I am trying to find an example of strictly stationary but not weakly stationary.
Plus, this example should not be i.i.d time series data.
How can I find this example?
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I want to improve step-by-step, whilst unevenly-sampled data are coming, the value of the first derivative at t = 0 s. For example, if you want to find the initial velocity in a projectile's motion, but you do not know its final position and velocity, however, you are receiving (slowly) the measurements of the projectile's current position and time.
Update - 26 Aug 2018
I would like to give you more details:
"Unevenly-sampled data" means the time intervals are not regular (irregular times between successive measurements). However, data have almost the same sampling frequency, i.e., it is about 15 min. Thus, there are some measurements without changes, because of the nature of the phenomenon (heat transfer). It gives an exponential tendency and I can fit data to a known model, but an important amount of information is required. For practical purposes, I only need to know the value of the very first slope for the whole process.
I tried a progresive Weighted Least Squares (WLS) fitting procedure, with a weight matrix such as
W = diag((0.5).^(1:kk)); % where kk is the last measurement id
But it was using preprocessed data (i.e., jitter-removing, smoothing, and fitting using the theoretical functional). I gave me the following result:
This is a real example of the problem and its "current solution"
It is good for me, but I would like to know if there is an optimal manner of doing that, but employing the raw data (or smoothed data).
IMO, additional data is not relevant to improve the estimate at zero. Because perturbations come into play and the correlation between the first and last samples goes decreasing.
Also, the asymptotic behavior of the phenomenon is probably not known rigorously (is it truly a first order linear model) ? And this can introduce a bias in the measurements.
I would stick to the first points (say up to t=20) and fit a simple model, say quadratic.
If in fact what you are trying to do is to fit a first order linear model to the data, then least-squares fitting on the raw data is fine. If there are significant outliers, robust fitting is preferable.
I'm reading a lot about Precision-Recall curves in order to evaluate my image retrieval system. In particular I'm reading this article about feature extractors in VLFeat and the wikipedia page about precision-recall.
I understand that this curve is useful to evaluate our system performance w.r.t. the number of elements retrieved. So we repeatedly compute precision-recall retrieving the top element, then top 2, top 3 and so on...but my question is: when do we stop?
My intuition is: we stop when our list of retrieved elements has recall equal to 1, so we retrieve all the relevant elements (i.e. there are no false negatives, only true positives).
Same question is for average precision: how many elements should be present in the retrieved result for computing it? If my previous intuition is correct, then we just need to find out what is the smallest list s.t. recall is 1 and use it for compute it AP.
I wonder why all the libraries for computing p-r curve don't show how this is implemented?
An information retrieval system with recall 1 means a perfect system which doesn't seem possible in practice! Precision-Recall curves are good when you need to compare two or more information retrieval systems. Its not about stopping when recall or precision reaches some value. Precision-Recall curve shows pairs of recall and precision values at each point (consider top 3 or 5 documents). You can draw the curve upto any reasonable point.
Curves close to the perfect Precision-Recall curve have a better performance level than the ones closes to the baseline. In other words, a curve above the other curve has a better performance level. Two Precision-Recall curves represent the performance levels of two IR systems: A and B. System A clearly outperforms system B according to the following figure.
Remember: Precision-Recall curve not only used for evaluating IR systems. It can be used to show how much good your classifier is! For example, you can compute precision, recall for a binary classification task and plot the Precision-Recall curve that can give you a good estimate of the performance of your classifier.
For example:
I would encourage you to see this tutorial from Coursera. I believe your idea will become more clear about Precision-Recall curve.
I have a time series that seems to be well described by a univariate local level model (a changing bias in human visual perception, sampled at regular intervals). I have a hunch, however, that the underlying random walk is at least partly driven by the measurements themselves. In order to test this hypothesis, I measure two types of time series. The first type is measured at regular steps, every 12 minutes:
u(0), u(12), u(24), u(36), u(48), u(60), u(72), ...
The second type is measured at partly irregular intervals, alternating between every 6 or 18 minutes:
v(0), v(6), v(24), v(30), v(48), v(54), v(72), ...
Of course I could compare the 6- and 18-minute steps in the v-series: if they're no different, then nothing really happens to the random walk between the measurements. The trouble is that the measurement noise is large compared to the random walk steps.
Is there some more principled way I could test this hypothesis? For instance, could I fit a modified local-level model in which the even and odd random-walk steps have different variances, and compare the two variances?
The post
Classification functions in linear discriminant analysis in R
from user Tyler provides a function to produce the classification functions (not discriminant functions!) from an LDA model generated with lda().
I used these classification functions to calculate all classification scores for my data. I want to use the additional information e.g. to find out which was the second most probable class and to understand the development in different time slices
Now I would like to ask you for your help to interpret the following scenarios:
scores close to/exactly zero (is it possible to claim that this exact class effectively was not recognized?)
single negative scores of higher absolute value than highest positive value (Does it mean anything at all?)
results with all negative scores (in the original interpretation, the highest score determines the classification. Is this intended by the LDA or does it mean that really none of the classifications is a good fit and one could say that no known pattern could be identified?)
single very low positive values while all others are high absolute negative values (can I argue that the "signal strength" is low in this case?)
I know this is more of a statistical than a programming problem. I thought of it as a follow-up of the post at the beginning of this entry.
Thank you very much for your help!
As a property of the markov chain, the stationary distribution has been widely used in many fields like page_rank etc.
However, since the distribution is just a property about the transition matrix and has nothing to do with the init state of the markov chain.
So what's the condition of the transition matrix make the init state has nothing to do with markov chain so it will finally arrive at the stationary distribution after nth iteration.
Markov chains aren't guaranteed to have unique stationary distributions. For example, consider a two state Markov Chain where the transition matrix is the identity matrix. That means that whatever the initial state is, it never changes. So in that case there is no stationary distribution that is independent of the initial case.
Where there is a stationary distribution, unless the initial state is the stationary distribution, the stationary distribution is only reached in the limit as n tends to infinity. So iteration n+1 will be closer to it that iteration n, but however large n is, it won't ever actually be the stationary distribution. However, for practical purposes (i.e. to the limit of the accuracy of floating point numbers in computers), the stationary state may well be reached after a handful of iterations.
You need the underlying graph to be strongly connected and aperiodic. If you want to find the stationary distribution of a periodic Markov chain just by running some chain, add "stay put" transitions with some constant probability to each node and scale the other transitions down appropriately.