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Gaussian data distributed in a single dimension requires two parameters to characterise it (mean, variance), and rumour has it that around 30 randomly-selected samples is usually sufficient to pin downestimate these parameters with reasonably high confidence. But what happens as the number of dimensions increases?

In two dimensions (e.g. height, weight) it takes 5 parameters to specifiy a "best-fit" ellipse. In three dimensions, this rises to 9 parameters to describe an ellipsoid, and in 4-D it takes 14 parameters. DoesI am interested to know if the number of samples required to estimate these parameters also riserises at a comparable rate? Indeed, isat a slower rate or (please no!) at a higher rate. Better still, if there was a broadly accepted rule of thumb that suggests how many samples are required to characterise a gaussian distribution in a given number of dimensions?, that would be good to know.

To be more precise, suppose we want to define a symmetrical "best-fit" boundary centred at the mean point inside which we can be confident that 95% of all samples will fall. I want to know how many samples it might take to find the parameters to approximate this boundary (interval in 1-D, ellipse in 2-D, etc) with suitably high (>95%) confidence, and how that number varies as the number of dimensions increases.

Gaussian data distributed in a single dimension requires two parameters to characterise it (mean, variance), and rumour has it that around 30 randomly-selected samples is usually sufficient to pin down these parameters. But what happens as the number of dimensions increases?

In two dimensions (e.g. height, weight) it takes 5 parameters to specifiy a "best-fit" ellipse. In three dimensions, this rises to 9 parameters to describe an ellipsoid, and in 4-D it takes 14 parameters. Does the number of samples required to estimate these parameters also rise at a comparable rate? Indeed, is there a broadly accepted rule of thumb that suggests how many samples are required to characterise a gaussian distribution in a given number of dimensions?

Gaussian data distributed in a single dimension requires two parameters to characterise it (mean, variance), and rumour has it that around 30 randomly-selected samples is usually sufficient to estimate these parameters with reasonably high confidence. But what happens as the number of dimensions increases?

In two dimensions (e.g. height, weight) it takes 5 parameters to specifiy a "best-fit" ellipse. In three dimensions, this rises to 9 parameters to describe an ellipsoid, and in 4-D it takes 14 parameters. I am interested to know if the number of samples required to estimate these parameters also rises at a comparable rate, at a slower rate or (please no!) at a higher rate. Better still, if there was a broadly accepted rule of thumb that suggests how many samples are required to characterise a gaussian distribution in a given number of dimensions, that would be good to know.

To be more precise, suppose we want to define a symmetrical "best-fit" boundary centred at the mean point inside which we can be confident that 95% of all samples will fall. I want to know how many samples it might take to find the parameters to approximate this boundary (interval in 1-D, ellipse in 2-D, etc) with suitably high (>95%) confidence, and how that number varies as the number of dimensions increases.

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omatai
omatai
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When data has a gaussian distribution, how many samples will characterise it?

Gaussian data distributed in a single dimension requires two parameters to characterise it (mean, variance), and rumour has it that around 30 randomly-selected samples is usually sufficient to pin down these parameters. But what happens as the number of dimensions increases?

In two dimensions (e.g. height, weight) it takes 5 parameters to specifiy a "best-fit" ellipse. In three dimensions, this rises to 9 parameters to describe an ellipsoid, and in 4-D it takes 14 parameters. Does the number of samples required to estimate these parameters also rise at a comparable rate? Indeed, is there a broadly accepted rule of thumb that suggests how many samples are required to characterise a gaussian distribution in a given number of dimensions?