The above image is the H2O GBM classification model lift chart for training and validation data sets. I am confused it with the other lift charts I have seen. Normally the baseline will be 45 degrees and the lift curve used to be somewhat convex shape from the baseline curve. In the above figure if the green line shows the lift curve, why is it constant and coming down and touches the baseline? Also why the baseline is not 45 degree? Can anyone help me to interpret the model using the above graph? Is my model perform well?
The black line is not the baseline, but the cumulative capture rate. The capture rate is the proportion of all the events that fall into the group/bin. E.g. if 90 out of total 100 positive outcomes/events fall into the first bin, then the capture rate for that bin is 0.9.
The green line is the cumulative lift curve, so by definition the two lines converge at 1.
Whether your model performs well or not depends on your goal. According to the validation metrics, you could capture about 80% of the events by targeting only 50% of the population, which means lift of about 1.6.
Related
I am trying to implement a Kalman Filter to estimate the position of my arm moving in the sagittal plane (2d). To do this, I have an IMU which as usually done, I use the gyro as input to my state model and the accelerometer as my observation.
Regarding the bias, I used 0.001 for the variances of my covariance matrix of the state estimation equation and 0.03 for the variance of the accelerometer (measurement).
This filter works really well if I move my arm slowly from 0 to 90º. But if I perform sudden movements, the accelerometer makes my estimation move downward and it is not very precise (i'm off about 15º), once I move slowly it works well again. But the response under high acceleration/sudden movement is not good.
For this reason, I've thought of having a variance switch which tracks the variance of the last 10-20 values of my accelerometer angle measurements and if the variance is above a certain level I would increase the variance of the accelerometer in the covariance matrix.
Would this be an accurate approach in a system with very high accelerations? What would be a more correct way to estimate the angle under sudden movements? As I mentioned, the result I get when the accelerometer has low variance is very good, but not when "shaken fast".
Also, I would assume that due to this behavior, the accelerometer's variance does not behave according to a gaussian distribution, but I would not know how to model this behavior.
You can run a "Bank of Filters", that is independent filters with different noise levels for the variance, and then compute a weighted average the estimates based on their likelihoodlink to a reference. You can find several references in literature, during my recent work I discovered Y.Bar-Shalom has documented such an approach.
In scientific terms what you are describing is an adaptive-stochastic state estimation problem΄ long story short there exist methods to change the modelled measurement noise on-line depending on performance indications from the filter.
All the best,
D.D.
Denmark
I'm working on a project in which a rod is attached at one end to a rotating shaft. So, as the shaft rotates from 0 to ~100 degrees back-and-forth (in the xy plane), so does the rod. I mounted a 3-axis accelerometer at the end of the moving rod, and I measured the distance of the accelerometer from the center of rotation (i.e., the length of the rod) to be about 38 cm. I have collected a lot of data, but I'm in need of help to find the best method to filter it. First, here's a plot of the raw data:
I think the data makes sense: if it's ramping up, then then I think at that point the acceleration should be linearly increasing, and then when it's ramping down, it should linearly decrease. If its moving constantly, the acceleration will be ~zero. Keep in mind though that sometimes the speed changes (is higher) from one "trial" to the other. In this case, there were ~120 "trials" or movements/sweeps, data sampled at 148 Hz.
For filtering, I've tried a low pass filter and then an exponentially decreasing moving average, and both plots weren't too hot. And although I'm not good at interpreting these: here is what I got when coding a power frequency plot:
What I was hoping to get help with here is, attain a really good method by which I can filter this data. The one thing that keeps coming up again time and time again (especially on this site) is the Kalman filter. While there's lots of code online that helps implementing these in MATLAB, I haven't been able to actually understand it that great, and therefore neglect to include my work on it here. So, is a kalman filter appropriate here, for rotational acceleration? If so, can someone help me implement one in matlab and interpret it? Is there something I'm not seeing that may be just as good/better that is relatively simple?
Here's the data I'm talking about. Looking at it more closely/zooming in gives a better appreciation for what's going on in the movement, I think:
http://cl.ly/433B1h3m1L0t?_ga=1.81885205.2093327149.1426657579
Edit: OK, here is the plot of both relavent dimensions collected from the accelerometer. I am neglecting to include the up and down dimension as the accelerometer shows a near constant ~1 G, so I think its safe to say its not capturing much rotational motion. Red is what I believe is the centripetal component, and blue is tangential. I have no idea how to combine them though, which is why I (maybe wrongfully?) ignored it in my post.
