Uzgodnienie to Math Behind Self- organing Maps: Obliczenia i wnioski

Self- organining maps (SOM) are a type of artificial neural nework used for data visualization and clustering. They y organize high-dimensional data into a low- dimensional grid, reserving topological relationships. Understanding the mathetical calculations behind SOMs is essential for their effective application.

Initialization of thee Map

Te procesy zaczynają się od with initializazing thee e weight vectors of each node in thee map. Typically, weights are assigned random or based on thee data distribution. Each walt vector has thee same dimension as thee input data.

Finding the Bess Matching Unit (BMU)

For each input vector, the algorithm calculates thee distance to o every node 's weight vector. The most concentrace distance metric is Euclideun distance, calculated as:

d = ΔΔ1; Xi1; FLT: 0 Xi3; Xi3; i Xi1; Xi1; FLT: 1 Xi3; Xi3; (x Xi1; Xi1; FLT: 2 Xi3; Xi1; FLT: 3 XI3; Xi3; - w XI1; FLT: 4 XI3; XI3; i Xi1; FLT: 5 XI3; XI3;) ²

were x present 1; inje1; FLT: 0 presenta3; i presenta1; FLT: 1 presenta3; Equita3; is the input data contesent and w presentation 1; Equipa1; FLT: 2 presenta3; Equipation 3; i extentation 1; FLT: 3 presentation 3; Is thee weight present of a node. The node with thee smaleste distance is identified as thee BMU.

Updating the Weights

Once thee BMU is identified, thee weights of thee BMU and it s nexts are adiusted to memore misilar the input vector. The update rule is:

W BELG1; BELG1; FLT: 0 XX3; XIR3; new XX1; XIR1; FLT: 1 XX3; XIR3; = w EFIR1; FLT: 2 XX3; FLT: XX3; ELD XI1; XI1; FLT: 3 XX3; XI3; + α (t) * h XX1; FLT: 4 XX3; XI3; ci XI1; CSI; FLT: 5 XXX3; XI3; (t) * (x - w 1; XIX1; FLT: 6 XX3; XIX3; old XI1; FLT: 7 XXX3; XIX3;

where α (t) is the learning rate, h has 1; Xi1; FLT: 0 supporte3; Xi3; ci supporte1; Xi1; FLT: 1 supporte3; Xi3; Xi3; Xi3; XiThe they neaghhood functionion, andd (x - w supporte1; Xi1; FLT: 2 supported 3; Xi1; Xi1; FLT: 3 supportex3; Xi3;) ites the differencete between the input vector and thee export weigt vector.

Wnioski o wydanie pozwolenia na dopuszczenie do obrotu

SOM are use in varioos fields for data analysis and visualization. They help identify Patterns, cluster similar data points, andd reduce dimensionality. Common applications include image analysis, market segmentation, andd bioinformatics.