first part of NCD

    
      
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 # Normalized compression distance (NCD)
 
+Normalized compression distance (NCD) is a way of measuring the similarity between two sequences.
+
+A [compression algorithm](https://en.wikipedia.org/wiki/Data_compression) looks for patterns and repetitions in the input sequence to compress it. For example, instead of "abababab" we can say "(ab)x4". This is how [RLE](https://en.wikipedia.org/wiki/Run-length_encoding) works and this is most simple and illustrative aexample of compression. Main idea of NCD is that a good compression algorithm compress concatenation of two sequences as good as they similar and much better than each sequence separately.
 
 ## Calculation
 
 $$NCD_{Z}(x,y)={\frac  {Z(xy)-\min\\{Z(x),Z(y)\\}}{\max\\{Z(x),Z(y)\\}}}.$$
 
++ `x` and `y` -- input sequences.
++ `Z(x)` -- size of compressed `x`.
+
+So, how it works:
+
+1. We calculate size of compressed concatenation of two sequences.
+1. We subtract from this size the minimal size of these compressed sequences. Some properties of value on this step:
+  1. If it is equal to the maximal size of these compressed sequences it means that size of concatenation's compression is equal to sum of sizes of separate compression of each sequences. That is, compression algorithm couldn't compress input data at all.
+  1. If it is equal to 0 then concatenation's compressed size is the same as the compressed size of each of these sequences. It means that these sequences are the same.
+1. We divide result from previous step by the maximal size of these compressed sequences to normalize results. So, for the same sequences we will get 0, and for different -- 1.
+
 ## Normal Compressor
 
+Ok, but what is `Z`? This is the size of compressed data by normal compressor (`C`). Yeah, you can use any compression algorithm, but for non-normal compressors you will get strange and non-comparable results. So, there are these properties:
+
+1. Idempotency: `C(xx) = C(x)`. Without it you wouldn't get 0 for the same sequences because `Z(xx) - Z(x) ≠ 0`.
+2. Monotonicity: `C(xy) ≥ C(x)`.
+3. Symmetry: `C(xy) = C(yx)`
+4. Distributivity: `C(xy) + C(z) ≤ C(xz) + C(yz)`
 
 ## Entropy
 
@@ -18,3 +38,8 @@ $$NCD_{Z}(x,y)={\frac  {Z(xy)-\min\\{Z(x),Z(y)\\}}{\max\\{Z(x),Z(y)\\}}}.$$
 
 
 ## Further reading
+
+1. [Clustering by Compression](https://homepages.cwi.nl/~paulv/papers/cluster.pdf)
+1. [Compression-based Similarity](https://homepages.cwi.nl/~paulv/papers/ccp11.pdf)
+1. [Article on the Wikipedia](https://en.wikipedia.org/wiki/Normalized_compression_distance)
+1. [Discussion about NCD with Rudi Cilibrasi](https://github.com/orsinium/textdistance/issues/21)