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Author: Subject: Snowfall Amounts Follow a Weibull Distribution?
AJKOER
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Snowfall Amounts Follow a Weibull Distribution?

Why does the amount of snowfall during NYC cold months (more precisely November to April) apparently follow a Weibull (or related like the Gamma or special cases which includes the exponential) distribution?

Data background, an examination of snowfall in inches for NYC (see, for example, https://www.currentresults.com/Weather/New-York/Places/new-y... and also https://www.weather.gov/media/okx/Climate/CentralPark/monthl... ) appears to be consistent with a hypothesis of a continuous two-parameter Weibull density function (see https://en.wikipedia.org/wiki/Weibull_distribution with parameters Lambda = 1, k =1.5, and here is a link to just the uploaded pdf at https://upl oad.wikimedia.org/wikipedia/commons/5/58/Weibull_PDF.svg . Here is a bar graph at https://www.weather.gov/images/okx/Climate/CentralPark_Snowf... ,

The possible significance relating to the appropriateness of employing a Weibull density function, for such things as snowfall amounts/days, is that it can be employed to simulate discrete scenarios for planning purposes.

Wikipedia, as one source, makes some interesting comment with respect to the Weibull distribution, for example:

“ first applied by Rosin & Rammler (1933) to describe a particle size distribution.”

Relatedly, on the mechanics of snowflake construction, one reference says (see https://blog.advancementcourses.com/classroom-activities/the... ):

“According to the National Oceanic and Atmospheric Administration (NOAA), snowflakes form when dust or pollen particles come into contact with water vapor and freeze high in the Earth’s atmosphere. When it freezes, the particle forms a crystal of ice and the process of building a snowflake begins. The water molecules inside each crystal arrange themselves into a six-sided structure.”

I would argue that the process of snowflake formation may involve a critical mass determination, at which point, precipitation of the ice crystal occurs. As particle size distribution can be modeled by a Weibull distribution, snowfall perhaps also. Supporting reference for the Weibull is based on its suggested use for rain distribution in this paper https://rmets.onlinelibrary.wiley.com/doi/pdf/10.1002/joc.11... . This paper involved a warmer region (Spain) and in other colder locales (like NYC), rain becomes ice/snow. Did find a source referencing the use of the Exponential distribution at https://www.geosci-model-dev.net/12/3759/2019/ for Artic snowfall.

Comments welcome including some of the science relating on how particle size may be material.
-----------------------------------------------------

Some assorted comments from the Wikipedia article:

“If the quantity X is a "time-to-failure", the Weibull distribution gives a distribution for which the failure rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:[3]“

“For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it.”

“A value of k>1 indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the context of the diffusion of innovations, this means positive word of mouth: the hazard function is a monotonically increasing function of the proportion of adopters. The function is first convex, then concave with an inflexion point"

[Edited on 19-11-2019 by AJKOER]
mayko
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well... it's not apparent to me that snowfall is "consistent" with a Weibull distribution, because you've just sort of tossed out a bunch of links to tabular data without providing any statistical analysis or formal hypothesis testing. You provide a visual aid ("Here is a bar graph") but it clouds the issue rather than clarifying it. Before I looked at it, I thought I understood what you were talking about, but now I think at least one of us is confused.
so, I'll ask: "How does that bar graph relate to the topic?" and in the meantime go ahead with what I think a reasonable approach here is.

I munged the data at https://www.weather.gov/media/okx/Climate/CentralPark/monthl... into a TSV flatfile; entries with T were rounded down to zero. (T stands for Trace, right? Is there a README for this data?), Snowfall between November and April inclusive were summed to get a winter total. The first year on record had incomplete data, and was excluded.

Here is the distribution of total snowfall amounts as a histogram:

Something that pops right out to me is that it's positively skewed: most winters fall between 0 and 40 inches, but there is a long tail of heavy snowfall. This skewness also appears to be significant (Of 1000 simulated normal distributions of the same length, only 2, ie 0.2%, had a greater skew.) Positive skewness is the main feature of the Weibull distribution under the parameters you've specified. So, in that sense, this distribution is "consistent" with a Weibull distribution.

