In theory clasees, we all have learnt about drainage density (DD), which is the ratio of 'total length of stream network in a basin (L)' to the 'basin area(A)'. However, when, we are trying to determine DD of some Indian river basins using Arc-GIS, we are facing some confusions. For instance,
We have to define some particular break value in Arc-GIS to get streams of differnt orders. Lesser the break value, higher is the stream order, therefore resulting in a higher value of L, and vice versa. For example, when we set the breakvalue as 500 for a particular basin, we are obatining a stream order upto 7, but if we increase the break value to 1500, the max. stream order reduces to 5 - automatically the L value also reduces. Thus, the same baisn may yeild two differnt DD values, under two aforesaid considertions of break values.
We also fixed the theoretically least possible break value , i.e., >0 and obtained a extremely dense stream network with high DD value.
So, my question is, what should be the threshold break value for a particular basin in order to get the DD?
From literature, we found that, there are five classes of DD with the following value ranges (km/km2), i.e., very coarse (<2), coarse (2-4), moderate (4-6), fine (6-8), and very fine (>8). However, for 20 river basins across differnt parts of India, we obtained DD values ranging between 1.03 to 1.29, which makes all those basins fall under very coarse category. But, from our visual inspection (one sample bain attached below), it seems to be very less to us.
We want some justification/ clarification/ comment on it.
Related
I need to monitor an AC Voltage waveform and record the RMS value when the breakdown happens. I roughly know how to acquire data from videos I have watched, however, it is difficult for me to produce a solution that reads the Breakdown Voltage Value. Ideally, I would also take a screenshot along with the breakdown voltage value,
In case you are not familiar with this topic, When a breakdown happens the voltage will drop immediately to zero. So what I need is to measure the voltage just before it falls to zero, and if possible take a screenshot. This is an image of a normal waveform (black) with a breakdown one (red).
Naive solution*:
Take the data and get the Y values (this would depend on the datatype you have, which would depend on how you acquire the data).
Find the breakdown point by iterating over the values and maintaining a couple of flags (I would probably say track "got higher than X" and once that's true, track "got lower than Y").
From that, I would just say take the last N points (Get Array Subset) and get the array max. Or just track the maximum value as you run.
Assuming you have the graph in a control, you can just right click and select Create>>Invoke Node>>Export Image.
I would suggest trying playing with that with a VI with static data which you can repeatedly run to check how your code behaves.
*I don't know the problem domain and an not overly familiar with the various analysis VIs that ship with LV, so there are quite possibly more efficient ways of doing this.
I have two identical gps modules running the same program with antennas side by side. I get different results. I don't understand why
One example is 4211.41545 from the first unit and 4211.41481 from the second the timestamp is the same.
You don't say what the values you quote are so I will assume that the value you are showing is the Lat or Long value from the NMEA output data. If this is the case then the difference between the values is 0.00064 minutes of arc. The maximum physical distance that this will represent is around 1.1 metres (along a great circle). At 60 degrees north this would correspond to around 55cm in an E/W direction.
You do not say how far apart the antennae are or whether there are any obstructions above the level of their horizons which will introduce varying multipath signals that will be different for the two antennae. The two receivers will not necessarily be sampling the satellite signals at the same instants and so can have marginally different signal timings resulting in a different position result.
A typical consumer grade GPS will have a CEP figure of 2.5 metres meaning that 50% of the values for position in an unobstructed sky view position will lie within a 2.5 metre radius circle of the true position.
Taking this into account you should not expect any two adjacent GPS devices to give identical results.
Im trying to analyse data from cycle accidents in the UK to find statistical black spots. Here is the example of the data from another website. http://www.cycleinjury.co.uk/map
I am currently using SQLite to ~100k store lat / lon locations. I want to group nearby locations together. This task is called cluster analysis.
I would like simplify the dataset by ignoring isolated incidents and instead only showing the origin of clusters where more than one accident have taken place in a small area.
There are 3 problems I need to overcome.
Performance - How do I ensure finding nearby points is quick. Should I use SQLite's implementation of an R-Tree for example?
Chains - How do I avoid picking up chains of nearby points?
