I have a complicated theoretical Probability Density Function (PDF) that I define in mathematica and that depends on some parameters that I need to estimate from comparison with real data. From a big simulation done on a cluster and not my laptop I have acquired a lot of events (over 10^9).
The way I understand things, given that I know what the PDF is I 'just' need to sum the probability that those events appear for a given set of parameters and maximise this quantity by adjusting the parameters.
However, given the number of events I would rather work with something less computer-time consuming and work for example with something easily generated like an histogram of my data. But then how would my log-likelihood estimator work?
Thanks a lot for your answers!
Related
I have a few questions regarding using a random effect in a GAM. First, how do you interpret and communicate the output graph?
I have fire modeled as a random effect in this GAM because it is largely a random occurrence at my different field sites and I only noted it as a binary. It wouldn't work as a normal variable since it has too few levels and there is also relatively few sites with fire. However, it greatly improved model variance capture when included so I don't want to simply exclude it. I don't know how to interpret the output and I am also not entirely confident that there wouldn't be another way to include it in the model other than as a random effect. Any help would be greatly appreciated!
The effect has been modelled as a random slope if you didn't code it as a factor in the data. The value on the y axis is the estimated slope; it will be a little smaller in absolute value than if you use Fire as a linear fixed effect in the model formula because it is being penalised (shrunk) towards zero.
This likely should have been fitted as a binary fixed effect; code Fire as a factor with two levels (Yes/No, or Burned / Unburned say). Just because a variable represents something that is random over the data doesn't mean it is a suitable random effect; fire here has some average effect and the fixed effect describes that well. There's nothing stopping you from using Fire coded as a factor as a random effect via the smooth, but with only two levels it's not going the two intercepts aren't going to be estimate that precisely.
Now, if you had repeated observations on n sites and you thought the Fire effect varied across the n sites then you could do s(Site, Fire, bs = 're') where both Site and Fire are factors and you'll get different Fire effects for each Site. Then the plot you show would have many points on it as it is a QQ-plot of the estimated values for the effect of Fire in each Site, hence 1 point per Site. Given the way this model is estimated, these are somewhat assumed to be distributed Gaussian with some variance that is inversely proportional to the smoothness parameter selected by gam() when fitting this random effect smoother. That's why the default plot is as it is; it's a QQ-plot comparing the observed distribution of estimate values of the random effects against the theoretical expectation.
So I have two subsets of data that represent two situations. The one that look more consistent needs to be filtered out (they are noise) while the one looks random are kept (they are motions). The method I was using was to define a moving window = 10 and whenever the standard deviation of the data within the window was smaller than some threshold, I suppressed them. However, this method could not filter out all "consistent" noise while also hurting the inconsistent one (real motion). I was hoping to use some kinds of statistical models and not machine learning to accomplish this. Any suggestions would be appreciated!
noise
real motion
The Kolmogorov–Smirnov test is used to compare two samples to determine if they come from the same distribution. I realized that real world data would never be uniform. So instead of comparing my noise data against the uniform distribution, I used scipy.stats.ks_2samp function to compare any bursts against one real motion burst. I then muted the motion if the return p-value is significantly small, meaning I can reject the hypothesis that two samples are from the same distribution.
I'm trying to work out how to solve what seems like a simple problem, but I can't convince myself of the correct method.
I have time-series data that represents the pdf of a Power output (P), varying over time, also the cdf and quantile functions - f(P,t), F(P,t) and q(p,t). I need to find the pdf, cdf and quantile function for the Energy in a given time interval [t1,t2] from this data - say e(), E(), and qe().
Clearly energy is the integral of the power over [t1,t2], but how do I best calculate e, E and qe ?
My best guess is that since q(p,t) is a power, I should generate qe by integrating q over the time interval, and then calculate the other distributions from that.
Is it as simple as that, or do I need to get to grips with stochastic calculus ?
Additional details for clarification
The data we're getting is a time-series of 'black-box' forecasts for f(P), F(P),q(P) for each time t, where P is the instantaneous power and there will be around 100 forecasts for the interval I'd like to get the e(P) for. By 'Black-box' I mean that there will be a function I can call to evaluate f,F,q for P, but I don't know the underlying distribution.
The black-box functions are almost certainly interpolating output data from the model that produces the power forecasts, but we don't have access to that. I would guess that it won't be anything straightforward, since it comes from a chain of non-linear transformations. It's actually wind farm production forecasts: the wind speeds may be normally distributed, but multiple terrain and turbine transformations will change that.
Further clarification
(I've edited the original text to remove confusing variable names in the energy distribution functions.)