However, there are a lot of distributions which are positively skewed, at least under some choices of parameter, and so this data is just as "consistent" with them in that sense. I don't see any evidence that any one of them is actually a better fit than any other (is it obvious that these data are less consistent with the beta distribution than with the Weibull?). There might be a theoretical reason to expect the Weibull distribution to model this distribution, but that isn't clear to me either.

It sounds like you're trying to investigate the process by which snowstorms begin, which is definitely interesting... but it seems to me that the amount of snowfall is an indirect and heavily confounded measure of snow initiation. (The amount of water vapor available to precipitate, ie humidity, is a major determinant!) If available, I'd probably start with the frequency of snowstorms, or better yet, the time in between them.

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AJKOER
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Mayko thanks for your work and interest!

If you could load the data into a worksheet and post it that may allow me and others to perform some statistical investigations.

I agree that is possible for there to be a wide range of possible selections for the best fitting distribution. My modeling/forecasting experience suggests always work with the more parsimonious model (to avoid overfitting and in my experience, inferior forecast) and select a model that has some underlying fundamental rationale (whether based on science or economic theories,...so that the model remains valid in changing conditions). In the current context, if particle size distribution is key, and there is a rationale for the weibull in this regard promoting its selection.

But having several distributions to select from is also informative as one could produce a random selection step for the choice of distribution in a simulation exercise. But, I feel there should be a rationale as to why several distributions relate to the phenomena being observed.

My guess is that data and insight in modeling rain patterns may be applicable also, as snow is the product of confounding of possible rain with a low-temperature distribution.

At least I think getting even rough parameter estimates may be insightful in predicting the odds of getting buried alive in a snow storm!

For some reason, I feel that our government is NOT always willing to reveal predictions that would inspire some of us to go south!

Changing weather conditions due to global warming (GW) likely just adds to the variability of estimates. Together with the total denial, by heads of governments, no less, of the very existent of GW, just adds to my angst.

[Edited on 22-11-2019 by AJKOER]
mayko
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sure thing!! (year ranges have been rounded down)

Attachment: CentralParkSnowfall.csv (7kB)

I noticed something else when I looked at the winter total as a time series: Although I haven't done any significance testing, there seems to be a depletion of heavy snowfall around 1970-1990:

This appears to be contributing some, but not all, of the distribution's asymmetry (excluding those two decades, only 1% of normals had a skew at least as positive).

Incidentally, there are only four years with snowfall to speak of outside Nov-Apr; all of them were early snow in October:

 Code:  year OCT nonWinter 1 1876 0.5 0.5 2 1925 0.8 0.8 3 1952 0.5 0.5 4 2011 2.9 2.90 

[Edited on 23-11-2019 by mayko]

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AJKOER
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Thanks for the data!

Did some work:

Total Snow Fall
OCT 4.7
NOV 125.4
DEC 817.9
JAN 1190.8
FEB 1337.7
MAR 752.6
APR 121.9

which shows a peak in February. However, on the data analysis, Wiki on the Weibull distribution employs a rank transformation, so working with total snowfall amounts transformed to rank:

Rank of Above Total Snow Fall:
OCT 1
NOV 3
DEC 5
JAN 6
FEB 7
MAR 4
APR 2

However, working with each year's rank transformed data, and taking the column average:

Col Avg on Rank Transformed Data:
OCT 2.742
NOV 3.358
DEC 4.960
JAN 5.722
FEB 5.669
MAR 5.013
APR 3.298

So, working with the rank transformed data appears to suggest a distribution also with more probability mass to the right but with a peak actually in the month of January, or is the peak actually in the middle of winter, February 5th! The latter is half way between the dates of the Winter Solstice and Spring (Vernal) Equinox (which can change by a day due to a leap year correction factor). The date is also nearly halfway between Jan 15th and Feb 15th with both months returning nearly the same average rank (differing by only 0.023).

[Edited on 24-11-2019 by AJKOER]
mayko
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First, as far as I can tell without context, the bar graph you linked isn't precisely the distribution of snowfall across the months, but the central tendency of a bunch of different distributions, one for each month. I wouldn't expect a huge difference when they were scaled down, but I wouldn't expect them to be identical, either.

