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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10,601 | Is there a "hello, world" for statistical graphics? | Not sure if it exactly qualifies as a hello world, but in R there are also demos built into many packages. e.g.
library(graphics)
demo(graphics)
will step the user through some basic graphics available in the package.
Just mouse click over each image to step through basic graphics illustrations.
With just two lines, t... | Is there a "hello, world" for statistical graphics? | Not sure if it exactly qualifies as a hello world, but in R there are also demos built into many packages. e.g.
library(graphics)
demo(graphics)
will step the user through some basic graphics availab | Is there a "hello, world" for statistical graphics?
Not sure if it exactly qualifies as a hello world, but in R there are also demos built into many packages. e.g.
library(graphics)
demo(graphics)
will step the user through some basic graphics available in the package.
Just mouse click over each image to step through ... | Is there a "hello, world" for statistical graphics?
Not sure if it exactly qualifies as a hello world, but in R there are also demos built into many packages. e.g.
library(graphics)
demo(graphics)
will step the user through some basic graphics availab |
10,602 | Is there a "hello, world" for statistical graphics? | I'd say there were two "Hello World" type programs for data visualization:
Print("Hello World"): Something like the histogram of a normally distributed variable, or perhaps a simple X,Y scatterplot.
For something slightly more complex, like the section where one takes the principles of Hello World and starts playing wi... | Is there a "hello, world" for statistical graphics? | I'd say there were two "Hello World" type programs for data visualization:
Print("Hello World"): Something like the histogram of a normally distributed variable, or perhaps a simple X,Y scatterplot.
F | Is there a "hello, world" for statistical graphics?
I'd say there were two "Hello World" type programs for data visualization:
Print("Hello World"): Something like the histogram of a normally distributed variable, or perhaps a simple X,Y scatterplot.
For something slightly more complex, like the section where one takes... | Is there a "hello, world" for statistical graphics?
I'd say there were two "Hello World" type programs for data visualization:
Print("Hello World"): Something like the histogram of a normally distributed variable, or perhaps a simple X,Y scatterplot.
F |
10,603 | Sanity check: how low can a p-value go? | P-values on standard computers (using IEEE double precision floats) can get as low as approximately $10^{-303}$. These can be legitimately correct calculations when effect sizes are large and/or standard errors are low. Your value, if computed with a T or normal distribution, corresponds to an effect size of about 31... | Sanity check: how low can a p-value go? | P-values on standard computers (using IEEE double precision floats) can get as low as approximately $10^{-303}$. These can be legitimately correct calculations when effect sizes are large and/or stan | Sanity check: how low can a p-value go?
P-values on standard computers (using IEEE double precision floats) can get as low as approximately $10^{-303}$. These can be legitimately correct calculations when effect sizes are large and/or standard errors are low. Your value, if computed with a T or normal distribution, c... | Sanity check: how low can a p-value go?
P-values on standard computers (using IEEE double precision floats) can get as low as approximately $10^{-303}$. These can be legitimately correct calculations when effect sizes are large and/or stan |
10,604 | Sanity check: how low can a p-value go? | There is nothing suspicious -- extremely low p-values like yours are pretty common when sample sizes are large (as yours is for comparing medians). As whuber mentioned, normally such p-values are reported as being less than some threshold (e.g. <0.001).
One thing to be careful about is that p-values only tells you whet... | Sanity check: how low can a p-value go? | There is nothing suspicious -- extremely low p-values like yours are pretty common when sample sizes are large (as yours is for comparing medians). As whuber mentioned, normally such p-values are repo | Sanity check: how low can a p-value go?
There is nothing suspicious -- extremely low p-values like yours are pretty common when sample sizes are large (as yours is for comparing medians). As whuber mentioned, normally such p-values are reported as being less than some threshold (e.g. <0.001).
One thing to be careful ab... | Sanity check: how low can a p-value go?
There is nothing suspicious -- extremely low p-values like yours are pretty common when sample sizes are large (as yours is for comparing medians). As whuber mentioned, normally such p-values are repo |
10,605 | Sanity check: how low can a p-value go? | A p-value can achieve a value of 0.
Suppose I am testing the composite hypothesis about the value of a range of a uniform 0, $\theta$ random variable. If I set $\mathcal{H}_0: \theta = 1$ and sample a value of $X=1.1$, you see it's impossible to observe such a value or higher under the null hypothesis. The p-value is 0... | Sanity check: how low can a p-value go? | A p-value can achieve a value of 0.
Suppose I am testing the composite hypothesis about the value of a range of a uniform 0, $\theta$ random variable. If I set $\mathcal{H}_0: \theta = 1$ and sample a | Sanity check: how low can a p-value go?
A p-value can achieve a value of 0.
Suppose I am testing the composite hypothesis about the value of a range of a uniform 0, $\theta$ random variable. If I set $\mathcal{H}_0: \theta = 1$ and sample a value of $X=1.1$, you see it's impossible to observe such a value or higher und... | Sanity check: how low can a p-value go?
A p-value can achieve a value of 0.
Suppose I am testing the composite hypothesis about the value of a range of a uniform 0, $\theta$ random variable. If I set $\mathcal{H}_0: \theta = 1$ and sample a |
10,606 | Is it a good idea to use CNN to classify 1D signal? | I guess that by 1D signal you mean time-series data, where you assume temporal dependence between the values. In such cases convolutional neural networks (CNN) are one of the possible approaches. The most popular neural network approach to such data is to use recurrent neural networks (RNN), but you can alternatively u... | Is it a good idea to use CNN to classify 1D signal? | I guess that by 1D signal you mean time-series data, where you assume temporal dependence between the values. In such cases convolutional neural networks (CNN) are one of the possible approaches. The | Is it a good idea to use CNN to classify 1D signal?
I guess that by 1D signal you mean time-series data, where you assume temporal dependence between the values. In such cases convolutional neural networks (CNN) are one of the possible approaches. The most popular neural network approach to such data is to use recurren... | Is it a good idea to use CNN to classify 1D signal?
I guess that by 1D signal you mean time-series data, where you assume temporal dependence between the values. In such cases convolutional neural networks (CNN) are one of the possible approaches. The |
10,607 | Is it a good idea to use CNN to classify 1D signal? | You can certainly use a CNN to classify a 1D signal. Since you are interested in sleep stage classification see this paper. Its a deep neural network called the DeepSleepNet, and uses a combination of 1D convolutional and LSTM layers to classify EEG signals into sleep stages.
Here is the architecture:
There are two p... | Is it a good idea to use CNN to classify 1D signal? | You can certainly use a CNN to classify a 1D signal. Since you are interested in sleep stage classification see this paper. Its a deep neural network called the DeepSleepNet, and uses a combination of | Is it a good idea to use CNN to classify 1D signal?
You can certainly use a CNN to classify a 1D signal. Since you are interested in sleep stage classification see this paper. Its a deep neural network called the DeepSleepNet, and uses a combination of 1D convolutional and LSTM layers to classify EEG signals into sleep... | Is it a good idea to use CNN to classify 1D signal?
You can certainly use a CNN to classify a 1D signal. Since you are interested in sleep stage classification see this paper. Its a deep neural network called the DeepSleepNet, and uses a combination of |
10,608 | Is it a good idea to use CNN to classify 1D signal? | FWIW, I'll recommend checking out the Temporal Convolutional Network from this paper (I am not the author). They have a neat idea for using CNN for time-series data, is sensitive to time order and can model arbitrarily long sequences (but doesn't have a memory). | Is it a good idea to use CNN to classify 1D signal? | FWIW, I'll recommend checking out the Temporal Convolutional Network from this paper (I am not the author). They have a neat idea for using CNN for time-series data, is sensitive to time order and can | Is it a good idea to use CNN to classify 1D signal?
FWIW, I'll recommend checking out the Temporal Convolutional Network from this paper (I am not the author). They have a neat idea for using CNN for time-series data, is sensitive to time order and can model arbitrarily long sequences (but doesn't have a memory). | Is it a good idea to use CNN to classify 1D signal?
FWIW, I'll recommend checking out the Temporal Convolutional Network from this paper (I am not the author). They have a neat idea for using CNN for time-series data, is sensitive to time order and can |
10,609 | Is it a good idea to use CNN to classify 1D signal? | I want to emphasis the use of a stacked hybrid approach (CNN + RNN) for processing long sequences:
As you may know, 1D CNNs are not sensitive to the order of timesteps (not further than a local scale); of course, by stacking lots of convolution and pooling layers on top of each other, the final layers are able to obse... | Is it a good idea to use CNN to classify 1D signal? | I want to emphasis the use of a stacked hybrid approach (CNN + RNN) for processing long sequences:
As you may know, 1D CNNs are not sensitive to the order of timesteps (not further than a local scale | Is it a good idea to use CNN to classify 1D signal?
I want to emphasis the use of a stacked hybrid approach (CNN + RNN) for processing long sequences:
As you may know, 1D CNNs are not sensitive to the order of timesteps (not further than a local scale); of course, by stacking lots of convolution and pooling layers on ... | Is it a good idea to use CNN to classify 1D signal?
I want to emphasis the use of a stacked hybrid approach (CNN + RNN) for processing long sequences:
As you may know, 1D CNNs are not sensitive to the order of timesteps (not further than a local scale |
10,610 | How to explain dropout regularization in simple terms? | The abstract of the dropout article seems perfectly serviceable.
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, Ruslan Salakhutdinov, "Dropout: A Simple Way to Prevent Neural Networks from Overfitting", Journal of Machine Learning Research, 2014.
Deep neural nets with a large number of parameters... | How to explain dropout regularization in simple terms? | The abstract of the dropout article seems perfectly serviceable.
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, Ruslan Salakhutdinov, "Dropout: A Simple Way to Prevent Neural Net | How to explain dropout regularization in simple terms?
The abstract of the dropout article seems perfectly serviceable.
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, Ruslan Salakhutdinov, "Dropout: A Simple Way to Prevent Neural Networks from Overfitting", Journal of Machine Learning Research, 20... | How to explain dropout regularization in simple terms?
The abstract of the dropout article seems perfectly serviceable.
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, Ruslan Salakhutdinov, "Dropout: A Simple Way to Prevent Neural Net |
10,611 | How to explain dropout regularization in simple terms? | This answer is a follow-up to Sycorax' great answer, for readers who would like to see how dropout is implemented.
When applying dropout in artificial neural networks, one needs to compensate for the fact that at training time a portion of the neurons were deactivated. To do so, there exist two common strategies:
Inve... | How to explain dropout regularization in simple terms? | This answer is a follow-up to Sycorax' great answer, for readers who would like to see how dropout is implemented.
When applying dropout in artificial neural networks, one needs to compensate for the | How to explain dropout regularization in simple terms?
This answer is a follow-up to Sycorax' great answer, for readers who would like to see how dropout is implemented.
When applying dropout in artificial neural networks, one needs to compensate for the fact that at training time a portion of the neurons were deactiva... | How to explain dropout regularization in simple terms?
This answer is a follow-up to Sycorax' great answer, for readers who would like to see how dropout is implemented.
When applying dropout in artificial neural networks, one needs to compensate for the |
10,612 | How to explain dropout regularization in simple terms? | Dropout momentarily (in a batch of input data) switches off some neurons in a layer so that they do not contribute any information or learn any information during those updates, and the onus falls on other active neurons to learn harder and reduce the error.
If I have to explain drop-out to a 6-year-old, this is how:
I... | How to explain dropout regularization in simple terms? | Dropout momentarily (in a batch of input data) switches off some neurons in a layer so that they do not contribute any information or learn any information during those updates, and the onus falls on | How to explain dropout regularization in simple terms?