The forecasts will be provided as follows:
The interval [t1,t2] that we need e, E and qe for is sub-divided into 100 (say) sub-intervals k=1...100. For each k we are given a distinct f(P), call them f_k(P). We need to calculate the energy distributions for the interval from this set of f_k(P).
Thanks for the clarification. From what I can tell, you don't have enough information to solve this problem properly. Specifically, you need to have some estimate of the dependence of power from one time step to the next. The longer the time step, the less the dependence; if the steps are long enough, power might be approximately independent from one step to the next, which would be good news because that would simplify the analysis quite a bit. So, how long are the time steps? An hour? A minute? A day?
If the time steps are long enough to be independent, the distribution of energy is the distribution of 100 variables, which will be very nearly normally distributed by the central limit theorem. It's easy to work out the mean and variance of the total energy in this case.
Otherwise, the distribution will be some more complicated result. My guess is that the variance as estimated by the independent-steps approach will be too big -- the actual variance would be somewhat less, I believe.
From what you say, you don't have any information about temporal dependence. Maybe you can find or derive from some other source or sources an estimate the autocorrelation function -- I wouldn't be surprised if that question has already been studied for wind power. I also wouldn't be surprised if a general version of this problem has already been studied -- perhaps you can search for something like "distribution of a sum of autocorrelated variables." You might get some interest in that question on stats.stackexchange.com.
I am running an experiment (it's an image processing experiment) in which I have a set of paper samples and each sample has a set of lines. For each line in the paper sample, its strength is calculated which is denoted by say 's'. For a given paper sample I have to find the variation amongst the strength values 's'. If the variation is above a certain limit, we have to discard that paper.
1) I started with the Standard Deviation of the values, but the problem I am facing is that for each sample, order of magnitude for s (because of various properties of line like its length, sharpness, darkness etc) might differ and also the calculated Standard Deviations values are also differing a lot in magnitude. So I can't really use this method for different samples.
Is there any way where I can find that suitable limit which can be applicable for all samples.
I am thinking that since I don't have any history of how the strength value should behave,( for a given sample depending on the order of magnitude of the strength value more variation could be tolerated in that sample whereas because the magnitude is less in another sample, there should be less variation in that sample) I first need to find a way of baselining the variation in different samples. I don't know what approaches I could try to get started.
Please note that I have to tell variation between lines within a sample whereas the limit should be applicable for any good sample.
Please help me out.
You seem to have a set of samples. Then, for each sample you want to do two things: 1) compute a descriptive metric and 2) perform outlier detection. Both of these are vast subjects that require some knowledge of the phenomenology and statistics of the underlying problem. However, below are some ideas to get you going.
Compute a metric
Median Absolute Deviation. If your sample strength s has values that can jump by an order of magnitude across a sample then it is understandable that the standard deviation was not a good metric. The standard deviation is notoriously sensitive to outliers. So, try a more robust estimate of dispersion in your data. For example, the MAD estimate uses the median in the underlying computations which is more robust to a large spread in the numbers.
Robust measures of scale. Read up on other robust measures like the Interquartile range.
Perform outlier detection
Thresholding. This is similar to what you are already doing. However, you have to choose a suitable threshold for the metric computed above. You might consider using another robust metric for thresholding the metric. You can compute a robust estimate of their mean (e.g., the median) and a robust estimate of their standard deviation (e.g., 1.4826 * MAD). Then identify outliers as metric values above some number of robust standard deviations above the robust mean.
Histogram Another simple method is to histogram your computed metrics from step #1. This is non-parametric so it doesn't require you to model your data. If can histogram your metric values and then use the top 1% (or some other value) as your threshold limit.
Triangle Method A neat and simple heuristic for thresholding is the triangle method to perform binary classification of a skewed distribution.
Anomaly detection Read up on other outlier detection methods.
I'm trying to find confidence intervals for the means of various variables in a database using SPSS, and I've run into a spot of trouble.
The data is weighted, because each of the people who was surveyed represents a different portion of the overall population. For example, one young man in our sample might represent 28000 young men in the general population. The problem is that SPSS seems to think that the young man's database entries each represent 28000 measurements when they actually just represent one, and this makes SPSS think we have much more data than we actually do. As a result SPSS is giving very very low standard error estimates and very very narrow confidence intervals.
I've tried fixing this by dividing every weight value by the mean weight. This gives plausible figures and an average weight of 1, but I'm not sure the resulting numbers are actually correct.
Is my approach sound? If not, what should I try?
I've been using the Explore command to find mean and standard error (among other things), in case it matters.
You do need to scale weights to the actual sample size, but only the procedures in the Complex Samples option are designed to account for sampling weights properly. The regular weight variable in Statistics is treated as a frequency weight.