Function fitting based on MATLAB

Function fitting

The whole idea

Divide the given coordinate points into 6 Section , Left 、 The lower left 、 Next 、 Right 、 The upper right 、 On ,6 part .

The lower left and upper right parts are fitted with quadratic functions , Other parts are fitted with a first-order function .

Observation data

The first 1-76 Points are linearly fitted as the left part

The first 67-96 Points as the lower left part of the quadratic fitting

The first 84-215 Points as the lower part of the linear fitting

The first 216-275 Points as the right part of the linear fitting

The first 269-298 Points as the upper right part of the quadratic fitting

The first 296-382 Points as the upper part of a fitting

Code implementation

First get x、y Scatter vector of data , And then call polyfit Library function , Return the function parameters of linear regression , Then initialize the denser x Abscissa as the abscissa of the sample , Use the obtained fitting function to substitute the abscissa of the sample point to obtain the ordinate of the sample point . At this time, more dense samples pass plot The function displays , You can get the fitting effect picture

• Final effect

It can be seen that , Fit by quadratic function at the bottom left and top right , Use linear fitting of the first-order function at other positions

• Problems encountered and solutions

When the fitting coordinates are initially selected , stay 6 Curve part , The point sets selected by each part do not intersect （ Mutually exclusive ）, At this time, the rendering is not ideal ： as follows

It can be seen that there is a large space in the middle of the curve , To solve this problem , Constantly adjust the interval of the point set （ Expand... Appropriately ）, Make the point sets of each interval partially intersect with each other , Point for example 1-76 It's the first part , spot 67-96 It's part two , At this point 67-76 Not only the points on the left part to fit linearly , It's the point in the lower left to fit twice , The effect will be much better , as follows ：

• Fitting results ：

On the left ( The first 1-76 A little bit )：y=0.0983x-381.095

The lower left ( The first 67-96 A little bit )：y=0.0055

+4.5245x+299.2073

Underside ( The first 84-215 A little bit )：y=0.0208x-600.9287

On the right ( The first 216-275 A little bit )：y=-0.0721x+383.0611

The upper right ( The first 269-298 A little bit )：y=-0.0056

+4.0934x-901.1372

above （ The first 296-382 A little bit ）：y=0.011x-181.6471

Function interpolation

linear interpolation

The whole idea

First mark as 1 Points of are stored separately v1 Matrix , Time of each action point 、 Horizontal ordinate , Then mark each with 0 Missing points Q, Find the nearest point before and after it AB, use A and B Linear interpolation , The missing point is on the linear interpolation line AB On , And by the Q The time of is known ,AB The arrival time of is also known . Suppose the proportion of time (tQ-tA)/(tB-tA) And abscissa scale （xQ-xA）/（xB-xA） Solve the abscissa of the missing point equally , Then substitute the interpolation function , Get the coordinates of the missing points and store them in v2 Matrix , The missing point information of each behavior is finally used plot Draw the image

Code implementation

utilize coord The second column of is the abscissa , The third column is the ordinate , The first column is the time. The missing points are stored in v1 in , Missing points are stored in v2 in , The abscissa corresponding to the missing time is obtained by assuming that the object moves at a uniform speed in the projection of the abscissa , Then using interpolation function （ linear interpolation ） Get ordinate , utilize plot Functions depict curves . The interpolation function obtains the slope k And intercept b that will do

Slope k=（y2-y1）/（x2-x1）

intercept b by （x1y2-x2y1）/（x1-x2）

Temp_x,temp_y Used to temporarily store the horizontal and vertical coordinates of the missing point , Finally stored in v2 Matrix

Final effect

Cubic spline interpolation

The whole idea

• about 309 There are no missing points , There is 308 Intervals , Every interval is interpolated by cubic function
• Total needs 4*308 Parameters to be determined
• You need to 4*408 An equation
• among 309 The first is the interpolation condition of the points satisfied by the interpolation function, and the inner points satisfy ： Function value 、 Derivative value 、 The second derivative is continuous
• It also needs to be 2 A boundary condition , Let the derivative at the first and last points be 0（ Natural boundary starting conditions ）
• The core idea is to solve this 4408 The first equation yields 4408 The result of coefficients .
• that M Is the coefficient matrix ,Y Is the result vector of the equation ,
• The coefficient matrix is filled with coefficients （ Corresponding to equation ）
• And then use it M\Y You can get X, namely 4*408 Values of undetermined coefficients , Then describe these interpolation functions .
• It is also assumed that the projection of the object on the abscissa moves at a uniform speed
• Then the abscissa of the missing point can be determined by the time proportion , And then according to its range , Substitute the corresponding cubic function to obtain the value of the ordinate

Code implementation

Also save the missing points v1 matrix ,M Used to store coefficient matrix ,Y Used to store the result vector ,X Is the vector of the result coefficient of the calculated interpolation function

• Final effect

Rendering of interpolation function ：

Interpolation rendering of missing points

Red is the interpolation result of missing points , Blue dots are not missing , Black is the interpolation curve

Polynomial interpolation

Realize the idea

The specific implementation considers the use of piecewise quadratic function interpolation , hold 309 Points of existence ,308 The intervals are divided into 154 Group , Each group has two intervals , Three points , Adjacent groups will share the existing interior points .

for example ：x1、x2、x3 For the first set of three interpolation points ,x3、x4、x5 Is the interpolation point of the second group , Adjacent pairing x3 Shared . altogether 154*3 Two parameters need to be determined , Each quadratic function determines 3 Parameters .

There are also equations that can be listed 154*3 individual , The interpolation condition equation can be obtained by using undetermined coefficients in each interval .

Empathy , Fill in M matrix , and Y The result vector , The result vector of quadratic function coefficient is obtained by operation X

Code implementation

Used to store known points

Vx1 and vy1 Store the horizontal and vertical coordinates of the known points

Vx2 and vy2 Horizontal and vertical coordinates of missing points

The abscissa of the missing point is calculated by assuming that the projection of the object on the abscissa moves at a uniform speed

The ordinate of the missing point is calculated by interpolation function

Final effect

Comparison of interpolation methods

linear interpolation ： The advantage is that the implementation is very simple , Just use the original points for linear interpolation , The disadvantage is that the connection at the segment is not smooth enough , It can only indicate the overall trend , The increase in detail is difficult to effectively express

Cubic spline interpolation ： The advantage is that the effect is good , Function value at segment 、 Slope 、 Concavity and convexity are very smooth

The disadvantage is that the number of equations is relatively large 、 complex , Once there is a missing point exception , The final result will be subject to great fluctuations ！

Piecewise polynomial interpolation （ secondary ）： The advantage is that the form is simple , And the overall curve is relatively smooth , It should be said that it integrates the advantages of linear interpolation and cubic spline interpolation , It is better to ！