Dropout momentarily (in a batch of input data) switches off some neurons in a layer so that they do not contribute any information or learn any information during those updates, and the onus falls on other active neurons to learn harder and reduce the error.
If I h... | How to explain dropout regularization in simple terms?
Dropout momentarily (in a batch of input data) switches off some neurons in a layer so that they do not contribute any information or learn any information during those updates, and the onus falls on |
10,613 | How to explain dropout regularization in simple terms? | You can look at drop-out as a prior probability on whether a feature (or latent feature in some intermediate layer) does not matter - i.e. a spike (point mass at zero = feature does not matter) and slab (flat = non-reglarized prior across the whole parameter space) prior.
Importantly, this allows you to not just regula... | How to explain dropout regularization in simple terms? | You can look at drop-out as a prior probability on whether a feature (or latent feature in some intermediate layer) does not matter - i.e. a spike (point mass at zero = feature does not matter) and sl | How to explain dropout regularization in simple terms?
You can look at drop-out as a prior probability on whether a feature (or latent feature in some intermediate layer) does not matter - i.e. a spike (point mass at zero = feature does not matter) and slab (flat = non-reglarized prior across the whole parameter space)... | How to explain dropout regularization in simple terms?
You can look at drop-out as a prior probability on whether a feature (or latent feature in some intermediate layer) does not matter - i.e. a spike (point mass at zero = feature does not matter) and sl |
10,614 | Why do we say the outcome variable "is regressed on" the predictor(s)? | I do not know what the etymology of "is regressed on" is but here is the interpretation that I have in mind when I am saying or hearing this expression. Consider the following figure from The Elements of Statistical Learning by Hastie et al.:
In its core, linear regression amounts to orthogonal projection of $\mathbf ... | Why do we say the outcome variable "is regressed on" the predictor(s)? | I do not know what the etymology of "is regressed on" is but here is the interpretation that I have in mind when I am saying or hearing this expression. Consider the following figure from The Elements | Why do we say the outcome variable "is regressed on" the predictor(s)?
I do not know what the etymology of "is regressed on" is but here is the interpretation that I have in mind when I am saying or hearing this expression. Consider the following figure from The Elements of Statistical Learning by Hastie et al.:
In it... | Why do we say the outcome variable "is regressed on" the predictor(s)?
I do not know what the etymology of "is regressed on" is but here is the interpretation that I have in mind when I am saying or hearing this expression. Consider the following figure from The Elements |
10,615 | Why do we say the outcome variable "is regressed on" the predictor(s)? | I've often used and heard this way of speaking. I'd guess that the sequence mentioning the outcome or response before the predictors follows from conventions in writing, using words or using notation or mixing the two, all the way up to
$Y = X\beta$
setting aside the equally interesting (or uninteresting!) question ... | Why do we say the outcome variable "is regressed on" the predictor(s)? | I've often used and heard this way of speaking. I'd guess that the sequence mentioning the outcome or response before the predictors follows from conventions in writing, using words or using notation | Why do we say the outcome variable "is regressed on" the predictor(s)?
I've often used and heard this way of speaking. I'd guess that the sequence mentioning the outcome or response before the predictors follows from conventions in writing, using words or using notation or mixing the two, all the way up to
$Y = X\bet... | Why do we say the outcome variable "is regressed on" the predictor(s)?
I've often used and heard this way of speaking. I'd guess that the sequence mentioning the outcome or response before the predictors follows from conventions in writing, using words or using notation |
10,616 | Why do we say the outcome variable "is regressed on" the predictor(s)? | 1) The term regression comes from the fact that in the usual simple linear regression model:
$y = \alpha + \beta x + \epsilon$
that unless the outcome, $y$, and predictor, $x$, variables are perfectly correlated, the fitted values, $\hat{y}$, are closer to the mean of the outcome, $\bar{y}$, (after standardization) th... | Why do we say the outcome variable "is regressed on" the predictor(s)? | 1) The term regression comes from the fact that in the usual simple linear regression model:
$y = \alpha + \beta x + \epsilon$
that unless the outcome, $y$, and predictor, $x$, variables are perfectly | Why do we say the outcome variable "is regressed on" the predictor(s)?
1) The term regression comes from the fact that in the usual simple linear regression model:
$y = \alpha + \beta x + \epsilon$
that unless the outcome, $y$, and predictor, $x$, variables are perfectly correlated, the fitted values, $\hat{y}$, are c... | Why do we say the outcome variable "is regressed on" the predictor(s)?
1) The term regression comes from the fact that in the usual simple linear regression model:
$y = \alpha + \beta x + \epsilon$
that unless the outcome, $y$, and predictor, $x$, variables are perfectly |
10,617 | Why do we say the outcome variable "is regressed on" the predictor(s)? | As the target predicted outcome y depends on the predictor x, you can say that "is regressed on" means "is dependent on".
The word "regressed" is used instead of "dependent" because we want to emphasise that we are using a regression technique to represent this dependency between x and y.
So, this sentence "y is regres... | Why do we say the outcome variable "is regressed on" the predictor(s)? | As the target predicted outcome y depends on the predictor x, you can say that "is regressed on" means "is dependent on".
The word "regressed" is used instead of "dependent" because we want to emphasi | Why do we say the outcome variable "is regressed on" the predictor(s)?
As the target predicted outcome y depends on the predictor x, you can say that "is regressed on" means "is dependent on".
The word "regressed" is used instead of "dependent" because we want to emphasise that we are using a regression technique to re... | Why do we say the outcome variable "is regressed on" the predictor(s)?
As the target predicted outcome y depends on the predictor x, you can say that "is regressed on" means "is dependent on".
The word "regressed" is used instead of "dependent" because we want to emphasi |
10,618 | Why do we say the outcome variable "is regressed on" the predictor(s)? | Personally, when it comes to explaining terminology, I find the definition of the term itself always helps, especially when explaining to students. The actual definition of the word regress is:
"return to a former or less developed state".
So one way to explain I guess would be the following:
"Thinking of the outcome a... | Why do we say the outcome variable "is regressed on" the predictor(s)? | Personally, when it comes to explaining terminology, I find the definition of the term itself always helps, especially when explaining to students. The actual definition of the word regress is:
"retur | Why do we say the outcome variable "is regressed on" the predictor(s)?
Personally, when it comes to explaining terminology, I find the definition of the term itself always helps, especially when explaining to students. The actual definition of the word regress is:
"return to a former or less developed state".
So one wa... | Why do we say the outcome variable "is regressed on" the predictor(s)?
Personally, when it comes to explaining terminology, I find the definition of the term itself always helps, especially when explaining to students. The actual definition of the word regress is:
"retur |
10,619 | R-squared in quantile regression | Koenker and Machado$^{[1]}$ describe $R^1$, a local measure of goodness of fit at the particular ($\tau$) quantile.
Let $V(\tau) = \min_{b}\sum \rho_\tau(y_i-x_i'b)$
Let $\hat{\beta}(\tau)$ and $\tilde{\beta}(\tau)$ be the coefficient estimates for the full model, and a restricted model, and let $\hat{V}$ and $\tilde{V... | R-squared in quantile regression | Koenker and Machado$^{[1]}$ describe $R^1$, a local measure of goodness of fit at the particular ($\tau$) quantile.
Let $V(\tau) = \min_{b}\sum \rho_\tau(y_i-x_i'b)$
Let $\hat{\beta}(\tau)$ and $\tild | R-squared in quantile regression
Koenker and Machado$^{[1]}$ describe $R^1$, a local measure of goodness of fit at the particular ($\tau$) quantile.
Let $V(\tau) = \min_{b}\sum \rho_\tau(y_i-x_i'b)$
Let $\hat{\beta}(\tau)$ and $\tilde{\beta}(\tau)$ be the coefficient estimates for the full model, and a restricted model... | R-squared in quantile regression
Koenker and Machado$^{[1]}$ describe $R^1$, a local measure of goodness of fit at the particular ($\tau$) quantile.
Let $V(\tau) = \min_{b}\sum \rho_\tau(y_i-x_i'b)$
Let $\hat{\beta}(\tau)$ and $\tild |
10,620 | R-squared in quantile regression | The pseudo-$R^2$ measure suggested by Koenker and Machado (1999) in JASA measures goodness of fit by comparing the sum of weighted deviations for the model of interest
with the same sum from a model in which only the intercept appears. It is calculated as
$$R_1(\tau) = 1 - \frac{\sum_{y_i \ge \hat y_i} \tau \cdot \vert... | R-squared in quantile regression | The pseudo-$R^2$ measure suggested by Koenker and Machado (1999) in JASA measures goodness of fit by comparing the sum of weighted deviations for the model of interest
with the same sum from a model i | R-squared in quantile regression
The pseudo-$R^2$ measure suggested by Koenker and Machado (1999) in JASA measures goodness of fit by comparing the sum of weighted deviations for the model of interest
with the same sum from a model in which only the intercept appears. It is calculated as
$$R_1(\tau) = 1 - \frac{\sum_{y... | R-squared in quantile regression
The pseudo-$R^2$ measure suggested by Koenker and Machado (1999) in JASA measures goodness of fit by comparing the sum of weighted deviations for the model of interest
with the same sum from a model i |
10,621 | What is the difference between bagging and random forest if only one explanatory variable is used? | The fundamental difference is that in Random forests, only a subset of features are selected at random out of the total and the best split feature from the subset is used to split each node in a tree, unlike in bagging where all features are considered for splitting a node. | What is the difference between bagging and random forest if only one explanatory variable is used? | The fundamental difference is that in Random forests, only a subset of features are selected at random out of the total and the best split feature from the subset is used to split each node in a tree, | What is the difference between bagging and random forest if only one explanatory variable is used?
The fundamental difference is that in Random forests, only a subset of features are selected at random out of the total and the best split feature from the subset is used to split each node in a tree, unlike in bagging wh... | What is the difference between bagging and random forest if only one explanatory variable is used?
The fundamental difference is that in Random forests, only a subset of features are selected at random out of the total and the best split feature from the subset is used to split each node in a tree, |
10,622 | What is the difference between bagging and random forest if only one explanatory variable is used? | I would like to provide clarification, there is a disctinction between bagging and bagged trees.
Bagging (bootstrap + aggregating) is using an ensemble of models where:
each model uses a bootstrapped data set (bootstrap part of bagging)
models' predictions are aggregated (aggregation part of bagging)
This means that ... | What is the difference between bagging and random forest if only one explanatory variable is used? | I would like to provide clarification, there is a disctinction between bagging and bagged trees.
Bagging (bootstrap + aggregating) is using an ensemble of models where:
each model uses a bootstrapped | What is the difference between bagging and random forest if only one explanatory variable is used?
I would like to provide clarification, there is a disctinction between bagging and bagged trees.
Bagging (bootstrap + aggregating) is using an ensemble of models where:
each model uses a bootstrapped data set (bootstrap ... | What is the difference between bagging and random forest if only one explanatory variable is used?
I would like to provide clarification, there is a disctinction between bagging and bagged trees.
Bagging (bootstrap + aggregating) is using an ensemble of models where:
each model uses a bootstrapped |
10,623 | What is the difference between bagging and random forest if only one explanatory variable is used? | Bagging in general is an acronym like work that is a portmanteau of Bootstrap and aggregation. In general if you take a bunch of bootstrapped samples of your original dataset, fit models $M_1, M_2, \dots, M_b$ and then average all $b$ model predictions this is bootstrap aggregation i.e. Bagging. This is done as a step ... | What is the difference between bagging and random forest if only one explanatory variable is used? | Bagging in general is an acronym like work that is a portmanteau of Bootstrap and aggregation. In general if you take a bunch of bootstrapped samples of your original dataset, fit models $M_1, M_2, \d | What is the difference between bagging and random forest if only one explanatory variable is used?
