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 |
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5,401 | AIC guidelines in model selection | I generally never use AIC or BIC objectively to describe adequate fit for a model. I do use these ICs to compare the relative fit of two predictive models. As far as whether an AIC of "2" or "4" is concerned, it's completely contextual. If you want to get a sense of how a "good" model fits, you can (should) always use ... | AIC guidelines in model selection | I generally never use AIC or BIC objectively to describe adequate fit for a model. I do use these ICs to compare the relative fit of two predictive models. As far as whether an AIC of "2" or "4" is co | AIC guidelines in model selection
I generally never use AIC or BIC objectively to describe adequate fit for a model. I do use these ICs to compare the relative fit of two predictive models. As far as whether an AIC of "2" or "4" is concerned, it's completely contextual. If you want to get a sense of how a "good" model ... | AIC guidelines in model selection
I generally never use AIC or BIC objectively to describe adequate fit for a model. I do use these ICs to compare the relative fit of two predictive models. As far as whether an AIC of "2" or "4" is co |
5,402 | AIC guidelines in model selection | Here is a related question when-is-it-appropriate-to-select-models-by-minimising-the-aic?. It gives you a general idea of what people not unrecognizable in academic world consider appropriate to write and what references to leave in as important.
Generally, it is the differences between the likelihoods or AICs that mat... | AIC guidelines in model selection | Here is a related question when-is-it-appropriate-to-select-models-by-minimising-the-aic?. It gives you a general idea of what people not unrecognizable in academic world consider appropriate to write | AIC guidelines in model selection
Here is a related question when-is-it-appropriate-to-select-models-by-minimising-the-aic?. It gives you a general idea of what people not unrecognizable in academic world consider appropriate to write and what references to leave in as important.
Generally, it is the differences betwee... | AIC guidelines in model selection
Here is a related question when-is-it-appropriate-to-select-models-by-minimising-the-aic?. It gives you a general idea of what people not unrecognizable in academic world consider appropriate to write |
5,403 | AIC guidelines in model selection | With regard to information criteria, here is what SAS says:
"Note that information criteria such as Akaike's (AIC), Schwarz's (SC, BIC), and QIC can be used to compare competing nonnested models, but do not provide a test of the comparison. Consequently, they cannot indicate whether one model is significantly better... | AIC guidelines in model selection | With regard to information criteria, here is what SAS says:
"Note that information criteria such as Akaike's (AIC), Schwarz's (SC, BIC), and QIC can be used to compare competing nonnested models, b | AIC guidelines in model selection
With regard to information criteria, here is what SAS says:
"Note that information criteria such as Akaike's (AIC), Schwarz's (SC, BIC), and QIC can be used to compare competing nonnested models, but do not provide a test of the comparison. Consequently, they cannot indicate whether... | AIC guidelines in model selection
With regard to information criteria, here is what SAS says:
"Note that information criteria such as Akaike's (AIC), Schwarz's (SC, BIC), and QIC can be used to compare competing nonnested models, b |
5,404 | Simple examples of uncorrelated but not independent $X$ and $Y$ | Let $X\sim U(-1,1)$.
Let $Y=X^2$.
The variables are uncorrelated but dependent.
Alternatively, consider a discrete bivariate distribution consisting of probability at 3 points (-1,1),(0,-1),(1,1) with probability 1/4, 1/2, 1/4 respectively. Then variables are uncorrelated but dependent.
Consider bivariate data uniform ... | Simple examples of uncorrelated but not independent $X$ and $Y$ | Let $X\sim U(-1,1)$.
Let $Y=X^2$.
The variables are uncorrelated but dependent.
Alternatively, consider a discrete bivariate distribution consisting of probability at 3 points (-1,1),(0,-1),(1,1) with | Simple examples of uncorrelated but not independent $X$ and $Y$
Let $X\sim U(-1,1)$.
Let $Y=X^2$.
The variables are uncorrelated but dependent.
Alternatively, consider a discrete bivariate distribution consisting of probability at 3 points (-1,1),(0,-1),(1,1) with probability 1/4, 1/2, 1/4 respectively. Then variables ... | Simple examples of uncorrelated but not independent $X$ and $Y$
Let $X\sim U(-1,1)$.
Let $Y=X^2$.
The variables are uncorrelated but dependent.
Alternatively, consider a discrete bivariate distribution consisting of probability at 3 points (-1,1),(0,-1),(1,1) with |
5,405 | Simple examples of uncorrelated but not independent $X$ and $Y$ | I think the essence of some of the simple counterexamples can be seen by starting with a continuous random variable $X$ centred on zero, i.e. $E[X]=0$. Suppose the pdf of $X$ is even and defined on an interval of the form $(-a,a)$, where $a>0$.
Now suppose $Y=f(X)$ for some function $f$. We now ask the question: for wh... | Simple examples of uncorrelated but not independent $X$ and $Y$ | I think the essence of some of the simple counterexamples can be seen by starting with a continuous random variable $X$ centred on zero, i.e. $E[X]=0$. Suppose the pdf of $X$ is even and defined on an | Simple examples of uncorrelated but not independent $X$ and $Y$
I think the essence of some of the simple counterexamples can be seen by starting with a continuous random variable $X$ centred on zero, i.e. $E[X]=0$. Suppose the pdf of $X$ is even and defined on an interval of the form $(-a,a)$, where $a>0$.
Now suppose... | Simple examples of uncorrelated but not independent $X$ and $Y$
I think the essence of some of the simple counterexamples can be seen by starting with a continuous random variable $X$ centred on zero, i.e. $E[X]=0$. Suppose the pdf of $X$ is even and defined on an |
5,406 | Simple examples of uncorrelated but not independent $X$ and $Y$ | We can define a discrete random variable $X\in\{-1,0,1\}$ with $\mathbb{P}(X=-1)=\mathbb{P}(X=0)=\mathbb{P}(X=1)=\frac{1}{3}$
and then define $Y=\begin{cases}1,\quad\text{if}\quad X=0\\0,\quad\text{otherwise}\end{cases}$
It can be easily verified that $X$ and $Y$ are uncorrelated but not independent. | Simple examples of uncorrelated but not independent $X$ and $Y$ | We can define a discrete random variable $X\in\{-1,0,1\}$ with $\mathbb{P}(X=-1)=\mathbb{P}(X=0)=\mathbb{P}(X=1)=\frac{1}{3}$
and then define $Y=\begin{cases}1,\quad\text{if}\quad X=0\\0,\quad\text{ot | Simple examples of uncorrelated but not independent $X$ and $Y$
We can define a discrete random variable $X\in\{-1,0,1\}$ with $\mathbb{P}(X=-1)=\mathbb{P}(X=0)=\mathbb{P}(X=1)=\frac{1}{3}$
and then define $Y=\begin{cases}1,\quad\text{if}\quad X=0\\0,\quad\text{otherwise}\end{cases}$
It can be easily verified that $X$... | Simple examples of uncorrelated but not independent $X$ and $Y$
We can define a discrete random variable $X\in\{-1,0,1\}$ with $\mathbb{P}(X=-1)=\mathbb{P}(X=0)=\mathbb{P}(X=1)=\frac{1}{3}$
and then define $Y=\begin{cases}1,\quad\text{if}\quad X=0\\0,\quad\text{ot |
5,407 | Simple examples of uncorrelated but not independent $X$ and $Y$ | Be the counterexample (i.e. hard-working student)! With that said:
I was trying to think of a real world example and this was the first that came to my mind. This will not be the mathematically simplest case (but if you understand this example, you should be able to find a simpler example with urns and balls or somethi... | Simple examples of uncorrelated but not independent $X$ and $Y$ | Be the counterexample (i.e. hard-working student)! With that said:
I was trying to think of a real world example and this was the first that came to my mind. This will not be the mathematically simple | Simple examples of uncorrelated but not independent $X$ and $Y$
Be the counterexample (i.e. hard-working student)! With that said:
I was trying to think of a real world example and this was the first that came to my mind. This will not be the mathematically simplest case (but if you understand this example, you should ... | Simple examples of uncorrelated but not independent $X$ and $Y$
Be the counterexample (i.e. hard-working student)! With that said:
I was trying to think of a real world example and this was the first that came to my mind. This will not be the mathematically simple |
5,408 | Simple examples of uncorrelated but not independent $X$ and $Y$ | I ran across the example of a short straddle in this "Mini-lesson" by Nassim Taleb.
The payoff has the shape of an inverted V with the peak when the price of the underlying security at expiration is the strike price at which both the call and the put are sold. The idea is that if at the last closing Microsoft shares we... | Simple examples of uncorrelated but not independent $X$ and $Y$ | I ran across the example of a short straddle in this "Mini-lesson" by Nassim Taleb.
The payoff has the shape of an inverted V with the peak when the price of the underlying security at expiration is t | Simple examples of uncorrelated but not independent $X$ and $Y$
I ran across the example of a short straddle in this "Mini-lesson" by Nassim Taleb.
The payoff has the shape of an inverted V with the peak when the price of the underlying security at expiration is the strike price at which both the call and the put are s... | Simple examples of uncorrelated but not independent $X$ and $Y$
I ran across the example of a short straddle in this "Mini-lesson" by Nassim Taleb.
The payoff has the shape of an inverted V with the peak when the price of the underlying security at expiration is t |
5,409 | Simple examples of uncorrelated but not independent $X$ and $Y$ | Try this (R code):
x=c(1,0,-1,0);
y=c(0,1,0,-1);
cor(x,y);
[1] 0
This is from the equation of circle $x^2+y^2-r^2=0$
$Y$ is not correlated with $x$, but it is functionally dependent (deterministic). | Simple examples of uncorrelated but not independent $X$ and $Y$ | Try this (R code):
x=c(1,0,-1,0);
y=c(0,1,0,-1);
cor(x,y);
[1] 0
This is from the equation of circle $x^2+y^2-r^2=0$
$Y$ is not correlated with $x$, but it is functionally dependent (determi | Simple examples of uncorrelated but not independent $X$ and $Y$
Try this (R code):
x=c(1,0,-1,0);
y=c(0,1,0,-1);
cor(x,y);
[1] 0
This is from the equation of circle $x^2+y^2-r^2=0$
$Y$ is not correlated with $x$, but it is functionally dependent (deterministic). | Simple examples of uncorrelated but not independent $X$ and $Y$
Try this (R code):
x=c(1,0,-1,0);
y=c(0,1,0,-1);
cor(x,y);
[1] 0
This is from the equation of circle $x^2+y^2-r^2=0$
$Y$ is not correlated with $x$, but it is functionally dependent (determi |
5,410 | Simple examples of uncorrelated but not independent $X$ and $Y$ | The only general case when lack of correlation implies independence is when the
joint distribution of X and Y is Gaussian. | Simple examples of uncorrelated but not independent $X$ and $Y$ | The only general case when lack of correlation implies independence is when the
joint distribution of X and Y is Gaussian. | Simple examples of uncorrelated but not independent $X$ and $Y$
The only general case when lack of correlation implies independence is when the
joint distribution of X and Y is Gaussian. | Simple examples of uncorrelated but not independent $X$ and $Y$
The only general case when lack of correlation implies independence is when the
joint distribution of X and Y is Gaussian. |
5,411 | Simple examples of uncorrelated but not independent $X$ and $Y$ | A two-sentence answer: the clearest case of uncorrelated statistical dependence is a non-linear function of a RV, say Y = X^n. The two RVs are clearly dependent but yet not correlated, because correlation is a linear relationship. | Simple examples of uncorrelated but not independent $X$ and $Y$ | A two-sentence answer: the clearest case of uncorrelated statistical dependence is a non-linear function of a RV, say Y = X^n. The two RVs are clearly dependent but yet not correlated, because correla | Simple examples of uncorrelated but not independent $X$ and $Y$
A two-sentence answer: the clearest case of uncorrelated statistical dependence is a non-linear function of a RV, say Y = X^n. The two RVs are clearly dependent but yet not correlated, because correlation is a linear relationship. | Simple examples of uncorrelated but not independent $X$ and $Y$
A two-sentence answer: the clearest case of uncorrelated statistical dependence is a non-linear function of a RV, say Y = X^n. The two RVs are clearly dependent but yet not correlated, because correla |
5,412 | Dice-coefficient loss function vs cross-entropy | One compelling reason for using cross-entropy over dice-coefficient or the similar IoU metric is that the gradients are nicer.
