根據(jù)氣壓對山體海拔高度的確定模型 (修改1)

1、根據(jù)氣壓對山體海拔高度的確定模型摘要：通過對該題的一個(gè)全面分析，我們看出這大致是一個(gè)數(shù)據(jù)的處理問題。首先對數(shù)據(jù)進(jìn)行描述性統(tǒng)計(jì)分析后我們大致看出p-h之間的關(guān)系，然后進(jìn)行模型的建立求解。第一問建立p-h的函數(shù)表達(dá)式，由熱力學(xué)上的“壓-高方程”作為我們的基本模型，對數(shù)據(jù)進(jìn)行擬合后得到我們的函數(shù)表達(dá)式；第二問是最少需要幾組數(shù)據(jù)可確定所得到的函數(shù)表達(dá)式，主要考察數(shù)據(jù)的有效性，先對數(shù)據(jù)進(jìn)行處理，排除問題數(shù)據(jù)后進(jìn)行排列組合,計(jì)算每一個(gè)組合得出的函數(shù)表達(dá)式（用什么方法。插值？），設(shè)定一個(gè)基本值（誤差？0.001）然后選出與第一問擬合出的模型的函數(shù)表達(dá)式近似度最大的數(shù)組即為所求；第三問是對模型本身的一個(gè)誤差分

2、析，計(jì)算誤差的均值、方差，進(jìn)行假設(shè)檢驗(yàn)等一系列對數(shù)據(jù)進(jìn)行的穩(wěn)定性分析；第四問是求出該山山腳處的海拔高度并估計(jì)誤差（求極限？證明連續(xù)性，排除斷點(diǎn)，可用插值法驗(yàn)證）一.問題的提出在一個(gè)登山記錄中，有若干點(diǎn)距離山腳的距離和對應(yīng)氣壓11組數(shù)據(jù)，需根據(jù)這些數(shù)據(jù)建立函數(shù)表達(dá)式，然后推算出該山山腳處的海拔高度以及該山體的海拔高度，最后對函數(shù)模型的誤差進(jìn)行分析二.模型的分析與假設(shè)模型分析因?yàn)槲覀冋w思路是對數(shù)據(jù)進(jìn)行統(tǒng)計(jì)分析與處理，所以首先對數(shù)據(jù)進(jìn)行了描述性統(tǒng)計(jì)分析，計(jì)算得出數(shù)據(jù)的偏度0,為左偏，峰度0,所以數(shù)據(jù)不是以正態(tài)函數(shù)進(jìn)行分布的；然后利用數(shù)據(jù)畫了箱圖，發(fā)現(xiàn)所給數(shù)據(jù)完好，沒有奇異值。在現(xiàn)實(shí)生活中，在其他條

3、件不變的情況下，氣壓與海拔高度成方向相關(guān)關(guān)系，任何地方的總是隨著海拔高度的增加而遞減，據(jù)實(shí)測，在地面層中，高度每升100米，氣壓平均降低12.Thpa,在高層則小于此數(shù)值，所以我們的模型自變量h與因變量p是反向相關(guān)的。模型假設(shè)該山體處的天氣、氣溫、空氣密度等能影響氣壓的因素保持不變，即該山體每一處的氣壓僅與該處的海拔高度有關(guān)設(shè)所用模型的三模型的建立與求解問題1根據(jù)提供的數(shù)據(jù)建立P關(guān)于h的函數(shù)表達(dá)式模型的建立：首先題目要求的是利用數(shù)據(jù)建立函數(shù)表達(dá)式，我們運(yùn)用的擬合的方法。根據(jù)靜力學(xué)原理所闡述的：假使大氣相對于地面處于靜止?fàn)顟B(tài)，則某一點(diǎn)的氣壓值等于該點(diǎn)單位面積上所承受空氣柱的重量，得到的壓強(qiáng)與高度