A more serious problem with what you seem to be trying to do, is that Weibull is a continuous distribution, whereas the months are a discrete domain. It's not an appropriate distribution for snowfall by month! (Questionable domain *would* be a reason to doubt the beta as a model, since it's defined on a closed interval, and there's no hard ceiling on total snowfall.) Now, there is an analogous distribution called the discrete Weibull:
https://en.wikipedia.org/wiki/Discrete_Weibull_distribution

But, that doesn't necessarily help much, because it's defined on the natural numbers (0,1,2,...), whereas the months are a finite number of non-numerical categories. I know how to divide 32 inches by a non-zero lambda...... but how do I divide the month of November?

Finally, the meaning of the distribution of snowfall across months wasn't obvious to me at first so I think it's worth thinking through. Imagine you save up all the snow that falls for a year, mix it up well, and pull out a single snowflake. The discrete by-month distribution tells you how likely it is that particular snowflake fell in, say, February. On the other hand, the continuous distribution of snowfall amounts describes how likely it is that, say, 32 inches of snow fall this year.

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AJKOER
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I have since examined and obtain parameters for the Weibull based on 150 data points for, separately, the months of Jan and Feb. I used the regression method discussed in the Wikipedia article (https://en.wikipedia.org/wiki/Weibull_distribution ) .

My results for data from the months of JAN and FEB are as follows:

Month /// k Parameter ///// Lambda
Jan /// 0.972 //// 8.031
Feb /// 1.374 //// 85.031

Focusing on a single month means that there is, on average, more control exercised on the variability in temperature, for example, across months.
------------------------------------------------

I actually tested the regression method by generating Weibull random deviates from a Uniform distribution of random numbers, indicated by Ui, as supplied by the RAND() worksheet function, using the inversion method with supplied values of the parameters, in an accuracy determination exercise. In the latter inversion approach employed in Monte Carlo methods (see https://en.wikipedia.org/wiki/Inverse_transform_sampling ), one replaces the F(x) by Ui and solves for Xi, assuming this can be performed. It can be in the case of the two-parameter Weibull. I tested the methodology for a pair of parameters, k=1 and Lambda=1.5, and was satisfied with the results. Here is a summary table over seven runs of 150 points each:

Parameter AVG: 0.98145883 1.448734814 (Note: Supplied parameter values are 1.0 and 1.5 )

Median: 0.9912668417 1.438871331

Avg Rank 1 & 7: 0.9716559998 1.416592451

Avg Rank 3 & 5: 0.9864466037 1.340333943

Avg Rank 2 & 6: 0.9813698805 1.444563933

Rank: k Parameter Simulation Rank Lambda
4/// 0.9912668417//// 2/// 1.320552039
5/// 1.017649631//// 1/// 1.241796556
6/// 1.042780621//// 6/// 1.568575826
1/// 0.8953570385//// 3/// 1.434384839
2/// 0.9199591396//// 7/// 1.591388347
7/// 1.047954961//// 5/// 1.54557476
3/// 0.955243576//// 4/// 1.438871331

[Edited on 26-11-2019 by AJKOER]
AJKOER
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My results for data from the months of DEC, JAN, FEB and MAR are as follows from fitting (by the regression method) a two-parameter Weibull distribution:

Month /// k Parameter ///// Lambda
Dec /// 0.555 //// 4.373
Jan /// 0.972 //// 8.031
Feb /// 0.840 //// 9.084
Mar /// 0.571 //// 4.097

[Edited on 28-11-2019 by AJKOER]
AJKOER
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Historical Based Simulation of Total Snow Fall for NYC Total for 4 Months (NOV, DEC, JAN and FEB)

Actual Historical for 4 Months// Avg of 7 Simulation Statistics of 150 Winters // Simulation Run # > 1 2

AVG: 27.2, 30.8 // 30.4 31.3
MEDIAN: 25.4, 25.9 // 25.2 26.6
MAX 72.0, 119.3 // 105.1 133.4
MIN 2.8, 4.0 // 3.0 4.9

Continued....