The gradients of cross-entropy wrt the logits is something like $p - t$, where $p$ is the softmax outputs and $t$ is the target. Meanwhile, if we try to write the dice coefficient in a differen... | Dice-coefficient loss function vs cross-entropy | One compelling reason for using cross-entropy over dice-coefficient or the similar IoU metric is that the gradients are nicer.
The gradients of cross-entropy wrt the logits is something like $p - t$, | Dice-coefficient loss function vs cross-entropy
One compelling reason for using cross-entropy over dice-coefficient or the similar IoU metric is that the gradients are nicer.
The gradients of cross-entropy wrt the logits is something like $p - t$, where $p$ is the softmax outputs and $t$ is the target. Meanwhile, if we... | Dice-coefficient loss function vs cross-entropy
One compelling reason for using cross-entropy over dice-coefficient or the similar IoU metric is that the gradients are nicer.
The gradients of cross-entropy wrt the logits is something like $p - t$, |
5,413 | Dice-coefficient loss function vs cross-entropy | As summarized by @shimao and @cherub, one cannot say apriori which one will work better on a particular dataset. The correct way is to try both and compare the results. Also, note that when it comes to segmentation, it is not so easy to "compare the results": IoU based measures like dice coefficient cover only some asp... | Dice-coefficient loss function vs cross-entropy | As summarized by @shimao and @cherub, one cannot say apriori which one will work better on a particular dataset. The correct way is to try both and compare the results. Also, note that when it comes t | Dice-coefficient loss function vs cross-entropy
As summarized by @shimao and @cherub, one cannot say apriori which one will work better on a particular dataset. The correct way is to try both and compare the results. Also, note that when it comes to segmentation, it is not so easy to "compare the results": IoU based me... | Dice-coefficient loss function vs cross-entropy
As summarized by @shimao and @cherub, one cannot say apriori which one will work better on a particular dataset. The correct way is to try both and compare the results. Also, note that when it comes t |
5,414 | Dice-coefficient loss function vs cross-entropy | I would recommend you to use Dice loss when faced with class imbalanced datasets, which is common in the medicine domain, for example. Also, Dice loss was introduced in the paper "V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation" and in that work the authors state that Dice loss work... | Dice-coefficient loss function vs cross-entropy | I would recommend you to use Dice loss when faced with class imbalanced datasets, which is common in the medicine domain, for example. Also, Dice loss was introduced in the paper "V-Net: Fully Convolu | Dice-coefficient loss function vs cross-entropy
I would recommend you to use Dice loss when faced with class imbalanced datasets, which is common in the medicine domain, for example. Also, Dice loss was introduced in the paper "V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation" and in... | Dice-coefficient loss function vs cross-entropy
I would recommend you to use Dice loss when faced with class imbalanced datasets, which is common in the medicine domain, for example. Also, Dice loss was introduced in the paper "V-Net: Fully Convolu |
5,415 | How to calculate a confidence level for a Poisson distribution? | For Poisson, the mean and the variance are both $\lambda$. If you want the confidence interval around lambda, you can calculate the standard error as $\sqrt{\lambda / n}$.
The 95-percent confidence interval is $\hat{\lambda} \pm 1.96\sqrt{\hat{\lambda} / n}$. | How to calculate a confidence level for a Poisson distribution? | For Poisson, the mean and the variance are both $\lambda$. If you want the confidence interval around lambda, you can calculate the standard error as $\sqrt{\lambda / n}$.
The 95-percent confidence i | How to calculate a confidence level for a Poisson distribution?
For Poisson, the mean and the variance are both $\lambda$. If you want the confidence interval around lambda, you can calculate the standard error as $\sqrt{\lambda / n}$.
The 95-percent confidence interval is $\hat{\lambda} \pm 1.96\sqrt{\hat{\lambda} / ... | How to calculate a confidence level for a Poisson distribution?
For Poisson, the mean and the variance are both $\lambda$. If you want the confidence interval around lambda, you can calculate the standard error as $\sqrt{\lambda / n}$.
The 95-percent confidence i |
5,416 | How to calculate a confidence level for a Poisson distribution? | 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.
Patil & Kulkarni (2012, "Comparison of Confidence Inte... | How to calculate a confidence level for a Poisson distribution? | 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 calculate a confidence level for a Poisson distribution?
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 calculate a confidence level for a Poisson distribution?
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.
|
5,417 | How to calculate a confidence level for a Poisson distribution? | In addition to the answers that others have provided, another approach to this problem is achieved through a model based approach. The central limit theorem approach is certainly valid, and the bootstrapped estimates offer a lot of protection from small sample and mode misspecification issues.
For sheer efficiency, yo... | How to calculate a confidence level for a Poisson distribution? | In addition to the answers that others have provided, another approach to this problem is achieved through a model based approach. The central limit theorem approach is certainly valid, and the bootst | How to calculate a confidence level for a Poisson distribution?
In addition to the answers that others have provided, another approach to this problem is achieved through a model based approach. The central limit theorem approach is certainly valid, and the bootstrapped estimates offer a lot of protection from small sa... | How to calculate a confidence level for a Poisson distribution?
In addition to the answers that others have provided, another approach to this problem is achieved through a model based approach. The central limit theorem approach is certainly valid, and the bootst |
5,418 | How to calculate a confidence level for a Poisson distribution? | Given an observation from a Poisson distribution,
the number of events counted is n.
the mean ($\lambda$) and variance ($\sigma^2$) are equal.
Step by step,
The estimate for the mean is $\hat \lambda = n \approx \lambda$
Assuming the number of events is big enough ($n \gt 20$), the standard error is the standard dev... | How to calculate a confidence level for a Poisson distribution? | Given an observation from a Poisson distribution,
the number of events counted is n.
the mean ($\lambda$) and variance ($\sigma^2$) are equal.
Step by step,
The estimate for the mean is $\hat \lamb | How to calculate a confidence level for a Poisson distribution?
Given an observation from a Poisson distribution,
the number of events counted is n.
the mean ($\lambda$) and variance ($\sigma^2$) are equal.
Step by step,
The estimate for the mean is $\hat \lambda = n \approx \lambda$
Assuming the number of events is... | How to calculate a confidence level for a Poisson distribution?
Given an observation from a Poisson distribution,
the number of events counted is n.
the mean ($\lambda$) and variance ($\sigma^2$) are equal.
Step by step,
The estimate for the mean is $\hat \lamb |
5,419 | Intuitive explanation for density of transformed variable? | PDFs are heights but they are used to represent probability by means of area. It therefore helps to express a PDF in a way that reminds us that area equals height times base.
Initially the height at any value $x$ is given by the PDF $f_X(x)$. The base is the infinitesimal segment $dx$, whence the distribution (that i... | Intuitive explanation for density of transformed variable? | PDFs are heights but they are used to represent probability by means of area. It therefore helps to express a PDF in a way that reminds us that area equals height times base.
Initially the height at | Intuitive explanation for density of transformed variable?
PDFs are heights but they are used to represent probability by means of area. It therefore helps to express a PDF in a way that reminds us that area equals height times base.
Initially the height at any value $x$ is given by the PDF $f_X(x)$. The base is the ... | Intuitive explanation for density of transformed variable?
PDFs are heights but they are used to represent probability by means of area. It therefore helps to express a PDF in a way that reminds us that area equals height times base.
Initially the height at |
5,420 | Intuitive explanation for density of transformed variable? | How about, if I manufacture objects that are always square and I know the distribution
of the side lengths of the squares; what can I say about the distribution of the areas
of the squares?
In particular, if I know the distribution of a random variable $X$, what can I say about $Y = X^{2}$? One thing that you can say ... | Intuitive explanation for density of transformed variable? | How about, if I manufacture objects that are always square and I know the distribution
of the side lengths of the squares; what can I say about the distribution of the areas
of the squares?
In particu | Intuitive explanation for density of transformed variable?
How about, if I manufacture objects that are always square and I know the distribution
of the side lengths of the squares; what can I say about the distribution of the areas
of the squares?
In particular, if I know the distribution of a random variable $X$, wha... | Intuitive explanation for density of transformed variable?
How about, if I manufacture objects that are always square and I know the distribution
of the side lengths of the squares; what can I say about the distribution of the areas
of the squares?
In particu |
5,421 | Intuitive explanation for density of transformed variable? | Imagine we have a population and $Y$ is a summary of that population. Then $P(Y \in (y, y + \Delta y))$ is counting the proportion of individuals that have variable $Y$ in the range $(y, y + \Delta y)$. You can consider this as a "bin" of size $\Delta y$ and we are counting how many individuals are inside that bin.
Now... | Intuitive explanation for density of transformed variable? | Imagine we have a population and $Y$ is a summary of that population. Then $P(Y \in (y, y + \Delta y))$ is counting the proportion of individuals that have variable $Y$ in the range $(y, y + \Delta y) | Intuitive explanation for density of transformed variable?
Imagine we have a population and $Y$ is a summary of that population. Then $P(Y \in (y, y + \Delta y))$ is counting the proportion of individuals that have variable $Y$ in the range $(y, y + \Delta y)$. You can consider this as a "bin" of size $\Delta y$ and we... | Intuitive explanation for density of transformed variable?
Imagine we have a population and $Y$ is a summary of that population. Then $P(Y \in (y, y + \Delta y))$ is counting the proportion of individuals that have variable $Y$ in the range $(y, y + \Delta y) |
5,422 | SVM, Overfitting, curse of dimensionality | In practice, the reason that SVMs tend to be resistant to over-fitting, even in cases where the number of attributes is greater than the number of observations, is that it uses regularization. They key to avoiding over-fitting lies in careful tuning of the regularization parameter, $C$, and in the case of non-linear S... | SVM, Overfitting, curse of dimensionality | In practice, the reason that SVMs tend to be resistant to over-fitting, even in cases where the number of attributes is greater than the number of observations, is that it uses regularization. They k | SVM, Overfitting, curse of dimensionality
In practice, the reason that SVMs tend to be resistant to over-fitting, even in cases where the number of attributes is greater than the number of observations, is that it uses regularization. They key to avoiding over-fitting lies in careful tuning of the regularization param... | SVM, Overfitting, curse of dimensionality
In practice, the reason that SVMs tend to be resistant to over-fitting, even in cases where the number of attributes is greater than the number of observations, is that it uses regularization. They k |
5,423 | SVM, Overfitting, curse of dimensionality | I will start with the second and last questions.
The problem of generalization is obviously important, because if the results of machine learning cannot be generalized, then they are completely useless.
The methods of ensuring generalization come from statistics. We usually assume, that data is generated from some prob... | SVM, Overfitting, curse of dimensionality | I will start with the second and last questions.
The problem of generalization is obviously important, because if the results of machine learning cannot be generalized, then they are completely useles | SVM, Overfitting, curse of dimensionality
I will start with the second and last questions.
The problem of generalization is obviously important, because if the results of machine learning cannot be generalized, then they are completely useless.
The methods of ensuring generalization come from statistics. We usually ass... | SVM, Overfitting, curse of dimensionality
I will start with the second and last questions.
The problem of generalization is obviously important, because if the results of machine learning cannot be generalized, then they are completely useles |
5,424 | SVM, Overfitting, curse of dimensionality | There are at least two major sources of overfitting you might wish to consider.
Overfitting from an algorithm which has inferred too much from the available training samples. This is best guarded against empirically by using a measure of the generalisation ability of the model. Cross validation is one such popular met... | SVM, Overfitting, curse of dimensionality | There are at least two major sources of overfitting you might wish to consider.
Overfitting from an algorithm which has inferred too much from the available training samples. This is best guarded aga | SVM, Overfitting, curse of dimensionality
There are at least two major sources of overfitting you might wish to consider.
Overfitting from an algorithm which has inferred too much from the available training samples. This is best guarded against empirically by using a measure of the generalisation ability of the model... | SVM, Overfitting, curse of dimensionality
There are at least two major sources of overfitting you might wish to consider.
