and Y are dependent on each other or the two variables are not identically di... | Simulating draws from a Uniform Distribution using draws from a Normal Distribution
Adding on to 5:
The trick of transforming random variables to bits works for any independent pair of absolutely continuous random variables X and Y, even if X and Y are dependent on each other or the |
12,389 | What is effect size... and why is it even useful? | That is one measure of effect size, but there are many others. It is certainly not the $t$ test statistic. Your measure of effect size is often called Cohen's $d$ (strictly speaking that is correct only if the SD is estimated via MLE—i.e., without Bessel's correction); more generically, it is called the 'standardized... | What is effect size... and why is it even useful? | That is one measure of effect size, but there are many others. It is certainly not the $t$ test statistic. Your measure of effect size is often called Cohen's $d$ (strictly speaking that is correct | What is effect size... and why is it even useful?
That is one measure of effect size, but there are many others. It is certainly not the $t$ test statistic. Your measure of effect size is often called Cohen's $d$ (strictly speaking that is correct only if the SD is estimated via MLE—i.e., without Bessel's correction)... | What is effect size... and why is it even useful?
That is one measure of effect size, but there are many others. It is certainly not the $t$ test statistic. Your measure of effect size is often called Cohen's $d$ (strictly speaking that is correct |
12,390 | What is effect size... and why is it even useful? | I expect someone with a background in a more relevant area (psychology or education, say) will chime in with a better answer, but I'll give it a shot.
"Effect size" is a term with more than one meaning -- which many years past led some some confused conversations until I eventually came to that realization. Here we're ... | What is effect size... and why is it even useful? | I expect someone with a background in a more relevant area (psychology or education, say) will chime in with a better answer, but I'll give it a shot.
"Effect size" is a term with more than one meanin | What is effect size... and why is it even useful?
I expect someone with a background in a more relevant area (psychology or education, say) will chime in with a better answer, but I'll give it a shot.
"Effect size" is a term with more than one meaning -- which many years past led some some confused conversations until ... | What is effect size... and why is it even useful?
I expect someone with a background in a more relevant area (psychology or education, say) will chime in with a better answer, but I'll give it a shot.
"Effect size" is a term with more than one meanin |
12,391 | What is effect size... and why is it even useful? | The formula above is how you calculate Cohen's d for related samples (which is probably what you have?), if they're unrelated you can use pooled variance instead. There are different stats that will tell you about effect size, but Cohen's d is a standardised measure that can vary between 0 and 3. If you have lots of ... | What is effect size... and why is it even useful? | The formula above is how you calculate Cohen's d for related samples (which is probably what you have?), if they're unrelated you can use pooled variance instead. There are different stats that will | What is effect size... and why is it even useful?
The formula above is how you calculate Cohen's d for related samples (which is probably what you have?), if they're unrelated you can use pooled variance instead. There are different stats that will tell you about effect size, but Cohen's d is a standardised measure th... | What is effect size... and why is it even useful?
The formula above is how you calculate Cohen's d for related samples (which is probably what you have?), if they're unrelated you can use pooled variance instead. There are different stats that will |
12,392 | What is effect size... and why is it even useful? | In fact, p-values are now finally 'out of fashion' as well: http://www.nature.com/news/psychology-journal-bans-p-values-1.17001. Null hypothesis significance testing (NHST) produces little more than a description of your sample size.(*) Any experimental intervention will have some effect, which is to say that the simpl... | What is effect size... and why is it even useful? | In fact, p-values are now finally 'out of fashion' as well: http://www.nature.com/news/psychology-journal-bans-p-values-1.17001. Null hypothesis significance testing (NHST) produces little more than a | What is effect size... and why is it even useful?
In fact, p-values are now finally 'out of fashion' as well: http://www.nature.com/news/psychology-journal-bans-p-values-1.17001. Null hypothesis significance testing (NHST) produces little more than a description of your sample size.(*) Any experimental intervention wil... | What is effect size... and why is it even useful?
In fact, p-values are now finally 'out of fashion' as well: http://www.nature.com/news/psychology-journal-bans-p-values-1.17001. Null hypothesis significance testing (NHST) produces little more than a |
12,393 | What is effect size... and why is it even useful? | What you wrote is not a test statistic. It's a measure used to define how different the two means are. Generally, effect sizes are used to quantify how far from the null hypotheses the something is. For example, if you are doing power analysis for the two sample $t$-test, you might quantify the power as a function of t... | What is effect size... and why is it even useful? | What you wrote is not a test statistic. It's a measure used to define how different the two means are. Generally, effect sizes are used to quantify how far from the null hypotheses the something is. F | What is effect size... and why is it even useful?