Based on a Fitted Weibull Two-Parameter Distribution Using 150 Years of Data

Run > 3 4 5 6 7
35.1 29.7 32.7 33.3 30.2
27.4 25.5 28.2 27.9 24.8
196.7 93.9 152.3 129.7 114.6
3.3 3.4 2.5 2.7 1.9

Some observations, using tailored monthly Weibull distributions appears to produce an excellent agreement to the actual historical median and average for the total 4 month snowfall for NYC. The only apparent deviation is at one tail relating to conceivable max values for snowfall which may be alleviated by increasing the number of simulations from just seven. However, this could also be representative of the appropriateness of the distribution selection (namely the Weibull), the fitting process itself (regression model method), or an independent assumption for extreme events, by summing four values and ascribing the formed max, versus real-world possibly monthly negative correlations (relating to the consumption of water) impacting the likelihood of an extreme snowfall total. Or, it could perhaps speak to the possibility of significantly heavier events that may yet to occur in the future.

[Edited on 28-11-2019 by AJKOER]
clearly_not_atara
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It's also not the only right-skewed distribution that could fit that snowfall graph. Usually, if you're just looking at a graph and picking a similar-looking distribution, you want to start with the simple ones, and the simple ones in this case are the log-normal distribution and the Poisson distribution:

https://en.wikipedia.org/wiki/Log-normal_distribution

https://en.wikipedia.org/wiki/Poisson_distribution

From my intuition alone, snowfall seems like a Poisson distribution: you're measuring the number of times some event (a snowflake hitting the ground) occurs (many) over a particular period of time (winter). These are sometimes called Poisson processes, which generally refers to anything where you measure the number of random events occurring in some period.

However, in fact the Poisson model is too simple, and doesn't quite capture the fact that precipitation is actually two random variables: whether a precipitation event occurs (which is Poisson distributed) and how much precipitation falls at once (which is gamma distributed). The gamma distribution captures the fact that it is more likely that many snowflakes hit the ground at once rather than their being totally uncorrelated (a silly oversimplification in my guess!). An analysis is given by Peter K Dunn:

https://rmets.onlinelibrary.wiley.com/doi/pdf/10.1002/joc.10...

So my guess would be, snowfall appears to follow a Weibull distribution just because the Weibull distribution is pretty similar to a Poisson-gamma distribution, and snowfall is a Poisson sum of gamma-distributed snowfall events.

[Edited on 04-20-1969 by clearly_not_atara]
AJKOER
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Found a reference referring to the historical use of the Weibull in the field of hydrology. To quote from 'On application of the Weibull distribution in hydrology' at

"The Weibull distribution, commonly used in hydrology, was derived using the principle of maximum entropy. The derivation required two constraints to be determined from data and yielded, in turn, a unique procedure for estimation of the distribution parameters."

Also, from the 1st and 2nd page:

"To this end, the Weibull distribution is commonly used for frequency analysis as well as risk and reliability analysis of the frequently in hydrology and meteorology. Grace and Eagleson (1966) fitted this distribution to the wet and dry sequences and obtained satisfactory results. Rao and Chenchayya (1974) applied it to short-time increment urban precipitation characteristics in various parts of the U.S.A. and found satisfactory fit to the durations of wet and dry periods as well as other characteristics.
.................
Entropy is a numerical measure of uncertainty or, conversely, the information content, associated with a pdf, f(x), used to describe a random variable x. I(f) can be thought of as the mean value of -lnf(x)....The principle of maximum entropy was formulated by Jaynes (1961, 1982) to produce the minimally biased distribution by maximizing entropy subject to given information."

Here is a link to a complete related advanced paper at https://bayes.wustl.edu/etj/articles/rational.pdf 'On The Rationale of Maximum-Entropy Methods', by ET JAYNES in PROCEEDINGS OF THE IEEE, 1982.

[Edited on 28-11-2019 by AJKOER]
Sulaiman
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I have been doing a little research on wind power to supplement solar power for my planned new home.
For the best data analyses that I've found so far,
the best fit for wind speed is a Weibull distribution

Maybe wind speed and snowfall are related ?

CAUTION : Hobby Chemist, not Professional or even Amateur
(suffering from separation of me and my chemistry stuff)
AJKOER
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 Quote: Originally posted by Sulaiman I have been doing a little research on wind power to supplement solar power for my planned new home. For the best data analyses that I've found so far, the best fit for wind speed is a Weibull distribution Maybe wind speed and snowfall are related ?

Per my extract above:

"Rao and Chenchayya (1974) applied it to short-time increment urban precipitation characteristics in various parts of the U.S.A. and found satisfactory fit to the durations of wet and dry periods as well as other characteristics."