Overfitting from an algorithm which has inferred too much from the available training samples. This is best guarded aga |
5,425 | When should I balance classes in a training data set? | The intuitive reasoning has been explained in the blogpost:
If our goal is Prediction, this will cause a definite bias. And worse,
it will be a permanent bias, in the sense that we will not have
consistent estimates as the sample size grows.
So, arguably the problem of (artificially) balanced data is worse than
... | When should I balance classes in a training data set? | The intuitive reasoning has been explained in the blogpost:
If our goal is Prediction, this will cause a definite bias. And worse,
it will be a permanent bias, in the sense that we will not have
| When should I balance classes in a training data set?
The intuitive reasoning has been explained in the blogpost:
If our goal is Prediction, this will cause a definite bias. And worse,
it will be a permanent bias, in the sense that we will not have
consistent estimates as the sample size grows.
So, arguably the pr... | When should I balance classes in a training data set?
The intuitive reasoning has been explained in the blogpost:
If our goal is Prediction, this will cause a definite bias. And worse,
it will be a permanent bias, in the sense that we will not have
|
5,426 | When should I balance classes in a training data set? | Consistent with @kjetil-b-halvorsen's comment, the rapid adoption of machine learning has confused researchers about prediction vs. classification. As I described in more detail here, classification is only appropriate in a minority of cases. When the outcome is rare (or too common), probabilities are everything bec... | When should I balance classes in a training data set? | Consistent with @kjetil-b-halvorsen's comment, the rapid adoption of machine learning has confused researchers about prediction vs. classification. As I described in more detail here, classification | When should I balance classes in a training data set?
Consistent with @kjetil-b-halvorsen's comment, the rapid adoption of machine learning has confused researchers about prediction vs. classification. As I described in more detail here, classification is only appropriate in a minority of cases. When the outcome is ... | When should I balance classes in a training data set?
Consistent with @kjetil-b-halvorsen's comment, the rapid adoption of machine learning has confused researchers about prediction vs. classification. As I described in more detail here, classification |
5,427 | When should I balance classes in a training data set? | Depends on what you want to achieve from the classification?
Say it is cancer v/s non cancer, then detecting cancer is vital. However since non-cancer will form majority of your data the classifier can essentially send all cases to non-cancer class and get very high accuracy. But we can't afford that, so we essentially... | When should I balance classes in a training data set? | Depends on what you want to achieve from the classification?
Say it is cancer v/s non cancer, then detecting cancer is vital. However since non-cancer will form majority of your data the classifier ca | When should I balance classes in a training data set?
Depends on what you want to achieve from the classification?
Say it is cancer v/s non cancer, then detecting cancer is vital. However since non-cancer will form majority of your data the classifier can essentially send all cases to non-cancer class and get very high... | When should I balance classes in a training data set?
Depends on what you want to achieve from the classification?
Say it is cancer v/s non cancer, then detecting cancer is vital. However since non-cancer will form majority of your data the classifier ca |
5,428 | When should I balance classes in a training data set? | When your data is balanced you can prefer to check the metric accuracy. But when such a situation your data is unbalanced your accuracy is not consistent for different iterations. You need to concentrate more metrics like Precision(PPR), Recall(sensitivity). This two metrics should be balanced when compare. Also you sh... | When should I balance classes in a training data set? | When your data is balanced you can prefer to check the metric accuracy. But when such a situation your data is unbalanced your accuracy is not consistent for different iterations. You need to concentr | When should I balance classes in a training data set?
When your data is balanced you can prefer to check the metric accuracy. But when such a situation your data is unbalanced your accuracy is not consistent for different iterations. You need to concentrate more metrics like Precision(PPR), Recall(sensitivity). This tw... | When should I balance classes in a training data set?
When your data is balanced you can prefer to check the metric accuracy. But when such a situation your data is unbalanced your accuracy is not consistent for different iterations. You need to concentr |
5,429 | Is a time series the same as a stochastic process? | Because many troubling discrepancies are showing up in comments and answers, let's refer to some authorities.
James Hamilton does not even define a time series, but he is clear about what one is:
... this set of $T$ numbers is only one possible outcome of the underlying stochastic process that generated the data. Ind... | Is a time series the same as a stochastic process? | Because many troubling discrepancies are showing up in comments and answers, let's refer to some authorities.
James Hamilton does not even define a time series, but he is clear about what one is:
... | Is a time series the same as a stochastic process?
Because many troubling discrepancies are showing up in comments and answers, let's refer to some authorities.
James Hamilton does not even define a time series, but he is clear about what one is:
... this set of $T$ numbers is only one possible outcome of the underlyi... | Is a time series the same as a stochastic process?
Because many troubling discrepancies are showing up in comments and answers, let's refer to some authorities.
James Hamilton does not even define a time series, but he is clear about what one is:
... |
5,430 | Is a time series the same as a stochastic process? | Defining a stochastic process
Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $S$ be another measurable space (such as the space of real numbers $\mathbb{R}$). Speaking somewhat imprecisely:
A random variable is a measurable function from $\Omega$ to $S$.
A stochastic process is a family of random variables... | Is a time series the same as a stochastic process? | Defining a stochastic process
Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $S$ be another measurable space (such as the space of real numbers $\mathbb{R}$). Speaking somewhat imprecisely | Is a time series the same as a stochastic process?
Defining a stochastic process
Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $S$ be another measurable space (such as the space of real numbers $\mathbb{R}$). Speaking somewhat imprecisely:
A random variable is a measurable function from $\Omega$ to $S$.
A... | Is a time series the same as a stochastic process?
Defining a stochastic process
Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $S$ be another measurable space (such as the space of real numbers $\mathbb{R}$). Speaking somewhat imprecisely |
5,431 | Is a time series the same as a stochastic process? | A stochastic process is a set or collection of random variables $\left\{X_t\right\}$ (not necessarily independent), where the index t takes values in a certain set, this set is ordered and corresponds to the moment of time. example random walk.
Time series Is the realisation of the stochastic process. | Is a time series the same as a stochastic process? | A stochastic process is a set or collection of random variables $\left\{X_t\right\}$ (not necessarily independent), where the index t takes values in a certain set, this set is ordered and corresponds | Is a time series the same as a stochastic process?
A stochastic process is a set or collection of random variables $\left\{X_t\right\}$ (not necessarily independent), where the index t takes values in a certain set, this set is ordered and corresponds to the moment of time. example random walk.
Time series Is the reali... | Is a time series the same as a stochastic process?
A stochastic process is a set or collection of random variables $\left\{X_t\right\}$ (not necessarily independent), where the index t takes values in a certain set, this set is ordered and corresponds |
5,432 | Is a time series the same as a stochastic process? | The difference between a stochastic process and a time series is somewhat like the difference between a cat on a keyboard and an answer on Stack Exchange: Cats on keyboards can produce answers, but cats on keyboards are not answers. Furthermore, not every answer is produced by a cat on a keyboard.
A time series can be ... | Is a time series the same as a stochastic process? | The difference between a stochastic process and a time series is somewhat like the difference between a cat on a keyboard and an answer on Stack Exchange: Cats on keyboards can produce answers, but ca | Is a time series the same as a stochastic process?
The difference between a stochastic process and a time series is somewhat like the difference between a cat on a keyboard and an answer on Stack Exchange: Cats on keyboards can produce answers, but cats on keyboards are not answers. Furthermore, not every answer is pro... | Is a time series the same as a stochastic process?
The difference between a stochastic process and a time series is somewhat like the difference between a cat on a keyboard and an answer on Stack Exchange: Cats on keyboards can produce answers, but ca |
5,433 | Is a time series the same as a stochastic process? | I appreciate all contributed discussions/comments on the subject of Time series vs Stochastic process. Here is my understanding of the difference: Time series is a phenomenon observed, recorded as a series of numbers that is indexed with the time at observation; it is most likely a series of observations of a real life... | Is a time series the same as a stochastic process? | I appreciate all contributed discussions/comments on the subject of Time series vs Stochastic process. Here is my understanding of the difference: Time series is a phenomenon observed, recorded as a s | Is a time series the same as a stochastic process?
I appreciate all contributed discussions/comments on the subject of Time series vs Stochastic process. Here is my understanding of the difference: Time series is a phenomenon observed, recorded as a series of numbers that is indexed with the time at observation; it is ... | Is a time series the same as a stochastic process?
I appreciate all contributed discussions/comments on the subject of Time series vs Stochastic process. Here is my understanding of the difference: Time series is a phenomenon observed, recorded as a s |
5,434 | Is a time series the same as a stochastic process? | A random variable is a random variable
A vector of random variables is a random vector
A set of random variables is a random field
Like random vector, we need to give the random variables an index to identify that variable. Using different indexing schemas result in different real life applications, for example:
If we... | Is a time series the same as a stochastic process? | A random variable is a random variable
A vector of random variables is a random vector
A set of random variables is a random field
Like random vector, we need to give the random variables an index to | Is a time series the same as a stochastic process?
A random variable is a random variable
A vector of random variables is a random vector
A set of random variables is a random field
Like random vector, we need to give the random variables an index to identify that variable. Using different indexing schemas result in di... | Is a time series the same as a stochastic process?
A random variable is a random variable
A vector of random variables is a random vector
A set of random variables is a random field
Like random vector, we need to give the random variables an index to |
5,435 | Combining probabilities/information from different sources | You ask about three things: (a) how to combine several forecasts to get single forecast, (b) if Bayesian approach can be used in here, and (c) how to deal with zero-probabilities.
Combining forecasts, is a common practice. If you have several forecasts than if you take average of those forecasts the resulting combined ... | Combining probabilities/information from different sources | You ask about three things: (a) how to combine several forecasts to get single forecast, (b) if Bayesian approach can be used in here, and (c) how to deal with zero-probabilities.
Combining forecasts, | Combining probabilities/information from different sources
You ask about three things: (a) how to combine several forecasts to get single forecast, (b) if Bayesian approach can be used in here, and (c) how to deal with zero-probabilities.
Combining forecasts, is a common practice. If you have several forecasts than if ... | Combining probabilities/information from different sources
You ask about three things: (a) how to combine several forecasts to get single forecast, (b) if Bayesian approach can be used in here, and (c) how to deal with zero-probabilities.
Combining forecasts, |
5,436 | Combining probabilities/information from different sources | There are two way to think of the problem.
One is to say that the sources observe a noisy version of the latent variable "it will rain / it will not rain".
For instance, we could say that each source draws its estimates from a $Beta(a+b,a)$ distribution if it will rain, and a $Beta(a,a+b)$ distribution if it will not.... | Combining probabilities/information from different sources | There are two way to think of the problem.
One is to say that the sources observe a noisy version of the latent variable "it will rain / it will not rain".
For instance, we could say that each source | Combining probabilities/information from different sources
There are two way to think of the problem.
One is to say that the sources observe a noisy version of the latent variable "it will rain / it will not rain".
For instance, we could say that each source draws its estimates from a $Beta(a+b,a)$ distribution if it ... | Combining probabilities/information from different sources
There are two way to think of the problem.
One is to say that the sources observe a noisy version of the latent variable "it will rain / it will not rain".
