1 生活中的立体图形名师教学设计1

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

1.

2. a)

See the Python code for details.

b)

Since labels are missing for some data points, we tested K = 2, 3 and 4 for each choice of p to compare. Nonetheless, for p = 0.5 and 0.75, when setting K = 4, there will pop up a “singular matrix” error, which is due to the fact that the difference between the results generated based on K = 3 and K = 4 are so small that the appearance of the new class is of very little significance; therefore, only K = 2 and K = 3 were tested for p = 0.5 and 0.75.

The estimated model parameters for each case are as follows.

-------- p = 0.0 -------- -------- Kfit = 2 -------- -- initial guess -- [ 0.49835133 0.50164867]

[[ 1.44894987 1.09492477] [ 1.4634363 1.0949032 ]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.69029654 0.30970346]

[[ 1.31804542 0.80829807] [ 1.75580906 1.72552368]]

[[[ 0.08762499 -0.03286065] [-0.03286065 0.1204758 ]]

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

[[ 0.12135553 -0.0747156 ] [-0.0747156 0.43062284]]]

-------- Kfit = 3 -------- -- initial guess -- [ 0.3339858 0.33371978 0.33229442]

[[ 1.46316153 1.10147044] [ 1.46680974 1.09544048] [ 1.45102202 1.10179691]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.27677887 0.24894146 0.47427967]

[[ 1.77938761 1.76358056] [ 1.47929918 0.4805992 ] [ 1.25003575 1.02176658]]

[[[ 0.12639369 -0.09855264] [-0.09855264 0.46413527]]

[[ 0.03796945 -0.02633576] [-0.02633576 0.03552184]]

[[ 0.09574792 0.02198109] [ 0.02198109 0.08192252]]]

-------- Kfit = 4 -------- -- initial guess -- [ 0.25109986 0.24886868 0.24992517 0.25010628]

[[ 1.46201632 1.09675925] [ 1.46346104 1.09209636] [ 1.45851911 1.08631457] [ 1.4594736 1.09645365]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.16667038 0.50719752 0.2499377 0.0761944 ]

[[ 1.77128953 1.77520577] [ 1.28577542 1.03266928] [ 1.47635909 0.48146457] [ 1.801457 1.99999414]]

[[[ 0.15381698 -0.17269732] [-0.17269732 0.6365188 ]]

[[ 0.11096153 0.02540617] [ 0.02540617 0.08348557]]

[[ 0.03909791 -0.02717554] [-0.02717554 0.03589123]]

[[ 0.0905151 0.01167074] [ 0.01167074 0.04488814]]]

-------- p = 0.25 -------- -------- Kfit = 2 -------- -- initial guess -- [ 0.50167135 0.49832865]

[[ 1.46252566 1.09330151] [ 1.45451775 1.10205348]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.73443703 0.26556297]

[[ 1.32358609 0.87682923] [ 1.7948676 1.84085281]]

[[[ 0.09228171 -0.01528174] [-0.01528174 0.12979229]]

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

[[ 0.12825835 -0.11222504] [-0.11222504 0.43531442]]]

-------- Kfit = 3 -------- -- initial guess -- [ 0.33299483 0.3340083 0.33299687]

[[ 1.44559642 1.10043981] [ 1.46380404 1.08549384] [ 1.46456933 1.0938217 ]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.5232781 0.24585777 0.23086414]

[[ 1.27409897 1.01752983] [ 1.79519383 1.83636295] [ 1.49677601 0.46967385]]

[[[ 0.10057622 0.02289809] [ 0.02289809 0.09103552]]

[[ 0.13188614 -0.11882957] [-0.11882957 0.45466389]]

[[ 0.03468481 -0.02622534] [-0.02622534 0.03652594]]]

-------- Kfit = 4 -------- -- initial guess -- [ 0.24901948 0.24873643 0.25155309 0.250691 ]