What you wrote is not a test statistic. It's a measure used to define how different the two means are. Generally, effect sizes are used to quantify how far from the null hypotheses the something is. For example, if you are doing power analysis for the two sample $t$-tes... | What is effect size... and why is it even useful?
What you wrote is not a test statistic. It's a measure used to define how different the two means are. Generally, effect sizes are used to quantify how far from the null hypotheses the something is. F |
12,394 | Does a univariate random variable's mean always equal the integral of its quantile function? | Let $F$ be the CDF of the random variable $X$, so the inverse CDF can be written $F^{-1}$. In your integral make the substitution $p = F(x)$, $dp = F'(x)dx = f(x)dx$ to obtain
$$\int_0^1F^{-1}(p)dp = \int_{-\infty}^{\infty}x f(x) dx = \mathbb{E}_F[X].$$
This is valid for continuous distributions. Care must be taken f... | Does a univariate random variable's mean always equal the integral of its quantile function? | Let $F$ be the CDF of the random variable $X$, so the inverse CDF can be written $F^{-1}$. In your integral make the substitution $p = F(x)$, $dp = F'(x)dx = f(x)dx$ to obtain
$$\int_0^1F^{-1}(p)dp = | Does a univariate random variable's mean always equal the integral of its quantile function?
Let $F$ be the CDF of the random variable $X$, so the inverse CDF can be written $F^{-1}$. In your integral make the substitution $p = F(x)$, $dp = F'(x)dx = f(x)dx$ to obtain
$$\int_0^1F^{-1}(p)dp = \int_{-\infty}^{\infty}x f... | Does a univariate random variable's mean always equal the integral of its quantile function?
Let $F$ be the CDF of the random variable $X$, so the inverse CDF can be written $F^{-1}$. In your integral make the substitution $p = F(x)$, $dp = F'(x)dx = f(x)dx$ to obtain
$$\int_0^1F^{-1}(p)dp = |
12,395 | Does a univariate random variable's mean always equal the integral of its quantile function? | An equivalent result is well known in survival analysis: the expected lifetime is $$\int_{t=0}^\infty S(t) \; dt$$ where the survival function is $S(t) = \Pr(T \gt t)$ measured from birth at $t=0$. (It can easily be extended to cover negative values of $t$.)
So we can rewrite this as $$\int_{t=0}^\infty (1-F(t)) \; d... | Does a univariate random variable's mean always equal the integral of its quantile function? | An equivalent result is well known in survival analysis: the expected lifetime is $$\int_{t=0}^\infty S(t) \; dt$$ where the survival function is $S(t) = \Pr(T \gt t)$ measured from birth at $t=0$. ( | Does a univariate random variable's mean always equal the integral of its quantile function?
An equivalent result is well known in survival analysis: the expected lifetime is $$\int_{t=0}^\infty S(t) \; dt$$ where the survival function is $S(t) = \Pr(T \gt t)$ measured from birth at $t=0$. (It can easily be extended t... | Does a univariate random variable's mean always equal the integral of its quantile function?
An equivalent result is well known in survival analysis: the expected lifetime is $$\int_{t=0}^\infty S(t) \; dt$$ where the survival function is $S(t) = \Pr(T \gt t)$ measured from birth at $t=0$. ( |
12,396 | Does a univariate random variable's mean always equal the integral of its quantile function? | For any real-valued random variable $X$ with cdf $F$ it is well-known that $F^{-1}(U)$ has the same law than $X$ when $U$ is uniform on $(0,1)$. Therefore the expectation of $X$, whenever it exists, is the same as the expectation of $F^{-1}(U)$: $$E(X)=E(F^{-1}(U))=\int_0^1 F^{-1}(u)\mathrm{d}u.$$
The representation $... | Does a univariate random variable's mean always equal the integral of its quantile function? | For any real-valued random variable $X$ with cdf $F$ it is well-known that $F^{-1}(U)$ has the same law than $X$ when $U$ is uniform on $(0,1)$. Therefore the expectation of $X$, whenever it exists, i | Does a univariate random variable's mean always equal the integral of its quantile function?
For any real-valued random variable $X$ with cdf $F$ it is well-known that $F^{-1}(U)$ has the same law than $X$ when $U$ is uniform on $(0,1)$. Therefore the expectation of $X$, whenever it exists, is the same as the expectati... | Does a univariate random variable's mean always equal the integral of its quantile function?
