I am trying to determine if the default t-test in sci-py is Welch's t-test or the student's t-test. I can't find an answer anywhere. I used the following code to do my t-test analysis. If it is not Welch's, can anyone advise me how to do Welch's
t_test = scipy.stats.ttest_ind(a, b, axis=0)
Thanks
Emma
The difference between those two tests is the assumption regarding equal variances. Welch's test does not assume equal variances. If you pass equal_var=False to your t-test (scipy.stats.ttest_ind(a, b, equal_var=False, axis=0) it will conduct Welch's test.
From the docs:
equal_var : bool, optional If True (default), perform a standard
independent 2 sample test that assumes equal population variances.
If False, perform Welch’s t-test, which does not assume equal
population variance.
Related
I looked into using xcov to calculate auto-covariances of a Tx1 time series vector. Using xcov in the first instance, it seemed as if this function would not devide by T as opposed to what the general formula for sample autocovariance would suggest: ?
Can someone confirm whether I need to devide by T to get a vector of sample auto-covariances? If I do not devide by T, the covariances seem to explode, as more and more terms are added. It is also not clear from the documentation of xcov.
For instance:
test = randn(10,1);
xcov(test)
Best
Marcel
I am taking an Econometrics course, and have been trying to use Python rather than the propreitry STATA and EVIEWS they set the assignments in.
In one of the questions, I have consumption data over time. I am asked to compute it in two ways.
The first way is calculating a model of the form consumption = Aexp(Bt), and the second way is to log both sides and do ordinary OLS on log(consumption) = alpha + Bt
I know how to do the second way. Howver, when I try to do the first way it goes wrong. Using statsmodels, I can exponentiate the time data (after normalising), but this calculates a regression in the form consumption = Aexp(t) + B, which is not what I want. (I want to specify where the parameters go). In sklearn I could find a polynomial regression, but not exponential.
Then I found scipy.curve_fit
However this seems to have two problems:
(1) It seems to rely on initial guesses for parameters, which means my output will end up being different from proprietry software (whereas output for things like OLS are the same) [as I assume initial guesses means some iterative solution is done which is helpful for very weird and wonderful functions, but I assume fairly standard results hold for exponential regression]
(2) every time I try to implement it, it just returns the guess parameters.
Here is my code
`consumption_data = pd.read_csv(......\consumption.csv")
def func(x,a,b):
return a * np.exp(b*x)
xdata = consumption_data.YEAR
ydata = consumption_data.CONSUMPTION
ydata = (ydata - 1948)/100
popt, pcov = curve_fit(func, xdata, ydata, (1,1))
print(popt)
plt.plot(xdata, func(xdata, *popt), 'g--',)
`
The scipy.optimize code is basically just copy-pasted from their tutorial
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html
short answer: use statsmodels GLM
statsmodels does not have nonlinear least squares. The best python library for that is lmfit https://pypi.org/project/lmfit/
curve_fit, lmfit and nonlinear least squares algorithm in general find an iterative solution to the optimization problem. Even when we have to provide starting values, the solution is in many cases the same across packages up to convergence tolerance, e.g. 1e-5 or 1e-6.
Many standard models in statistics and econometrics have a single global maximum with well behaved data. However, in other cases like mixture models, there might be many local optima and the estimation might converge to one of them.
To the specific case:
consumption = A exp(B t)
can be rewritten as
consumption = exp(a + B t)
So this is just a single index model or a generalized linear model with an exponential mean function.
The general version has the expectation of the dependent variable as a nonlinear function of a linear combination of the explanatory variables:
E(y | x) = g(x b)
This can be estimated with statsmodels with GLM with family Gaussian and the log-link.
Aside: In econometrics, there is a literature to use Poisson quasi-likelihood as an estimator for exp models instead of taking the log of the dependent variable.
Poisson usually uses the log-link function as in the above.
However, using GLM allows us to use log-link, i.e. exponential mean function, with any of the supported distribution families. The main difference is in the underlying variance assumption. Gaussian assumes constant variance, Poisson assumes that the variance is proportional to the mean and Gamma assumes that the variance is quadratic in the mean.
If we use a robust sandwich covariance estimator for parameter inference, then standard errors and inference are correct even if the variance function is misspecified.
Consider 2 sets
A = randi(1000,100,7);
B = randi(700,300,7);
I would like to find a function : B# = optimf(A,B) and gives me B# = {100x7} which is a collection of rows from B such that some attribute( eg. mean ) is minimum.