For instance, we could say that each source |
5,437 | Combining probabilities/information from different sources | In the framework of Transferable Belief Model (TBM), it is possible to combine different predictions using for instance the "conjunctive rule of combination". In order to apply this rule, you need to transform the probabilities of the predictions into basic belief assignments. This can be achieved with the so-called Le... | Combining probabilities/information from different sources | In the framework of Transferable Belief Model (TBM), it is possible to combine different predictions using for instance the "conjunctive rule of combination". In order to apply this rule, you need to | Combining probabilities/information from different sources
In the framework of Transferable Belief Model (TBM), it is possible to combine different predictions using for instance the "conjunctive rule of combination". In order to apply this rule, you need to transform the probabilities of the predictions into basic bel... | Combining probabilities/information from different sources
In the framework of Transferable Belief Model (TBM), it is possible to combine different predictions using for instance the "conjunctive rule of combination". In order to apply this rule, you need to |
5,438 | Combining probabilities/information from different sources | I think it's worthwhile to look at the weighting scheme based on inverse errors mentioned in one of the answers. If the sources are truly independent and we constrain the weights to sum to one, the weights are given by $$ w_1 = {{\sigma_2^2 \sigma_3^2} \over {\sigma_1^2 \sigma_2^2 + \sigma_1^2 \sigma_3^2 + \sigma_2^2 \... | Combining probabilities/information from different sources | I think it's worthwhile to look at the weighting scheme based on inverse errors mentioned in one of the answers. If the sources are truly independent and we constrain the weights to sum to one, the we | Combining probabilities/information from different sources
I think it's worthwhile to look at the weighting scheme based on inverse errors mentioned in one of the answers. If the sources are truly independent and we constrain the weights to sum to one, the weights are given by $$ w_1 = {{\sigma_2^2 \sigma_3^2} \over {\... | Combining probabilities/information from different sources
I think it's worthwhile to look at the weighting scheme based on inverse errors mentioned in one of the answers. If the sources are truly independent and we constrain the weights to sum to one, the we |
5,439 | Combining probabilities/information from different sources | There are a lot of complicated answers given to this question, but what about the Inverse Variance Weighted Mean: https://en.wikipedia.org/wiki/Inverse-variance_weighting
Instead of n repeated measurements with one instrument, if the
experimenter makes n of the same quantity with n different instruments
with varyi... | Combining probabilities/information from different sources | There are a lot of complicated answers given to this question, but what about the Inverse Variance Weighted Mean: https://en.wikipedia.org/wiki/Inverse-variance_weighting
Instead of n repeated measur | Combining probabilities/information from different sources
There are a lot of complicated answers given to this question, but what about the Inverse Variance Weighted Mean: https://en.wikipedia.org/wiki/Inverse-variance_weighting
Instead of n repeated measurements with one instrument, if the
experimenter makes n of ... | Combining probabilities/information from different sources
There are a lot of complicated answers given to this question, but what about the Inverse Variance Weighted Mean: https://en.wikipedia.org/wiki/Inverse-variance_weighting
Instead of n repeated measur |
5,440 | Combining probabilities/information from different sources | Their numbers for rain likelihood is only half the story, as we'd have to temper their predictions with the probability that they are accurate when making guesses.
Because something like rain is mutually exclusive(it's either raining or isn't, in this setup), they cannot all simultaneously be correct with 75% probabili... | Combining probabilities/information from different sources | Their numbers for rain likelihood is only half the story, as we'd have to temper their predictions with the probability that they are accurate when making guesses.
Because something like rain is mutua | Combining probabilities/information from different sources
Their numbers for rain likelihood is only half the story, as we'd have to temper their predictions with the probability that they are accurate when making guesses.
Because something like rain is mutually exclusive(it's either raining or isn't, in this setup), t... | Combining probabilities/information from different sources
Their numbers for rain likelihood is only half the story, as we'd have to temper their predictions with the probability that they are accurate when making guesses.
Because something like rain is mutua |
5,441 | Combining probabilities/information from different sources | For combining reliability, my go-to formula is r1xr2xr3÷(r1xr2xr3+(1-r1)x(1-r2)x(1-r3). So for the 3 sources of reliability 75% all saying the same thing, i would have .75^3 ÷ (.75^3 + .25^3) => 96% reliability of the combined response | Combining probabilities/information from different sources | For combining reliability, my go-to formula is r1xr2xr3÷(r1xr2xr3+(1-r1)x(1-r2)x(1-r3). So for the 3 sources of reliability 75% all saying the same thing, i would have .75^3 ÷ (.75^3 + .25^3) => 96% r | Combining probabilities/information from different sources
For combining reliability, my go-to formula is r1xr2xr3÷(r1xr2xr3+(1-r1)x(1-r2)x(1-r3). So for the 3 sources of reliability 75% all saying the same thing, i would have .75^3 ÷ (.75^3 + .25^3) => 96% reliability of the combined response | Combining probabilities/information from different sources
For combining reliability, my go-to formula is r1xr2xr3÷(r1xr2xr3+(1-r1)x(1-r2)x(1-r3). So for the 3 sources of reliability 75% all saying the same thing, i would have .75^3 ÷ (.75^3 + .25^3) => 96% r |
5,442 | Taking the expectation of Taylor series (especially the remainder) | You are right to be skeptical of this approach. The Taylor series method does not work in general, although the heuristic contains a kernel of truth. To summarize the technical discussion below,
Strong concentration implies that the Taylor series method works for nice functions
Things can and will go dramatically wron... | Taking the expectation of Taylor series (especially the remainder) | You are right to be skeptical of this approach. The Taylor series method does not work in general, although the heuristic contains a kernel of truth. To summarize the technical discussion below,
Stro | Taking the expectation of Taylor series (especially the remainder)
You are right to be skeptical of this approach. The Taylor series method does not work in general, although the heuristic contains a kernel of truth. To summarize the technical discussion below,
Strong concentration implies that the Taylor series metho... | Taking the expectation of Taylor series (especially the remainder)
You are right to be skeptical of this approach. The Taylor series method does not work in general, although the heuristic contains a kernel of truth. To summarize the technical discussion below,
Stro |
5,443 | Taking the expectation of Taylor series (especially the remainder) | Although my answer will nowhere approach the level of mathematical sophistication of the other answers, I decided to post it because I believe it has something to contribute -although the result will be "negative", as they say.
In a light tone, I would say that the OP is "risk-averse", (as most people are, as well as... | Taking the expectation of Taylor series (especially the remainder) | Although my answer will nowhere approach the level of mathematical sophistication of the other answers, I decided to post it because I believe it has something to contribute -although the result will | Taking the expectation of Taylor series (especially the remainder)
Although my answer will nowhere approach the level of mathematical sophistication of the other answers, I decided to post it because I believe it has something to contribute -although the result will be "negative", as they say.
In a light tone, I woul... | Taking the expectation of Taylor series (especially the remainder)
Although my answer will nowhere approach the level of mathematical sophistication of the other answers, I decided to post it because I believe it has something to contribute -although the result will |
5,444 | Taking the expectation of Taylor series (especially the remainder) | Not an actual answer, but an example to show that things are not so nice, and that extra hypotheses are needed to make this result true.
Define $X_n$ as a mixture between a uniform $U\left( \left[ -{1\over n} ; {1\over n} \right] \right)$ and a normal $\mathcal N({n \over n-1}, {1\over n})$, the uniform component being... | Taking the expectation of Taylor series (especially the remainder) | Not an actual answer, but an example to show that things are not so nice, and that extra hypotheses are needed to make this result true.
Define $X_n$ as a mixture between a uniform $U\left( \left[ -{1 | Taking the expectation of Taylor series (especially the remainder)
Not an actual answer, but an example to show that things are not so nice, and that extra hypotheses are needed to make this result true.
Define $X_n$ as a mixture between a uniform $U\left( \left[ -{1\over n} ; {1\over n} \right] \right)$ and a normal $... | Taking the expectation of Taylor series (especially the remainder)
Not an actual answer, but an example to show that things are not so nice, and that extra hypotheses are needed to make this result true.
Define $X_n$ as a mixture between a uniform $U\left( \left[ -{1 |
5,445 | Taking the expectation of Taylor series (especially the remainder) | This is not a complete answer, just a different way of arriving at the second order approximation.
I think the best way to go is to use Cauchy's mean value theorem, rather than work with the remainder term of a Taylor series. If we apply it one time we have
$$f(X)=f(\mu)+f'(\xi_1)(X-\mu)$$
for some $X\leq\xi_1 \leq \... | Taking the expectation of Taylor series (especially the remainder) | This is not a complete answer, just a different way of arriving at the second order approximation.
I think the best way to go is to use Cauchy's mean value theorem, rather than work with the remainder | Taking the expectation of Taylor series (especially the remainder)
This is not a complete answer, just a different way of arriving at the second order approximation.
I think the best way to go is to use Cauchy's mean value theorem, rather than work with the remainder term of a Taylor series. If we apply it one time we... | Taking the expectation of Taylor series (especially the remainder)
This is not a complete answer, just a different way of arriving at the second order approximation.
I think the best way to go is to use Cauchy's mean value theorem, rather than work with the remainder |
5,446 | What is the derivative of the ReLU activation function? | The derivative is:
$$ f(x)=
\begin{cases}
0 & \text{if } x < 0 \\
1 & \text{if } x > 0 \\
\end{cases}
$$
And undefined in $x=0$.
The reason for it being undefined at $x=0$ is that its left- and right derivative are not equal. | What is the derivative of the ReLU activation function? | The derivative is:
$$ f(x)=
\begin{cases}
0 & \text{if } x < 0 \\
1 & \text{if } x > 0 \\
\end{cases}
$$
And undefined in $x=0$.
The reason for it being undefined at $x=0$ is that its left- and r | What is the derivative of the ReLU activation function?
The derivative is:
$$ f(x)=
\begin{cases}
0 & \text{if } x < 0 \\
1 & \text{if } x > 0 \\
\end{cases}
$$
And undefined in $x=0$.
The reason for it being undefined at $x=0$ is that its left- and right derivative are not equal. | What is the derivative of the ReLU activation function?
The derivative is:
$$ f(x)=
\begin{cases}
0 & \text{if } x < 0 \\
1 & \text{if } x > 0 \\
\end{cases}
$$
And undefined in $x=0$.
The reason for it being undefined at $x=0$ is that its left- and r |
5,447 | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Be sure to read the full example on the UCLA site that you linked.
Regarding 1:
Using a multivariate model helps you (formally, inferentially) compare coefficients across outcomes.
In that linked example, they use the multivariate model to test whether the write coefficient is significantly different for the locus_of_c... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Be sure to read the full example on the UCLA site that you linked.
Regarding 1:
Using a multivariate model helps you (formally, inferentially) compare coefficients across outcomes.
In that linked exam | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Be sure to read the full example on the UCLA site that you linked.
Regarding 1:
Using a multivariate model helps you (formally, inferentially) compare coefficients across outcomes.
In that linked example, they use the multivariate... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Be sure to read the full example on the UCLA site that you linked.
Regarding 1:
Using a multivariate model helps you (formally, inferentially) compare coefficients across outcomes.
In that linked exam |
5,448 | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Think about all the false and sometimes dangerous conclusions that come from simply multiplying probabilities, thinking events are independent. Because of all the built-in redundant safeguards, we put into our nuclear power plants experts using the independence assumption told us that the chance of a major nuclear acc... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Think about all the false and sometimes dangerous conclusions that come from simply multiplying probabilities, thinking events are independent. Because of all the built-in redundant safeguards, we pu | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Think about all the false and sometimes dangerous conclusions that come from simply multiplying probabilities, thinking events are independent. Because of all the built-in redundant safeguards, we put into our nuclear power plant... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Think about all the false and sometimes dangerous conclusions that come from simply multiplying probabilities, thinking events are independent. Because of all the built-in redundant safeguards, we pu |
5,449 | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Consider this quote from p. 36 of Darcy Olsen's book The Right to Try [1]:
But about sixteen weeks after the [eteplirsen] infusions began, Jenn started noticing changes in [her son] Max. "The kid stopped wanting to use his wheelchair," she says. A few weeks later, he was asking to play outside — something he had not d... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Consider this quote from p. 36 of Darcy Olsen's book The Right to Try [1]:
But about sixteen weeks after the [eteplirsen] infusions began, Jenn started noticing changes in [her son] Max. "The kid sto | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Consider this quote from p. 36 of Darcy Olsen's book The Right to Try [1]:
But about sixteen weeks after the [eteplirsen] infusions began, Jenn started noticing changes in [her son] Max. "The kid stopped wanting to use his wheelc... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Consider this quote from p. 36 of Darcy Olsen's book The Right to Try [1]:
But about sixteen weeks after the [eteplirsen] infusions began, Jenn started noticing changes in [her son] Max. "The kid sto |
5,450 | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Let's make a simple analogy, since that's all I can really try to contribute. Instead of univariate versus multivariate regression, let's consider univariate (marginal) versus multivariate (joint) distributions. Say I have the following data and I want to find "outliers". As a first approach, I might use the two margin... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | Let's make a simple analogy, since that's all I can really try to contribute. Instead of univariate versus multivariate regression, let's consider univariate (marginal) versus multivariate (joint) dis | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Let's make a simple analogy, since that's all I can really try to contribute. Instead of univariate versus multivariate regression, let's consider univariate (marginal) versus multivariate (joint) distributions. Say I have the fol... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
Let's make a simple analogy, since that's all I can really try to contribute. Instead of univariate versus multivariate regression, let's consider univariate (marginal) versus multivariate (joint) dis |
5,451 | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | 1) Nature isn't always simple. In fact, most phenomena (outcome) we study depend on multiple variables, and in a complex manner. An inferential model based on one variable at a time will most likely have a high bias.
