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使用MATLAB建立前饋神經網絡主要會使用到下面3個函數：
newff：前饋神經網絡建立函數；
train：訓練一個神經網絡；
sim：使用神經網絡進行仿真。
下面簡要介紹這3個函數的用法。
newff函數newff函數文法
文法：net=newff（P，T，［S1 S2…S（N-l）］，｛TF1 TF2…TFN｝，BTF，BLF，PF，IPF，OPF，DDF）
P：輸入矩陣向量；
T：目标矩陣向量；
［S1 S2…S（N-l）］：神經網絡前N-1層每層神經元數；
｛TF1 TF2…TFN｝：神經網絡激活函數，預設為‘tansig’；
BTF：學習規則采用的訓練算法，預設為‘trainlm’；
BLF：BP權值/偏差學習函數，預設為‘learngdm’；
PF：性能函數，預設為‘mse’；
IPF：輸入處理函數；
OPF：輸出處理函數；
DDF：驗證資料劃分函數。一般在使用過程中設定前7個參數，後3個參數采用系統預設參數即可
常用的激活函數
| 常用的激活函數有 | |
|---|---|
| 線性函數 f ( x ) = x f(x)=x f(x)=x | 該函數的字元串為‘purelin’ |
| 對數S形轉移函數 f ( x ) = 1 1 + e − x f(x)=\frac{1}{1+e^{-x}} f(x)=1+e−x1 ( 0 < f ( x ) < 1 ) (0 < f(x) < 1) (0<f(x)<1) | 該函數的字元串為‘logsig’ |
| 雙曲正切S形函數 f ( x ) = 2 1 + e − x − 1 f(x)=\frac{2}{1+e^{-x}}-1 f(x)=1+e−x2−1 ( − 1 < f ( x ) < 1 ) (-1 < f(x) < 1) (−1<f(x)<1) | 該函數的字元串為‘tansig’ |
隻有當希望對網絡的輸出進行限制（如限制在0和1之間）時在輸出層才應當包含S形激活函數。在一般情況下，均是在隐含層采用雙極S形激活函數，而輸出層采用線性激活函數。
| 常用的訓練函數有： | |
|---|---|
| trainbfg | BFGS拟牛頓BP算法訓練函數 |
| trainbr | 貝葉斯正則化算法的BP算法訓練函數 |
| traingd | 梯度下降的BP算法訓練函數 |
| traingda | 梯度下降自适應lr的BP算法訓練函數 |
| traingdm | 梯度下降動量的BP算法訓練函數 |
| traingdx | 梯度下降動量和自适應lr的BP算法訓練函數 |
| trainlm | Levenberg-Marquardt的BP算法訓練函數 |
| trainrp | 具有彈性的BP算法訓練函數 |
| trains | 順序遞增BP訓練函數 |
| trainscg | 量化連接配接梯度BP訓練函數 |
| 常用的學習函數 | |
|---|---|
| learngd | BP學習規則 |
| learngdm | 帶動量項的BP學習規則 |
| 常用的性能函數 | |
|---|---|
| mse | 均方誤差函數 |
| msereg | 均方誤差規範化函數 |
| 配置參數 | 一些重要的網絡配置參數如下 |
|---|---|
| net.trainparam.goal | 神經網絡訓練的目标誤差 |
| net.trainparam.show | 顯示中間結果的周期 |
| net.trainparam.epochs | 最大疊代次數 |
| net.trainParam.lr | 學習率 |
train函數，train函數即神經網絡訓練學習函數。
文法：［net，tr，Y1，E］=train（net，X，Y）
X：網絡輸入矩陣；Y：網絡輸出矩陣；tr：訓練跟蹤資訊；Y1：網絡實際輸出；E：誤差矩陣。
sim函數，sim函數即神經網絡仿真計算函數。
文法：Y=sim（net，X）
net：訓練好的神經網絡；X：網絡輸入矩陣；Y：網絡輸出矩陣。
MATLAB仿真執行個體
采用貝葉斯正則化算法提高BP網絡的推廣能力。用來訓練BP網絡，使其能夠拟合某一附加有白噪聲的正弦樣本資料。
解：仿真過程如下：
（1）建構一個3層BP神經網絡：輸入層結點數為1個，隐含層結點數為3個，隐含層的激活函數為‘tansig’；輸出層結點數為1個，輸出層的激活函數為‘logsig’。
（2）采用貝葉斯正則化算法‘trainbr’訓練BP網絡，目标誤差goal=1×10-3，學習率lr=0.05，最大疊代次數epochs=500，拟合附加有白噪聲的正弦樣本資料，拟合資料均方根誤差為0.0054，拟合後的圖形如圖所示。
BP網絡拟合附加白噪聲的正弦曲線
MATLAB源程式如下：
%%%%%%%%%%%%%運用BP網絡拟合白噪聲的正弦樣本資料%%%%%%%%%%%%%%%%
clear all; %清除所有變量
close all; %清圖
clc; %清屏