[[ 1.44264802 1.0829949 ] [ 1.44997213 1.08827416] [ 1.4440786 1.09842781] [ 1.4606242 1.08865406]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.52499859 0.22770079 0.23107068 0.01622994]

[[ 1.27610634 1.01772101] [ 1.79496426 1.83487046] [ 1.49668602 0.46970855] [ 1.79379985 1.95481904]]

[[[ 0.10155772 0.02324225] [ 0.02324225 0.09016549]]

[[ 0.13221097 -0.13064201] [-0.13064201 0.48499905]]

[[ 0.03468129 -0.02619838] [-0.02619838 0.03648439]]

[[ 0.13704276 0.02369029] [ 0.02369029 0.00424113]]]

-------- p = 0.5 -------- -------- Kfit = 2 -------- -- initial guess -- [ 0.49979459 0.50020541]

[[ 1.45798963 1.08662836] [ 1.44076968 1.09967133]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.69909577 0.30090423]

[[ 1.30373199 0.9071445 ] [ 1.77985228 1.8079245 ]]

[[[ 0.09378547 -0.00650109] [-0.00650109 0.11804495]]

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

[[ 0.13154108 -0.10353047] [-0.10353047 0.4452401 ]]]

-------- Kfit = 3 -------- -- initial guess -- [ 0.33528381 0.33045046 0.33426573]

[[ 1.44393711 1.08726044] [ 1.44150335 1.09256627] [ 1.45569313 1.09152687]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.51347788 0.25865312 0.227869 ]

[[ 1.27360261 0.99427891] [ 1.78155952 1.82376029] [ 1.48703702 0.48319212]]

[[[ 0.099533 0.0169684 ] [ 0.0169684 0.0918917 ]]

[[ 0.13365188 -0.10857936] [-0.10857936 0.43209066]]

[[ 0.03797137 -0.03105717] [-0.03105717 0.04203827]]]

-------- p = 0.75 -------- -------- Kfit = 2 -------- -- initial guess -- [ 0.50032696 0.49967304]

[[ 1.4667117 1.09393747] [ 1.45026071 1.08489352]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

-- estimated parameters ----- [ 0.68527136 0.31472864]

[[ 1.28807946 0.95753566] [ 1.78665649 1.82385889]]

[[[ 0.09740422 0.00608739] [ 0.00608739 0.10421715]]

[[ 0.13120358 -0.11080516] [-0.11080516 0.44340947]]]

-------- Kfit = 3 -------- -- initial guess -- [ 0.3314372 0.33548 0.3330828]

[[ 1.44236618 1.09971416] [ 1.46498039 1.10005968] [ 1.45810187 1.09013843]]

[[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]

[[ 0.13931964 0.04009849] [ 0.04009849 0.39718358]]]

-- estimated parameters ----- [ 0.51768417 0.25404774 0.22826809]

[[ 1.27458659 0.99500506] [ 1.78703804 1.83802019] [ 1.48858326 0.48330352]]

[[[ 0.10017068 0.01600791] [ 0.01600791 0.09164531]]

[[ 0.13234515 -0.11283545] [-0.11283545 0.42743208]]

[[ 0.03700917 -0.03034101] [-0.03034101 0.04171527]]]

c)

See next page for the plots.

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

d)

There are a few things worth noting from the plots above.

STA2536 MINI PROJECT 1 Name: Ke (Frieda) Tong Student #: 999614092

i.

ii.

For each choice of K, as we increased the percentage p, the plots gradually converged to the one with full information as more labels became available. For p = 0, when setting K = 4, there appeared to be 4 classes that can be identified distinctively; nonetheless, for the remaining choices of p, setting K = 4 seems to be overfitting and thus it was not necessary to consider a fourth class for such cases. As mentioned before, the plots for p = 0.5 are 0.75 when K = 4 are missing because those 2 plots are almost the same as those when K = 3; in other words, the fourth class is well embedded and hidden in the rest of classes. iii.