2) Univariate models are the simplest model you can build, by definition. It's fine if you are investi... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | 1) Nature isn't always simple. In fact, most phenomena (outcome) we study depend on multiple variables, and in a complex manner. An inferential model based on one variable at a time will most likely | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
1) Nature isn't always simple. In fact, most phenomena (outcome) we study depend on multiple variables, and in a complex manner. An inferential model based on one variable at a time will most likely have a high bias.
2) Univariat... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
1) Nature isn't always simple. In fact, most phenomena (outcome) we study depend on multiple variables, and in a complex manner. An inferential model based on one variable at a time will most likely |
5,452 | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | My answer depends on what you want to do with the regression. If you are trying to compare the effect of different coefficients, then regression may not be the right tool for you. If you are trying to make predictions using different coefficients that you have proven are independent, then maybe multiple regression is... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)? | My answer depends on what you want to do with the regression. If you are trying to compare the effect of different coefficients, then regression may not be the right tool for you. If you are trying | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
My answer depends on what you want to do with the regression. If you are trying to compare the effect of different coefficients, then regression may not be the right tool for you. If you are trying to make predictions using diff... | Why do we need multivariate regression (as opposed to a bunch of univariate regressions)?
My answer depends on what you want to do with the regression. If you are trying to compare the effect of different coefficients, then regression may not be the right tool for you. If you are trying |
5,453 | Entropy of an image | “What is the most information/physics-theoretical correct way to compute the entropy of an image?“
An excellent and timely question.
Contrary to popular belief, it is indeed possible to define an intuitively (and theoretically) natural information-entropy for an image.
Consider the following figure:
We can see that th... | Entropy of an image | “What is the most information/physics-theoretical correct way to compute the entropy of an image?“
An excellent and timely question.
Contrary to popular belief, it is indeed possible to define an intu | Entropy of an image
“What is the most information/physics-theoretical correct way to compute the entropy of an image?“
An excellent and timely question.
Contrary to popular belief, it is indeed possible to define an intuitively (and theoretically) natural information-entropy for an image.
Consider the following figure:... | Entropy of an image
“What is the most information/physics-theoretical correct way to compute the entropy of an image?“
An excellent and timely question.
Contrary to popular belief, it is indeed possible to define an intu |
5,454 | Entropy of an image | There is none, it all depends on the context and your prior information. Entropy has many interpretations such as "measurement of order" or "measurement of information", but instead of looking at the interpretations you could just look at what it actually is. Entropy is just a way of expressing the number of states of ... | Entropy of an image | There is none, it all depends on the context and your prior information. Entropy has many interpretations such as "measurement of order" or "measurement of information", but instead of looking at the | Entropy of an image
There is none, it all depends on the context and your prior information. Entropy has many interpretations such as "measurement of order" or "measurement of information", but instead of looking at the interpretations you could just look at what it actually is. Entropy is just a way of expressing the ... | Entropy of an image
There is none, it all depends on the context and your prior information. Entropy has many interpretations such as "measurement of order" or "measurement of information", but instead of looking at the |
5,455 | Entropy of an image | Essentially the idea of entropy is something like "number of micro-states consistent with the macrostate".
I think the comment by sean507 and the answer by bottiger both point to a common framework. If you represent the image space by a generative model, $p[\,I,h\,]$, then for a given image $I$ you can (in principle) c... | Entropy of an image | Essentially the idea of entropy is something like "number of micro-states consistent with the macrostate".
I think the comment by sean507 and the answer by bottiger both point to a common framework. I | Entropy of an image
Essentially the idea of entropy is something like "number of micro-states consistent with the macrostate".
I think the comment by sean507 and the answer by bottiger both point to a common framework. If you represent the image space by a generative model, $p[\,I,h\,]$, then for a given image $I$ you ... | Entropy of an image
Essentially the idea of entropy is something like "number of micro-states consistent with the macrostate".
I think the comment by sean507 and the answer by bottiger both point to a common framework. I |
5,456 | Entropy of an image | $$
H = - \sum_k p_k log_2(p_k)
$$
does NOT work in practice, for the simple reason that it's almost impossible to determine $P_k$. You think that you can do it, as you've done by considering the number of grey levels. Pk is not that. Pk is all possible combinations of grey levels. So you have to create a multi dime... | Entropy of an image | $$
H = - \sum_k p_k log_2(p_k)
$$
does NOT work in practice, for the simple reason that it's almost impossible to determine $P_k$. You think that you can do it, as you've done by considering the numb | Entropy of an image
$$
H = - \sum_k p_k log_2(p_k)
$$
does NOT work in practice, for the simple reason that it's almost impossible to determine $P_k$. You think that you can do it, as you've done by considering the number of grey levels. Pk is not that. Pk is all possible combinations of grey levels. So you have to... | Entropy of an image
$$
H = - \sum_k p_k log_2(p_k)
$$
does NOT work in practice, for the simple reason that it's almost impossible to determine $P_k$. You think that you can do it, as you've done by considering the numb |
5,457 | Entropy of an image | The entropy of an image can be computed generalizing classical 1D
methods to the 2D case.
Obviously it must take into account “spatial information”, not just gray levels.
Classical approaches used in 1D systems take into account, at least potentially, configurations at all the scales (all the n-grams, also for very lar... | Entropy of an image | The entropy of an image can be computed generalizing classical 1D
methods to the 2D case.
Obviously it must take into account “spatial information”, not just gray levels.
Classical approaches used in | Entropy of an image
The entropy of an image can be computed generalizing classical 1D
methods to the 2D case.
Obviously it must take into account “spatial information”, not just gray levels.
Classical approaches used in 1D systems take into account, at least potentially, configurations at all the scales (all the n-gram... | Entropy of an image
The entropy of an image can be computed generalizing classical 1D
methods to the 2D case.
Obviously it must take into account “spatial information”, not just gray levels.
Classical approaches used in |
5,458 | Where to start with statistics for an experienced developer | I would suggest you a basic road-map about how to go about it:
You can brush up basic math and stats at Khan Academy, and/or take the Intro to Statistics course by Udacity.
Then, you can take these two nice courses of Udacity. Descriptive Statistics and Inferential Statistics
Then, you can dive into some Bayesian stat... | Where to start with statistics for an experienced developer | I would suggest you a basic road-map about how to go about it:
You can brush up basic math and stats at Khan Academy, and/or take the Intro to Statistics course by Udacity.
Then, you can take these t | Where to start with statistics for an experienced developer
I would suggest you a basic road-map about how to go about it:
You can brush up basic math and stats at Khan Academy, and/or take the Intro to Statistics course by Udacity.
Then, you can take these two nice courses of Udacity. Descriptive Statistics and Infer... | Where to start with statistics for an experienced developer
I would suggest you a basic road-map about how to go about it:
You can brush up basic math and stats at Khan Academy, and/or take the Intro to Statistics course by Udacity.
Then, you can take these t |
5,459 | Where to start with statistics for an experienced developer | Have you checked out either Think Stats or Think Bayes--they are both (free) stats books geared towards programmers and with plenty of Python code.
Also, if you're interested in learning R then CRAN has a lot of (free) pdfs that you might want to check out, such as Introduction to Probability and Statistics Using R. Th... | Where to start with statistics for an experienced developer | Have you checked out either Think Stats or Think Bayes--they are both (free) stats books geared towards programmers and with plenty of Python code.
Also, if you're interested in learning R then CRAN h | Where to start with statistics for an experienced developer
Have you checked out either Think Stats or Think Bayes--they are both (free) stats books geared towards programmers and with plenty of Python code.
Also, if you're interested in learning R then CRAN has a lot of (free) pdfs that you might want to check out, su... | Where to start with statistics for an experienced developer
Have you checked out either Think Stats or Think Bayes--they are both (free) stats books geared towards programmers and with plenty of Python code.
Also, if you're interested in learning R then CRAN h |
5,460 | Where to start with statistics for an experienced developer | If you were ever, even in distant past, able to solve problems in this list, then you should attempt to study applied stats "properly". I'll give you a simple two step algorithm.
First, get up to speed with probability theory. There are many great books. My favorite is the classic book by Feller. It's called "Introduct... | Where to start with statistics for an experienced developer | If you were ever, even in distant past, able to solve problems in this list, then you should attempt to study applied stats "properly". I'll give you a simple two step algorithm.
First, get up to spee | Where to start with statistics for an experienced developer
If you were ever, even in distant past, able to solve problems in this list, then you should attempt to study applied stats "properly". I'll give you a simple two step algorithm.
First, get up to speed with probability theory. There are many great books. My fa... | Where to start with statistics for an experienced developer
If you were ever, even in distant past, able to solve problems in this list, then you should attempt to study applied stats "properly". I'll give you a simple two step algorithm.
First, get up to spee |
5,461 | Where to start with statistics for an experienced developer | Everyone is recommending Casella & Berger, which is almost universally used in graduate statistics programs. It's not a bad reference book, but I'm not sure I'd do more than scan the first 4-5 chapters. I don't think you need the theory of how to construct a Neyman-Pearson type test before delving into "statistics" i.e... | Where to start with statistics for an experienced developer | Everyone is recommending Casella & Berger, which is almost universally used in graduate statistics programs. It's not a bad reference book, but I'm not sure I'd do more than scan the first 4-5 chapter | Where to start with statistics for an experienced developer
Everyone is recommending Casella & Berger, which is almost universally used in graduate statistics programs. It's not a bad reference book, but I'm not sure I'd do more than scan the first 4-5 chapters. I don't think you need the theory of how to construct a N... | Where to start with statistics for an experienced developer
Everyone is recommending Casella & Berger, which is almost universally used in graduate statistics programs. It's not a bad reference book, but I'm not sure I'd do more than scan the first 4-5 chapter |
5,462 | Where to start with statistics for an experienced developer | This is not intended to be a complete answer, it's just a suggestion.
If you want to learn more about statistics (the foundation), you could read:
Casella, G. and R. L. Berger (2002): Statistical Inference, Duxbury
This is a pretty standard book for statisticians and it has a lot of interesting results. You don't need... | Where to start with statistics for an experienced developer | This is not intended to be a complete answer, it's just a suggestion.
If you want to learn more about statistics (the foundation), you could read:
Casella, G. and R. L. Berger (2002): Statistical Infe | Where to start with statistics for an experienced developer
This is not intended to be a complete answer, it's just a suggestion.
If you want to learn more about statistics (the foundation), you could read:
Casella, G. and R. L. Berger (2002): Statistical Inference, Duxbury
This is a pretty standard book for statistic... | Where to start with statistics for an experienced developer
This is not intended to be a complete answer, it's just a suggestion.
If you want to learn more about statistics (the foundation), you could read:
Casella, G. and R. L. Berger (2002): Statistical Infe |
5,463 | Where to start with statistics for an experienced developer | I'd suggest a new book that came out since the original question: Statistical Rethinking: A Bayesian Course with Examples in R and Stan by Richard McElreath, CRC Press.
It's very well written and uses a Bayesian approach. It's very interactive, and you'll want to work the problems or you may get halfway through and beg... | Where to start with statistics for an experienced developer | I'd suggest a new book that came out since the original question: Statistical Rethinking: A Bayesian Course with Examples in R and Stan by Richard McElreath, CRC Press.
It's very well written and uses | Where to start with statistics for an experienced developer
I'd suggest a new book that came out since the original question: Statistical Rethinking: A Bayesian Course with Examples in R and Stan by Richard McElreath, CRC Press.
It's very well written and uses a Bayesian approach. It's very interactive, and you'll want... | Where to start with statistics for an experienced developer
I'd suggest a new book that came out since the original question: Statistical Rethinking: A Bayesian Course with Examples in R and Stan by Richard McElreath, CRC Press.