Density - How to take cycle population density into account? There is a far greater population density of cyclist in london then say Bristol, therefore there appears to be a greater number of backstops in London.
I would like to avoid 'chain' scenarios like this:
Instead I would like to find clusters:
London screenshot (I hand drew some clusters)...
Bristol screenshot - Much lower density - the same program ran over this area might not find any blackspots if relative density was not taken into account.
Any pointers would be great!
Well, your problem description reads exactly like the DBSCAN clustering algorithm (Wikipedia). It avoids chain effects in the sense that it requires them to be at least minPts objects.
As for the differences in densities across, that is what OPTICS (Wikipedia) is supposed do solve. You may need to use a different way of extracting clusters though.
Well, ok, maybe not 100% - you maybe want to have single hotspots, not areas that are "density connected". When thinking of an OPTICS plot, I figure you are only interested in small but deep valleys, not in large valleys. You could probably use the OPTICS plot an scan for local minima of "at least 10 accidents".
Update: Thanks for the pointer to the data set. It's really interesting. So I did not filter it down to cyclists, but right now I'm using all 1.2 million records with coordinates. I've fed them into ELKI for analysis, because it's really fast, and it actually can use the geodetic distance (i.e. on latitude and longitude) instead of Euclidean distance, to avoid bias. I've enabled the R*-tree index with STR bulk loading, because that is supposed to help to get the runtime down a lot. I'm running OPTICS with Xi=.1, epsilon=1 (km) and minPts=100 (looking for large clusters only). Runtime was around 11 Minutes, not too bad. The OPTICS plot of course would be 1.2 million pixels wide, so it's not really good for full visualization anymore. Given the huge threshold, it identified 18 clusters with 100-200 instances each. I'll try to visualize these clusters next. But definitely try a lower minPts for your experiments.
So here are the major clusters found:
51.690713 -0.045545 a crossing on A10 north of London just past M25
51.477804 -0.404462 "Waggoners Roundabout"
51.690713 -0.045545 "Halton Cross Roundabout" or the crossing south of it
51.436707 -0.499702 Fork of A30 and A308 Staines By-Pass
53.556186 -2.489059 M61 exit to A58, North-West of Manchester
55.170139 -1.532917 A189, North Seaton Roundabout
55.067229 -1.577334 A189 and A19, just south of this, a four lane roundabout.
51.570594 -0.096159 Manour House, Picadilly Line
53.477601 -1.152863 M18 and A1(M)
53.091369 -0.789684 A1, A17 and A46, a complex construct with roundabouts on both sides of A1.
52.949281 -0.97896 A52 and A46
50.659544 -1.15251 Isle of Wight, Sandown.
...
Note, these are just random points taken from the clusters. It may be sensible to compute e.g. cluster center and radius instead, but I didn't do that. I just wanted to get a glimpse of that data set, and it looks interesting.
Here are some screenshots, with minPts=50, epsilon=0.1, xi=0.02:
Notice that with OPTICS, clusters can be hierarchical. Here is a detail:
First, your example is quite misleading. You have two different sets of data, and you don't control the data. If it appears in a chain, then you will get a chain out.
This problem is not exactly suitable for a database. You'll have to write code or find a package that implements this algorithm on your platform.
There are many different clustering algorithms. One, k-means, is an iterative algorithm where you look for a fixed number of clusters. k-means requires a few complete scans of the data, and voila, you have your clusters. Indexes are not particularly helpful.
Another, which is usually appropriate on slightly smaller data sets, is hierarchical clustering -- you put the two closest things together, and then build the clusters. An index might be helpful here.
I recommend though that you peruse a site such as kdnuggets in order to see what software -- free and otherwise -- is available.
I have read through various papers on the 'Balls and Bins' problem and it seems that if a hash function is working right (ie. it is effectively a random distribution) then the following should/must be true if I hash n values into a hash table with n slots (or bins):
Probability that a bin is empty, for large n is 1/e.
Expected number of empty bins is n/e.
Probability that a bin has k balls is <= 1/ek! (corrected).
Probability that a bin has at least k collisions is <= ((e/k)**k)/e (corrected).
