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y = Ax + b



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y = Ax

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        .   .

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        ,   , :    20,   40.      ,    .      .   ,     .  ,  =40 (=40),         = 0,4,   :



y = Ax = (0,4) * (40) =16

     y = 16,    y =20.

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 =       



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 = 20  16 = 4



      4       :












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 = 21  16 = 5



    :








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y = Ax

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T = ( + ?) 

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       = T  :



 = T   = ( + ?)    = Ax + (?)    = (?)

 = (?)



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? =  / 

!              ?   .

  !  x = 40    = 0,4,  E = 5,    ?:

?= / = 5 / 40 = 0,125

    :

 = +? = 0,4 +0,125 = 0,525

 , ,    = 0,525.    ,       :



y =   = 0,525 * 40 = 21



 !

        ,    ,        .

     y = 40,  x = 20.   ,         ,         y = 39.

 x = 20   ,       =0,525:

 = Ax = 0,525 * 20 = 10,5

   = 10,5,    y = 39.

       ,          ,        .

 = T  y = 39  10,5 = 28,5



  ?   :



?= / = 28,5 / 20 = 1,425



   :



 = +? = 0,525 +1,425 = 1,95

  :

y =   = 1,95 * 20 = 39



 ,  x = 20, A = 1,95  ? = 1,425        39,      .

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? = L * ( / X)

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  ,    =0,4.       = Ax = ,4 * 40 = 16.  x = 40    = 0,4,  E = T  y = 21  16 = 5.   ,       ,    ,      T = 21.

 : ?= L ( / ) = 0,5*(5 / 40) = 0,0625.  :  = A + ? = 0,4 + 0,0625= 0,4625.

 : y = Ax = 0,4625 * 40 = 18,5.

      .

      ,      = Ax = ,4625 * 20 = 9,25.

,  = 9,25        y = 39,        ,     .

 x = 20    = 0, 4625,  E = T  y = 39  9,25 = 29,75.    ,   ,       ,    ,      T = 39.   ? = L ( / ) = 0,5*(29,75 / 20) = 0,74375.    = A + ? = 0,4625+ 0,74375 = 1,20625.

  y = = Ax = 1,20625 * 20 = 24,125.

      , ,      :










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import numpy



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import numpy as np



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#   at ()

class at:

#    !



def says (self):

print (!)

pass

pass



 class at      at (),   def says(),       at,        .       says()     !.

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classcat = at () #  classat,  at

classcat.says () #  says (),  classat

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   ,   ,   .

  ,     ,     ,       =0.4.

A = 0.4

    :

A_vis = A

   :

print(' : ', A, '* X')

   :

lr = 0.001

  :



epochs = 3000



       .      ,             .

   ,  .   .         x,      ()  Y.

    :

arr_x = [1, 2, 3, 3.5, 4, 6, 7.5, 8.5, 9]



    ( Y):

arr_y = [2.4, 4.5, 5.5, 6.4, 8.5, 11.7, 16.1, 16.5, 18.3]



   ,    for i in range(len(arr)),       ,    .    for e in range(epochs),        ():



for e in range(epochs):

for i in range(len(arr)):



 len(arr)   ,     .



 x       x:



x = arr_x[i]



        :



#   y,  

y = A * x

#   Y,  

target_Y = arr_y[i]

#  E =     

E = target_Y  y

#    x,     A+A = A

A += lr*(E/x)



,        ,    .



   :



print(' : y = ', A, '* X')



  :



#       

A = 0.4

A_vis = A #     

#    

print(' : ', A, '* X')



#  

lr = 0.001

#   

epochs = 3000



#     x

arr_x = [1, 2, 3, 3.5, 4, 6, 7.5, 8.5, 9]

#     ( Y)

arr_y = [2.4, 4.5, 5.5, 6.4, 8.5, 11.7, 16.1, 16.5, 18.3]



#   

for e in range(epochs):

for i in range(len(arr_x)): #len(arr)     

#  x  

x = arr_x[i]



#   y,  

y = A * x



#   Y,  

target_Y = arr_y[i]



#  E =     

E = target_Y  y



#    x,     A+A = A

A += lr*(E/x)



#    

print(' : y = ', A, '* X')



      :



y = 2.0562708725692196 * X



  ,        ,        y = 2x.    ,        .