It's very well written and uses |
5,464 | Where to start with statistics for an experienced developer | Figured I'd throw this answer in for posterity, even if it's likely too late to be useful to you. Larry Wasserman's All Of Statistics was conceived as a course for people with a background in machine learning, other comp sci disciplines, or math who didn't have any formal statistics training -- i.e., people in pretty ... | Where to start with statistics for an experienced developer | Figured I'd throw this answer in for posterity, even if it's likely too late to be useful to you. Larry Wasserman's All Of Statistics was conceived as a course for people with a background in machine | Where to start with statistics for an experienced developer
Figured I'd throw this answer in for posterity, even if it's likely too late to be useful to you. Larry Wasserman's All Of Statistics was conceived as a course for people with a background in machine learning, other comp sci disciplines, or math who didn't ha... | Where to start with statistics for an experienced developer
Figured I'd throw this answer in for posterity, even if it's likely too late to be useful to you. Larry Wasserman's All Of Statistics was conceived as a course for people with a background in machine |
5,465 | Latent Class Analysis vs. Cluster Analysis - differences in inferences? | Latent Class Analysis is in fact an Finite Mixture Model (see here). The main difference between FMM and other clustering algorithms is that FMM's offer you a "model-based clustering" approach that derives clusters using a probabilistic model that describes distribution of your data. So instead of finding clusters with... | Latent Class Analysis vs. Cluster Analysis - differences in inferences? | Latent Class Analysis is in fact an Finite Mixture Model (see here). The main difference between FMM and other clustering algorithms is that FMM's offer you a "model-based clustering" approach that de | Latent Class Analysis vs. Cluster Analysis - differences in inferences?
Latent Class Analysis is in fact an Finite Mixture Model (see here). The main difference between FMM and other clustering algorithms is that FMM's offer you a "model-based clustering" approach that derives clusters using a probabilistic model that ... | Latent Class Analysis vs. Cluster Analysis - differences in inferences?
Latent Class Analysis is in fact an Finite Mixture Model (see here). The main difference between FMM and other clustering algorithms is that FMM's offer you a "model-based clustering" approach that de |
5,466 | Latent Class Analysis vs. Cluster Analysis - differences in inferences? | The difference is Latent Class Analysis would use hidden data (which is usually patterns of association in the features) to determine probabilities for features in the class. Then inferences can be made using maximum likelihood to separate items into classes based on their features.
Cluster analysis plots the features... | Latent Class Analysis vs. Cluster Analysis - differences in inferences? | The difference is Latent Class Analysis would use hidden data (which is usually patterns of association in the features) to determine probabilities for features in the class. Then inferences can be ma | Latent Class Analysis vs. Cluster Analysis - differences in inferences?
The difference is Latent Class Analysis would use hidden data (which is usually patterns of association in the features) to determine probabilities for features in the class. Then inferences can be made using maximum likelihood to separate items i... | Latent Class Analysis vs. Cluster Analysis - differences in inferences?
The difference is Latent Class Analysis would use hidden data (which is usually patterns of association in the features) to determine probabilities for features in the class. Then inferences can be ma |
5,467 | Latent Class Analysis vs. Cluster Analysis - differences in inferences? | A latent class model (or latent profile, or more generally, a finite mixture model) can be thought of as a probablistic model for clustering (or unsupervised classification). The goal is generally the same - to identify homogenous groups within a larger population. I think the main differences between latent class mode... | Latent Class Analysis vs. Cluster Analysis - differences in inferences? | A latent class model (or latent profile, or more generally, a finite mixture model) can be thought of as a probablistic model for clustering (or unsupervised classification). The goal is generally the | Latent Class Analysis vs. Cluster Analysis - differences in inferences?
A latent class model (or latent profile, or more generally, a finite mixture model) can be thought of as a probablistic model for clustering (or unsupervised classification). The goal is generally the same - to identify homogenous groups within a l... | Latent Class Analysis vs. Cluster Analysis - differences in inferences?
A latent class model (or latent profile, or more generally, a finite mixture model) can be thought of as a probablistic model for clustering (or unsupervised classification). The goal is generally the |
5,468 | What is it meant with the $\sigma$-algebra generated by a random variable? | Consider a random variable $X$. We know that $X$ is nothing but a measurable function from $\left(\Omega, \mathcal{A} \right)$ into $\left(\mathbb{R}, \mathcal{B}(\mathbb{R}) \right)$, where $\mathcal{B}(\mathbb{R})$ are the Borel sets of the real line. By definition of measurability we know that we have
$$X^{-1} \lef... | What is it meant with the $\sigma$-algebra generated by a random variable? | Consider a random variable $X$. We know that $X$ is nothing but a measurable function from $\left(\Omega, \mathcal{A} \right)$ into $\left(\mathbb{R}, \mathcal{B}(\mathbb{R}) \right)$, where $\mathcal | What is it meant with the $\sigma$-algebra generated by a random variable?
Consider a random variable $X$. We know that $X$ is nothing but a measurable function from $\left(\Omega, \mathcal{A} \right)$ into $\left(\mathbb{R}, \mathcal{B}(\mathbb{R}) \right)$, where $\mathcal{B}(\mathbb{R})$ are the Borel sets of the re... | What is it meant with the $\sigma$-algebra generated by a random variable?
Consider a random variable $X$. We know that $X$ is nothing but a measurable function from $\left(\Omega, \mathcal{A} \right)$ into $\left(\mathbb{R}, \mathcal{B}(\mathbb{R}) \right)$, where $\mathcal |
5,469 | What is it meant with the $\sigma$-algebra generated by a random variable? | I will attempt to illustrate the intuition from a different perspective, less technically detailed.
Assume 4 random variables $X_1,X_2,X_3$ and $Y=f(X_1,X_2)$ for an arbitrary function $f$. Notice that $Y$ is random, but it's determined completely for fixed $X_1, X_2$, while $X_3$ is not determined for fixed $X_1, X_2... | What is it meant with the $\sigma$-algebra generated by a random variable? | I will attempt to illustrate the intuition from a different perspective, less technically detailed.
Assume 4 random variables $X_1,X_2,X_3$ and $Y=f(X_1,X_2)$ for an arbitrary function $f$. Notice th | What is it meant with the $\sigma$-algebra generated by a random variable?
I will attempt to illustrate the intuition from a different perspective, less technically detailed.
Assume 4 random variables $X_1,X_2,X_3$ and $Y=f(X_1,X_2)$ for an arbitrary function $f$. Notice that $Y$ is random, but it's determined complet... | What is it meant with the $\sigma$-algebra generated by a random variable?
I will attempt to illustrate the intuition from a different perspective, less technically detailed.
Assume 4 random variables $X_1,X_2,X_3$ and $Y=f(X_1,X_2)$ for an arbitrary function $f$. Notice th |
5,470 | How are Random Forests not sensitive to outliers? | Your intuition is correct. This answer merely illustrates it on an example.
It is indeed a common misconception that CART/RF are somehow robust to outliers.
To illustrate the lack of robustness of RF to the presence of a single outliers, we can (lightly) modify the code used in Soren Havelund Welling's answer above to... | How are Random Forests not sensitive to outliers? | Your intuition is correct. This answer merely illustrates it on an example.
It is indeed a common misconception that CART/RF are somehow robust to outliers.
To illustrate the lack of robustness of RF | How are Random Forests not sensitive to outliers?
Your intuition is correct. This answer merely illustrates it on an example.
It is indeed a common misconception that CART/RF are somehow robust to outliers.
To illustrate the lack of robustness of RF to the presence of a single outliers, we can (lightly) modify the cod... | How are Random Forests not sensitive to outliers?
Your intuition is correct. This answer merely illustrates it on an example.
It is indeed a common misconception that CART/RF are somehow robust to outliers.
To illustrate the lack of robustness of RF |
5,471 | How are Random Forests not sensitive to outliers? | It is not the Random Forest algorithm itself that is robust to outliers, but the base learner it is based on: the decision tree. Decision trees isolate atypical observations into small leaves (i.e., small subspaces of the original space). Furthermore, decision trees are local models. Unlike linear regression, where the... | How are Random Forests not sensitive to outliers? | It is not the Random Forest algorithm itself that is robust to outliers, but the base learner it is based on: the decision tree. Decision trees isolate atypical observations into small leaves (i.e., s | How are Random Forests not sensitive to outliers?
It is not the Random Forest algorithm itself that is robust to outliers, but the base learner it is based on: the decision tree. Decision trees isolate atypical observations into small leaves (i.e., small subspaces of the original space). Furthermore, decision trees are... | How are Random Forests not sensitive to outliers?
It is not the Random Forest algorithm itself that is robust to outliers, but the base learner it is based on: the decision tree. Decision trees isolate atypical observations into small leaves (i.e., s |
5,472 | How are Random Forests not sensitive to outliers? | outlier 1a: This outlier has one or more extreme feature values and is placed distant to any other sample. The outlier will influence the initial splits of the trees as any other sample, so no strong influence. It will have low proximity to any other sample, and will only define the model structure in a remote part of ... | How are Random Forests not sensitive to outliers? | outlier 1a: This outlier has one or more extreme feature values and is placed distant to any other sample. The outlier will influence the initial splits of the trees as any other sample, so no strong | How are Random Forests not sensitive to outliers?
outlier 1a: This outlier has one or more extreme feature values and is placed distant to any other sample. The outlier will influence the initial splits of the trees as any other sample, so no strong influence. It will have low proximity to any other sample, and will on... | How are Random Forests not sensitive to outliers?
outlier 1a: This outlier has one or more extreme feature values and is placed distant to any other sample. The outlier will influence the initial splits of the trees as any other sample, so no strong |
5,473 | Pitfalls in time series analysis | Extrapolating a linear regression on a time series, where time is one of the independent variables in the regression. A linear regression may approximate a time series on a short time scale, and may be useful in an analysis, but extrapolating a straight line is foolish. (Time is infinite and ever-increasing.)
EDIT: In ... | Pitfalls in time series analysis | Extrapolating a linear regression on a time series, where time is one of the independent variables in the regression. A linear regression may approximate a time series on a short time scale, and may b | Pitfalls in time series analysis
Extrapolating a linear regression on a time series, where time is one of the independent variables in the regression. A linear regression may approximate a time series on a short time scale, and may be useful in an analysis, but extrapolating a straight line is foolish. (Time is infinit... | Pitfalls in time series analysis
Extrapolating a linear regression on a time series, where time is one of the independent variables in the regression. A linear regression may approximate a time series on a short time scale, and may b |
5,474 | Pitfalls in time series analysis | Paying attention to correlation between two non-stationary time series. (It is not unexpected that they will have a high correlation coefficient: search on "non-sense correlation" and "cointegration".)
For example, on google correlate, dogs and ear piercings have a correlation coefficient of 0.84.
For an older analysi... | Pitfalls in time series analysis | Paying attention to correlation between two non-stationary time series. (It is not unexpected that they will have a high correlation coefficient: search on "non-sense correlation" and "cointegration". | Pitfalls in time series analysis
Paying attention to correlation between two non-stationary time series. (It is not unexpected that they will have a high correlation coefficient: search on "non-sense correlation" and "cointegration".)
For example, on google correlate, dogs and ear piercings have a correlation coefficie... | Pitfalls in time series analysis
Paying attention to correlation between two non-stationary time series. (It is not unexpected that they will have a high correlation coefficient: search on "non-sense correlation" and "cointegration". |
5,475 | Pitfalls in time series analysis | At the top level, Kolmogorov identified independence as a key assumption in statistics - without i.i.d assumption, many important results in statistics aren't true, whether applied to time series or more general analysis tasks.
Successive or nearby samples in most real-world discrete-time signals are not independent, s... | Pitfalls in time series analysis | At the top level, Kolmogorov identified independence as a key assumption in statistics - without i.i.d assumption, many important results in statistics aren't true, whether applied to time series or m | Pitfalls in time series analysis
At the top level, Kolmogorov identified independence as a key assumption in statistics - without i.i.d assumption, many important results in statistics aren't true, whether applied to time series or more general analysis tasks.
