Windows 10 appears to be randomly starting systems that have been shut-down without user input.
When a PC is shut-down from Windows 10 "Power" Start menu option ("Start » Power » Shut-down"), systems may restart on their own without user input (the user pressing the power button) anywhere from 30 minutes to several hours after the action. There may be no immediate or apparent reason(s) or causes for the computer to start-up unassisted.
There are potentially a number of reasons why a computer may boot and start-up unassisted.
If the PC successfully boots into Windows, to find out what initiated the boot call, from the "Command Prompt" ("Start » All apps » Windows System » Command Prompt" (right-click "Run as administrator") type "powercfg -lastwake". This will report the last item to initialise the computer and narrow down the device or service making the call.
Wake on LAN
If nothing is listed using "powercfg - lastwake" check to make sure the system is not set to "Wake on Lan". This may need to be done in both Windows 10 "Device Manager" and the systems BIOS (usually "F2" or "F10" to access on start-up). To access "Device Manager" click "Start » Settings » System » About (bottom-left menu) » Device Manager (bottom-right panel)". In Device Manager click the arrow (">") to the left of "Network adapters" to access the available hardware, right-click each entry in turn, select "Properties" and in their respective "Power Management" settings check that "Allow this device to wake the computer" is DISABLED (NO checkbox).
Additionally (and depending on availability, optionally), when checking the PCIe Lan properties (settings associated with a hardwired connection to the PC and not the Wireless device) click "Advanced" and scroll down the options shown to "Wake on Magic Packet"; set the "Value:" to "Disabled". Similarly for "Wake on pattern match"; set it to "Disabled". This essentially turns off 'wake' calls received over Lan.
If the above fail it may be Windows 10 "Fast Boot" ("Fast start-up") feature design is causing the issue (may also applicable to Windows 8.1). With this enabled when a system is shut-down from the Power menu options it does not fully power-off the machine but instead goes into a state similar to "Hibernation" (as opposed to "Sleep" mode) where certain features, services and devices are kept alive in memory to speed to the boot process - from a 'cold boot', all system devices have to be initialised which can take some time to do, "Fast Boot" relieves this by keeping some services in memory requiring fewer to be started on boot, thus reducing boot time. For "Fast Boot" to work, PC's are shut-down but not completely powered-off (so memory can remain active). This 'state' may result in systems randomly booting up without user input or prompting.
The "Fast Boot" ("Fast start-up") feature is part of Windows 10 "Power & sleep" settings and can be disabled (admin rights required). To access click "Start » Settings » System » Power & sleep (left menu)" then under "Related Settings" to the right click "Additional power settings". When "Power Options" opens click "Choose what the power buttons do" on the left then at the top of the page click "Change settings that are currently unavailable" (admin rights required). Scroll to the bottom of the page and under "Shut-down settings" disable "Turn on fast start-up (recommended)" (no checkbox).
Windows Update Reboots
It's also possible that apparent random system reboots are the result of Windows 10 automatically rebooting on schedule or after downloading and installing updates. Depending on the options set this cannot be fully disabled as Windows 10 (currently) downloads and installs updates automatically, rebooting from which can be delayed but not completely disabled (as is possible with Windows 7). To check, click "Settings » Update & security » Advanced options".