Since the probability distribution for Brownian Motion is Gaussian (white noise) you can simply take that as your null hypothesis. You then have many statistical tools to assess whether or not the actual distributions depart significantly from your null hypothesis.

You may need to deal with spatial autocorrelation in two dimensions. A software package such as Idrisi from Clark Labs may provide the tools for that. However, if you have time series data, your experimental data may be 3-dimensional . I understand that the latest version of Idrisi may be able to cope but am not certain.

Thanks for your quick and informative answer.

For your first point, it is about the fact to test or not if the Brownian motion is representative of the real yields uncertainty if I get it. Do you confirm? But if I can characterize the distribution, can I then find some parameter that we can consider as the level of uncertainty (that can be understood as the marginal effect of the Brownian motion at each period on the yields for instance)? And then could I consider it like a variable, and then proceed to a regression with the land conversion as a dependent variable and uncertainty as an independent variable (account being taken of all potential bias)? To sum up my question, how can we proxy the level of uncertainty of yields and input it in a regression as a independent variable?

For the autocorrelation matter, I do not have spatial data. I think I will only make a panel regression because of the time series.

S'il vous plaît écrivez ceci encore en français.

9OK, after reading again, I think I have got your model. You have a database that classifies farms by land use. For land use you have cattle (1), corn (maize) 2, permanent pasture (3), temporary pasture (4). For maize you have yields (P) for 100 (96?) departments for 15 years.

(Since Y is usually reserved for the dependent variable, and both (T) taux and R rendement are also reserved, I am using (P) productivity for yield because it is the same in French and English.)

You have not mentioned what data is in the second database, but I infer that you have for the same 100 departments 15 years of data for the same time period that describes some sort of change in land use. Possibly change from permanent pasture to temporary pasture or to maize.

Your hypothesis is that uncertainty in yield (risk) for maize motivates some farmers to change land use. At least for now, you are not testing for geographical differences in uncertainty or geographical differences in changes in land use. I assume that land use can change both from and to maize cultivation from both types of pasture and between both types of pasture. But I suggest that you keep your preliminary calculations simple.

In my opinion, you have an interesting project.

My first impression is that uncertainty is best represented by variance in time. But you have only 15 data points in your time series (n=15) which means that you have only 14 degrees of freedom. In comparing the mean values of P (yield).

I would set aside preliminary analysis to become intimate with my data:

1. start by calculating the means and variances.

2. code the departments with numerical codes for the 22 regions.

3. determine which if any of these farm activities is related to "annual forage" which seems to me to be a potential way for a farmer to reduce risk instead of switchng to pasture.

4. There appear to be trends in farm activities possibly related to EU-wide changes in agricultural policy or as a response by French farmers to competition from other EU states. You don't want to do all this work and have somebody falsify your results by showing that it is competition from outside France that is driving changes in land use within France.

I would test for trends in the data and explore ways to remove the trends for inter-departmental comparison.

http://www.fao.org/ag/agp/agpc/doc/Counprof/France/france.htm

5. As a separate issue, test to determine if there are trends in production in the regions and if so are there inter-regional differences in trend.

6. examine the potential for using relative changes instead of absolute changes in both yields and land-use to develop variables that are standardized for France (z-scores).

7. Use the t-test to compare means and the F-test to compare variances first on date aggregated by regions and then by departments.

The easiest way might be to use the programming language R because it is free and has an extensive library of statistical packages.

Brownian motion

To respond to you original query about using Brownian motion as your key concept, Brownian motion is white noise. Virtually all of econometrics and parametric statistics are based on the assumption of the normal (Gaussian) distribution.

Instead of Gaussian, we could say Brownian distribution, except that we wish to honour Mr Gauss instead of Mr Brown.

One way to test for normality do this is to test for skewness and heteroscedasticity, not easy to do with only 15 data points. If you aggregate all your departments and then bin (categorize) all your values you could then run chi-square tests of each department against the aggregate for departments. Better to do this within and among regions first to get a feel for the data.

There are R-packages that allow you to plot the data and this should allow you to reduce the time required for the work. You might find that the data itself suggests ways to aggregate departments to defined your own ad hoc "agricultural regions" instead of political regions.

The t-test for means is based on the variances being equal across all samples. So you might aggregate departments based on similarity in means and variances. Then you might find there are 16 or 25 ad hoc regions/ Test this by using the t-test. To test if two variances are equal, use the F-test.

For both of these tests e you will get 96 x 96 matrices, which is part of the reason that I suggest you start with regions which will give you 22 x 22 matrices. Or start with departments then aggregate them to make your own ad hoc regions.

Part of the reason I suggest you use R instead of a commercial statistical package such as SPSS is that R is an interpreted language and interactive. You can play with the data until you come to "own the data".

Thank you very much for your help !

If I get well your point, I should aggregated department by regional unit, unit that I identify myself regarding to their differences in mean and variance (by observing the level of heteroskedasticity and comparing skewness), in order to have sufficient variability between observation and make a regression. I put in the same "adhoc" region all the department which have significantly the same level of means and variance in yields. Is it correct?

I just have a question:

I thought that I should make a logit regression of the land use (relative or absolute terms), in order to explain the distribution of the proportion of lands dedicated to each use by my independant variables (yields mean, yields variance, control variables). What do you thing of that idea?

And the variance should be calculated with an Brownian motion equation, if I am correct (as in Schatzki 2003 for instance)

Thank you again

Youlr difficulty with any regression, logit or otherwise is the limited number of observations (15 years).