For any real-valued random variable $X$ with cdf $F$ it is well-known that $F^{-1}(U)$ has the same law than $X$ when $U$ is uniform on $(0,1)$. Therefore the expectation of $X$, whenever it exists, i |
12,397 | Does a univariate random variable's mean always equal the integral of its quantile function? | We are evaluating:
Let's try with a simple change of variable:
And we notice that, by definition of PDF and CDF:
almost everywhere. Thus we have, by definition of expected value: | Does a univariate random variable's mean always equal the integral of its quantile function? | We are evaluating:
Let's try with a simple change of variable:
And we notice that, by definition of PDF and CDF:
almost everywhere. Thus we have, by definition of expected value: | Does a univariate random variable's mean always equal the integral of its quantile function?
We are evaluating:
Let's try with a simple change of variable:
And we notice that, by definition of PDF and CDF:
almost everywhere. Thus we have, by definition of expected value: | Does a univariate random variable's mean always equal the integral of its quantile function?
We are evaluating:
Let's try with a simple change of variable:
And we notice that, by definition of PDF and CDF:
almost everywhere. Thus we have, by definition of expected value: |
12,398 | Does a univariate random variable's mean always equal the integral of its quantile function? | Note that $F(x)$ is defined as $P(X\le x)$ and is a right-continuous function. $F^{-1}$ is defined as
\begin{equation}
F^{-1}(p)=\min(x|F(x)\ge p).
\end{equation}
The $\min$ makes sense because of the right continuity. Let $U$ be a uniform distribution on $[0, 1]$. You can easily verify that $F^{-1}(U)$ has the same CD... | Does a univariate random variable's mean always equal the integral of its quantile function? | Note that $F(x)$ is defined as $P(X\le x)$ and is a right-continuous function. $F^{-1}$ is defined as
\begin{equation}
F^{-1}(p)=\min(x|F(x)\ge p).
\end{equation}
The $\min$ makes sense because of the | Does a univariate random variable's mean always equal the integral of its quantile function?
Note that $F(x)$ is defined as $P(X\le x)$ and is a right-continuous function. $F^{-1}$ is defined as
\begin{equation}
F^{-1}(p)=\min(x|F(x)\ge p).
\end{equation}
The $\min$ makes sense because of the right continuity. Let $U$ ... | Does a univariate random variable's mean always equal the integral of its quantile function?
Note that $F(x)$ is defined as $P(X\le x)$ and is a right-continuous function. $F^{-1}$ is defined as
\begin{equation}
F^{-1}(p)=\min(x|F(x)\ge p).
\end{equation}
The $\min$ makes sense because of the |
12,399 | The proof of equivalent formulas of ridge regression | The classic Ridge Regression (Tikhonov Regularization) is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{2}^{2} $$
The claim above is that the following problem is equivalent:
$$\begin{align*}
\arg \min_{x} \quad & \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} \\
... | The proof of equivalent formulas of ridge regression | The classic Ridge Regression (Tikhonov Regularization) is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{2}^{2} $$
The claim above is that the | The proof of equivalent formulas of ridge regression
The classic Ridge Regression (Tikhonov Regularization) is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{2}^{2} $$
The claim above is that the following problem is equivalent:
$$\begin{align*}
\arg \min_{x} \qu... | The proof of equivalent formulas of ridge regression
The classic Ridge Regression (Tikhonov Regularization) is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{2}^{2} $$
The claim above is that the |
12,400 | The proof of equivalent formulas of ridge regression | A less mathematically rigorous, but possibly more intuitive, approach to understanding what is going on is to start with the constraint version (equation 3.42 in the question) and solve it using the methods of "Lagrange Multiplier" (https://en.wikipedia.org/wiki/Lagrange_multiplier or your favorite multivariable calcul... | The proof of equivalent formulas of ridge regression | A less mathematically rigorous, but possibly more intuitive, approach to understanding what is going on is to start with the constraint version (equation 3.42 in the question) and solve it using the m | The proof of equivalent formulas of ridge regression
A less mathematically rigorous, but possibly more intuitive, approach to understanding what is going on is to start with the constraint version (equation 3.42 in the question) and solve it using the methods of "Lagrange Multiplier" (https://en.wikipedia.org/wiki/Lagr... | The proof of equivalent formulas of ridge regression
A less mathematically rigorous, but possibly more intuitive, approach to understanding what is going on is to start with the constraint version (equation 3.42 in the question) and solve it using the m |
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