Successive or nearby samples in most real-world discrete-ti... | Pitfalls in time series analysis
At the top level, Kolmogorov identified independence as a key assumption in statistics - without i.i.d assumption, many important results in statistics aren't true, whether applied to time series or m |
5,476 | Pitfalls in time series analysis | Defining Trend as a Linear growth over time .
Although some trends are somehow linear (see Apple stock price), and although time series chart looks like a line chart where you can find linear regression, most trends are not linear.
There are Step changes like changes when something happened in a specific point in tim... | Pitfalls in time series analysis | Defining Trend as a Linear growth over time .
Although some trends are somehow linear (see Apple stock price), and although time series chart looks like a line chart where you can find linear regress | Pitfalls in time series analysis
Defining Trend as a Linear growth over time .
Although some trends are somehow linear (see Apple stock price), and although time series chart looks like a line chart where you can find linear regression, most trends are not linear.
There are Step changes like changes when something ha... | Pitfalls in time series analysis
Defining Trend as a Linear growth over time .
Although some trends are somehow linear (see Apple stock price), and although time series chart looks like a line chart where you can find linear regress |
5,477 | Pitfalls in time series analysis | Being too certain of your model's results because you use a technique/model (such as OLS) that does not account for a time series' autocorrelation.
I don't have a nice graph, but the book "Introductory Time Series with R" (2009, Cowpertwait, et al) gives a reasonable intuitive explanation: If there is a positive autoco... | Pitfalls in time series analysis | Being too certain of your model's results because you use a technique/model (such as OLS) that does not account for a time series' autocorrelation.
I don't have a nice graph, but the book "Introductor | Pitfalls in time series analysis
Being too certain of your model's results because you use a technique/model (such as OLS) that does not account for a time series' autocorrelation.
I don't have a nice graph, but the book "Introductory Time Series with R" (2009, Cowpertwait, et al) gives a reasonable intuitive explanati... | Pitfalls in time series analysis
Being too certain of your model's results because you use a technique/model (such as OLS) that does not account for a time series' autocorrelation.
I don't have a nice graph, but the book "Introductor |
5,478 | Pitfalls in time series analysis | The impact of level shifts , seasonal pulses and local time trends ... in addition to one-time pulses. Changes in parameters over time are important to investigate/model. Possible changes in variance of the errors over time have to be investigated. How to determine how Y is impacted by contemporaneous and lagged values... | Pitfalls in time series analysis | The impact of level shifts , seasonal pulses and local time trends ... in addition to one-time pulses. Changes in parameters over time are important to investigate/model. Possible changes in variance | Pitfalls in time series analysis
The impact of level shifts , seasonal pulses and local time trends ... in addition to one-time pulses. Changes in parameters over time are important to investigate/model. Possible changes in variance of the errors over time have to be investigated. How to determine how Y is impacted by ... | Pitfalls in time series analysis
The impact of level shifts , seasonal pulses and local time trends ... in addition to one-time pulses. Changes in parameters over time are important to investigate/model. Possible changes in variance |
5,479 | Pitfalls in time series analysis | In addition to some great points that have already been mentioned, I would add:
Failure to spot long cycles or seasonality - by examining only data over 'an insufficiently long'
period of time
Failure to evaluate the forecasting error for past
periods (backtesting)
Failure to detect and deal with regime changes
The... | Pitfalls in time series analysis | In addition to some great points that have already been mentioned, I would add:
Failure to spot long cycles or seasonality - by examining only data over 'an insufficiently long'
period of time
Failu | Pitfalls in time series analysis
In addition to some great points that have already been mentioned, I would add:
Failure to spot long cycles or seasonality - by examining only data over 'an insufficiently long'
period of time
Failure to evaluate the forecasting error for past
periods (backtesting)
Failure to detect ... | Pitfalls in time series analysis
In addition to some great points that have already been mentioned, I would add:
Failure to spot long cycles or seasonality - by examining only data over 'an insufficiently long'
period of time
Failu |
5,480 | Pitfalls in time series analysis | Avoid Aliasing in sampled time series. If you are analyzing time series data that is sampled at regular intervals, then the sampling rate must be twice the frequency of the highest frequency component in the data you are sampling. This is the Nyquist sampling theory, and it applies to digital audio, but also to any ... | Pitfalls in time series analysis | Avoid Aliasing in sampled time series. If you are analyzing time series data that is sampled at regular intervals, then the sampling rate must be twice the frequency of the highest frequency compone | Pitfalls in time series analysis
Avoid Aliasing in sampled time series. If you are analyzing time series data that is sampled at regular intervals, then the sampling rate must be twice the frequency of the highest frequency component in the data you are sampling. This is the Nyquist sampling theory, and it applies t... | Pitfalls in time series analysis
Avoid Aliasing in sampled time series. If you are analyzing time series data that is sampled at regular intervals, then the sampling rate must be twice the frequency of the highest frequency compone |
5,481 | Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom | Since
$$\begin{align*}
\hat\beta &= (X^TX)^{-1}X^TY \\
&= (X^TX)^{-1}X^T(X\beta + \varepsilon) \\
&= \beta + (X^TX)^{-1}X^T\varepsilon
\end{align*}$$
we know that
$$\hat\beta-\beta \sim \mathcal{N}(0,\sigma^2 (X^TX)^{-1})$$
and thus we know that for each component $k$ of $\hat\beta$,
$$\hat\beta_k -\beta_k \sim \mathca... | Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom | Since
$$\begin{align*}
\hat\beta &= (X^TX)^{-1}X^TY \\
&= (X^TX)^{-1}X^T(X\beta + \varepsilon) \\
&= \beta + (X^TX)^{-1}X^T\varepsilon
\end{align*}$$
we know that
$$\hat\beta-\beta \sim \mathcal{N}(0, | Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom
Since
$$\begin{align*}
\hat\beta &= (X^TX)^{-1}X^TY \\
&= (X^TX)^{-1}X^T(X\beta + \varepsilon) \\
&= \beta + (X^TX)^{-1}X^T\varepsilon
\end{align*}$$
we know that
$$\hat\beta-\beta \sim \mathcal{N}(0,\sigma^2 (X^TX)^{-1})$... | Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom
Since
$$\begin{align*}
\hat\beta &= (X^TX)^{-1}X^TY \\
&= (X^TX)^{-1}X^T(X\beta + \varepsilon) \\
&= \beta + (X^TX)^{-1}X^T\varepsilon
\end{align*}$$
we know that
$$\hat\beta-\beta \sim \mathcal{N}(0, |
5,482 | Bayesian statistics tutorial | Here's a place to start:
ftp://selab.janelia.org/pub/publications/Eddy-ATG3/Eddy-ATG3-reprint.pdf
http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/
http://yudkowsky.net/rational/bayes
http://www.math.umass.edu/~lavine/whatisbayes.pdf
http://en.wikipedia.org/wiki/Bayesian_inference
http://en.wikipedia.org/... | Bayesian statistics tutorial | Here's a place to start:
ftp://selab.janelia.org/pub/publications/Eddy-ATG3/Eddy-ATG3-reprint.pdf
http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/
http://yudkowsky.net/rational/bayes
ht | Bayesian statistics tutorial
Here's a place to start:
ftp://selab.janelia.org/pub/publications/Eddy-ATG3/Eddy-ATG3-reprint.pdf
http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/
http://yudkowsky.net/rational/bayes
http://www.math.umass.edu/~lavine/whatisbayes.pdf
http://en.wikipedia.org/wiki/Bayesian_infer... | Bayesian statistics tutorial
Here's a place to start:
ftp://selab.janelia.org/pub/publications/Eddy-ATG3/Eddy-ATG3-reprint.pdf
http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/
http://yudkowsky.net/rational/bayes
ht |
5,483 | Bayesian statistics tutorial | If you'd like to try a few learn by examples, you may be interested in "Bayesian Computation in R" by Jim Albert.
Its related R package is called LearnBayes. | Bayesian statistics tutorial | If you'd like to try a few learn by examples, you may be interested in "Bayesian Computation in R" by Jim Albert.
Its related R package is called LearnBayes. | Bayesian statistics tutorial
If you'd like to try a few learn by examples, you may be interested in "Bayesian Computation in R" by Jim Albert.
Its related R package is called LearnBayes. | Bayesian statistics tutorial
If you'd like to try a few learn by examples, you may be interested in "Bayesian Computation in R" by Jim Albert.
Its related R package is called LearnBayes. |
5,484 | Bayesian statistics tutorial | Some more depth:
http://math.tut.fi/~piche/bayes/notes01.pdf covers Bayes' theorem
https://ccrma.stanford.edu/~jos/bayes/bayes.pdf and
http://www-personal.une.edu.au/~jvanderw/Introduction_to_Bayesian_Statistics1.pdf are more about statistical applications | Bayesian statistics tutorial | Some more depth:
http://math.tut.fi/~piche/bayes/notes01.pdf covers Bayes' theorem
https://ccrma.stanford.edu/~jos/bayes/bayes.pdf and
http://www-personal.une.edu.au/~jvanderw/Introduction_to_Bayesi | Bayesian statistics tutorial
Some more depth:
http://math.tut.fi/~piche/bayes/notes01.pdf covers Bayes' theorem
https://ccrma.stanford.edu/~jos/bayes/bayes.pdf and
http://www-personal.une.edu.au/~jvanderw/Introduction_to_Bayesian_Statistics1.pdf are more about statistical applications | Bayesian statistics tutorial
Some more depth:
http://math.tut.fi/~piche/bayes/notes01.pdf covers Bayes' theorem
https://ccrma.stanford.edu/~jos/bayes/bayes.pdf and
http://www-personal.une.edu.au/~jvanderw/Introduction_to_Bayesi |
5,485 | Bayesian statistics tutorial | These aren't complete tutorials on Bayesian statistics, but rather isolated explanations of individual concepts that I like. Just thought I'd add in case it helps.
http://lesswrong.com/lw/2b0/bayes_theorem_illustrated_my_way - Graphical explanation of Bayes' theorem.
What's the difference between a confidence interval... | Bayesian statistics tutorial | These aren't complete tutorials on Bayesian statistics, but rather isolated explanations of individual concepts that I like. Just thought I'd add in case it helps.
http://lesswrong.com/lw/2b0/bayes_t | Bayesian statistics tutorial
These aren't complete tutorials on Bayesian statistics, but rather isolated explanations of individual concepts that I like. Just thought I'd add in case it helps.
http://lesswrong.com/lw/2b0/bayes_theorem_illustrated_my_way - Graphical explanation of Bayes' theorem.
What's the difference ... | Bayesian statistics tutorial
These aren't complete tutorials on Bayesian statistics, but rather isolated explanations of individual concepts that I like. Just thought I'd add in case it helps.
http://lesswrong.com/lw/2b0/bayes_t |
5,486 | Bayesian statistics tutorial | I wrote a post on getting started with JAGS for Bayesian modelling. If you're keen to get started quickly then playing around with some variant of BUGS, such as JAGS, is a practical way to get started.
To quote the abstract of the post
This post provides links to various resources on getting started with
Bayesian ... | Bayesian statistics tutorial | I wrote a post on getting started with JAGS for Bayesian modelling. If you're keen to get started quickly then playing around with some variant of BUGS, such as JAGS, is a practical way to get started | Bayesian statistics tutorial
I wrote a post on getting started with JAGS for Bayesian modelling. If you're keen to get started quickly then playing around with some variant of BUGS, such as JAGS, is a practical way to get started.
To quote the abstract of the post
This post provides links to various resources on get... | Bayesian statistics tutorial
I wrote a post on getting started with JAGS for Bayesian modelling. If you're keen to get started quickly then playing around with some variant of BUGS, such as JAGS, is a practical way to get started |
5,487 | Bayesian statistics tutorial | You could try 'Teaching Bayesian Reasoning In Less Than Two Hours'. | Bayesian statistics tutorial | You could try 'Teaching Bayesian Reasoning In Less Than Two Hours'. | Bayesian statistics tutorial
You could try 'Teaching Bayesian Reasoning In Less Than Two Hours'. | Bayesian statistics tutorial
You could try 'Teaching Bayesian Reasoning In Less Than Two Hours'. |
5,488 | Bayesian statistics tutorial | The Bayes' rule guide on Arbital is the best resource I've ever found by a good margin.