%%%%%%%%%%%%%%%%%%%%%定義訓練樣本矢量%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%P 為輸入矢量%%%%%%%%%%%%%%%%%%%%%%%%%%%%
P = [-1:0.05:1];
%%%%%%%%%%%%%%%%%%%%%%T 為目标矢量%%%%%%%%%%%%%%%%%%%%%%%%%%%%
T = sin(2*pi*P)+0.1*randn(size(P));
%%%%%%%%%%%%%%%%%%%%%%繪制樣本資料點%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
plot(P,T,'+');
hold on;
plot(P,sin(2*pi*P),':');
%%%%%%%%%%%%%%%%%%%%%繪制不含噪聲的正弦曲線%%%%%%%%%%%%%%%%%%%
net=newff(P,T,20,{'tansig','purelin'});
%%%%%%%%%%%%%%%%%%%采用貝葉斯正則化算法 TRAINBR%%%%%%%%%%%%%%%
net.trainFcn='trainbr';
%%%%%%%%%%%%%%%%%%%%%%設定訓練參數%%%%%%%%%%%%%%%%%%%%%%%%%%%%
net.trainParam.show = 50; %顯示中間結果的周期
net.trainParam.lr = 0.05; %學習率
net.trainParam.epochs = 500; %最大疊代次數
net.trainParam.goal = 1e-3; %目标誤差
net.divideFcn = ''; %清除樣本資料分為訓練集、驗證集和測試集指令
%%%%%%%%%%%%%%%%%%%%用相應算法訓練 BP 網絡%%%%%%%%%%%%%%%%%%%%
[net,tr]=train(net,P,T);
%%%%%%%%%%%%%%%%%%%%%對 BP 網絡進行仿真%%%%%%%%%%%%%%%%%%%%%%%
A = sim(net,P);
%%%%%%%%%%%%%%%%%%%%%%計算仿真誤差%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E = T - A;
MSE=mse(E);
%%%%%%%%%%%%%%%%%%%%%%繪制比對結果曲線%%%%%%%%%%%%%%%%%%%%%%%%
plot(P,A,P,T,'+',P,sin(2*pi*P),':');
legend('樣本點','标準正弦曲線','拟合正弦曲線');
表中所示為某藥品的月度銷售情況，利用BP神經網絡對藥品的銷售進行預測，預測方法采用滾動預測方式，即用前3個月的銷售量來預測第4個月的銷售量。如用1、2、3月的銷售量為輸入，預測第4個月的銷售量；用2、3、4月的銷售量為輸入，預測第5個月的銷售量。反複疊代，直至滿足預測精度要求為止。
| 月份 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 銷量 | 2056 | 2395 | 2600 | 2298 | 1634 | 1600 |
| 月份 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|
| 銷量 | 1873 | 1478 | 1900 | 1500 | 2046 | 1556 |
解：仿真過程如下：
（1）建構一個3層BP神經網絡對藥品的銷售進行預測：輸入層結點數為3個，隐含層結點數為5，隐含層的激活函數為‘tansig’；輸出層結點數為1個，輸出層的激活函數為‘logsig’。
（2）采用梯度下降動量和自适應lr算法‘traingdx’訓練BP網絡，目标誤差goal= 1 × 1 0 − 3 1×10^{-3} 1×10−3，學習率lr=0.05，最大疊代次數epochs=1000，其銷售實際值和預測值對比曲線如圖所示。
藥品銷售實際值和預測值對比曲線
MATLAB源程式如下：
%%%%%%%%%%%%%%%%%%%%%%運用BP網絡預測資料%%%%%%%%%%%%%%%%%%%%%%%%
clear all; %清除所有變量
close all; %清圖
clc; %清屏
%%%%%%%%%%%%%%%%%%%%%%%%%%%%原始資料%%%%%%%%%%%%%%%%%%%%%%%%%%%%
p=[2056 2395 2600 2298 1634 1600 1873 1478 1900
2395 2600 2298 1634 1600 1873 1478 1900 1500
2600 2298 1634 1600 1873 1478 1900 1500 2046];
t=[2298 1634 1600 1873 1478 1900 1500 2046 1556];
%%%%%%%%%%%%%%%%%%%%%%%%%%原始資料歸一化%%%%%%%%%%%%%%%%%%%%%%%%%