          Python.

       ,      .



   ,     Python    .       ,  ,   ,      .

  ,            :



import matplotlib.pyplot as plt



 ,     Python  ,      ,     .     :



%matplotlib inline



       ,        .     Anaconda Prompt,      Anaconda.

  Windows,  Anaconda Prompt  :



conda install matplotlib



  .        .

     ,         .

 :



import matplotlib.pyplot as plt

%matplotlib inline



#     

def func_data(x_data):

return [arr_y[i] for i in range(len(arr_y))]



#     

def func_begin(x_begin):

return [A_vis*i for i in x_begin]



#     

def func(x):

return [A*i for i in x]



#   X  

x_data = arr_x



#   X   ( )

x_begin = [i for i in range(0, 11)]



#   X   ( )

x = [i for i in range(0, 11)]

#x = np.arange(0,11,1)



#   Y  

y_data = func_data(x_data)



#   Y  

y_begin = func_begin(x_begin)



#   Y  

y = func(x)



#      

plt.title("Neuron")

plt.xlabel("X")

plt.ylabel("Y")



#      

plt.plot(x,y, label=' ', color = 'g')

plt.plot(x,y, label=' ', color = 'r')

plt.plot(x,y, label=' ', color = 'b')

plt.legend(loc=2) #loc   , 2    



#    (,)   10

plt.scatter(x_data, y_data, color ='g', s=10)

#  

plt.plot(x_begin, y_begin, 'b')

#  

plt.plot(x, y, 'r')

#      

plt.grid(True, linestyle='-', color='0.75')

#  

plt.show()



  ,    :








       : https://github.com/CaniaCan/neuralmaster (https://github.com/CaniaCan/neuralmaster)



     ,       matplotlib.

      def func_data(x_data), def func_data(x_begin), def func_data(x),   y,       .



    plt.title(),     plt.xlabel():



plt.title("Neuron")

plt.xlabel("X")

plt.ylabel("Y")



       plt.plot(),      , plt.legend(loc=2)       :



plt.plot(x,y, label=' ', color = 'g')

plt.plot(x,y, label=' ', color = 'r')

plt.plot(x,y, label=' ', color = 'b')

plt.legend(loc=2) #loc   , 2    



 scatter       :



plt.scatter(x_data, y_data, color ='g', s=10)



 plot       :



plt.plot(x, y, 'r')



      ,  plt.show().



   .     ,      .  ,   A,    = 2.05.     y = Ax = 2.05*x,    ,         (    ).

  ,           ?   ,           ,      :



x = input("   : ")

x = int(x)

T = input("   Y: ")

T = int(T)

y = A * x



# 

if T > y:

print(' !')

else:

print(' !')

 input   ,  .   :    ( )      (  ),      ,      .

   ,  :

   : 4

   Y: 15

 !



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,   ,        x (   )  x2 = 1.  ,       ,    : A*x+b*x2.   ,       y = A*x+b*x2 = A*x+b,   .

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      ()   (b),    ,    .    .        ,    (w2),        ().

 ,     ,    :

#   

for e in range(epochs):

for i in range(len(arr)): #len(arr)     

#  x  

x = arr[i]



#   y,  

y = w1 * x + w2



#   Y,  

target_Y = arr_y[i]



#  E =     

E = target_Y  y



#     x

w1 += lr*(E/x)



#     x2 = 1, w2 += lr*(E/x2) = lr*E

w2 += lr*E



  , ,         

  ,   (w2) (  (b)),      (w1) (  ())  .      ,          .

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               ,       ,             .  ,        .  ,         ,        .