These look easy to check. But the max-load test (the maximum number of collisions with high probability) is usually stated vaguely.
Most texts state that the maximum number of collisions in any bin is O( ln(n) / ln(ln(n)) ).
Some say it is 3*ln(n) / ln(ln(n)). Other papers mix ln and log - usually without defining them, or state that log is log base e and then use ln elsewhere.
Is ln the log to base e or 2 and is this max-load formula right and how big should n be to run a test?
This lecture seems to cover it best, but I am no mathematician.
http://pages.cs.wisc.edu/~shuchi/courses/787-F07/scribe-notes/lecture07.pdf
BTW, with high probability seems to mean 1 - 1/n.
That is a fascinating paper/lecture-- makes me wish I had taken some formal algorithms class.
I'm going to take a stab at some answers here, based on what I've just read from that, and feel free to vote me down. I'd appreciate a correction, though, rather than just a downvote :) I'm also going to use n and N interchangeably here, which is a big no-no in some circles, but since I'm just copy-pasting your formulae, I hope you'll forgive me.
First, the base of the logs. These numbers are given as big-O notation, not as absolute formulae. That means that you're looking for something 'on the order of ln(n) / ln(ln(n))', not with an expectation of an absolute answer, but more that as n gets bigger, the relationship of n to the maximum number of collisions should follow that formula. The details of the actual curve you can graph will vary by implementation (and I don't know enough about the practical implementations to tell you what's a 'good' curve, except that it should follow that big-O relationship). Those two formulae that you posted are actually equivalent in big-O notation. The 3 in the second formula is just a constant, and is related to a particular implementation. A less efficient implementation would have a bigger constant.
With that in mind, I would run empirical tests, because I'm a biologist at heart and I was trained to avoid hard-and-fast proofs as indications of how the world actually works. Start with N as some number, say 100, and find the bin with the largest number of collisions in it. That's your max-load for that run. Now, your examples should be as close as possible to what you expect actual users to use, so maybe you want to randomly pull words from a dictionary or something similar as your input.
Run that test many times, at least 30 or 40. Since you're using random numbers, you'll need to satisfy yourself that the average max-load you're getting is close to the theoretical 'expectation' of your algorithm. Expectation is just the average, but you'll still need to find it, and the tighter your std dev/std err about that average, the more you can say that your empirical average matches the theoretical expectation. One run is not enough, because a second run will (most likely) give a different answer.
Then, increase N, to say, 1000, 10000, etc. Increase it logarithmically, because your formula is logarithmic. As your N increases, your max-load should increase on the order of ln(n) / ln(ln(n)). If it increases at a rate of 3*ln(n) / ln(ln(n)), that means that you're following the theory that they put forth in that lecture.
This kind of empirical test will also show you where your approach breaks down. It may be that your algorithm works well for N < 10 million (or some other number), but above that, it starts to collapse. Why could that be? Maybe you have some limitation to 32 bits in your code without realizing it (ie, using a 'float' instead of a 'double'), or some other implementation detail. These kinds of details let you know where your code will work well in practice, and then as your practical needs change, you can modify your algorithm. Maybe making the algorithm work for very large datasets makes it very inefficient for very small ones, or vice versa, so pinpointing that tradeoff will help you further characterize how you could adapt your algorithm to particular situations. Always a useful skill to have.
EDIT: a proof of why the base of the log function doesn't matter with big-O notation:
log N = log_10 (N) = log_b (N)/log_b (10)= (1/log_b(10)) * log_b(N)
1/log_b(10) is a constant, and in big-O notation, constants are ignored. Base changes are free, which is why you're encountering such variation in the papers.
Here is a rough start to the solution of this problem involving uniform distributions and maximum load.
Instead of bins and balls or urns or boxes or buckets or m and n, people (p) and doors (d) will be used as designations.
There is an exact expected value for each of the doors given a certain number of people. For example, with 5 people and 5 doors, the expected maximum door is exactly 1.2864 {(1429-625) / 625} above the mean (p/d) and the minimum door is exactly -0.9616 {(24-625) / 625} below the mean. The absolute value of the highest door's distance from the mean is a little larger than the smallest door's because all of the people could go through one door, but no less than zero can go through one of the doors. With large numbers of people (p/d > 3000), the difference between the absolute value of the highest door's distance from the mean and the lowest door's becomes negligible.