For eg: B# = optimf(A,B) such that mean(B#) - mean(A) is minimum.
Any ideas?
According to me Optimization is finding the best suitable value of a function. For example, if you have an equation and you need to find minimum value at which the equation satisfies criteria, then this is an optimization problem.
There is no such optimization function AFAIK for your function. But, you can take help of optimization algorithms like Least Square Errors
Or simply use some filter of MATLAB.
I hope it helps.
UPDATE
(Not a sophisticated solution but just works in some cases.)
Step 1-
Make a for-loop and select randomly some values from input vector. So you get a random subset.
Step 2
Define a cost function. A function that can measure how good is the subset. The function will take input as vector and gives output a numerical quantity such as % of quality.
Step 3
Go on taking these readings. Take the max value of output function and its corresponding vector. That should be a solution.
OR
use algorithms like ACO
I performed PCA on a 63*2308 matrix and obtained a score and a co-efficient matrix. The score matrix is 63*2308 and the co-efficient matrix is 2308*2308 in dimensions.
How do i extract the column names for the top 100 features which are most important so that i can perform regression on them?
PCA should give you both a set of eigenvectors (your co-efficient matrix) and a vector of eigenvalues (1*2308) often referred to as lambda). You might been to use a different PCA function in matlab to get them.
The eigenvalues indicate how much of your data each eigenvector explains. A simple method for selecting features would be to select the 100 features with the highest eigen values. This gives you a set of feature which explain most of the variance in the data.
If you need to justify your approach for a write up you can actually calculate the amount of variance explained per eigenvector and cut of at, for example, 95% variance explained.
Bear in mind that selecting based solely on eigenvalue, might not correspond to the set of features most important to your regression, so if you don't get the performance you expect you might want to try a different feature selection method such as recursive feature selection. I would suggest using google scholar to find a couple of papers doing something similar and see what methods they use.
A quick matlab example of taking the top 100 principle components using PCA.
[eigenvectors, projected_data, eigenvalues] = princomp(X);
[foo, feature_idx] = sort(eigenvalues, 'descend');
selected_projected_data = projected(:, feature_idx(1:100));
Have you tried with
B = sort(your_matrix,2,'descend');
C = B(:,1:100);
Be careful!
With just 63 observations and 2308 variables, your PCA result will be meaningless because the data is underspecified. You should have at least (rule of thumb) dimensions*3 observations.
With 63 observations, you can at most define a 62 dimensional hyperspace!
I would like to measure the goodness-of-fit to an exponential decay curve. I am using the lsqcurvefit MATLAB function. I have been suggested by someone to do a chi-square test.
I would like to use the MATLAB function chi2gof but I am not sure how I would tell it that the data is being fitted to an exponential curve
The chi2gof function tests the null hypothesis that a set of data, say X, is a random sample drawn from some specified distribution (such as the exponential distribution).
From your description in the question, it sounds like you want to see how well your data X fits an exponential decay function. I really must emphasize, this is completely different to testing whether X is a random sample drawn from the exponential distribution. If you use chi2gof for your stated purpose, you'll get meaningless results.
The usual approach for testing the goodness of fit for some data X to some function f is least squares, or some variant on least squares. Further, a least squares approach can be used to generate test statistics that test goodness-of-fit, many of which are distributed according to the chi-square distribution. I believe this is probably what your friend was referring to.
EDIT: I have a few spare minutes so here's something to get you started. DISCLAIMER: I've never worked specifically on this problem, so what follows may not be correct. I'm going to assume you have a set of data x_n, n = 1, ..., N, and the corresponding timestamps for the data, t_n, n = 1, ..., N. Now, the exponential decay function is y_n = y_0 * e^{-b * t_n}. Note that by taking the natural logarithm of both sides we get: ln(y_n) = ln(y_0) - b * t_n. Okay, so this suggests using OLS to estimate the linear model ln(x_n) = ln(x_0) - b * t_n + e_n. Nice! Because now we can test goodness-of-fit using the standard R^2 measure, which matlab will return in the stats structure if you use the regress function to perform OLS. Hope this helps. Again I emphasize, I came up with this off the top of my head in a couple of minutes, so there may be good reasons why what I've suggested is a bad idea. Also, if you know the initial value of the process (ie x_0), then you may want to look into constrained least squares where you bind the parameter ln(x_0) to its known value.