And here is the data for the other dimension:
http://cl.ly/1u133033182V?_ga=1.74069905.2093327149.1426657579
Forget the Kalman filter, see the note at the end of the answer for the reason why.
Using a simple moving average filter (like I showed you on an earlier reply if i recall) which is in essence a low-pass filter :
n = 30 ; %// length of the filter
kernel = ones(1,n)./n ;
ysm = filter( kernel , 1 , flipud(filter( kernel , 1 , flipud(y) )) ) ;
%// assuming your data "y" are in COLUMN (otherwise change 'flipud' to 'fliplr')
note: if you have access to the curvefit toolbox, you can simply use: ys = smooth(y,30) ; to get nearly the same result.
I get:
which once zoomed look like:
You can play with the parameter n to increase or decrease the smoothing.
The gray signal is your original signal. I strongly suspect that the noise spikes you are getting are just due to the vibrations of your rod. (depending on the ratio length/cross section of your rod, you can get significant vibrations at the end of your 38 cm rod. These vibrations will take the shape of oscillations around the main carrier signal, which definitely look like what I am seeing in your signal).
Note:
The Kalman filter is way overkill to do a simple filtering of noisy data. Kalman filter is used when you want to calculate a value (a position if I follow your example) based on some noisy measurement, but to refine the calculations, the Kalman filter will also use a prediction of the position based on the previous state (position) and the inertial data (how fast you were rotating for example). For that prediction you need a "model" of the behavior of your system, which you do not seem to have.
In your case, you would need to calculate the acceleration seen by the accelerometer based on the (known or theoretical) rotation speed of the shaft at any point of time, the distance of the accell to the center of rotation, and probably to make it more precise, a dynamic model of the main vibration modes of your rod. Then for each step, compare that to the actual measurement... seems a bit heavy for your case.
Look at the quick figure explaining the Kalman filter process in this wikipedia entry : Kalman filter, and read on if you want to understand it more.
I will propose for you low-pass filter, but ordinary first-order inertial model instead of Kalman. I designed filter with pass-band till 10 Hz (~~0,1 of your sample frequency). Discrete model has following equation:
y[k] = 0.9418*y[k-1] + 0.05824*u[k-1]
where u is your measured vector, and y is vector after filtering. This equation starts at sample number 1, so you can just assign 0 to the sample number 0.
I am working on a software (Ruby - Sketchup) to calculate the radiation (sun, sky and surrounding buildings) within urban development at pedestrian level. The final goal is to be able to create a contour map that shows the level of total radiation. With total radiation I mean shortwave (light) and longwave(heat). (To give you an idea: http://www.iaacblog.com/maa2011-2012-digitaltools/files/2012/01/Insolation-Analysis-All-Year.jpg)
I know there are several existing software that do this, but I need to write my own as this calculation is only part of a more complex workflow.
The (obvious) pseudo code is the following:
Select and mesh surface for analysis
From each point of the mesh
Cast n (see below) rays in the upper hemisphere (precalculated)
For each ray check whether it is in shade
If in shade => Extract properties from intersected surface
If not in shade => Flag it
loop
loop
loop
The approach above is brute force, but it is the only I can think of. The calculation time increases with the fourth power of the accuracy (Dx,Dy,Dazimth, Dtilt). I know that software like radiance use a Montecarlo approach to reduce the number of rays.
As you can imagine, the accuracy of the calculation for a specific point of the mesh is strongly dependent by the accuracy of the skydome subdivision. Similarly the accuracy on the surface depends on the coarseness of the mesh.
I was thinking to a different approach using adaptive refinement based on the results of the calculations. The refinement could work for the surface analyzed and the skydome. If the results between two adjacent points differ more than a threshold value, than a refinement will be performed. This is usually done in fluid simulation, but I could not find anything about light simulation.
Also i wonder whether there are are algorithms, from computer graphics for example, that would allow to minimize the number of calculations. For example: check the maximum height of the surroundings so to exclude certain part of the skydome for certain points.
I don't need extreme accuracy as I am not doing rendering. My priority is speed at this moment.
Any suggestion on the approach?
Thanks
n rays
At the moment I subdivide the sky by constant azimuth and tilt steps; this causes irregular solid angles. There are other subdivisions (e.g. Tregenza) that maintain a constant solid angle.