For an odd number of doors, the center door is essentially zero and is not scalable, but all of the other doors are scalable from certain values representing p=d. These rounded values for d=5 are:
-1.163 -0.495 0* 0.495 1.163
* slowly approaching zero from -0.12
From these values, you can compute the expected number of people for any count of people going through each of the 5 doors, including the maximum door. Except for the middle ordered door, the difference from the mean is scalable by sqrt(p/d).
So, for p=50,000 and d=5:
Expected number of people going through the maximum door, which could be any of the 5 doors, = 1.163 * sqrt(p/d) + p/d.
= 1.163 * sqrt(10,000) + 10,000 = 10,116.3
For p/d < 3,000, the result from this equation must be slightly increased.
With more people, the middle door slowly becomes closer and closer to zero from -0.11968 at p=100 and d=5. It can always be rounded up to zero and like the other 4 doors has quite a variance.
The values for 6 doors are:
-1.272 -0.643 -0.202 0.202 0.643 1.272
For 1000 doors, the approximate values are:
-3.25, -2.95, -2.79 … 2.79, 2.95, 3.25
For any d and p, there is an exact expected value for each of the ordered doors. Hopefully, a good approximation (with a relative error < 1%) exists. Some professor or mathematician somewhere must know.
For testing uniform distribution, you will need a number of averaged ordered sessions (750-1000 works well) rather than a greater number of people. No matter what, the variances between valid sessions are great. That's the nature of randomness. Collisions are unavoidable. *
The expected values for 5 and 6 doors were obtained by sheer brute force computation using 640 bit integers and averaging the convergence of the absolute values of corresponding opposite doors.
For d=5 and p=170:
-6.63901 -2.95905 -0.119342 2.81054 6.90686
(27.36099 31.04095 33.880658 36.81054 40.90686)
For d=6 and p=108:
-5.19024 -2.7711 -0.973979 0.734434 2.66716 5.53372
(12.80976 15.2289 17.026021 18.734434 20.66716 23.53372)
I hope that you may evenly distribute your data.
It's almost guaranteed that all of George Foreman's sons or some similar situation will fight against your hash function. And proper contingent planning is the work of all good programmers.
After some more research and trial-and-error I think I can provide something part way to to an answer.
To start off, ln and log seem to refer to log base-e if you look into the maths behind the theory. But as mmr indicated, for the O(...) estimates, it doesn't matter.
max-load can be defined for any probability you like. The typical formula used is
1-1/n**c
Most papers on the topic use
1-1/n
An example might be easiest.
Say you have a hash table of 1000 slots and you want to hash 1000 things. Say you also want to know the max-load with a probability of 1-1/1000 or 0.999.
The max-load is the maximum number of hash values that end up being the same - ie. collisions (assuming that your hash function is good).
Using the formula for the probability of getting exactly k identical hash values
Pr[ exactly k ] = ((e/k)**k)/e
then by accumulating the probability of exactly 0..k items until the total equals or exceeds 0.999 tells you that k is the max-load.
eg.
Pr[0] = 0.37
Pr[1] = 0.37
Pr[2] = 0.18
Pr[3] = 0.061
Pr[4] = 0.015
Pr[5] = 0.003 // here, the cumulative total is 0.999
Pr[6] = 0.0005
Pr[7] = 0.00007
So, in this case, the max-load is 5.
So if my hash function is working well on my set of data then I should expect the maxmium number of identical hash values (or collisions) to be 5.
If it isn't then this could be due to the following reasons:
Your data has small values (like short strings) that hash to the same value. Any hash of a single ASCII character will pick 1 of 128 hash values (there are ways around this. For example you could use multiple hash functions, but slows down hashing and I don't know much about this).
Your hash function doesn't work well with your data - try it with random data.
Your hash function doesn't work well.
The other tests I mentioned in my question also are helpful to see that your hash function is running as expected.