Various approaches to Brownian motion refer to random walks / stationarity etc. But with only 15 observations, I think the tests would not be robust.

I understand the idea of using uncertainty as you independent variable. What is your dependent variable? Some measure of change in land-use but precisely what?

Your other issues.

1. I put in the same "ad hoc" region all the department which have significantly the same level of means and variance in yields.

In my opinion, with 15 years data you have limited scope for the standard tests and models. So you need to focus on the substantive question about farmer behaviour and develop your own methodology. You hypothesize that farmers change their approach to land use based on risk and risk is revealed by variability in yield of maize. (I note that the output from maize in France is both silage and grain but I infer from the context that the data for yield refers to grain.)

Maize as grain has specific climatic and soil needs that may be revealed more by mean yield than variance in yield. So you could try using mean yield to group your departments. You would test the difference in means as a criterion using the F-test and you might impose a distance restriction to avoid disparities in environmental variables. If it is possible to group departments in this way, then you have another method to test Brownian motion. Within a group of departments that appear to form an ad hoc region, what is the pattern of change in yield from year 1 to year 15, trend, random walk, mean reverting? As I understand the theory. Brownian motion would give a random walk. If you get a similar pattern for the departments in an ad hoc region, the there would be a low probability that random walks would have the same profile over time. You could aggregate the data for a ""region" and apply various tests to determine if the patterns differ significantly.

2. Testing the proportions of land in various uses.

Let us say that you have 4 uses. If you attempt to assess all 4 land uses that would mean you have only 3 degrees of freedom. In my opinion, you should begin by testing each land use independently. If you hypothesize that excess risk may drive farmers away from maize as a grain crop, then you are not obliged, at least in the first instance, to say which alternative activity they adopt. Instead, you can break the analysis into two steps.

1. Do farmers shift out of maize-as-grain production in (a) a systematic way over time in groups of nearby or contiguous departments, or (b) does the process appear to be based on the variance in yield and lack of coherence in the pattern of grain yields across departments over time. A shift based on (a) might indicate environmental factors ( We are all in the same boat / Le soleil brille pour tout le monde) , while (b) might indicate pure risk random or without specific cause (Everybody for hinself / Sauve qui peut).

2. When farmers shift away from grain production do they shift to the same suite of alternative productive activities? When they shift away from maize as grain, do they shift to the same activities in the northeast, northwest, central, southeast or southwest. If not, why not?

I am biased by my training as a geographer, In my opinion, the FAO map that I referenced previously, indicates that France is large enough in extent, both latitudinally and longitudinally, that soil, temperature, rainfall, elevation, etc. vary widely. My experiences in Ontario Canada and Ohio and other parts of the American Midwest lead me to think that you may need to use explicitly some weather data for the 15 years of your study.

As I recall maize-as-grain has specific requirements (though hybrid varieties modify the particulars): 200 days growing season, 1000 mm of rainfall per annum concentrated in summer. At the latitude of France, elevation above a couple of hundred meters would be a negative,. The same for dry summers in the south. Thus, the marine climate of the northwest and the Mediterranean climate of the south coast would also be negatives. If the micro-climates of France rate as a risk factor, I would expect that farmers would switch from maize-as-grain to maize-as-silage. especially in the northwest.

Actually, you might approach your study by the inverse method. The evidence of the direction of land-use change might lead you to determine if and where the process is driven by environment and where it is driven by pure risk.

You have chosen a fascinating study. In my opinion, the most interesting aspect is that the process driving land-use change may be partly Brownian (pure risk with no specific cause) and partly systematic (climatic variability). There may be no standard way to approach the question, so you have to invent your own approach,

For climates of France, this map may be a good start and might be a way to develop ad hoc regions. My suggestion would be to form study regions on the basis of climate categories and to determine if land-use changes within these regions is based on pure risk (Brownian i.e. not coherence across departments over space and time) or if there are residual systematic drivers of shifts in land use, i.e. land use shifts from maize-as-grain to some preferred land use.

Koppen climates of France:

https://en.climate-data.org/country/216/

Belda et al have updated the Koppen / Trewartha classification. There is a little wet bias in the data they used, but probably not significant for France.

Article Climate classification revisited: From Köppen to Trewartha

_www.int-res.com/abstracts/cr/v59/n1/p1-13/

To sum up all of this, I suggest from experience of my own hypothesis-driven research, to focus rather on the economic issues. I suggest that you explore the data to determine to what extent various models might "explain" what we see in the real world.

This way all the work that you will do has a better chance of producing something useful that if you decide upon the hypothesis and then try to see if the data fits. If the data does not fit, you end up with a negative, a risky way to spend your time.

Pr Colbourne, thank you very much, I really appreciate your help. I was focused on another article from many day, sorry if I did not answer in time. I received my data and I have a 27 years time serie, for approximately 100 departments. I will try to apply your first advices and observe for the differences/convergences between department mean and variance in yields. Actually I have three kind of land : forrage crops (especially maize, but it is not specifically precized), temporary/artificial grassland, and permanent grassland (+6 years, legally forbidden to remove it, thus totally irreversibles). I think your idea to separate potentiel climate related issues that directly explain the land use specialization by local regions, and pure random uncertainty, possibly Brownian, is a really good point. Still I have to find a way to do it econometrically... I will continue to work and integrate your suggestions, thank you again, and if I have any advices to ask I will come back to you! Hope my topic is technically feasible to study! If I have good results I'll upload them on my profile.