Bagging in general is an acronym like work that is a portmanteau of Bootstrap and aggregation. In general if you take a bunch of bootstrapped samples of your original dataset, fit models $M_1, M_2, \dots, M_b$ and then av... | What is the difference between bagging and random forest if only one explanatory variable is used?
Bagging in general is an acronym like work that is a portmanteau of Bootstrap and aggregation. In general if you take a bunch of bootstrapped samples of your original dataset, fit models $M_1, M_2, \d |
10,624 | Comparing and contrasting, p-values, significance levels and type I error | The question looks simple, but your reflection around it shows that it is not that simple.
Actually, p-values are a relatively late addition to the theory of statistics. Computing a p-value without a computer is very tedious; this is why the only way to perform a statistical test until recently was to use tables of sta... | Comparing and contrasting, p-values, significance levels and type I error | The question looks simple, but your reflection around it shows that it is not that simple.
Actually, p-values are a relatively late addition to the theory of statistics. Computing a p-value without a | Comparing and contrasting, p-values, significance levels and type I error
The question looks simple, but your reflection around it shows that it is not that simple.
Actually, p-values are a relatively late addition to the theory of statistics. Computing a p-value without a computer is very tedious; this is why the only... | Comparing and contrasting, p-values, significance levels and type I error
The question looks simple, but your reflection around it shows that it is not that simple.
Actually, p-values are a relatively late addition to the theory of statistics. Computing a p-value without a |
10,625 | Comparing and contrasting, p-values, significance levels and type I error | You are getting good answers here from @MansT & @gui11aume (+1 to each). Let me see if I can get more explicitly at something in both of their answers.
When working with discrete data, there are only certain p-values possible, and the problem is worse with fewer possibilities / smaller data sets. For example, imagin... | Comparing and contrasting, p-values, significance levels and type I error | You are getting good answers here from @MansT & @gui11aume (+1 to each). Let me see if I can get more explicitly at something in both of their answers.
When working with discrete data, there are onl | Comparing and contrasting, p-values, significance levels and type I error
You are getting good answers here from @MansT & @gui11aume (+1 to each). Let me see if I can get more explicitly at something in both of their answers.
When working with discrete data, there are only certain p-values possible, and the problem i... | Comparing and contrasting, p-values, significance levels and type I error
You are getting good answers here from @MansT & @gui11aume (+1 to each). Let me see if I can get more explicitly at something in both of their answers.
When working with discrete data, there are onl |
10,626 | Comparing and contrasting, p-values, significance levels and type I error | The concepts are indeed intimately linked to each other.
The significance level is the probability of a type I error, or rather, the presumed probability of such an event. ${\rm P}({\rm type~I~error})= \alpha$ can generally only be obtained when working with continuous distributions, so in classic test theory a test i... | Comparing and contrasting, p-values, significance levels and type I error | The concepts are indeed intimately linked to each other.
The significance level is the probability of a type I error, or rather, the presumed probability of such an event. ${\rm P}({\rm type~I~error} | Comparing and contrasting, p-values, significance levels and type I error
The concepts are indeed intimately linked to each other.
The significance level is the probability of a type I error, or rather, the presumed probability of such an event. ${\rm P}({\rm type~I~error})= \alpha$ can generally only be obtained when... | Comparing and contrasting, p-values, significance levels and type I error
The concepts are indeed intimately linked to each other.
The significance level is the probability of a type I error, or rather, the presumed probability of such an event. ${\rm P}({\rm type~I~error} |
10,627 | Comparing and contrasting, p-values, significance levels and type I error | Summary. Significance level is the approximately same as the probability of getting a Type I errors for discrete distributions. We will show below with basic probability and empirical validation.
-- And they're probably exactly the same for continuous distribution, although I didn't do careful math proof for it.
Theory... | Comparing and contrasting, p-values, significance levels and type I error | Summary. Significance level is the approximately same as the probability of getting a Type I errors for discrete distributions. We will show below with basic probability and empirical validation.
-- A | Comparing and contrasting, p-values, significance levels and type I error
Summary. Significance level is the approximately same as the probability of getting a Type I errors for discrete distributions. We will show below with basic probability and empirical validation.
-- And they're probably exactly the same for conti... | Comparing and contrasting, p-values, significance levels and type I error
Summary. Significance level is the approximately same as the probability of getting a Type I errors for discrete distributions. We will show below with basic probability and empirical validation.
-- A |
10,628 | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | Asymptotic unbiasedness $\impliedby$ consistency + bounded variance
Consider an estimator $\hat{\theta}_n$ for a parameter $\theta$. Asymptotic unbiasedness means that the bias of the estimator goes to zero as $n \rightarrow \infty$, which means that the expected value of the estimator converges to the true value of t... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | Asymptotic unbiasedness $\impliedby$ consistency + bounded variance
Consider an estimator $\hat{\theta}_n$ for a parameter $\theta$. Asymptotic unbiasedness means that the bias of the estimator goes | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
Asymptotic unbiasedness $\impliedby$ consistency + bounded variance
Consider an estimator $\hat{\theta}_n$ for a parameter $\theta$. Asymptotic unbiasedness means that the bias of the estimator goes to zero as $n \righ... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
Asymptotic unbiasedness $\impliedby$ consistency + bounded variance
Consider an estimator $\hat{\theta}_n$ for a parameter $\theta$. Asymptotic unbiasedness means that the bias of the estimator goes |
10,629 | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent.
For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with mean $\mu$ and variance $\sigma^2$. As an estimator of $\mu$ consider $T = X_1 + 1/n$.
(Edit: Note the $X_1$ there, not $\b... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent.
For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with mea | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent.
For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with mean $\mu$ and varianc... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent.
For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with mea |
10,630 | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | I would like to clarify that consistency in general does not imply asymptotic unbiasedness. Consider an estimator for $0$ taking value $0$ with probability $(n-1)/n$ and value $n$ with probability $1/n$. It is a biased estimator since the expected value is always equal to $1$ and the bias does not disappear even if $n\... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | I would like to clarify that consistency in general does not imply asymptotic unbiasedness. Consider an estimator for $0$ taking value $0$ with probability $(n-1)/n$ and value $n$ with probability $1/ | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
I would like to clarify that consistency in general does not imply asymptotic unbiasedness. Consider an estimator for $0$ taking value $0$ with probability $(n-1)/n$ and value $n$ with probability $1/n$. It is a biased ... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
I would like to clarify that consistency in general does not imply asymptotic unbiasedness. Consider an estimator for $0$ taking value $0$ with probability $(n-1)/n$ and value $n$ with probability $1/ |
10,631 | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | There are "unbiased but not consistent" estimators as well as "biased but consistent" estimators:
https://en.wikipedia.org/wiki/Consistent_estimator#Unbiased_but_not_consistent
So, they are not the same thing.
Also, there is a long discussion about this topic here:
What is the difference between a consistent estimator ... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | There are "unbiased but not consistent" estimators as well as "biased but consistent" estimators:
https://en.wikipedia.org/wiki/Consistent_estimator#Unbiased_but_not_consistent
So, they are not the sa | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
There are "unbiased but not consistent" estimators as well as "biased but consistent" estimators:
https://en.wikipedia.org/wiki/Consistent_estimator#Unbiased_but_not_consistent
So, they are not the same thing.
Also, the... | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
There are "unbiased but not consistent" estimators as well as "biased but consistent" estimators:
https://en.wikipedia.org/wiki/Consistent_estimator#Unbiased_but_not_consistent
So, they are not the sa |
10,632 | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | If the estimator is bounded then consistency implies asymptotic unbiasness by the dominated convergence theorem. | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | If the estimator is bounded then consistency implies asymptotic unbiasness by the dominated convergence theorem. | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
If the estimator is bounded then consistency implies asymptotic unbiasness by the dominated convergence theorem. | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
If the estimator is bounded then consistency implies asymptotic unbiasness by the dominated convergence theorem. |
10,633 | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | Asymptotic unbiased: As $n \rightarrow \infty$, bias converges to $0$.
Consistent: As $n \rightarrow \infty$, variance of the estimator converges to $0$. | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate] | Asymptotic unbiased: As $n \rightarrow \infty$, bias converges to $0$.
Consistent: As $n \rightarrow \infty$, variance of the estimator converges to $0$. | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
Asymptotic unbiased: As $n \rightarrow \infty$, bias converges to $0$.
Consistent: As $n \rightarrow \infty$, variance of the estimator converges to $0$. | Intuitive understanding of the difference between consistent and asymptotically unbiased [duplicate]
Asymptotic unbiased: As $n \rightarrow \infty$, bias converges to $0$.
Consistent: As $n \rightarrow \infty$, variance of the estimator converges to $0$. |
10,634 | When to use the Wilcoxon rank-sum test instead of the unpaired t-test? | Yes, there is. For example, any sampling from distributions with infinite variance will wreck the t-test, but not the Wilcoxon. Referring to Nonparametric Statistical Methods (Hollander and Wolfe), I see that the asymptotic relative efficiency (ARE) of the Wilcoxon relative to the t test is 1.0 for the Uniform distri... | When to use the Wilcoxon rank-sum test instead of the unpaired t-test? | Yes, there is. For example, any sampling from distributions with infinite variance will wreck the t-test, but not the Wilcoxon. Referring to Nonparametric Statistical Methods (Hollander and Wolfe), | When to use the Wilcoxon rank-sum test instead of the unpaired t-test?
Yes, there is. For example, any sampling from distributions with infinite variance will wreck the t-test, but not the Wilcoxon. Referring to Nonparametric Statistical Methods (Hollander and Wolfe), I see that the asymptotic relative efficiency (AR... | When to use the Wilcoxon rank-sum test instead of the unpaired t-test?
Yes, there is. For example, any sampling from distributions with infinite variance will wreck the t-test, but not the Wilcoxon. Referring to Nonparametric Statistical Methods (Hollander and Wolfe), |
10,635 | When to use the Wilcoxon rank-sum test instead of the unpaired t-test? | Let me bring you back to our discussion in comments to this your question. Wilcoxon sum-rank test is equivalent to Mann-Whitney U test (and its direct extension for more-than-two samples is called Kruskal-Wallis test). You can see in Wikipedia as well as in this text that Mann-Whitney (or Kruskal-Wallis) generally comp... | When to use the Wilcoxon rank-sum test instead of the unpaired t-test? | Let me bring you back to our discussion in comments to this your question. Wilcoxon sum-rank test is equivalent to Mann-Whitney U test (and its direct extension for more-than-two samples is called Kru | When to use the Wilcoxon rank-sum test instead of the unpaired t-test?
Let me bring you back to our discussion in comments to this your question. Wilcoxon sum-rank test is equivalent to Mann-Whitney U test (and its direct extension for more-than-two samples is called Kruskal-Wallis test). You can see in Wikipedia as we... | When to use the Wilcoxon rank-sum test instead of the unpaired t-test?
