In her book Pain and Grace: A Study of Two Mystical Writers of Eighteenth-Century Muslim India, Dr. Annemarie Schimmel writes:

Khwaja Mir Dards family, like many nobles, from Bukhara; led their pedigree back to Bahauddin Naqshband, after whom the Naqshbandi order is named, and who was a descendant, in the 11th generation of the 11th Shia imam al-Hasan al-Askari. Although Shiite historians generally reject the claim that Hasan al-Askari fathered children other than Muhammad al-Mahdi, Bab Mawlid Abi Muhammad al-Hasan writes, in the Shiite hadith book Usul al-Kafi:

When the caliph got news of Hasan Askaris illness, he instructed his agents to keep a constant watch over the house of the Imam...he sent some of these midwives to examine the slave girls of the Imam to determine if they were pregnant. If a woman was found pregnant she was detained and imprisoned....

Africa
Most of the Muslim historians claimed that three of the descendants of Ali ibn Abu Talib migrated into Somalia and Ethiopia. The two Ashrafs migrated to Ethiopia and the remaining sayyid settled in Somalia. Ethiopia
Most of the Muslim historians and geologists claimed that one of the Ashrafs called Hajji Ali migrated into southern part of Ethiopia. After he migrated there, he got a baby and named him Gen-Silte. His children then called by their fathers name "Silte". according to the Silte tribesmen, the father of Hajji Aliyye(Hajji Ali) was Hajji Omar bin Osman, who was an Arab. He used to live in Hijaz, now part of Saudi Arabia.
Hence it is even problematic in a judicial point of view to call a patrilineal descendant of Ali Ibn Abi Talib a Sayyid, when he does not meet the above-mentioned criteria. See also
Family tree of Muhammad
Kohen, a similar status in Judaism
Sharīf, another term with a very similar meaning

Notes
References
Sources
This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Sayad". Encyclopædia Britannica (11th ed.). Cambridge University Press. Van Arendonk, C.; Graham, W.A. (1960–2007). "Sharīf". In Bearman, P. J.; Bianquis, Th. ; Bosworth, C. E.; van Donzel, E.; Heinrichs, W. P. (eds.). Encyclopaedia of Islam, Second Edition.