pmax=max(max(p));·
pmin=min(min(p));
P=(p-pmin)./(pmax-pmin); %輸入資料矩陣
tmax=max(t);
tmin=min(t);
T=(t-tmin)./(tmax-tmin); %目标資料向量
%%%%%%%%%%%%%%%%%%%建立一個新的前向神經網絡%%%%%%%%%%%%%%%%%%%%%%%
net=newff(P,T,5,{'tansig','purelin'},'traingdx');
%%%%%%%%%%%%%%%%%%%%%%%設定訓練參數%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
net.trainParam.show = 50; %顯示中間結果的周期
net.trainParam.lr = 0.01; %學習率
net.trainParam.epochs = 1000; %最大疊代次數
net.trainParam.goal = 1e-3; %目标誤差
net.divideFcn = ''; %清除樣本資料分為訓練集、驗證集和測試集指令
%%%%%%%%%%%%%%%%%調用 TRAINGDM 算法訓練 BP 網絡%%%%%%%%%%%%%%%%%%
[net,tr]=train(net,P,T);
%%%%%%%%%%%%%%%%%%%%%對 BP 網絡進行仿真%%%%%%%%%%%%%%%%%%%%%%%%%%
A = sim(net,P);
%%%%%%%%%%%%%%%%%%%%%%計算預測資料原始值%%%%%%%%%%%%%%%%%%%%%%%%%
a = A.*(tmax-tmin)+tmin;
%%%%%%%%%%%%%%%%%%%%%繪制實際值和預測值曲線%%%%%%%%%%%%%%%%%%%%%%
x=4:12;
figure
plot(x,t,'+');
hold on;
plot(x,a,'or');
hold off
xlabel('月份')
ylabel('銷量')
legend('實際銷量 ','預測銷量');
表中所示為某地區公路運力的曆史統計資料表，請建立相應的BP神經網絡預測模型，并根據給出的2010年和2011年的資料，預測相應的公路客運量和貨運量。
解：仿真過程如下：
（1）建構一個3層BP神經網絡對該地區公路運力進行預測：輸入層結點數為3個，隐含層結點數為8，隐含層的激活函數為‘tansig’；輸出層結點數為2個，輸出層的激活函數為‘purelin’。
（2）采用梯度下降動量和自适應lr算法‘traingdx’訓練BP網絡，目标誤差goal=1×10-3，學習率lr=0.035，最大疊代次數epochs=2000。拟合的曆年公路客運量曲線和曆年公路貨運量曲線分别如圖9.6和圖9.7所示。預測結果為：2010年公路客運量為4.5277億人，公路貨運量為2.2290億噸；2011年公路客運量為4.5308億人，公路貨運量為2.2296億噸。
曆年公路客運量拟合曲線
曆年公路貨運量拟合曲線
MATLAB源程式如下：
%%%%%%%%%%%%%%%%%%%%%%運用BP網絡預測資料%%%%%%%%%%%%%%%%%%%%%%%%
clear all; %清除所有變量
close all; %清圖
clc; %清屏
%%%%%%%%%%%%%%%%%%%%%%%%%%%%原始資料%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%人數(機關：萬人)%%%%%%%%%%%%%%%%%%%%%%%%%
sqrs=[20.55 22.44 25.37 27.13 29.45 30.10 30.96 34.06 36.42 38.09...
39.13 39.99 41.93 44.59 47.30 52.89 55.73 56.76 59.17 60.63];
%%%%%%%%%%%%%%%%%%%%%%機動車數(機關：萬輛)%%%%%%%%%%%%%%%%%%%%%%%%
sqjdcs=[0.6 0.75 0.85 0.9 1.05 1.35 1.45 1.6 1.7 1.85 2.15 2.2...
2.25 2.35 2.5 2.6 2.7 2.85 2.95 3.1];
%%%%%%%%%%%%%%%%%%%%公路面積(機關：萬平方公裡)%%%%%%%%%%%%%%%%%%%%
sqglmj=[0.09 0.11 0.11 0.14 0.20 0.23 0.23 0.32 0.32 0.34 0.36...
0.36 0.38 0.49 0.56 0.59 0.59 0.67 0.69 0.79];
%%%%%%%%%%%%%%%%%%%%%公路客運量(機關：萬人)%%%%%%%%%%%%%%%%%%%%%%%
glkyl=[5126 6217 7730 9145 10460 11387 12353 15750 18304 19836 ...