Let me bring you back to our discussion in comments to this your question. Wilcoxon sum-rank test is equivalent to Mann-Whitney U test (and its direct extension for more-than-two samples is called Kru |
10,636 | Clustering quality measure | The choice of metric rather depends on what you consider the purpose of clustering to be. Personally I think clustering ought to be about identifying different groups of observations that were each generated by a different data generating process. So I would test the quality of a clustering by generating data from kn... | Clustering quality measure | The choice of metric rather depends on what you consider the purpose of clustering to be. Personally I think clustering ought to be about identifying different groups of observations that were each g | Clustering quality measure
The choice of metric rather depends on what you consider the purpose of clustering to be. Personally I think clustering ought to be about identifying different groups of observations that were each generated by a different data generating process. So I would test the quality of a clustering... | Clustering quality measure
The choice of metric rather depends on what you consider the purpose of clustering to be. Personally I think clustering ought to be about identifying different groups of observations that were each g |
10,637 | Clustering quality measure | The Silhouette can be used to evaluate clustering results. It does so by comparing the average distance within a cluster with the average distance to the points in the nearest cluster. | Clustering quality measure | The Silhouette can be used to evaluate clustering results. It does so by comparing the average distance within a cluster with the average distance to the points in the nearest cluster. | Clustering quality measure
The Silhouette can be used to evaluate clustering results. It does so by comparing the average distance within a cluster with the average distance to the points in the nearest cluster. | Clustering quality measure
The Silhouette can be used to evaluate clustering results. It does so by comparing the average distance within a cluster with the average distance to the points in the nearest cluster. |
10,638 | Clustering quality measure | Since clustering is unsupervised, it's hard to know a priori what the best clustering is. This is research topic. Gary King, a well-known quantitative social scientist, has a forthcoming article on this topic. | Clustering quality measure | Since clustering is unsupervised, it's hard to know a priori what the best clustering is. This is research topic. Gary King, a well-known quantitative social scientist, has a forthcoming article on th | Clustering quality measure
Since clustering is unsupervised, it's hard to know a priori what the best clustering is. This is research topic. Gary King, a well-known quantitative social scientist, has a forthcoming article on this topic. | Clustering quality measure
Since clustering is unsupervised, it's hard to know a priori what the best clustering is. This is research topic. Gary King, a well-known quantitative social scientist, has a forthcoming article on th |
10,639 | Clustering quality measure | Here you have a couple of measures, but there are many more:
SSE: sum of the square error from the items of each cluster.
Inter cluster distance: sum of the square distance between each cluster centroid.
Intra cluster distance for each cluster: sum of the square distance from the items of each cluster to its centroid.
... | Clustering quality measure | Here you have a couple of measures, but there are many more:
SSE: sum of the square error from the items of each cluster.
Inter cluster distance: sum of the square distance between each cluster centro | Clustering quality measure
Here you have a couple of measures, but there are many more:
SSE: sum of the square error from the items of each cluster.
Inter cluster distance: sum of the square distance between each cluster centroid.
Intra cluster distance for each cluster: sum of the square distance from the items of eac... | Clustering quality measure
Here you have a couple of measures, but there are many more:
SSE: sum of the square error from the items of each cluster.
Inter cluster distance: sum of the square distance between each cluster centro |
10,640 | Clustering quality measure | As others have pointed out, there are many measures of clustering "quality";
most programs minimize SSE.
No single number can tell much about noise in the data,
or noise in the method,
or flat minima — low points in Saskatchewan.
So first try to visualize, get a feel for,
a given clustering, before reducing it to "41".... | Clustering quality measure | As others have pointed out, there are many measures of clustering "quality";
most programs minimize SSE.
No single number can tell much about noise in the data,
or noise in the method,
or flat minima | Clustering quality measure
As others have pointed out, there are many measures of clustering "quality";
most programs minimize SSE.
No single number can tell much about noise in the data,
or noise in the method,
or flat minima — low points in Saskatchewan.
So first try to visualize, get a feel for,
a given clustering, ... | Clustering quality measure
As others have pointed out, there are many measures of clustering "quality";
most programs minimize SSE.
No single number can tell much about noise in the data,
or noise in the method,
or flat minima |
10,641 | Clustering quality measure | You ran into the Clustering Validation area. My student did validation using techniques described in:
A. Banerjee and R. N. Dave. Validating clusters using the hopkins statistic. 2004 IEEE International Conference on Fuzzy Systems IEEE Cat No04CH37542, 1:p. 149–153, 2004.
It is based on the principle, that if a cluster... | Clustering quality measure | You ran into the Clustering Validation area. My student did validation using techniques described in:
A. Banerjee and R. N. Dave. Validating clusters using the hopkins statistic. 2004 IEEE Internation | Clustering quality measure
You ran into the Clustering Validation area. My student did validation using techniques described in:
A. Banerjee and R. N. Dave. Validating clusters using the hopkins statistic. 2004 IEEE International Conference on Fuzzy Systems IEEE Cat No04CH37542, 1:p. 149–153, 2004.
It is based on the p... | Clustering quality measure
You ran into the Clustering Validation area. My student did validation using techniques described in:
A. Banerjee and R. N. Dave. Validating clusters using the hopkins statistic. 2004 IEEE Internation |
10,642 | Clustering quality measure | If the clustering algorithm isn't deterministic, then try to measure "stability" of clusterings - find out how often each two observations belongs to the same cluster. That's generaly interesting method, useful for choosing k in kmeans algorithm. | Clustering quality measure | If the clustering algorithm isn't deterministic, then try to measure "stability" of clusterings - find out how often each two observations belongs to the same cluster. That's generaly interesting met | Clustering quality measure
If the clustering algorithm isn't deterministic, then try to measure "stability" of clusterings - find out how often each two observations belongs to the same cluster. That's generaly interesting method, useful for choosing k in kmeans algorithm. | Clustering quality measure
If the clustering algorithm isn't deterministic, then try to measure "stability" of clusterings - find out how often each two observations belongs to the same cluster. That's generaly interesting met |
10,643 | Clustering quality measure | A method such as that used in unsupervised random forest could be used.
Random Forest algorithms treat unsupervised classification as a two class problem, were a whole different artificial and random data set is created from the first data set by removing the dependency structure in the data (randomization).
You could ... | Clustering quality measure | A method such as that used in unsupervised random forest could be used.
Random Forest algorithms treat unsupervised classification as a two class problem, were a whole different artificial and random | Clustering quality measure
A method such as that used in unsupervised random forest could be used.
Random Forest algorithms treat unsupervised classification as a two class problem, were a whole different artificial and random data set is created from the first data set by removing the dependency structure in the data ... | Clustering quality measure
A method such as that used in unsupervised random forest could be used.
Random Forest algorithms treat unsupervised classification as a two class problem, were a whole different artificial and random |
10,644 | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | There are simulations that are not Monte Carlo:
Basically, all Monte Carlo methods use the (weak) law of large numbers: The mean converges to its expectation.
Then there are Quasi Monte Carlo methods. These are simulated with a compromise of random numbers and equally spaced grids to yield faster convergece.
Simulatio... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | There are simulations that are not Monte Carlo:
Basically, all Monte Carlo methods use the (weak) law of large numbers: The mean converges to its expectation.
Then there are Quasi Monte Carlo methods | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
There are simulations that are not Monte Carlo:
Basically, all Monte Carlo methods use the (weak) law of large numbers: The mean converges to its expectation.
Then there are Quasi Monte Carlo methods. These are simulated with a c... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
There are simulations that are not Monte Carlo:
Basically, all Monte Carlo methods use the (weak) law of large numbers: The mean converges to its expectation.
Then there are Quasi Monte Carlo methods |
10,645 | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | Nicholas Metropolis claimed in 1987 that
It was at that
time that I suggested an obvious name
for the statistical method - a suggestion
not unrelated to the fact that Stan[islaw Ulam] had an
uncle who would borrow money from relatives because he “just had to go to Monte
Carlo.”
"Monte Carlo"... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | Nicholas Metropolis claimed in 1987 that
It was at that
time that I suggested an obvious name
for the statistical method - a suggestion
not unrelated to the fact that Stan[islaw Ul | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
Nicholas Metropolis claimed in 1987 that
It was at that
time that I suggested an obvious name
for the statistical method - a suggestion
not unrelated to the fact that Stan[islaw Ulam] had an
uncle who would ... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
Nicholas Metropolis claimed in 1987 that
It was at that
time that I suggested an obvious name
for the statistical method - a suggestion
not unrelated to the fact that Stan[islaw Ul |
10,646 | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | I have sometimes heard of people who distinguish between Monte Carlo algorithms and Las Vegas algorithms. Unlike a Monte Carlo algorithm—which will always terminate, but has a chance of giving wildly inaccurate results—a Las Vegas algorithm has a chance of running for an arbitrarily long time, but will always give accu... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | I have sometimes heard of people who distinguish between Monte Carlo algorithms and Las Vegas algorithms. Unlike a Monte Carlo algorithm—which will always terminate, but has a chance of giving wildly | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
I have sometimes heard of people who distinguish between Monte Carlo algorithms and Las Vegas algorithms. Unlike a Monte Carlo algorithm—which will always terminate, but has a chance of giving wildly inaccurate results—a Las Vegas... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
I have sometimes heard of people who distinguish between Monte Carlo algorithms and Las Vegas algorithms. Unlike a Monte Carlo algorithm—which will always terminate, but has a chance of giving wildly |
10,647 | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | You can read about the history of the Monte Carlo name in the other answers and comments. So this answer will provide a complementary perspective.
In sophisticated company, it's referred to as stochastic simulation. See for example, the book "Stochastic Simulation: Algorithms and Analysis", Asmussen and Glynn. http://... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | You can read about the history of the Monte Carlo name in the other answers and comments. So this answer will provide a complementary perspective.
In sophisticated company, it's referred to as stochas | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
You can read about the history of the Monte Carlo name in the other answers and comments. So this answer will provide a complementary perspective.
In sophisticated company, it's referred to as stochastic simulation. See for examp... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
You can read about the history of the Monte Carlo name in the other answers and comments. So this answer will provide a complementary perspective.
In sophisticated company, it's referred to as stochas |
10,648 | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | This is actually a really good question, and it has provoked some fine answers. I'm adding this because I wondered about this too, and believe that Monte Carlo was used and became popular, because the process employed by gambling casinos, and the statistical process of estimation have specific, similar characteristics.... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate] | This is actually a really good question, and it has provoked some fine answers. I'm adding this because I wondered about this too, and believe that Monte Carlo was used and became popular, because the | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
This is actually a really good question, and it has provoked some fine answers. I'm adding this because I wondered about this too, and believe that Monte Carlo was used and became popular, because the process employed by gambling ... | Why is the term "Monte Carlo simulation" used instead of "Random simulation"? [duplicate]
This is actually a really good question, and it has provoked some fine answers. I'm adding this because I wondered about this too, and believe that Monte Carlo was used and became popular, because the |
10,649 | Do all machine learning algorithms separate data linearly? | The answer is No. user20160 has a perfect answer, I will add 3 examples with visualization to illustrate the idea. Note, these plots may not be helpful for you to see if the "final decision" is in linear form but give you some sense about tree, boosting and KNN.
We will start with decision trees. With many splits, it ... | Do all machine learning algorithms separate data linearly? | The answer is No. user20160 has a perfect answer, I will add 3 examples with visualization to illustrate the idea. Note, these plots may not be helpful for you to see if the "final decision" is in lin | Do all machine learning algorithms separate data linearly?
The answer is No. user20160 has a perfect answer, I will add 3 examples with visualization to illustrate the idea. Note, these plots may not be helpful for you to see if the "final decision" is in linear form but give you some sense about tree, boosting and KNN... | Do all machine learning algorithms separate data linearly?
The answer is No. user20160 has a perfect answer, I will add 3 examples with visualization to illustrate the idea. Note, these plots may not be helpful for you to see if the "final decision" is in lin |
10,650 | Do all machine learning algorithms separate data linearly? | Some algorithms use a hyperplane (i.e. linear function) to separate the data. A prominent example is logistic regression. Others use a hyperplane to separate the data after a nonlinear transformation (e.g. neural networks and support vector machines with nonlinear kernels). In this case, the decision boundary is nonlin... | Do all machine learning algorithms separate data linearly? | Some algorithms use a hyperplane (i.e. linear function) to separate the data. A prominent example is logistic regression. Others use a hyperplane to separate the data after a nonlinear transformation | Do all machine learning algorithms separate data linearly?
