numerical come station checked p
me deal number four we now look at
nonlinear least squeeze method we will take
two teams the first example
nonlinear least-square method his son what we called
pricing that means after some
there we will change some manipulation we can change the problem
into heaney from so consider now we have a dataset
XK way K and we want to feeding
with the function that is why
equals to pee times be to the power hacks
so this means we need
we would like to find the coefficient
PNB such that this function y
thanks with the best feet the date notice now
in the function a times be to the power
hacks the parameters eighty be the
enter the function leniently peace to the power
he's not to play so before we proceed
we would like to do some manipulation
change this in some way that PNB
to see function me
so if we take the natural law humble site
and we see that now we have the humiliation
which his equipment you one that is natural log
why you close to log P plus X times
be now we see
he we shall call the first time here
email hence week or the
L up 8 to be mine keep I'm
endl b2b my the block
and the function s depends home pay by
Bieber in the knee we feed
he would half in the regression okay so
now we have a new problem so standing from the tax KY Katie
I can generate to the data XK
SKT we're
the ass case simply equal to natural of Waikiki
for pay here for this dataset here
I would do the linear regression so this we already know how to do
week you know it in the least square method and we can
plywood we have learned valium purcell
and we would find the coefficients a bar
people and once the a bar
be boss fund weekend go back
the company the B&B for the many
prop using just the relation the must relations OP were equal to
exponentially by be to exponential
people so
this is an kind at least square method
to so the original problem
so I would like to mention that hmm one has to be a little bit careful here
because if you do this manipulation
any sold these
me at least square method you Harrell is defined
has least square have no foolish SKT
EMTs evaluating attacks key and then for the original problem
if you really want the least squares approximation
for the original problem then you will have to minimize the hero
between y que here and why
XK using PC expression now
that a row his different from the camera that be
used so be careful you actually solving
it different but you keep you minimum
in a different things and it isn't me okay our second example he he's actually
namely me cell again given the data
X ke y que me and 12 feeling the phone
action so the function is y ax equal to
px times sign BX to just to motivate a little bit at this discussion
can you think of I'm any application
dysfunction trying to feed him so if you recall
which you have none he knew 200 differential equation course
we sold second differential equations
with peace Western in size from teen
to forced migration without
NP when the frequency of course
matches the system frequency in
your system article something called residents
way it oscillates was certain frequency
and DC time the amplitude up the assassination
grows me time so you see the function we have here
is exactly an oscillation he think axes time
he and the npt that grows me capsule
you could be measured the upset mechanical structure that and
maintenance you mentioned
certain points at certain time the want to understand
when to print be his
preparation okay so
I'm engine peace I'm linear functions
berries not always
leave which can change it
to any so we just say we can
as truly them sim
comment that's solution with resin
looking now we define
the so you sunny he be
from the square the have reported
some no to let its plant in the expression poor
why hacks which is AX sign
BX put to XK -1 K
school year sup they'll be making function
so the moon we will know that
passion riveted with respect
be shambles be see that has to be signs
so if we differentiate this expression
he mod weekend week by chain rule
weekend two times this expression and then have to differentiate this
expression
he cheating knows has constant
which means I was just have this to me and p
XK times sign be axed and the partial derivative per side
with respect p whoops
to papal that should be passion to repeated with respect
to be get so and
to change you say he's dead the first part his seen
and then you should be multiplied by differentiating this expression
be thinking everything else a constant so why keep is not contributing
him this term will give me a
he XK times
differentiates fine again cosine the XK and then missed you have plenty one more
time
I will have to differentiate PXP be which
mix K we see that now we have
to buy two system me questions too so
p
so we know general theories no exact method
one console young systems
possible way would be to use
Newton's method for systems to bring
K so this week haven't talked about
there are some party my lecture notes that
included this point you might take a look if you're interested in how to
carry
out one comment on that would be
theoretically roots
may not be unique might be your route at all
they may be one something maybe more to put them
me media and second comment will be
if you use to Newton's method and find a rude you are not guaranteed
his me because are you finding his critical point
could be alchemy could be maximum
could be said as well so you have to want some
additional test possibly some second limited test
to figure out exactly this is me IMAX
so the conclusion would be
namely near problems he's a lot
hard much more complicated okay so
that's all for this hope you 20 see you next time