Incidentally, my hash function worked nicely - except on short (1..4 character) strings.
I also implemented a simple split-table version which places the hash value into the least used slot from a choice of 2 locations. This more than halves the number of collisions and means that adding and searching the hash table is a little slower.
I hope this helps.
Problem:
A relational database (Postgres) storing timeseries data of various measurement values. Each measurement value can have a specific "measurement type" (e.g. temperature, dissolved oxygen, etc) and can have specific "measurement units" (e.g. Fahrenheit/Celsius/Kelvin, percent/milligrams per liter, etc).
Question:
Has anyone built a similar database such that dimensional integrity is conserved? Have any suggestions?
I'm considering building a measurement_type and a measurement_unit table, both of these would have text two columns, ID and text. Then I would create foreign keys to these tables in the measured_value table. Text worries me somewhat because there's the possibility for non-unique duplicates (e.g. 'ug/l' vs 'µg/l' for micrograms per liter).
The purpose of this would be so that I can both convert and verify units on queries, or via programming externally. Ideally, I would have the ability later to include strict dimensional analysis (e.g. linking µg/l to the value 'M/V' (mass divided by volume)).
Is there a more elegant way to accomplish this?
I produced a database sub-schema for handling units an aeon ago (okay, I exaggerate slightly; it was about 20 years ago, though). Fortunately, it only had to deal with simple mass, length, time dimensions - not temperature, or electric current, or luminosity, etc. Rather less simple was the currency side of the game - there were a myriad different ways of converting between one currency and another depending on date, currency, and period over which conversion rate was valid. That was handled separately from the physical units.
Fundamentally, I created a table 'measures' with an 'id' column, a name for the unit, an abbreviation, and a set of dimension exponents - one each for mass, length, time. This gets populated with names such as 'volume' (length = 3, mass = 0, time = 0), 'density' (length = 3, mass = -1, time = 0) - and the like.
There was a second table of units, which identified a measure and then the actual units used by a particular measurement. For example, there were barrels, and cubic metres, and all sorts of other units of relevance.
There was a third table that defined conversion factors between specific units. This consisted of two units and the multiplicative conversion factor that converted unit 1 to unit 2. The biggest problem here was the dynamic range of the conversion factors. If the conversion from U1 to U2 is 1.234E+10, then the inverse is a rather small number (8.103727714749e-11).
The comment from S.Lott about temperatures is interesting - we didn't have to deal with those. A stored procedure would have addressed that - though integrating one stored procedure into the system might have been tricky.
The scheme I described allowed most conversions to be described once (including hypothetical units such as furlongs per fortnight, or less hypothetical but equally obscure ones - outside the USA - like acre-feet), and the conversions could be validated (for example, both units in the conversion factor table had to have the same measure). It could be extended to handle most of the other units - though the dimensionless units such as angles (or solid angles) present some interesting problems. There was supporting code that would handle arbitrary conversions - or generate an error when the conversion could not be supported. One reason for this system was that the various international affiliate companies would report their data in their locally convenient units, but the HQ system had to accept the original data and yet present the resulting aggregated data in units that suited the managers - where different managers each had their own idea (based on their national background and length of duty in the HQ) about the best units for their reports.
"Text worries me somewhat because there's the possibility for non-unique duplicates"
Right. So don't use text as a key. Use the ID as a key.
"Is there a more elegant way to accomplish this?"
Not really. It's hard. Temperature is it's own problem because temperature is itself an average, and doesn't sum like distance does; plus F to C conversion is not a multiply (as it is with every other unit conversion.)
A note about conversions: a lot of units are linearly related, and can be converted using a formula like "y = A + Bx", where A and B are constants which could be stored in the database for each pair of units that you need to convert between. For example, for Celsius to Farenheit the constants are A=32, B=1.8.
However, there are also rare exceptions. Converting between logarithmic and non-logarithmic units, for example. Or converting between mass-per-volume and molar-mass-per-volume (in which case you would need to know the molar mass of the compound being measured).
Of course, if you are sure that all the conversions required by the system are linear, then there's no need for over-engineering, just store the two constants. You can then extract standardized results from the database using straight SQL joins with calculated fields.