Some algorithms use a hyperplane (i.e. linear function) to separate the data. A prominent example is logistic regression. Others use a hyperplane to separate the data after a nonlinear transformation (e.g. neural networks and support vector machines with nonlin... | Do all machine learning algorithms separate data linearly?
Some algorithms use a hyperplane (i.e. linear function) to separate the data. A prominent example is logistic regression. Others use a hyperplane to separate the data after a nonlinear transformation |
10,651 | How is interpolation related to the concept of regression? | The main difference between interpolation and regression, is the definition of the problem they solve.
Given $n$ data points, when you interpolate, you look for a function that is of some predefined form that has the values in that points exactly as specified. That means given pairs $(x_i, y_i)$ you look for $F$ of som... | How is interpolation related to the concept of regression? | The main difference between interpolation and regression, is the definition of the problem they solve.
Given $n$ data points, when you interpolate, you look for a function that is of some predefined f | How is interpolation related to the concept of regression?
The main difference between interpolation and regression, is the definition of the problem they solve.
Given $n$ data points, when you interpolate, you look for a function that is of some predefined form that has the values in that points exactly as specified. ... | How is interpolation related to the concept of regression?
The main difference between interpolation and regression, is the definition of the problem they solve.
Given $n$ data points, when you interpolate, you look for a function that is of some predefined f |
10,652 | How is interpolation related to the concept of regression? | The two previous answers have explained the relationship between linear interpolation and linear regression (or even general interpolation and polynomial regression). But an important connection is that once you fit a regression model you can use it to interpolate between the given data points. | How is interpolation related to the concept of regression? | The two previous answers have explained the relationship between linear interpolation and linear regression (or even general interpolation and polynomial regression). But an important connection is t | How is interpolation related to the concept of regression?
The two previous answers have explained the relationship between linear interpolation and linear regression (or even general interpolation and polynomial regression). But an important connection is that once you fit a regression model you can use it to interpo... | How is interpolation related to the concept of regression?
The two previous answers have explained the relationship between linear interpolation and linear regression (or even general interpolation and polynomial regression). But an important connection is t |
10,653 | How is interpolation related to the concept of regression? | Hopefully this will come rather quickly with a simple example and visualization.
Suppose you have the following data:
X Y
1 6
10 15
20 25
30 35
40 45
50 55
We may use regression to model Y as a response to X. Using R:
lm(y ~ x)
The results are an intercept of 5, and a coefficent for x of 1. Which means an arbitrar... | How is interpolation related to the concept of regression? | Hopefully this will come rather quickly with a simple example and visualization.
Suppose you have the following data:
X Y
1 6
10 15
20 25
30 35
40 45
50 55
We may use regression to model Y as a res | How is interpolation related to the concept of regression?
Hopefully this will come rather quickly with a simple example and visualization.
Suppose you have the following data:
X Y
1 6
10 15
20 25
30 35
40 45
50 55
We may use regression to model Y as a response to X. Using R:
lm(y ~ x)
The results are an intercept... | How is interpolation related to the concept of regression?
Hopefully this will come rather quickly with a simple example and visualization.
Suppose you have the following data:
X Y
1 6
10 15
20 25
30 35
40 45
50 55
We may use regression to model Y as a res |
10,654 | How is interpolation related to the concept of regression? | Regression is the process of finding the line of best fit[1]. Interpolation is the process of using the line of best fit to estimate the value of one variable from the value of another, provided that the value you are using is within the range of your data. If it's outside the range, then you would be using Extrapolat... | How is interpolation related to the concept of regression? | Regression is the process of finding the line of best fit[1]. Interpolation is the process of using the line of best fit to estimate the value of one variable from the value of another, provided that | How is interpolation related to the concept of regression?
Regression is the process of finding the line of best fit[1]. Interpolation is the process of using the line of best fit to estimate the value of one variable from the value of another, provided that the value you are using is within the range of your data. If... | How is interpolation related to the concept of regression?
Regression is the process of finding the line of best fit[1]. Interpolation is the process of using the line of best fit to estimate the value of one variable from the value of another, provided that |
10,655 | How is interpolation related to the concept of regression? | the basic difference b/w Interpolation and regression is as follows:
Interpolation:suppose there are n points (eg:10 data points),in interpolation we will fit the curve passing through all the data points (i.e here 10 data points) with a degree of the polynomial (no.of data points -1; i.e here it is 9).where as in regr... | How is interpolation related to the concept of regression? | the basic difference b/w Interpolation and regression is as follows:
Interpolation:suppose there are n points (eg:10 data points),in interpolation we will fit the curve passing through all the data po | How is interpolation related to the concept of regression?
the basic difference b/w Interpolation and regression is as follows:
Interpolation:suppose there are n points (eg:10 data points),in interpolation we will fit the curve passing through all the data points (i.e here 10 data points) with a degree of the polynomia... | How is interpolation related to the concept of regression?
the basic difference b/w Interpolation and regression is as follows:
Interpolation:suppose there are n points (eg:10 data points),in interpolation we will fit the curve passing through all the data po |
10,656 | How is interpolation related to the concept of regression? | With interpolation or spline fitting what we get is a numeric data (interpolated bet ween each pair of original data) of larger size, which when plotted generates the effect of a smooth curve. In actuality, between each pair of original data a different polynomial is fitted, therefore the entire curve after interpolati... | How is interpolation related to the concept of regression? | With interpolation or spline fitting what we get is a numeric data (interpolated bet ween each pair of original data) of larger size, which when plotted generates the effect of a smooth curve. In actu | How is interpolation related to the concept of regression?
With interpolation or spline fitting what we get is a numeric data (interpolated bet ween each pair of original data) of larger size, which when plotted generates the effect of a smooth curve. In actuality, between each pair of original data a different polynom... | How is interpolation related to the concept of regression?
With interpolation or spline fitting what we get is a numeric data (interpolated bet ween each pair of original data) of larger size, which when plotted generates the effect of a smooth curve. In actu |
10,657 | How is interpolation related to the concept of regression? | Both regression and interpolation are used to predict values of a variable(Y) for a given value of another variable(X).
In Regression we can predict any value of the dependent variable(Y) for a given value of the independent variable(X) Even if it is outside the range of tabulated values.But in case of Interpolation we... | How is interpolation related to the concept of regression? | Both regression and interpolation are used to predict values of a variable(Y) for a given value of another variable(X).
In Regression we can predict any value of the dependent variable(Y) for a given | How is interpolation related to the concept of regression?
Both regression and interpolation are used to predict values of a variable(Y) for a given value of another variable(X).
In Regression we can predict any value of the dependent variable(Y) for a given value of the independent variable(X) Even if it is outside th... | How is interpolation related to the concept of regression?
Both regression and interpolation are used to predict values of a variable(Y) for a given value of another variable(X).
In Regression we can predict any value of the dependent variable(Y) for a given |
10,658 | How is interpolation related to the concept of regression? | Interpolation is the process of fitting a number of points between x=a and x=b exactly to an interpolating polynomial.
Interpolation can be used to find the approximate value (or the missing value) of y in the domain x=[a,b] with better accuracy than regression technique.
On the other hand, regression is a process of f... | How is interpolation related to the concept of regression? | Interpolation is the process of fitting a number of points between x=a and x=b exactly to an interpolating polynomial.
Interpolation can be used to find the approximate value (or the missing value) of | How is interpolation related to the concept of regression?
Interpolation is the process of fitting a number of points between x=a and x=b exactly to an interpolating polynomial.
Interpolation can be used to find the approximate value (or the missing value) of y in the domain x=[a,b] with better accuracy than regression... | How is interpolation related to the concept of regression?
Interpolation is the process of fitting a number of points between x=a and x=b exactly to an interpolating polynomial.
Interpolation can be used to find the approximate value (or the missing value) of |
10,659 | How is interpolation related to the concept of regression? | Compared to interpolation, regression takes the uncertainty of measurements into consideration. The pairs of observed values may be noisy. | How is interpolation related to the concept of regression? | Compared to interpolation, regression takes the uncertainty of measurements into consideration. The pairs of observed values may be noisy. | How is interpolation related to the concept of regression?
Compared to interpolation, regression takes the uncertainty of measurements into consideration. The pairs of observed values may be noisy. | How is interpolation related to the concept of regression?
Compared to interpolation, regression takes the uncertainty of measurements into consideration. The pairs of observed values may be noisy. |
10,660 | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | Bishop is a great book. I hope these suggestions help with your study:
The author himself has posted some slides for Chapters 1, 2, 3 & 8, as well as many solutions.
A reading group at INRIA have posted their own slides covering every chapter.
João Pedro Neto has posted some notes and workings in R here. (Scroll down ... | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | Bishop is a great book. I hope these suggestions help with your study:
The author himself has posted some slides for Chapters 1, 2, 3 & 8, as well as many solutions.
A reading group at INRIA have pos | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
Bishop is a great book. I hope these suggestions help with your study:
The author himself has posted some slides for Chapters 1, 2, 3 & 8, as well as many solutions.
A reading group at INRIA have posted their own slides cover... | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
Bishop is a great book. I hope these suggestions help with your study:
The author himself has posted some slides for Chapters 1, 2, 3 & 8, as well as many solutions.
A reading group at INRIA have pos |
10,661 | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | I would recommend these resources to you:
Tom Mitchell: Carnegie Mellon University
(Only for Supervised Learning and follows Bishop) Pattern Recognition: Indian Institute of Science (I personally like this course as I have attended it, but this course requires you to know probability theory.)
Both the courses are mat... | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | I would recommend these resources to you:
Tom Mitchell: Carnegie Mellon University
(Only for Supervised Learning and follows Bishop) Pattern Recognition: Indian Institute of Science (I personally lik | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
I would recommend these resources to you:
Tom Mitchell: Carnegie Mellon University
(Only for Supervised Learning and follows Bishop) Pattern Recognition: Indian Institute of Science (I personally like this course as I have at... | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
I would recommend these resources to you:
Tom Mitchell: Carnegie Mellon University
(Only for Supervised Learning and follows Bishop) Pattern Recognition: Indian Institute of Science (I personally lik |
10,662 | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | https://www.cs.toronto.edu/~rsalakhu/STA4273_2015/
This course closely follows part of Bishop's. It has lecture videos with it. | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | https://www.cs.toronto.edu/~rsalakhu/STA4273_2015/
This course closely follows part of Bishop's. It has lecture videos with it. | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
https://www.cs.toronto.edu/~rsalakhu/STA4273_2015/
This course closely follows part of Bishop's. It has lecture videos with it. | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
https://www.cs.toronto.edu/~rsalakhu/STA4273_2015/
This course closely follows part of Bishop's. It has lecture videos with it. |
10,663 | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | jupyter notebooks with python implementations and scikit-learn usage at PRML | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | jupyter notebooks with python implementations and scikit-learn usage at PRML | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
jupyter notebooks with python implementations and scikit-learn usage at PRML | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
jupyter notebooks with python implementations and scikit-learn usage at PRML |
10,664 | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | I think an often overlooked book is Information Theory, Inference, and Learning Algorithms by David MacKay.
It follows the general framework of PRML, since the authors seem to have a similar (at least in my view) perspective. Depending on your background -- whether or not you enjoy concepts like information theory/codi... | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources? | I think an often overlooked book is Information Theory, Inference, and Learning Algorithms by David MacKay.
It follows the general framework of PRML, since the authors seem to have a similar (at least | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
I think an often overlooked book is Information Theory, Inference, and Learning Algorithms by David MacKay.
It follows the general framework of PRML, since the authors seem to have a similar (at least in my view) perspective. ... | I am learning from Pattern Recognition and Machine Learning, Chris Bishop any good resources?