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 ꠖ (https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82%D0%BD%D1%8B%D0%B5_%D0%BC%D0%B5%D1%82%D0%BE%D0%B4%D1%8B)    (https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BB%D0%B5%D0%B2%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F)    (https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82)   (https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)),        ,         ,    ()        .












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,    ,   1,5 /.     0,2   .  ,     :

 = 1,5 + (0,2 * )

S = 1,5 + 0,2t



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s = t?



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s(t) = ,    

? = 0

 ?       .



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    ,       :

s = t?

   :

s(t) = t?

?s = s(t+?t)  s(t) = (t+?t) ?  t? = t? + 2t?t + ?t?  t? = ?t(2t+?t)










      !       .  ,      t = 3.

s(t) = t?

s?(t) = 2*3 = 6

 ,    ,        .



  . ,        :

s(t) = t?

  :

s(t) = t?

?s = s(t+?t)  s(t) = t? + 3 t??t+ 3t? t? + ? t?  t? = ?t(3 t? + 3t?t + ?t?)










    (   s(t) = t?  s(t) =t?) ,    ,   ,     :

s(t) = t?










      ?  



s(t) = t

:



?s = s(t+?t)  s(t) = t + ?t  t = ?t



:










,    :



t? = 0




     


 

(u+v)? = u? + v?,  u  v  .

 f(x) = u(x) + v(x). :

?f = f(x+?x)  f(x) = u(x+?x) + v(x+?x)  u(x)  v(x) = u(x) + ?u + v(x) + ?v  u(x)  v(x) = ?u + ?v

 :








 ?u/?  ?v/?  ?->0    u?(x)  v? (x).       u?(x) + v? (x).

f?(x)= u? (x) + v? (x)

 

(u*v)? = u? v + v?u,  u  v  

,   .  f(x) = u(x) * v(x). :

?f = f(x+?x)  f(x) = u(x+?x) * v(x+?x)  u(x) * v(x) = (u(x) + ?u) * (v(x) + ?v)  u(x) * v(x) = u(x)v(x) + v(x)?u + u(x)?v + ?u?v  u(x)v(x) = v(x)?u + u(x)?v + ?u?v

 :








    u?(x) v(x).     v?(x)* u(x).  ,   ?u/?x,     u?(x),    ?v   ,        .  ,   :

f?(x)= u? (x) v(x) + v? (x) u(x)

  ,  , :

(c*u)? = c? u + cu? = cu?

,   ,      (c? = 0).

     ,     .

     :

s? (t) = (0,2t) ? + (1,5) ?

  ,    (0,2t) ?.         :

(0,2t) ? = 0,2

       (1,5) ? = 0.

,   ,  :

s? (t) = 0,2

    ,     .



  

,    , y   :

f = y?

y = x?+x

     :








    ,    .








 ,    : dy?/dy = 2y

  ,    : ? +  = (?)? + ? = 2+1

:

2(x?+x) * (2+1) = (2?+2) * (2+1) = 4?+6?+2

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   ,      :





















   


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Ij   ,     .    x1         w11,    2    -w21.    ,     ,   .



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   :








     f(x) = y,      = ?Iwij*xi,  xi    (),    wij  ,   ,     ,         :








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  ,         ,   (w2),     (w1)   (?w), : (w2 ==w1+?w).  (?w)    (   ),         .

  ,  (E2  E1 = -?E)  (w2 w1 = ?w),   :

?w = -?E/?w

  ?    ,      (? = E/),           .   ,      ( = +?),  ,   :

wij=  wij-( ?E/?w)

 ,        ,   .      w1  w2.

     :

wij=  wij dE/dwij

  , ,    ,  ,      .

,      :








 ,  (E2  E1 = ?E)  (w2 w1 = ?w),   :

?w = ?E/?w

  ,    ,     ,       (w1),    (w1)   (?w):

wij=  wij- ?E/?w

,    ,         ,        ,       .    ,   ,       .

  ,      :

wij=  wij dE/dwij

        !     ,     .   ,          !   ,    .

         : ? = L*(/).  (L)   ,   ,    , ,  .

