Step 1 - Modeling with trig function.sagews

%md

Here we have some data:

Syracuse weather data (month, max temp):  (1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)

We'll make a scatter plot of the data first and call the plot 'p'.

Syracuse weather data (month, max temp): (1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)

data=[(1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)]
p=scatter_plot(data)
p

data=[(1, 32),(2,34),(3,43),(4,57),(5,69),(6,78),(7,82),(8,80),(9,72),(10,60),(11,48),(12,36)]#you will edit this with new data
a,b,c,d=var('a,b,c,d')
model(x)=a * sin(b*(x - c))+d
values=find_fit(data,model,initial_guess=(25,pi/6,7,60))#you will edit the guess values based on the scatterplot
params=[a,b,c,d]
for i in range(4):
    params[i]=values[i].rhs()

model(x)=params[0] * sin(params[1]*(x - params[2]))+params[3]
model(x)
p=scatter_plot(data,xmin=-2,xmax=14,ymin=-10,ymax=100)
p+=plot(model(x),xmin=-2,xmax=14,ymin=-10,ymax=100)
p

25.61099628075631*sin(0.5089522520223427*x - 2.068485038219847) + 56.87966137474419

find_fit(data,model,initial_guess=(25,pi/6,7,60))

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/misc/lazy_import.pyx", line 354, in sage.misc.lazy_import.LazyImport.__call__ (/ext/sage/sage-8.0/src/build/cythonized/sage/misc/lazy_import.c:3559) return self.get_object()(*args, **kwds) File "/ext/sage/sage-8.0/local/lib/python2.7/site-packages/sage/numerical/optimize.py", line 685, in find_fit raise ValueError("no variables given") ValueError: no variables given

a = 25.61099628075631; b = 0.5089522520223427; c = 4.064202545524923; d = 56.87966137474419

y=a * sin(b*(x - c))+d
show(y)

$\displaystyle 25.6109962807563 \, \sin\left(0.508952252022343 \, x - 2.06848503821985\right) + 56.8796613747442$

p=plot(y,0,12)
p+s

Your turn!

1. Follow the link http://mathbits.com/MathBits/TISection/Statistics2/sinusoidal.html to find data for the two other cities. Type in the data and create separate scatter plots. Then find the sinusoidal model for each.

2. Plot the function: f(x)=sin(2x-π)-2, g(x)=-3sec(πx); h(x)=tan(x-π/2) in a single graphing window from [-2π,2π]. Use different colors for each graph!

data=[(1, 43),(2,47),(3,56),(4,67),(5,75),(6,84),(7,88),(8,87),(9,80),(10,68),(11,58),(12,47)]
p=scatter_plot(data)
p


data=[(1, 43),(2,47),(3,56),(4,67),(5,75),(6,84),(7,88),(8,87),(9,80),(10,68),(11,58),(12,47)]
a,b,c,d=var('a,b,c,d')
model(x)=a * sin(b*(x - c))+d
values=find_fit(data,model,initial_guess=(22,pi/6,7,68))
params=[a,b,c,d]
for i in range(4):
    params[i]=values[i].rhs()

model(x)=params[0] * sin(params[1]*(x - params[2]))+params[3]
model(x)
p=scatter_plot(data,xmin=-2,xmax=14,ymin=-10,ymax=100)
p+=plot(model(x),xmin=-2,xmax=14,ymin=-10,ymax=100)
p

22.750416739876204*sin(0.4941622370155175*x - 1.947020081527723) + 65.36784787240074

y=a * sin(b*(x - c))+d
show(y)

$\displaystyle a \sin\left(-b {\left(c - x\right)}\right) + d$

p=plot(y,0,12)
p+s

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'y' is not defined

data=[(1, 62),(2,65),(3,72),(4,80),(5,87),(6,92),(7,96),(8,97),(9,91),(10,82),(11,71),(12,63)]
p=scatter_plot(data)
p


data=[(1, 62),(2,65),(3,72),(4,80),(5,87),(6,92),(7,96),(8,97),(9,91),(10,82),(11,71),(12,63)]
a,b,c,d=var('a,b,c,d')
model(x)=a * sin(b*(x - c))+d
values=find_fit(data,model,initial_guess=(17,pi/6,7,82))
params=[a,b,c,d]
for i in range(4):
    params[i]=values[i].rhs()

model(x)=params[0] * sin(params[1]*(x - params[2]))+params[3]
model(x)
p=scatter_plot(data,xmin=-2,xmax=14,ymin=-10,ymax=100)
p+=plot(model(x),xmin=-2,xmax=14,ymin=-10,ymax=100)
p

17.757748190683763*sin(0.5034534732269098*x - 2.004683736962059) + 79.1495989202177

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/misc/lazy_import.pyx", line 354, in sage.misc.lazy_import.LazyImport.__call__ (/ext/sage/sage-8.0/src/build/cythonized/sage/misc/lazy_import.c:3559) return self.get_object()(*args, **kwds) File "/ext/sage/sage-8.0/local/lib/python2.7/site-packages/sage/numerical/optimize.py", line 685, in find_fit raise ValueError("no variables given") ValueError: no variables given