I think an often overlooked book is Information Theory, Inference, and Learning Algorithms by David MacKay.
It follows the general framework of PRML, since the authors seem to have a similar (at least |
10,665 | Not normalizing data before PCA gives better explained variance ratio | Depends on the goal of your analysis. Some common practices, some of which are mentioned in whuber's link:
Standardizing is usually done when the variables on which the PCA is performed are not measured on the same scale. Note that standardizing implies assigning equal importance to all variables.
If they are not meas... | Not normalizing data before PCA gives better explained variance ratio | Depends on the goal of your analysis. Some common practices, some of which are mentioned in whuber's link:
Standardizing is usually done when the variables on which the PCA is performed are not measu | Not normalizing data before PCA gives better explained variance ratio
Depends on the goal of your analysis. Some common practices, some of which are mentioned in whuber's link:
Standardizing is usually done when the variables on which the PCA is performed are not measured on the same scale. Note that standardizing imp... | Not normalizing data before PCA gives better explained variance ratio
Depends on the goal of your analysis. Some common practices, some of which are mentioned in whuber's link:
Standardizing is usually done when the variables on which the PCA is performed are not measu |
10,666 | Algorithms for Time Series Anomaly Detection | Twitter algorithm is based on
Rosner, B., (May 1983), "Percentage Points for a Generalized ESD
Many-Outlier Procedure" , Technometrics, 25(2), pp. 165-172
I'm sure there have been many techniques and advances since 1983!. I have tested on my internal data, and Twitter's anomaly detection does not identify obvious ... | Algorithms for Time Series Anomaly Detection | Twitter algorithm is based on
Rosner, B., (May 1983), "Percentage Points for a Generalized ESD
Many-Outlier Procedure" , Technometrics, 25(2), pp. 165-172
I'm sure there have been many techniques | Algorithms for Time Series Anomaly Detection
Twitter algorithm is based on
Rosner, B., (May 1983), "Percentage Points for a Generalized ESD
Many-Outlier Procedure" , Technometrics, 25(2), pp. 165-172
I'm sure there have been many techniques and advances since 1983!. I have tested on my internal data, and Twitter's... | Algorithms for Time Series Anomaly Detection
Twitter algorithm is based on
Rosner, B., (May 1983), "Percentage Points for a Generalized ESD
Many-Outlier Procedure" , Technometrics, 25(2), pp. 165-172
I'm sure there have been many techniques |
10,667 | Algorithms for Time Series Anomaly Detection | Here are the options for Anomaly Detection in R as of 2017.
Twitter's AnomalyDetection Package
Works by using Seasonal Hybrid ESD (S-H-ESD);
Builds upon the Generalized ESD test for detecting anomalies;
Can detect both local and global anomalies;
Employing time series decomposition and robust statistical metrics (e.g... | Algorithms for Time Series Anomaly Detection | Here are the options for Anomaly Detection in R as of 2017.
Twitter's AnomalyDetection Package
Works by using Seasonal Hybrid ESD (S-H-ESD);
Builds upon the Generalized ESD test for detecting anomali | Algorithms for Time Series Anomaly Detection
Here are the options for Anomaly Detection in R as of 2017.
Twitter's AnomalyDetection Package
Works by using Seasonal Hybrid ESD (S-H-ESD);
Builds upon the Generalized ESD test for detecting anomalies;
Can detect both local and global anomalies;
Employing time series deco... | Algorithms for Time Series Anomaly Detection
Here are the options for Anomaly Detection in R as of 2017.
Twitter's AnomalyDetection Package
Works by using Seasonal Hybrid ESD (S-H-ESD);
Builds upon the Generalized ESD test for detecting anomali |
10,668 | Algorithms for Time Series Anomaly Detection | I've come across a few sources that may help you but they won't be as easy/convenient as running an R script over your data:
Numenta have a open-sourced their NuPIC platform that is used for many things including anomaly detection.
Netflix's Atlas Project will soon release an open-source outlier/anomaly detection tool... | Algorithms for Time Series Anomaly Detection | I've come across a few sources that may help you but they won't be as easy/convenient as running an R script over your data:
Numenta have a open-sourced their NuPIC platform that is used for many thi | Algorithms for Time Series Anomaly Detection
I've come across a few sources that may help you but they won't be as easy/convenient as running an R script over your data:
Numenta have a open-sourced their NuPIC platform that is used for many things including anomaly detection.
Netflix's Atlas Project will soon release ... | Algorithms for Time Series Anomaly Detection
I've come across a few sources that may help you but they won't be as easy/convenient as running an R script over your data:
Numenta have a open-sourced their NuPIC platform that is used for many thi |
10,669 | Algorithms for Time Series Anomaly Detection | Autobox(my company) provides outlier detection. Twitter's algorithm gets the big outliers, but misses the smaller ones compared to Autobox.
It takes a long time to run, but the results are better for finding the smaller outliers and also changes in the seasonality which are also outliers. Below is the model finding 79... | Algorithms for Time Series Anomaly Detection | Autobox(my company) provides outlier detection. Twitter's algorithm gets the big outliers, but misses the smaller ones compared to Autobox.
It takes a long time to run, but the results are better for | Algorithms for Time Series Anomaly Detection
Autobox(my company) provides outlier detection. Twitter's algorithm gets the big outliers, but misses the smaller ones compared to Autobox.
It takes a long time to run, but the results are better for finding the smaller outliers and also changes in the seasonality which are... | Algorithms for Time Series Anomaly Detection
Autobox(my company) provides outlier detection. Twitter's algorithm gets the big outliers, but misses the smaller ones compared to Autobox.
It takes a long time to run, but the results are better for |
10,670 | Algorithms for Time Series Anomaly Detection | In Python, the Anomaly Detection Toolkit (ADTK) provides really a nice interface and suit of functions. This talk from 2019 provides a walkthrough of the features, but essentially the same material can be found in the examples in the docs. The package provides 13 built-in methods for detection ranging from the very sim... | Algorithms for Time Series Anomaly Detection | In Python, the Anomaly Detection Toolkit (ADTK) provides really a nice interface and suit of functions. This talk from 2019 provides a walkthrough of the features, but essentially the same material ca | Algorithms for Time Series Anomaly Detection
In Python, the Anomaly Detection Toolkit (ADTK) provides really a nice interface and suit of functions. This talk from 2019 provides a walkthrough of the features, but essentially the same material can be found in the examples in the docs. The package provides 13 built-in me... | Algorithms for Time Series Anomaly Detection
In Python, the Anomaly Detection Toolkit (ADTK) provides really a nice interface and suit of functions. This talk from 2019 provides a walkthrough of the features, but essentially the same material ca |
10,671 | How to whiten the data using principal component analysis? | First, you get the mean zero by subtracting the mean $\boldsymbol \mu = \frac{1}{N}\sum \mathbf{x}$.
Second, you get the covariances zero by doing PCA. If $\boldsymbol \Sigma$ is the covariance matrix of your data, then PCA amounts to performing an eigendecomposition $\boldsymbol \Sigma = \mathbf{U} \boldsymbol \Lambda... | How to whiten the data using principal component analysis? | First, you get the mean zero by subtracting the mean $\boldsymbol \mu = \frac{1}{N}\sum \mathbf{x}$.
Second, you get the covariances zero by doing PCA. If $\boldsymbol \Sigma$ is the covariance matrix | How to whiten the data using principal component analysis?
First, you get the mean zero by subtracting the mean $\boldsymbol \mu = \frac{1}{N}\sum \mathbf{x}$.
Second, you get the covariances zero by doing PCA. If $\boldsymbol \Sigma$ is the covariance matrix of your data, then PCA amounts to performing an eigendecompo... | How to whiten the data using principal component analysis?
First, you get the mean zero by subtracting the mean $\boldsymbol \mu = \frac{1}{N}\sum \mathbf{x}$.
Second, you get the covariances zero by doing PCA. If $\boldsymbol \Sigma$ is the covariance matrix |
10,672 | What to learn after Casella & Berger? | I do not think I will be able to give regular time investment to continue learning data analysis
I don't think Casella & Berger is a place to learn data much in the way of data analysis. It's a place to learn some of the tools of statistical theory.
My experience so far telling me to be a statistican one needs to be... | What to learn after Casella & Berger? | I do not think I will be able to give regular time investment to continue learning data analysis
I don't think Casella & Berger is a place to learn data much in the way of data analysis. It's a place | What to learn after Casella & Berger?
I do not think I will be able to give regular time investment to continue learning data analysis
I don't think Casella & Berger is a place to learn data much in the way of data analysis. It's a place to learn some of the tools of statistical theory.
My experience so far telling ... | What to learn after Casella & Berger?
I do not think I will be able to give regular time investment to continue learning data analysis
I don't think Casella & Berger is a place to learn data much in the way of data analysis. It's a place |
10,673 | What to learn after Casella & Berger? | My advice, coming from the opposite perspective (Stats PhD student) is to work through a regression textbook. This seems a natural starting point for someone with a solid theoretical background without any applied experience. I know many graduate students from outside our department start in a regression course.
A goo... | What to learn after Casella & Berger? | My advice, coming from the opposite perspective (Stats PhD student) is to work through a regression textbook. This seems a natural starting point for someone with a solid theoretical background withou | What to learn after Casella & Berger?
My advice, coming from the opposite perspective (Stats PhD student) is to work through a regression textbook. This seems a natural starting point for someone with a solid theoretical background without any applied experience. I know many graduate students from outside our departmen... | What to learn after Casella & Berger?
My advice, coming from the opposite perspective (Stats PhD student) is to work through a regression textbook. This seems a natural starting point for someone with a solid theoretical background withou |
10,674 | What to learn after Casella & Berger? | I'm trying in a roundabout way to be more of a statistician myself, but I'm primarily a psychologist who happens to have some quantitative and methodological interests. To do psychometric work properly, I've been studying advanced (for a psychologist) methods that I wouldn't dream of calculating manually (much less wou... | What to learn after Casella & Berger? | I'm trying in a roundabout way to be more of a statistician myself, but I'm primarily a psychologist who happens to have some quantitative and methodological interests. To do psychometric work properl | What to learn after Casella & Berger?
I'm trying in a roundabout way to be more of a statistician myself, but I'm primarily a psychologist who happens to have some quantitative and methodological interests. To do psychometric work properly, I've been studying advanced (for a psychologist) methods that I wouldn't dream ... | What to learn after Casella & Berger?
I'm trying in a roundabout way to be more of a statistician myself, but I'm primarily a psychologist who happens to have some quantitative and methodological interests. To do psychometric work properl |
10,675 | What to learn after Casella & Berger? | I stumbled upon this one in 2019. My two cents.
I'm a statistics professor with an inclination to do data analysis of various kinds (that's why I chose statistics!). To pick up some practical knowledge, I recommend James, Witten, Hastie and Tibshirani "An Introduction to Statistical Learning". They even have a MOOC ba... | What to learn after Casella & Berger? | I stumbled upon this one in 2019. My two cents.
I'm a statistics professor with an inclination to do data analysis of various kinds (that's why I chose statistics!). To pick up some practical knowled | What to learn after Casella & Berger?
I stumbled upon this one in 2019. My two cents.
I'm a statistics professor with an inclination to do data analysis of various kinds (that's why I chose statistics!). To pick up some practical knowledge, I recommend James, Witten, Hastie and Tibshirani "An Introduction to Statistic... | What to learn after Casella & Berger?
I stumbled upon this one in 2019. My two cents.