EDIT: Response to the great questions from Spektre
Time frame. I run one simulation for each hour of the year. The weather data is extracted from an epw weather file. It contains, for each hour, solar altitude and azimuth, direct radiation, diffuse radiation, cloudiness (for atmospheric longwave diffuse). My algorithm calculates the shadow mask separately then it uses this shadow mask to calculate the radiation on the surface (and on a typical pedestrian) for each hour of the year. It is in this second step that I add the actual radiation. In the the first step I just gather information on the geometry and properties of the various surfaces.
Sun paths. No, i don't. See point 1
Include reflection from buildings? Not at the moment, but I plan to include it as an overall diffuse shortwave reflection based on sky view factor. I consider only shortwave reflection from the ground now.
Include heat dissipation from buildings? Absolutely yes. That is the reason why I wrote this code myself. Here in Dubai this is key as building surfaces gets very, very hot.
Surfaces albedo? Yes, I do. In Skethcup I have associated a dictionary to every surface and in this dictionary I include all the surface properties: temperature, emissivity, etc.. At the moment the temperatures are fixed (ambient temperature if not assigned), but I plan, in the future, to combine this with the results from a building dynamic thermal simulation that already calculates all the surfaces temperatures.
Map resolution. The resolution is chosen by the user and the mesh generated by the algorithm. In terms of scale, I use this for masterplans. The scale goes from 100mx100m up to 2000mx2000m. I usually tend to use a minimum resolution of 2m. The limit is the memory and the simulation time. I also have the option to refine specific areas with a much finer mesh: for example areas where there are restaurants or other amenities.
Framerate. I do not need to make an animation. Results are exported in a VTK file and visualized in Paraview and animated there just to show off during presentations :-)
Heat and light. Yes. Shortwave and longwave are handled separately. See point 4. The geolocalization is only used to select the correct weather file. I do not calculate all the radiation components. The weather files I need have measured data. They are not great, but good enough for now.
https://www.lucidchart.com/documents/view/5ca88b92-9a21-40a8-aa3a-0ff7a5968142/0
visible light
for relatively flat global base ground light map I would use projection shadow texture techniques instead of ray tracing angular integration. It is way faster with almost the same result. This will not work on non flat grounds (many bigger bumps which cast bigger shadows and also change active light absorbtion area to anisotropic). Urban areas are usually flat enough (inclination does not matter) so the technique is as follows:
camera and viewport
the ground map is a target screen so set the viewpoint to underground looking towards Sun direction upwards. Resolution is at least your map resolution and there is no perspective projection.
rendering light map 1st pass
first clear map with the full radiation (direct+diffuse) (light blue) then render buildings/objects but with diffuse radiation only (shadow). This will make the base map without reflections and or soft shadows in the Magenta rendering target
rendering light map 2nd pass
now you need to add building faces (walls) reflections for that I would take every outdoor face of the building facing Sun or heated enough and compute reflection points onto light map and render reflection directly to map
in tis parts you can add ray tracing for vertexes only to make it more precise and also for including multiple reflections (bu in that case do not forget to add scattering)
project target screen to destination radiation map
just project the Magenta rendering target image to ground plane (green). It is only simple linear affine transform ...
post processing
you can add soft shadows by blurring/smoothing the light map. To make it more precise you can add info to each pixel if it is shadow or wall. Actual walls are just pixels that are at 0m height above ground so you can use Z-buffer values directly for this. Level of blurring depends on the scattering properties of the air and of coarse pixels at 0m ground height are not blurred at all
IR
this can be done in similar way but temperature behaves a bit differently so I would make several layers of the scene in few altitudes above ground forming a volume render and then post process the energy transfers between pixels and layers. Also do not forget to add the cooling effect of green plants and water vaporisation.
I do not have enough experience in this field to make more suggestions I am more used to temperature maps for very high temperature variances in specific conditions and material not the outdoor conditions.
PS. I forgot albedo for IR and visible light is very different for many materials especially aluminium and some wall paintings
Not sure if this may or may not be valid here on SO, but I was hoping someone can advise of the correct algorithm to use.
I have the following RAW data.
In the image you can see "steps". Essentially I wish to get these steps, but then get a moving average of all the data between. In the following image, you can see the moving average:
However you will notice that at the "steps", the moving average decreases the gradient where I wish to keep the high vertical gradient.
Is there any smoothing technique that will take into account a large vertical "offset", but smooth the other data?
Yup, I had to do something similar with images from a spacecraft.