I like how they emphasize the odds form, include good visualizations, talk about how Bayesianism relates to philosophy, and include different learning paths depending on your background and interests. | Bayesian statistics tutorial | The Bayes' rule guide on Arbital is the best resource I've ever found by a good margin.
I like how they emphasize the odds form, include good visualizations, talk about how Bayesianism relates to phil | Bayesian statistics tutorial
The Bayes' rule guide on Arbital is the best resource I've ever found by a good margin.
I like how they emphasize the odds form, include good visualizations, talk about how Bayesianism relates to philosophy, and include different learning paths depending on your background and interests. | Bayesian statistics tutorial
The Bayes' rule guide on Arbital is the best resource I've ever found by a good margin.
I like how they emphasize the odds form, include good visualizations, talk about how Bayesianism relates to phil |
5,489 | First R packages source code to study in preparation for writing own package | I would suggest looking at the zoo package for the following reasons:
It has several well-written vignettes;
It uses a namespace using useDynLib, import, export, and S3method;
It has several unit tests using RUnit;
It provides good examples of how to create/document S3 methods;
It has some calls to C code via the .Cal... | First R packages source code to study in preparation for writing own package | I would suggest looking at the zoo package for the following reasons:
It has several well-written vignettes;
It uses a namespace using useDynLib, import, export, and S3method;
It has several unit tes | First R packages source code to study in preparation for writing own package
I would suggest looking at the zoo package for the following reasons:
It has several well-written vignettes;
It uses a namespace using useDynLib, import, export, and S3method;
It has several unit tests using RUnit;
It provides good examples o... | First R packages source code to study in preparation for writing own package
I would suggest looking at the zoo package for the following reasons:
It has several well-written vignettes;
It uses a namespace using useDynLib, import, export, and S3method;
It has several unit tes |
5,490 | First R packages source code to study in preparation for writing own package | I do not consider myself an established R package developer but have recently undergone the process of writing and maintaining a package for my work environment.
I had previously been writing / maintaining / updating a set of scripts that I would pass from project to project via the source() function. The end result of... | First R packages source code to study in preparation for writing own package | I do not consider myself an established R package developer but have recently undergone the process of writing and maintaining a package for my work environment.
I had previously been writing / mainta | First R packages source code to study in preparation for writing own package
I do not consider myself an established R package developer but have recently undergone the process of writing and maintaining a package for my work environment.
I had previously been writing / maintaining / updating a set of scripts that I wo... | First R packages source code to study in preparation for writing own package
I do not consider myself an established R package developer but have recently undergone the process of writing and maintaining a package for my work environment.
I had previously been writing / mainta |
5,491 | First R packages source code to study in preparation for writing own package | Why not take an empirically-driven random sampling approach? Just pick a few and see which work for you.
Kidding aside, just look at a few packages you yourself use and are familiar with. Downloading them is easy, or if you prefer you can also view them via a web interface at R-Forge, RForge, or Github.
You will most... | First R packages source code to study in preparation for writing own package | Why not take an empirically-driven random sampling approach? Just pick a few and see which work for you.
Kidding aside, just look at a few packages you yourself use and are familiar with. Downloadin | First R packages source code to study in preparation for writing own package
Why not take an empirically-driven random sampling approach? Just pick a few and see which work for you.
Kidding aside, just look at a few packages you yourself use and are familiar with. Downloading them is easy, or if you prefer you can al... | First R packages source code to study in preparation for writing own package
Why not take an empirically-driven random sampling approach? Just pick a few and see which work for you.
Kidding aside, just look at a few packages you yourself use and are familiar with. Downloadin |
5,492 | First R packages source code to study in preparation for writing own package | Another piece of advise might be to look at packages yours will be depending on or interacting with, especially if these implement some items Joshua Ulrich mentioned or have been written by renowned authors. It might be helpful to learn how things are done in your field, to ensure some compatibility. Often people will ... | First R packages source code to study in preparation for writing own package | Another piece of advise might be to look at packages yours will be depending on or interacting with, especially if these implement some items Joshua Ulrich mentioned or have been written by renowned a | First R packages source code to study in preparation for writing own package
Another piece of advise might be to look at packages yours will be depending on or interacting with, especially if these implement some items Joshua Ulrich mentioned or have been written by renowned authors. It might be helpful to learn how th... | First R packages source code to study in preparation for writing own package
Another piece of advise might be to look at packages yours will be depending on or interacting with, especially if these implement some items Joshua Ulrich mentioned or have been written by renowned a |
5,493 | First R packages source code to study in preparation for writing own package | i would recommend hadley's reshape package. you can find the source at https://github.com/hadley/reshape | First R packages source code to study in preparation for writing own package | i would recommend hadley's reshape package. you can find the source at https://github.com/hadley/reshape | First R packages source code to study in preparation for writing own package
i would recommend hadley's reshape package. you can find the source at https://github.com/hadley/reshape | First R packages source code to study in preparation for writing own package
i would recommend hadley's reshape package. you can find the source at https://github.com/hadley/reshape |
5,494 | Does the reciprocal of a probability represent anything? | Yes, it provides a 1-in-$n$ scale for probabilities. For example, the reciprocal of .01 is 100, so an event with probability .01 has a 1 in 100 chance of happening. This is a useful way to represent small probabilities, such as .0023, which is about 1 in 435. | Does the reciprocal of a probability represent anything? | Yes, it provides a 1-in-$n$ scale for probabilities. For example, the reciprocal of .01 is 100, so an event with probability .01 has a 1 in 100 chance of happening. This is a useful way to represent | Does the reciprocal of a probability represent anything?
Yes, it provides a 1-in-$n$ scale for probabilities. For example, the reciprocal of .01 is 100, so an event with probability .01 has a 1 in 100 chance of happening. This is a useful way to represent small probabilities, such as .0023, which is about 1 in 435. | Does the reciprocal of a probability represent anything?
Yes, it provides a 1-in-$n$ scale for probabilities. For example, the reciprocal of .01 is 100, so an event with probability .01 has a 1 in 100 chance of happening. This is a useful way to represent |
5,495 | Does the reciprocal of a probability represent anything? | $\frac 1p$ does not mean anything in general (but for a particular meaning
for a specific random variable see the answer by Alex R.). However,
the
logarithm of $\frac 1p$ to base 2, viz., $\log_2 \frac 1p = -\log_2 p$ is the amount of information (measured in bits) that you receive when you are told that
the event (of... | Does the reciprocal of a probability represent anything? | $\frac 1p$ does not mean anything in general (but for a particular meaning
for a specific random variable see the answer by Alex R.). However,
the
logarithm of $\frac 1p$ to base 2, viz., $\log_2 \fr | Does the reciprocal of a probability represent anything?
$\frac 1p$ does not mean anything in general (but for a particular meaning
for a specific random variable see the answer by Alex R.). However,
the
logarithm of $\frac 1p$ to base 2, viz., $\log_2 \frac 1p = -\log_2 p$ is the amount of information (measured in bi... | Does the reciprocal of a probability represent anything?
$\frac 1p$ does not mean anything in general (but for a particular meaning
for a specific random variable see the answer by Alex R.). However,
the
logarithm of $\frac 1p$ to base 2, viz., $\log_2 \fr |
5,496 | Does the reciprocal of a probability represent anything? | In the case of a geometric distribution, the reciprocal $1/p$ represents the expected number of throws you need to make to see one success. For example if a coin has probability $0.2$ of landing on heads, then you'd need to throw it around 5 times to see one head. | Does the reciprocal of a probability represent anything? | In the case of a geometric distribution, the reciprocal $1/p$ represents the expected number of throws you need to make to see one success. For example if a coin has probability $0.2$ of landing on he | Does the reciprocal of a probability represent anything?
In the case of a geometric distribution, the reciprocal $1/p$ represents the expected number of throws you need to make to see one success. For example if a coin has probability $0.2$ of landing on heads, then you'd need to throw it around 5 times to see one head... | Does the reciprocal of a probability represent anything?
In the case of a geometric distribution, the reciprocal $1/p$ represents the expected number of throws you need to make to see one success. For example if a coin has probability $0.2$ of landing on he |
5,497 | Does the reciprocal of a probability represent anything? | What are sometimes called European odds or decimal odds if fair are the reciprocal of the probability of winning, which might be a Bernoulli random variable $P(X=1)$.
For example if the quoted odds are "1.25" and you bet $8$ then you get $8 \times 1.25=10$ back if you win (including your original stake, so a gain of $... | Does the reciprocal of a probability represent anything? | What are sometimes called European odds or decimal odds if fair are the reciprocal of the probability of winning, which might be a Bernoulli random variable $P(X=1)$.
For example if the quoted odds a | Does the reciprocal of a probability represent anything?
What are sometimes called European odds or decimal odds if fair are the reciprocal of the probability of winning, which might be a Bernoulli random variable $P(X=1)$.
For example if the quoted odds are "1.25" and you bet $8$ then you get $8 \times 1.25=10$ back ... | Does the reciprocal of a probability represent anything?
What are sometimes called European odds or decimal odds if fair are the reciprocal of the probability of winning, which might be a Bernoulli random variable $P(X=1)$.
For example if the quoted odds a |
5,498 | Does the reciprocal of a probability represent anything? | In the context of survey design, the inverse of the probability of being included in the sample is called sampling weight.
For example, in a representative sample of some population, a respondent with the weight of 100 has 1/100 chance to be included in the sample, in other words, this respondent represents 100 similar... | Does the reciprocal of a probability represent anything? | In the context of survey design, the inverse of the probability of being included in the sample is called sampling weight.
For example, in a representative sample of some population, a respondent with | Does the reciprocal of a probability represent anything?
In the context of survey design, the inverse of the probability of being included in the sample is called sampling weight.
For example, in a representative sample of some population, a respondent with the weight of 100 has 1/100 chance to be included in the sampl... | Does the reciprocal of a probability represent anything?
In the context of survey design, the inverse of the probability of being included in the sample is called sampling weight.
For example, in a representative sample of some population, a respondent with |
5,499 | Does the reciprocal of a probability represent anything? | In statistical mechanics, a system has a large number of microstates, and it is a fundamental principle that these are all assumed to be equally likely. The reciprocal of the probability of a particular microstate is therefore the number of possible microstates, and this has a name in physics; it is (confusingly) calle... | Does the reciprocal of a probability represent anything? | In statistical mechanics, a system has a large number of microstates, and it is a fundamental principle that these are all assumed to be equally likely. The reciprocal of the probability of a particul | Does the reciprocal of a probability represent anything?
In statistical mechanics, a system has a large number of microstates, and it is a fundamental principle that these are all assumed to be equally likely. The reciprocal of the probability of a particular microstate is therefore the number of possible microstates, ... | Does the reciprocal of a probability represent anything?
In statistical mechanics, a system has a large number of microstates, and it is a fundamental principle that these are all assumed to be equally likely. The reciprocal of the probability of a particul |
5,500 | Understanding regressions - the role of the model | It helps to view regression as a linear approximation of the true form. Suppose the true relationship is
$$y=f(x_1,...,x_k)$$
with $x_1,...,x_k$ factors explaining the $y$. Then first order Taylor approximation of $f$ around zero is:
$$f(x_1,...,x_k)=f(0,...,0)+\sum_{i=1}^{k}\frac{\partial f(0)}{\partial x_k}x_k+\vare... | Understanding regressions - the role of the model | It helps to view regression as a linear approximation of the true form. Suppose the true relationship is
$$y=f(x_1,...,x_k)$$
with $x_1,...,x_k$ factors explaining the $y$. Then first order Taylor ap | Understanding regressions - the role of the model
It helps to view regression as a linear approximation of the true form. Suppose the true relationship is
$$y=f(x_1,...,x_k)$$
with $x_1,...,x_k$ factors explaining the $y$. Then first order Taylor approximation of $f$ around zero is:
$$f(x_1,...,x_k)=f(0,...,0)+\sum_{i... | Understanding regressions - the role of the model
It helps to view regression as a linear approximation of the true form. Suppose the true relationship is
$$y=f(x_1,...,x_k)$$
with $x_1,...,x_k$ factors explaining the $y$. Then first order Taylor ap |
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