I'm a statistics professor with an inclination to do data analysis of various kinds (that's why I chose statistics!). To pick up some practical knowled |
10,676 | What to learn after Casella & Berger? | Answering for others who come to this question later…
real life data analysis
Learn databases (SQL), dplyr/pandas, unix tools (sed, grep), scraping, scripting, data cleaning, and software testing. The various specialised distributions have little value in industry.
An applied regression book like Angrist & Pischke, ... | What to learn after Casella & Berger? | Answering for others who come to this question later…
real life data analysis
Learn databases (SQL), dplyr/pandas, unix tools (sed, grep), scraping, scripting, data cleaning, and software testing. | What to learn after Casella & Berger?
Answering for others who come to this question later…
real life data analysis
Learn databases (SQL), dplyr/pandas, unix tools (sed, grep), scraping, scripting, data cleaning, and software testing. The various specialised distributions have little value in industry.
An applied re... | What to learn after Casella & Berger?
Answering for others who come to this question later…
real life data analysis
Learn databases (SQL), dplyr/pandas, unix tools (sed, grep), scraping, scripting, data cleaning, and software testing. |
10,677 | Why using Newton's method for logistic regression optimization is called iterative re-weighted least squares? | Summary: GLMs are fit via Fisher scoring which, as Dimitriy V. Masterov notes, is Newton-Raphson with the expected Hessian instead (i.e. we use an estimate of the Fisher information instead of the observed information). If we are using the canonical link function it turns out that the observed Hessian equals the expect... | Why using Newton's method for logistic regression optimization is called iterative re-weighted least | Summary: GLMs are fit via Fisher scoring which, as Dimitriy V. Masterov notes, is Newton-Raphson with the expected Hessian instead (i.e. we use an estimate of the Fisher information instead of the obs | Why using Newton's method for logistic regression optimization is called iterative re-weighted least squares?
Summary: GLMs are fit via Fisher scoring which, as Dimitriy V. Masterov notes, is Newton-Raphson with the expected Hessian instead (i.e. we use an estimate of the Fisher information instead of the observed info... | Why using Newton's method for logistic regression optimization is called iterative re-weighted least
Summary: GLMs are fit via Fisher scoring which, as Dimitriy V. Masterov notes, is Newton-Raphson with the expected Hessian instead (i.e. we use an estimate of the Fisher information instead of the obs |
10,678 | what happens when a model is having more parameters than training samples | When talking about neural networks (nowadays especially deep neural networks), it is nearly always the case that the network has far more parameters than training samples.
Theoretically, a simple two-layer neural network with $2n+d$ parameters is capable of perfectly fitting any dataset of $n$ samples of dimension $d$ ... | what happens when a model is having more parameters than training samples | When talking about neural networks (nowadays especially deep neural networks), it is nearly always the case that the network has far more parameters than training samples.
Theoretically, a simple two- | what happens when a model is having more parameters than training samples
When talking about neural networks (nowadays especially deep neural networks), it is nearly always the case that the network has far more parameters than training samples.
Theoretically, a simple two-layer neural network with $2n+d$ parameters is... | what happens when a model is having more parameters than training samples
When talking about neural networks (nowadays especially deep neural networks), it is nearly always the case that the network has far more parameters than training samples.
Theoretically, a simple two- |
10,679 | How the embedding layer is trained in Keras Embedding layer | Embedding layers in Keras are trained just like any other layer in your network architecture: they are tuned to minimize the loss function by using the selected optimization method. The major difference with other layers, is that their output is not a mathematical function of the input. Instead the input to the layer i... | How the embedding layer is trained in Keras Embedding layer | Embedding layers in Keras are trained just like any other layer in your network architecture: they are tuned to minimize the loss function by using the selected optimization method. The major differen | How the embedding layer is trained in Keras Embedding layer
Embedding layers in Keras are trained just like any other layer in your network architecture: they are tuned to minimize the loss function by using the selected optimization method. The major difference with other layers, is that their output is not a mathemat... | How the embedding layer is trained in Keras Embedding layer
Embedding layers in Keras are trained just like any other layer in your network architecture: they are tuned to minimize the loss function by using the selected optimization method. The major differen |
10,680 | How the embedding layer is trained in Keras Embedding layer | The embedding layer is just a projection from discrete and sparse 1-hot-vector into a continuous and dense latent space. It is a matrix of (n,m) where n is your vocabulary size and n is your desired latent space dimensions. Only in practice, there's no need to actually do the matrix multiplication, and instead you can ... | How the embedding layer is trained in Keras Embedding layer | The embedding layer is just a projection from discrete and sparse 1-hot-vector into a continuous and dense latent space. It is a matrix of (n,m) where n is your vocabulary size and n is your desired l | How the embedding layer is trained in Keras Embedding layer
The embedding layer is just a projection from discrete and sparse 1-hot-vector into a continuous and dense latent space. It is a matrix of (n,m) where n is your vocabulary size and n is your desired latent space dimensions. Only in practice, there's no need to... | How the embedding layer is trained in Keras Embedding layer
The embedding layer is just a projection from discrete and sparse 1-hot-vector into a continuous and dense latent space. It is a matrix of (n,m) where n is your vocabulary size and n is your desired l |
10,681 | What is the difference between pooled cross sectional data and panel data? | When I see panel data, I think longitudinal data, so observations collected on the same individuals at multiple times, on the same topics. Repeated cross sections should be the same topics, but you get different samples of individuals at each observation. I'd welcome other descriptions. | What is the difference between pooled cross sectional data and panel data? | When I see panel data, I think longitudinal data, so observations collected on the same individuals at multiple times, on the same topics. Repeated cross sections should be the same topics, but you ge | What is the difference between pooled cross sectional data and panel data?
When I see panel data, I think longitudinal data, so observations collected on the same individuals at multiple times, on the same topics. Repeated cross sections should be the same topics, but you get different samples of individuals at each ob... | What is the difference between pooled cross sectional data and panel data?
When I see panel data, I think longitudinal data, so observations collected on the same individuals at multiple times, on the same topics. Repeated cross sections should be the same topics, but you ge |
10,682 | What is the difference between pooled cross sectional data and panel data? | The answer here is pretty straight forward:
Both pooled cross sectional data and pure panel data collect data over time (this can range from 2 time periods to any large number). The key difference between the two is the "units" we follow. I am defining units as households, countries, or whatever we are collecting data ... | What is the difference between pooled cross sectional data and panel data? | The answer here is pretty straight forward:
Both pooled cross sectional data and pure panel data collect data over time (this can range from 2 time periods to any large number). The key difference bet | What is the difference between pooled cross sectional data and panel data?
The answer here is pretty straight forward:
Both pooled cross sectional data and pure panel data collect data over time (this can range from 2 time periods to any large number). The key difference between the two is the "units" we follow. I am d... | What is the difference between pooled cross sectional data and panel data?
The answer here is pretty straight forward:
Both pooled cross sectional data and pure panel data collect data over time (this can range from 2 time periods to any large number). The key difference bet |
10,683 | What is the difference between pooled cross sectional data and panel data? | Cross-sectional data, or a cross section of a study population, in statistics and econometrics is a type of one- dimensional data set. Cross-sectional data refers to data collected by
observing many subjects (such as
individuals, firms or countries/regions)
at the same point of time, or without
regard to differences in... | What is the difference between pooled cross sectional data and panel data? | Cross-sectional data, or a cross section of a study population, in statistics and econometrics is a type of one- dimensional data set. Cross-sectional data refers to data collected by
observing many s | What is the difference between pooled cross sectional data and panel data?
Cross-sectional data, or a cross section of a study population, in statistics and econometrics is a type of one- dimensional data set. Cross-sectional data refers to data collected by
observing many subjects (such as
individuals, firms or countr... | What is the difference between pooled cross sectional data and panel data?
Cross-sectional data, or a cross section of a study population, in statistics and econometrics is a type of one- dimensional data set. Cross-sectional data refers to data collected by
observing many s |
10,684 | What is the difference between pooled cross sectional data and panel data? | Based on the definition of Corey, we have following methodology to estimate the model with the pooled cross-sectional data and panel data.
Pooled cross section: one way fixed effects or random effects (only time) or just pooled OLS.
Panel data: two (or one) way fixed effects/random effects (either time or individual ... | What is the difference between pooled cross sectional data and panel data? | Based on the definition of Corey, we have following methodology to estimate the model with the pooled cross-sectional data and panel data.
Pooled cross section: one way fixed effects or random effect | What is the difference between pooled cross sectional data and panel data?
Based on the definition of Corey, we have following methodology to estimate the model with the pooled cross-sectional data and panel data.
Pooled cross section: one way fixed effects or random effects (only time) or just pooled OLS.
Panel data... | What is the difference between pooled cross sectional data and panel data?
Based on the definition of Corey, we have following methodology to estimate the model with the pooled cross-sectional data and panel data.
Pooled cross section: one way fixed effects or random effect |
10,685 | What is the difference between pooled cross sectional data and panel data? | This is from "Basic Econometrics" by Gujarati (4th Edition, P28):
Panel, Longitudinal, or Micropanel Data This is a special type of
pooled data in which the same cross-sectional unit (say, a family or a firm)
is surveyed over time. For example, the U.S. Department of Commerce carries
out a census of housing at periodic... | What is the difference between pooled cross sectional data and panel data? | This is from "Basic Econometrics" by Gujarati (4th Edition, P28):
Panel, Longitudinal, or Micropanel Data This is a special type of
pooled data in which the same cross-sectional unit (say, a family or | What is the difference between pooled cross sectional data and panel data?
This is from "Basic Econometrics" by Gujarati (4th Edition, P28):
Panel, Longitudinal, or Micropanel Data This is a special type of
pooled data in which the same cross-sectional unit (say, a family or a firm)
is surveyed over time. For example, ... | What is the difference between pooled cross sectional data and panel data?
This is from "Basic Econometrics" by Gujarati (4th Edition, P28):
Panel, Longitudinal, or Micropanel Data This is a special type of
pooled data in which the same cross-sectional unit (say, a family or |
10,686 | What is the difference between pooled cross sectional data and panel data? | Pooled data is also panel data but the inverse is not true. | What is the difference between pooled cross sectional data and panel data? | Pooled data is also panel data but the inverse is not true. | What is the difference between pooled cross sectional data and panel data?
Pooled data is also panel data but the inverse is not true. | What is the difference between pooled cross sectional data and panel data?
Pooled data is also panel data but the inverse is not true. |
10,687 | Regression modelling with unequal variance | Pills against the "megaphone effect" include (among others):
Use log or square root transform $Y$. This is not exact but sometimes it tames the widening.
Use weighted least square regression. In this approach, each observation is given its own variance factor. This answer shows how to use WLSR in R (for instance if t... | Regression modelling with unequal variance | Pills against the "megaphone effect" include (among others):
Use log or square root transform $Y$. This is not exact but sometimes it tames the widening.
Use weighted least square regression. In thi | Regression modelling with unequal variance
Pills against the "megaphone effect" include (among others):
Use log or square root transform $Y$. This is not exact but sometimes it tames the widening.
Use weighted least square regression. In this approach, each observation is given its own variance factor. This answer sh... | Regression modelling with unequal variance
Pills against the "megaphone effect" include (among others):
Use log or square root transform $Y$. This is not exact but sometimes it tames the widening.