4、的靜力學(xué)方程：dp=-pgdz,其中z是氣壓高度差、p是氣壓(m/hpa)。再結(jié)合理想氣體狀態(tài)方程式：PV=nRT推導(dǎo)出的熱力學(xué)上的大氣壓與高度關(guān)系的玻爾茲曼公式(也就是著名的“壓-高方程”)：p二Poe一哪/RT其中Po是標(biāo)準(zhǔn)面(一般為海平面)的氣壓，P是對應(yīng)于高度z的氣壓,卩為空氣的摩爾質(zhì)量，R為普適氣體常數(shù)，T為氣體的溫度可轉(zhuǎn)化為：In(P2/Pl)=-pg(h2-hl)/RT,其中Pl、P2分別是高度hl、h2對應(yīng)的氣壓值，作為我們的模型的來源。于是我們據(jù)此建立了指數(shù)衰減型函數(shù)模型：P=aebh(1)其中a、b為待定系數(shù)，a0,b0,h為氣壓P對應(yīng)的海拔與山腳海拔的差值，即該點(diǎn)與山腳的

5、高度差對方程兩邊取對數(shù)得：InP二Ina+bh則線性化為方程：y二n+mx(2)其中m、n為待定系數(shù)根據(jù)表中數(shù)據(jù)以及(2)式，應(yīng)用線性最小二乘方程(線性回歸分析)則可求出參數(shù)a、b的值模型求解。(把你們求出的貼上來，最好加文字?jǐn)⑹銮宄?，把得到完整的表達(dá)式寫出來，代碼也貼在這兒后面好修改)利用Matlab軟件程序代入數(shù)據(jù)求解得：b二-1.426176817502167e-004a=7.841813039203729e+002取b=-0.0001,a=784.1813帶入(1)式得到模型為:p=784.1813Xe-00001xh+Epsilon(3)Epsilon為該擬合模型的誤差項(xiàng)，它表示除h

6、外其他影響p的因素模型估計(jì)值Phat:123456891011784.755.715.688.660.632.605.urrUif.557.528.514.18133127672383430781973559846745122856187985估計(jì)值與真實(shí)值的差R“=P-Phat：1234568910112.881.31-0.32-1.0-1.5-1.6-1.4-0.82-0.171.061.791303267177472656821942654016355026720941804845292037712447801206771820940781750734552206975385736369

7、041985688014175057411423338163512719855殘差均值：E(Rn)=0.0098殘差方差:Var(Rn)=2.3268殘差分布：p-h指數(shù)衰減型模型的顯著性檢驗(yàn)(這個(gè)能做也把它做了，然后貼上來)模型的分析5問題一的結(jié)論問題2最少需要幾組數(shù)據(jù)可確定所得到的函數(shù)表達(dá)式問題分析：該問題：“最少需要兒組數(shù)據(jù)能得到表達(dá)式”，理論上是數(shù)據(jù)越多表達(dá)式擬合的效果越好，但相對來說，設(shè)定一個(gè)標(biāo)準(zhǔn)值和選擇規(guī)則就可以找出在該條件下最接近原函數(shù)值的新函數(shù)的幾組數(shù)據(jù)。方法步驟：由于數(shù)據(jù)的描述性分析中沒有異常值，所以直接對數(shù)據(jù)組進(jìn)行排列組合,因?yàn)閿?shù)據(jù)越多擬合效果越好，所以直接從兩組數(shù)據(jù)進(jìn)行排

8、列組合，一共有2n-l1=2037種組合對每一種組合進(jìn)行擬合回歸，計(jì)算回歸后的函數(shù)與原函數(shù)求得的殘差平方和選定標(biāo)準(zhǔn)值為p,選出殘差平方和小于p且最小時(shí)，即認(rèn)定為所選組合，組成該組合的數(shù)據(jù)有幾組即為題目所求。問題求解：通過matlab軟件對上述程序編程，設(shè)定p二25(此時(shí)相關(guān)系數(shù)R的平方越接近于1),當(dāng)計(jì)算到三組數(shù)據(jù)中的一組時(shí)己達(dá)設(shè)定的標(biāo)準(zhǔn)，就沒有進(jìn)行四組、五組及后面的計(jì)算。最后求出數(shù)組為：(xl=263(x2=641(x3=2397yl=754.0&2=716.03=578.5擬合的殘差平方和為22.155177751836515pMat1ab程序?yàn)椋篽=0263641909120815021

