Assuming they really are histograms - i.e. the x axis is not categorical - then the Kolmogrov-Smirnov test may be an option. If they are categorical then I'm not sure a statistical test would be meaningful, the absolute difference is what it is for each land cover.

Thank you for your answer. I believe the test of Kolmogrov-Smirnov is for continuous variables, which is not the case of my histogram. I have keeped searching and I think the Wilcoxon test is the good one, but I am still testing it. If someone has another idea or knows if this test is the good one or not, it will be nice to have his/her feedback.

Example of the type of histograms I work with: H1(0, 0.2, 0.3, 0.5, 0), and H2(0.2,0.2,0.2,0,0.4). The sum is allways 1 (they are percentages), and the average value of the two histograms is also the same for both.

This sounds to me like a two way table---with factors (Land Type) x (Region) which can be analyzed as a multimomial or with what I'm guessing are giant sample sizes, treated as Poisson Counts. But you will also want to account for the spatial dependenci.e. each pixel in the map is likely to be spatially dependent on it's neighbors), So you would want to add this complexity into the analysis by setting up a generalized mixed model.

But practically speaking, given the giant sample sizes (I assume) you will undoubtedly get significant differences, so I would focus more on estimating odds ratios to see if the magnitude of differences are physically meaningful ....

Please, don't call them histograms (Pearson will be turning in its grave), they looks like compositional data.

No test can be used if the number of pixels are not available (the size of the samples).

So, I believe that John is right.

Firstly, similarity of the land cover in the two regions may require not only comparison of the percentages of areas covered by each LC, but also the spatial arrangement of LC patches. You may want to take a look at Fragstats or similar software for computing metrics for categorical map patterns.

Secondly, you seem to be comparing maps, which are imperfect representations of reality. Maybe, your "significance" refers to whether the difference between the two maps is large enough to be distinguished from uncertainty owing to the unknown map errors. If so you may want to look into error propagation methods.

I agree with Antonio, your histograms look like compositional data, and the difference can be measured using the Aitchison distance. The problem is the zeros (probably due to zero counts) for computing this distance. You might be interested in looking at publications on zero replacement for compositional data by Martin-Fernandez and by Palarea-Albaladejo.

Chi Square would be the simplest test to use with your data. This link http://wps.pearsoned.co.uk/ema_uk_he_howitt_spsspsych_4/82/21003/5377014.cw/content/index.htmllink gives a good overview of Chi Square in a series of bullet point. Good luck

Thank you everybody for the responses. Actually, my major problem is that I have frequencies, and I want to have frecuencies, not number of pixels, since I am interested to know the predominant (or distribution of) classes of each region and comparing to the distribution of a neighbour region. And having frequencies is a problem for numerous tests.

For those who say that this is not a histogram, could you please explain me why? or give the definition of histogram? Moreover, this made me think that I could obtain a linear regression between the frequencies of class ocupation of the two regions, this gives a correlation coefficient for which I can easily obtain the significance, do you think this is the correct way to analyse it? The results I obtain for the test I have done give the results that I excpected. The question if I do like that, is if I can ignore the data of the classes with low or null occupacy in the two regions I am comparing, since this classes are clearly not important.

You need to define the term frequency in your terms as I would refer to the number of pixels in a particular class synonymous with the frequency of that class in the area of interest. In statistics, frequencies are counts, the number of observations in a group. If this is what you are working with then the comments people have added seem applicable.

A histogram is, in the Pearson sense (i.e. the original sense), a graphical representation of a density distribution using a set of non overlapping rectangles. The base of the rectangle (on the X axes) is an interval of measures (in a distribution for grouped data the interval is identified by the class boundaries), the height is the density of that class, namely the frequency (relative frequency) divided by the length of the base. This is because the area of the rectangle must be proportional to the frequency counts. Thus, a histogram is an "approximation" of a frequency or of a probability density function. This is relatively different from the bar diagram, that is useful for qualitative or discrete statistical variables. For example the Histogram graph proposed in EXCEL is not a histogram for a statistician, but a bar diagram (this is one of the misuse of statistical concepts done by EXCEL). Also the histogram used in the image analysis are not technically histograms, they are bar diagrams on a discrete variable (255 levels of tone of a color).

Here http://en.wikipedia.org/wiki/Histogram the concept of histogram is explained in a plain way.

Hi Mar, I would not do correlation analysis with the type of data you have, you will get what Pearson called "spurious correlation" back in 1897. Look better for methods specific for compositional data.

I am assuming you want to measure the difference between all classes in the two regions. You might look at the Kappa statistic which is often used to compare the classes of known to classified results at specific locations. This would give a measure of how related the two areas are. I do not know if it is often used in judging the 'similarity' of images so am not sure if there would be a specific cut off to say if they are or are not similar. Others may have comments on this method.

I think I have finally well understood the problem, as Antonio and Vera said, what I have are not histograms, but compositional data, thank both, and so I should use the Aitchison distance. I have been reading some bibliography but for the moment I have not found how to obtain the significance of this distance that I will need if I want to be objective when saying if two sets of data can be considered 'the same' or not.