Use weighted least square regression. In thi |
10,688 | Regression modelling with unequal variance | The gls function in the nlme package for R can estimate the regression and the relationship with the variance at the same time. See the weights argument and the 2nd example on the help page. | Regression modelling with unequal variance | The gls function in the nlme package for R can estimate the regression and the relationship with the variance at the same time. See the weights argument and the 2nd example on the help page. | Regression modelling with unequal variance
The gls function in the nlme package for R can estimate the regression and the relationship with the variance at the same time. See the weights argument and the 2nd example on the help page. | Regression modelling with unequal variance
The gls function in the nlme package for R can estimate the regression and the relationship with the variance at the same time. See the weights argument and the 2nd example on the help page. |
10,689 | Regression modelling with unequal variance | With the gamlss package you can model the error distribution of the response as a linear, a non-linear, or a smooth function of the explanatory variables. This seems to be a quite powerful approach (I learned a lot about all the possibilities that might arise during the model selection process) and everything is nicely... | Regression modelling with unequal variance | With the gamlss package you can model the error distribution of the response as a linear, a non-linear, or a smooth function of the explanatory variables. This seems to be a quite powerful approach (I | Regression modelling with unequal variance
With the gamlss package you can model the error distribution of the response as a linear, a non-linear, or a smooth function of the explanatory variables. This seems to be a quite powerful approach (I learned a lot about all the possibilities that might arise during the model ... | Regression modelling with unequal variance
With the gamlss package you can model the error distribution of the response as a linear, a non-linear, or a smooth function of the explanatory variables. This seems to be a quite powerful approach (I |
10,690 | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | You are perfectly right, probability is the measure of uncertainty. Coin flip is a nice example, as discussed in another thread. Tossing a coin is a physical, deterministic process. In fact there are people who have learned to flip the coin in such way to get the outcome they want and are machines that produce determin... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | You are perfectly right, probability is the measure of uncertainty. Coin flip is a nice example, as discussed in another thread. Tossing a coin is a physical, deterministic process. In fact there are | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
You are perfectly right, probability is the measure of uncertainty. Coin flip is a nice example, as discussed in another thread. Tossing a coin is a physical, deterministic process. In fact there are people who have learn... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
You are perfectly right, probability is the measure of uncertainty. Coin flip is a nice example, as discussed in another thread. Tossing a coin is a physical, deterministic process. In fact there are |
10,691 | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | There is a long and deep history of uncertainty and the quantification of uncertainty, with terms like "subjective probability." A key result is Cox's Theorem. He posited three properties of any measure or representation of uncertainty:
Divisibility and comparability – The plausibility of a proposition is a real num... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | There is a long and deep history of uncertainty and the quantification of uncertainty, with terms like "subjective probability." A key result is Cox's Theorem. He posited three properties of any mea | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
There is a long and deep history of uncertainty and the quantification of uncertainty, with terms like "subjective probability." A key result is Cox's Theorem. He posited three properties of any measure or representatio... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
There is a long and deep history of uncertainty and the quantification of uncertainty, with terms like "subjective probability." A key result is Cox's Theorem. He posited three properties of any mea |
10,692 | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | The short answer is yes. The first chapter of this phd thesis has an example with a simulation of flipping a throwing pin. The outcome 'pin-up' or 'pin-down' depends on a number of variables (like rotation speed and size), which we do not usually control in everyday life. So in the simulation the system is deterministi... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | The short answer is yes. The first chapter of this phd thesis has an example with a simulation of flipping a throwing pin. The outcome 'pin-up' or 'pin-down' depends on a number of variables (like rot | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
The short answer is yes. The first chapter of this phd thesis has an example with a simulation of flipping a throwing pin. The outcome 'pin-up' or 'pin-down' depends on a number of variables (like rotation speed and size)... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
The short answer is yes. The first chapter of this phd thesis has an example with a simulation of flipping a throwing pin. The outcome 'pin-up' or 'pin-down' depends on a number of variables (like rot |
10,693 | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | Talking quantum physics might nevertheless help to appreciate certain issues and paradoxes. Take for example lemur’s comment:
..., but these hurt my philosophical feelings: QM is Nature’s way of having to avoid dealing with an infinite number of bits
But there is a paradox here, since it seems that Nature still requi... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)? | Talking quantum physics might nevertheless help to appreciate certain issues and paradoxes. Take for example lemur’s comment:
..., but these hurt my philosophical feelings: QM is Nature’s way of havi | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
Talking quantum physics might nevertheless help to appreciate certain issues and paradoxes. Take for example lemur’s comment:
..., but these hurt my philosophical feelings: QM is Nature’s way of having to avoid dealing w... | Is everyday probability just a way of dealing with the unknown (not talking quantum physics here)?
Talking quantum physics might nevertheless help to appreciate certain issues and paradoxes. Take for example lemur’s comment:
..., but these hurt my philosophical feelings: QM is Nature’s way of havi |
10,694 | What intuitively is "bias"? | Bias is the difference between the expected value of an estimator and the true value being estimated. For example the sample mean for a simple random sample (SRS) is an unbiased estimator of the population mean because if you take all the possible SRS's find their means, and take the mean of those means then you will ... | What intuitively is "bias"? | Bias is the difference between the expected value of an estimator and the true value being estimated. For example the sample mean for a simple random sample (SRS) is an unbiased estimator of the popu | What intuitively is "bias"?
Bias is the difference between the expected value of an estimator and the true value being estimated. For example the sample mean for a simple random sample (SRS) is an unbiased estimator of the population mean because if you take all the possible SRS's find their means, and take the mean o... | What intuitively is "bias"?
Bias is the difference between the expected value of an estimator and the true value being estimated. For example the sample mean for a simple random sample (SRS) is an unbiased estimator of the popu |
10,695 | What intuitively is "bias"? | Bias means that the expected value of the estimator is not equal to the population parameter.
Intuitively in a regression analysis, this would mean that the estimate of one of the parameters is too high or too low. However, ordinary least squares regression estimates are BLUE, which stands for best linear unbiased esti... | What intuitively is "bias"? | Bias means that the expected value of the estimator is not equal to the population parameter.
Intuitively in a regression analysis, this would mean that the estimate of one of the parameters is too hi | What intuitively is "bias"?
Bias means that the expected value of the estimator is not equal to the population parameter.
Intuitively in a regression analysis, this would mean that the estimate of one of the parameters is too high or too low. However, ordinary least squares regression estimates are BLUE, which stands f... | What intuitively is "bias"?
Bias means that the expected value of the estimator is not equal to the population parameter.
Intuitively in a regression analysis, this would mean that the estimate of one of the parameters is too hi |
10,696 | What intuitively is "bias"? | In Linear regression analysis, bias refer to the error that is introduced by approximating a real-life problem, which may be complicated, by a much simpler model. In simple terms, you assume a simple linear model such as y*=(a*)x+b* where as in real life the business problem could be y = ax^3 + bx^2+c.
It can be said t... | What intuitively is "bias"? | In Linear regression analysis, bias refer to the error that is introduced by approximating a real-life problem, which may be complicated, by a much simpler model. In simple terms, you assume a simple | What intuitively is "bias"?
In Linear regression analysis, bias refer to the error that is introduced by approximating a real-life problem, which may be complicated, by a much simpler model. In simple terms, you assume a simple linear model such as y*=(a*)x+b* where as in real life the business problem could be y = ax^... | What intuitively is "bias"?
In Linear regression analysis, bias refer to the error that is introduced by approximating a real-life problem, which may be complicated, by a much simpler model. In simple terms, you assume a simple |
10,697 | Why Lasso or ElasticNet perform better than Ridge when the features are correlated | Suppose you have two highly correlated predictor variables $x,z$, and suppose both are centered and scaled (to mean zero, variance one). Then the ridge penalty on the parameter vector is $\beta_1^2 + \beta_2^2$ while the lasso penalty term is $ \mid \beta_1 \mid + \mid \beta_2 \mid$. Now, since the model is supposed ... | Why Lasso or ElasticNet perform better than Ridge when the features are correlated | Suppose you have two highly correlated predictor variables $x,z$, and suppose both are centered and scaled (to mean zero, variance one). Then the ridge penalty on the parameter vector is $\beta_1^2 + | Why Lasso or ElasticNet perform better than Ridge when the features are correlated
Suppose you have two highly correlated predictor variables $x,z$, and suppose both are centered and scaled (to mean zero, variance one). Then the ridge penalty on the parameter vector is $\beta_1^2 + \beta_2^2$ while the lasso penalty t... | Why Lasso or ElasticNet perform better than Ridge when the features are correlated
Suppose you have two highly correlated predictor variables $x,z$, and suppose both are centered and scaled (to mean zero, variance one). Then the ridge penalty on the parameter vector is $\beta_1^2 + |
10,698 | Why Lasso or ElasticNet perform better than Ridge when the features are correlated | most important difference between lasso and ridge is that lasso naturally makes a selection, expecially where covariates are very correlated. it's impossible to be really sure without seeing the fitted coefficients, but it's easy to think that among those correlated features, many were simply useless. | Why Lasso or ElasticNet perform better than Ridge when the features are correlated | most important difference between lasso and ridge is that lasso naturally makes a selection, expecially where covariates are very correlated. it's impossible to be really sure without seeing the fitte | Why Lasso or ElasticNet perform better than Ridge when the features are correlated
most important difference between lasso and ridge is that lasso naturally makes a selection, expecially where covariates are very correlated. it's impossible to be really sure without seeing the fitted coefficients, but it's easy to thin... | Why Lasso or ElasticNet perform better than Ridge when the features are correlated
most important difference between lasso and ridge is that lasso naturally makes a selection, expecially where covariates are very correlated. it's impossible to be really sure without seeing the fitte |
10,699 | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$ | Expanding on @whuber's point in the comments, if $Y$ and $Z$ are orthogonal, you have the Pythagorean Theorem:
$$ \|Y\|^2 + \|Z\|^2 = \|Y + Z\|^2 $$
Observe that $\langle Y, Z \rangle \equiv \mathrm{E}[YZ]$ is a valid inner product and that $\|Y\| = \sqrt{\mathrm{E}[Y^2]}$ is the norm induced by that inner product.
Let... | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$ | Expanding on @whuber's point in the comments, if $Y$ and $Z$ are orthogonal, you have the Pythagorean Theorem:
$$ \|Y\|^2 + \|Z\|^2 = \|Y + Z\|^2 $$
Observe that $\langle Y, Z \rangle \equiv \mathrm{E | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$
Expanding on @whuber's point in the comments, if $Y$ and $Z$ are orthogonal, you have the Pythagorean Theorem:
$$ \|Y\|^2 + \|Z\|^2 = \|Y + Z\|^2 $$
Observe that $\langle Y, Z \rangle \equiv \mathrm{E}[YZ]$ is a valid inner product and that $\|Y\| = \sqrt{\... | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$
Expanding on @whuber's point in the comments, if $Y$ and $Z$ are orthogonal, you have the Pythagorean Theorem:
$$ \|Y\|^2 + \|Z\|^2 = \|Y + Z\|^2 $$
Observe that $\langle Y, Z \rangle \equiv \mathrm{E |
10,700 | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$ | I will go for a purely geometric approach for a very specific scenario. Let us consider a discrete valued random variable $X$ taking values $\{x_1,x_2\}$ with probabilities $(p_1,p_2)$. We will further assume that this random variable can be represented in $\mathbb{R}^2$ as a vector, $\mathbf{X} = \left(x_1\sqrt{p_1},x... | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$ | I will go for a purely geometric approach for a very specific scenario. Let us consider a discrete valued random variable $X$ taking values $\{x_1,x_2\}$ with probabilities $(p_1,p_2)$. We will furthe | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$
I will go for a purely geometric approach for a very specific scenario. Let us consider a discrete valued random variable $X$ taking values $\{x_1,x_2\}$ with probabilities $(p_1,p_2)$. We will further assume that this random variable can be represented in ... | Intuition (geometric or other) of $Var(X) = E[X^2] - (E[X])^2$
I will go for a purely geometric approach for a very specific scenario. Let us consider a discrete valued random variable $X$ taking values $\{x_1,x_2\}$ with probabilities $(p_1,p_2)$. We will furthe |
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