  ,         :

wij=  wij L*(dE/dwij)

   ,    ,     .       ,      .

 

      ,  ,         ,   ,        ,    , ..      .








       ,       .












!            .   ,  ,     ,       .




   


      ,   ,     ,   .      Python,       ,      :








      ,   ,          y = 2x + 2,5.



import random

#       w1 = A

w1 = 0.4

w1_vis = w1 #     

#   w2 = b         Y

w2 = random.uniform(-4, 4)

w2_vis = w2 #    

#    

print(' : ', w1, '* X + ', w2)



#  

lr = 0.001

#   

epochs = 3000

#   (  )   x1

arr_x1 = [1, 2, 3, 3.5, 4, 6, 7.5, 8.5, 9]



#        1

x2 = 1

#    ( )

arr_y = [4.3, 7, 8.0, 10.1, 11.3, 14.2, 18.5, 19.3, 21.4]

#   

for e in range(epochs):

for i in range(len(arr_x1)): #len(arr)     

#  x  

x1 = arr_x1[i]



#   y,  

y = w1 * x1 + w2



#   Y,  

target_Y = arr_y[i]



#  E = -(    )

E =  (target_Y  y)



#    x,      

w1 -= lr * E * x1



#    x2 = 1

#w2 -= rate * E * x2 # .. x2 = 1,      

w2 -= lr * E



#    

print(' : ', w1, '* X + ', w2)

 ,    ,     : https://github.com/CaniaCan/neuralmaster (https://github.com/CaniaCan/neuralmaster)

  :

          :

import random

  ,  ,     (w2 = b)         Y:

w2 = random.uniform(-4, 4)

  random  uniform(from, to),     from  to .

  ,  ,    ,         .     (2 = 1),     (w2).  ()      (w1),  (b)     (w2).    ,     ,        .

  ,   ,       .        ,            Y,       .

       :

 : 0.4 * X + 0.3652477754014445



 : 2.058410130422831 * X + 2.5013583972057263










 !        .      ,     .

  ,    .      ,    ,      .      , ,        ,    ,       .

x1 = input("   : ")

x1 = int(x1)

T = input("   Y: ")

T = int(T)

y = w1 * x1 + w2



# 

if T > y:

print(' !')

else:

print(' !')

   ,  ,  ,    ,   ,      .

   : 4

   Y: 15

 !



  :

 ,    ,  ,    ,     (  ),     ,     ,    ,     (y = Ax). , ,         ,         .             ,      .   ,      ,        .           .  ,   ,    (),     ,     , ,     .

    !        .          ,     !

 ,   , ,   .        (y = Ax).      ,    .        (),  , ,       .     ,    ,      .

     (b),        Y,    ,        (y = Ax+b)  .

    (b),    (y = Ax1+b),     ,    ,   ,    (x2 = 1).      : y = Ax1+bx2.  2 = 1,    y = Ax1+b.  ,  ,      ,     w1 = ,  w2 = b,  : y = w1x1+w2.

   ,    ,       . ,    ,   ,       ,  .     .  ,        .  ,   ,        .    ,       ,     .

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        ,     ,  ,       .  ,     ,   ,    ,     ,        .

  , ,       ,     .   ,    .     ,   ,     ,    .




 5





  



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    .  , ,      ,      .       ,        ,        .

  ?          ,    .




 




,      ,      .  ,        .

      .  ,  ,  .        (0  1).

  ().         ,  (1 = 1, 2 = 1):








  (),  ,     .   :

1 * 2 = 0 * 0 = 0

1 * 2 = 1 * 0 = 0

1 * 2 = 0 * 1 = 0

1 * 2 = 1 * 1 = 1

         ,      ?      ,      ,     ,        (y = w1x1 + w2),        .       y = w1x1 + w2x2 + w3.    ,   ,      :








 ,         ,   .

       ,   ,   , ,            :




  .


   .

   ,     (https://www.litres.ru/kania-alekseevich-kan/neyronnyy-seti-evoluciya/)  .

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