Simple technique #1: use a median filter with a modest width - say about 5 samples, or 7. This provides an output value that is the median of the corresponding input value and several of its immediate neighbors on either side. It will get rid of those spikes, and do a good job preserving the step edges.
The median filter is provided in all number-crunching toolkits that I know of such as Matlab, Python/Numpy, IDL etc., and libraries for compiled languages such as C++, Java (though specific names don't come to mind right now...)
Technique #2, perhaps not quite as good: Use a Savitzky-Golay smoothing filter. This works by effectively making least-square polynomial fits to the data, at each output sample, using the corresponding input sample and a neighborhood of points (much like the median filter). The SG smoother is known for being fairly good at preserving peaks and sharp transistions.
The SG filter is usually provided by most signal processing and number crunching packages, but might not be as common as the median filter.
Technique #3, the most work and requiring the most experience and judgement: Go ahead and use a smoother - moving box average, Gaussian, whatever - but then create an output that blends between the original with the smoothed data. The blend, controlled by a new data series you create, varies from all-original (blending in 0% of the smoothed) to all-smoothed (100%).
To control the blending, start with an edge detector to detect the jumps. You may want to first median-filter the data to get rid of the spikes. Then broaden (dilation in image processing jargon) or smooth and renormalize the the edge detector's output, and flip it around so it gives 0.0 at and near the jumps, and 1.0 everywhere else. Perhaps you want a smooth transition joining them. It is an art to get this right, which depends on how the data will be used - for me, it's usually images to be viewed by Humans. An automated embedded control system might work best if tweaked differently.
The main advantage of this technique is you can plug in whatever kind of smoothing filter you like. It won't have any effect where the blend control value is zero. The main disadvantage is that the jumps, the small neighborhood defined by the manipulated edge detector output, will contain noise.
I recommend first detecting the steps and then smoothing each step individually.
You know how to do the smoothing, and edge/step detection is pretty easy also (see here, for example). A typical edge detection scheme is to smooth your data and then multiply/convolute/cross-corelate it with some filter (for example the array [-1,1] that will show you where the steps are). In a mathematical context this can be viewed as studying the derivative of your plot to find inflection points (for some of the filters).
An alternative "hackish" solution would be to do a moving average but exclude outliers from the smoothing. You can decide what an outlier is by using some threshold t. In other words, for each point p with value v, take x points surrounding it and find the subset of those points which are between v - t and v + t, and take the average of these points as the new value of p.
I'm writing an application that averages/combines/stacks a series of exposures. This is commonly used to reduce noise in the resultant image.
However, it seems, to optimize the average/stack the exposures are usually first normalized. It seems that this process assigns weights to each of the exposures and then proceeds to combine them. I am guessing that the process computes the overall intensity of each image as the purpose is to match the intensities of all the images in the stack.
My question is, how can I incorporate an algorithm that will allow me to normalize a series of images? I guess the question be generalized by instead asking "How can I normalize a series of readings?"
An outline in my head appears as follows:
Compute the average of a reference image.
Divide the average of each frame by the average of the the reference frame.
The result of each division is the weight for each frame.
Scale/Multiply each pixel in a frame by the weight found for that particular frame.
Does this seem to make sense to anyone? I have tried to google for the past hour but didn't found anything. Also took at the indices of various image processing books on Amazon but that didn't turn up anything either.
Each integration consists of signal and assorted noise - some is time-independent (e.g. bias or CCD readout noise), some time-dependent (e.g dark current), and some is random (shot noise). The aim is to remove the noise, and leave the signal. So you would first subtract the 'fixed' sources using dark frames (which will include dark current, readout and bias) leaving signal plus shot noise. Signal scales as flux times exposure time, shot noise as the square root of the signal
http://en.wikipedia.org/wiki/Shot_noise
so overall your signal/noise scales as the square root of the integration time (assuming your integrations are short enough that they are not saturated). So by adding frames you are simply increasing the exposure time, and hence the signal/noise ratio. You don't need to normalize first.
To complicate matters, transient non-Gaussian noise is also present (e.g. cosmic ray hits). There are many techniques for dealing with these, but a common one is 'sigma-clipping', where you have an extra pass to calculate the mean and standard deviation of each pixel, and then reject outliers that are many standard deviations from the mean. Real signal will show Gaussian fluctuations around the mean value, whereas transients will show a large deviation in one frame of the stack. Maybe that's what you are thinking of?