21024 19490 20433 22598 25107 33442 36836 40548 42927 43462];
%%%%%%%%%%%%%%%%%%%%%公路貨運量(機關：萬噸)%%%%%%%%%%%%%%%%%%%%%%%
glhyl=[1237 1379 1385 1399 1663 1714 1834 4322 8132 8936 11099 ...
11203 10524 11115 13320 16762 18673 20724 20803 21804];
%%%%%%%%%%%%%%%%%%%%%%%%%%輸入資料矩陣%%%%%%%%%%%%%%%%%%%%%%%%%%%
p=[sqrs;sqjdcs;sqglmj];
%%%%%%%%%%%%%%%%%%%%%%%%%%目标資料矩陣%%%%%%%%%%%%%%%%%%%%%%%%%%%
t=[glkyl;glhyl];
%%%%%%%%%%%%%%%%%%%%%%%%%原始樣本歸一化%%%%%%%%%%%%%%%%%%%%%%%%%%
[P,PSp] = mapminmax(p);
[T,PSt] = mapminmax(t);
%%%%%%%%%%%%%%%%%%%建立一個新的前向神經網絡%%%%%%%%%%%%%%%%%%%%%%%
net=newff(P,T,8,{'tansig','purelin'},'traingdx');
%%%%%%%%%%%%%%%%%%%%%%%設定訓練參數%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
net.trainParam.show = 50; %顯示中間結果的周期
net.trainParam.lr = 0.035; %學習率
net.trainParam.epochs = 1000; %最大疊代次數
net.trainParam.goal = 1e-3; %目标誤差
net.divideFcn = ''; %清除樣本資料分為訓練集、驗證集和測試集指令
%%%%%%%%%%%%%%%%%調用 TRAINGDM 算法訓練 BP 網絡%%%%%%%%%%%%%%%%%%
[net,tr]=train(net,P,T);
%%%%%%%%%%%%%%%%%%%%%對 BP 網絡進行仿真%%%%%%%%%%%%%%%%%%%%%%%%%%
A = sim(net,P);
a=mapminmax('reverse',A,PSt);
%%%%%%%%%%%%%%%%%%%%優化後輸入層權值和門檻值%%%%%%%%%%%%%%%%%%%%%%%
inputWeights=net.IW{1,1};
inputbias=net.b{1};
%%%%%%%%%%%%%%%%%%%%優化後網絡層權值和門檻值%%%%%%%%%%%%%%%%%%%%%%%
layerWeights=net.LW{2,1};
layerbias=net.b{2};
%%%%%%%%%%%%%%%%%%%%%%%%%時間軸刻度%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=1990:2009;
%%%%%%%%%%%%%%%%%%%%%%%網絡輸出客運量%%%%%%%%%%%%%%%%%%%%%%%%%%%%
newk=a(1,:);
%%%%%%%%%%%%%%%%%%%%%%%網絡輸出貨運量%%%%%%%%%%%%%%%%%%%%%%%%%%%%
newh=a(2,:);
%%%%%%%%%%%%%%%%%%%%繪值公路客運量對比圖%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
plot(x,newk,'r-o',x,glkyl,'b--+')
legend('網絡輸出客運量','實際客運量');
xlabel('年份');ylabel('客運量/萬人');
%%%%%%%%%%%%%%%%%%%%%%繪制公路貨運量對比圖%%%%%%%%%%%%%%%%%%%%%%%
figure
plot(x,newh,'r-o',x,glhyl,'b--+')
legend('網絡輸出貨運量','實際貨運量');
xlabel('年份');ylabel('貨運量/萬噸');
%%%%%%%%%%%%%%%%%%%%%利用訓練好的網絡進行預測%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%2010年和2011年的相關資料%%%%%%%%%%%%%%%%%%%%%
pnew=[73.39 75.55;3.9 4.1;0.98 1.02];
SamNum=size(pnew,2);
%%%%%%%%%%%利用原始輸入資料的歸一化參數對新資料進行歸一化%%%%%%%%%%%
pnewn=mapminmax('apply',pnew,PSp);
%%%%%%%%%%%%%%%%%%%%%%%隐含層輸出預測結果%%%%%%%%%%%%%%%%%%%%%%%%%
HiddenOut=tansig(inputWeights*pnewn+repmat(inputbias,1,SamNum));
%%%%%%%%%%%%%%%%%%%%%%%輸出層輸出預測結果%%%%%%%%%%%%%%%%%%%%%%%%%
anewn=purelin(layerWeights*HiddenOut+repmat(layerbias,1,SamNum));
%%%%%%%%%%%%%%把網絡預測得到的資料還原為原始的數量級%%%%%%%%%%%%%%%
anew=mapminmax('reverse',anewn,PSt);