I can't find how to obtain the significance of the Aithchison distance, can Vera or someone else tell me how to do it? I have seen that this distance is implemented in R, but it doesn't give the significance and I don't know how to use this distance or how to obtain the significance. For example I have made two tests, one comparint two compositions which are similar and I obtain a distance of 1.13, and the other one comparing to compositions which are different and I obtain a value of 3. But where is the limit? Is it possible to establish a limit objectively?

The Aitchison distance can be analysed like all other distances. But, as I see it, the problem is that you have not a sample, but just one composition in each case. If you have a sample, you can do a MANOVA on ilr coordinates.

If the two cases constitute your population and the maps are errorless, then any difference in the composition between the cases is statistically significant. However, (1)such difference may not be meaningful and (2) your maps are not error free. Regarding case 2: Having a sample of ground truth points you can use a geostatistical model or design-based inference to estimate confidence intervals for the relative area of land cover types for each case.

I have followed your question and the answers you got. I was wondering how a simple question generates more and more related questions and problems, which touch the principles of statistics. As a consequence, to get a clear, rigorous, and definitive answer may be very difficult. I would recommend to stop at that approach satisfying your needs: if you only need a test (a p-value) to report it in a publication, the answer may be simple; if you want to go further and get inside of a methodology the answer may be complicated.

When somebody asks for getting a p-value (significance), one is referred to a hypothesis testing. In your case, the null hypothesis appears clear: " H_0: the proportions of soil cover type in the two regions are equal " against the contrary. To go ahead with the hypothesis testing, the next step is to identify which are the random variables involved and their (credible) distributions. Here, we find the first problem.

If you have "only" two vectors of proportions (compositions), are they realisations of a random composition? I guess they are. However, we need to know which is the source of uncertainty. Frequently, these proportions can be assumed to be probabilities in a multinomial sampling. If the sampling can be assumed multinomial, a testing procedure can be performed in the following steps:

A.- compute the Aitchison square-distance between the two sample compositions of soil cover for the two regions. Let it be d2.

B.- Simulate Aitchison distances assuming H0, i.e. consider the pixels from both regions mixed, and then extract two samples with replacement (bootstrap), one for each region. Then, compute the simulated Aitchison square-distance. This can be repeated a large number of times (say 10,000 or more).

C.- Identify the probability p of the quantile d2 within the simulated sample of square-Aitchison distances. Then 1-p is a bootstrap p-value for H0.

Three comments:

(1) the same procedure could be applied to any proper divergence between the two compositions.

(2) In this kind of problems, the proportions of different covers can differ in some orders of magnitude. In this situation, any asymptotic distribution (e.g. chi2) is not reliable, as asymptotic conditions are unlike to hold.

(3) In the resampling procedure, you may find some zero-counts which prevents computation of Aitchison distances. Then I would recommend hot-deck zero-replacement. For instance, you can compute frequencies using

freq_i = ( n_i + 1/k ) / (n + 1) (or similar)

where n is the total of pixels per region, n_i pixels in the i-cover-class and k the number of possible cover classes.

However, your case seems to be spatial, and therefore, some spatial dependence is expected, even within an individual region. If so, resampling can be carried out using connected subsets of pixels (for instance, 25 pixels in a square) better than individual pixels.

Other approaches related to multipoint geostatistics or Kriging may get more and more involved.

Hope, this long comment is useful.

Thank you everybody for your answers. And thanks Juan Jose for your definitive and summarising answer, it was really usefull, this is exactly what I wanted. I am going to try that.

Well done, Mar! I think the proposal of Juanjo (Juan Jose) is a reasonable approach if you do resampling of connected subsets of pixels. For that, besides the spatial information on each data point you need a rough idea of the spatial range of influence of your cathegories.

The Kolmogorov-Smirnov (K-S) test is the statistic method that is usually used for this.

Hi Vera, I want to know how one would go about using MANOVA to compare the ilr coordinates? If I have compositional data for n=x and n=y samples of a control and treated group respectively, How are all the coordinates compared? 

Aaron,

I can answer your question addressed to Vera. The summary is brief but details are longer:  Take log-ratio transforms (ilr) to obtain coordinates of proportions. Then proceed with standard MANOVA using Euclidean distance. The main assumption may be multivariate normal distribution of the coordinates; we call this distribution Normal distribution in the Simplex. I recomend the following two papers (open source):

Josep-Antoni Martın-Fernandez, Josep Daunis-i-Estadella and Gl oria Mateu-Figueras: On the interpretation of differences between groups for compositional data. SORT 39 (2) July-December 2015, 231-252

Statistics & Operations Research

Transactions © Institut d’Estad´ıstica de Catalunya

ISSN: 1696-2281 [email protected]

eISSN: 2013-8830

www.idescat.cat/sort/

To do a visual comparison  of two compositions, you can use CoDa-dendrogram

Pawlowsky-Glahn, V. and Egozcue, J. J.: Exploring Compositional Data with

the CoDa-Dendrogram. Austrian Journal of Statistics, 40, 1& 2, 103{113,

2011.

You can find the general procedures in

V. Pawlowsky-Glahn, J. J. Egozcue and R. Tolosana-Delgado: Modeling and

Analysis of Compositional Data, John Wiley & Sons, Chichester, UK, 272pp.,

ISBN: 9781118443064, 2015