One could modify the statement for wind as follows, "and found satisfactory fit to the durations of windy and calm periods as well as other characteristics.'

which would be seemingly in accord with your observation. For support, a google search return several articles, for example, 'Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis' at https://www.sciencedirect.com/science/article/pii/S016761059... to quote the abstract:

"Three methods for calculating the parameters of the Weibull wind speed distribution for wind energy analysis are presented: the maximum likelihood method, the proposed modified maximum likelihood method, and the commonly used graphical method. The application of each method is demonstrated using a sample wind speed data set, and a comparison of the accuracy of each method is also performed. The maximum likelihood method is recommended for use with time series wind data, and the modified maximum likelihood method is recommended for use with wind data in frequency distribution format."

and this abstract also at https://journals.ametsoc.org/doi/abs/10.1175/1520-0450(1978)017%3C0350:MFEWSF%3E2.0.CO;2 :

"The Weibull function is discussed for representation of the wind speed frequency distribution. Methods are presented for estimating the two Weibull parameters (scale factor c and shape factor k) from simple wind statistics. Comparison is made with a recently proposed method based on the “square-root-normal” distribution with mean wind speed and fastest mile data as input statistics. The Weibull distribution is shown to give smaller root-mean-square errors than the square-root-normal distribution when fitting actual distributions of observed wind speed. Another advantage of the Weibull distribution is the available methodology for projecting to another height the observed Weibull distribution parameters at anemometer height."

This work 'The Estimation of the Parameters of the Weibull Wind Speed Distribution for Wind Energy Utilization Purposes' at https://www.jstor.org/stable/43749134?seq=1 actually likes the regression/graphical method. The first paper of the paper discusses a comparative study among several alternative distributions and concludes the Weibull distribution renders the best fit to experimental data.

But why the Weibull? Because, to quote from my prior source above, the Weibull is able 'to produce the minimally biased distribution by maximizing entropy subject to given information' or constraints, as could arise from the presence of rugged terrain.
--------------------------------------------

With respect to the use of wind energy for a home, I recall an instance where one 'stored' the wind energy by pumping a large pond of water to a higher pond (providing a waterfall indoors), to be released as needed to generate energy. Both ponds were located in the expansive home, as likely in some locales during cold months, ponds can be mostly turned into ice!

My wish list would have three possible heating (or cooling) sources of energy. One being conventional fireplaces/furnaces in areas with ample trees. The last two are wind and a solar system, with the first two being able to fill the gap when there is no sunlight available. A smart home may be able to switch between systems based on cost and expected/current supply inputs.

[Edited on 29-11-2019 by AJKOER]
Sulaiman
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 Quote: Originally posted by AJKOER My wish list would have three possible heating (or cooling) sources of energy. One being conventional fireplaces/furnaces in areas with ample trees. The last two are wind and a solar system, with the first two being able to fill the gap when there is no sunlight available. A smart home may be able to switch between systems based on cost and expected/current supply inputs.

That is almost exactly my plan
except
- I will have no need for space heating, burning fuel will be mostly for cooking
- the fuel will be oil palm tree fronds (the hill which the house will be on is my brother-in-law's oil palm plantation)
- I plan to also have solar heated water
- photovoltaic power will be used to pump stored rainwater to a header tank for a decent outlet pressure

My big problem will be air-conditioning during darkness ...

We will have access to the internet, 3-phase electricity, bottled propane gas,
and at the bottom of the hill we will have piped mains water to a tank,
from which which we can pump water up to the house if required,
We will not be off-grid, just aiming for reduced grid usage,
with the option of being completely off-grid except for the air-conditioning.

All of this will casue a significant delay in me getting back to doing chemistry (and other hobbies)
and will change some aspects of how I do chemistry.

CAUTION : Hobby Chemist, not Professional or even Amateur
(suffering from separation of me and my chemistry stuff)
AJKOER
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A house on a hill in a region possibly characterized by significant rainfall season in a new century of likely significant global warming....what could go wrong?

Unfortunately, I would be modeling rainfall distributions and probabilities of mudslides! Perhaps having access to a second home with low landslide exposure should be considered in the planning as well.

My thoughts, even though a plantation on a hill has survived for over a 100 years, it may still be a matter for my peace of mind. Speaking to an unbiased expert on earth movements may be advised.

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