9、8122143239727662951;P二781.3754.0716.0689.9661.6634.6607.0578.5557.3527.5513.0;%所有數(shù)據(jù)Y=log(P);a=polyfit(h,Y,1)kl=a(l);k2二exp(a(2);z=k2.*exp(h.*kl);%殘差均值方差eps二z-P;mean(eps)var(eps)RMS=sqrt(sum(eps)2)/n);mean(abs(eps-mean(eps)err=eps/z;mepsmaxteps);%選取殘差平方和最小組合hrand二combntns(h,3);Prand二combntns(P,3);2.Y

10、rand=log(Prand);a二；k=;z二；n=length(Yrand);fori=l:na(i,:)=polyfit(hrand(i,:),Yrand(i,:),1)k(i,1)=a(i,1)k(i,2)=exp(a(i,2)z(i,:)=k(i,2).*exp(h.*k(i,1);eps(i,:)=z(i,:)Pendb=sum(eps.2,2)bmin=find(b=min(b)hrand(bmin,:)Prand(bmin,:)得到的殘差及殘差平方和數(shù)據(jù)在附錄2問題二結(jié)論：根據(jù)我們設(shè)定的標(biāo)準(zhǔn)值，我們求出最少需要三組數(shù)據(jù)可確定所得到的函數(shù)表達(dá)式。問題3估計(jì)所得到的表達(dá)式的誤差1.

11、針對該問題利用所求的模型求出問題4確定該山山腳處的海拔高度并估計(jì)誤差四模型檢驗(yàn)五.模型的評價(jià)與改進(jìn)參考文獻(xiàn)附錄1.（問題一模型擬合的代碼：）xlx2x302636410.074-0.1202639090.15-0.2026312080.235-0.29026315020.319-0.37026318120.409-0.46026321430.505-0.56026323970.579-0.63026327660.69-0.74026329510.746-0.7906419090.066-0.23064112080.235-0.25064115020.437-0.24064118120.661-

12、0.21064121430.905-0093-0371-0.11064129511.512-0.09090912080.086-0.36090915020.293-0.33090918120.577-0.25090921430.911-0178-0.05090927661.5760.101090929511.7780.1780120815020.087-0.510120818120.333-0.440120821430.702-0.290120823971.026-0.140120827661.5350.104012

13、0829511.8010.2360150218120.1-0.660150221430.394-0.550150223970.713-0.390150227661.269-0.10150229511.5760.0720181221430.116-0.820.050.4811.232.23.48-0.20.0540.661.52.67-0.6-0.40.060.81.8-0.9-0.85-0.50.10.95-1.2-1.32-1.2-10.05-1.5-1.83-1.8-2-0.9-1.8-2.21-2.3-2-1.6-2.2-2.79-3.1-3-2.7-2.4-3.09-3.5-4-3.3

14、-0.20.1390.81.72.89-0.5-0.270.2312.03-0.7-0.67-0.30.31.17-0.9-1.08-0.9-00.28_1.2-1.5-1.5-1-0.7-1.4-1.83-2-2-1.4-1.6-2.29-2.7-3-2.4-1.8-2.53-3-3-2.9-0.5-0.310.2212.09-0.7-0.66-0.30.41.31-0.9-1.02-0.8-00.45-1.1-1.4-1.4-1-0.5_1.2-1.67-1.8-2-1.2-1.4-2.07-2.4-2-2.2-1.5-2.26-2.7-3-2.7-0.9-0.78-0.40.31.26-

15、1-1.09-0.8-00.52_1.2-1.41-1.4-1-0.3_1.3-1.65-1.7-2-1-1.4-1.97-2.3-2-2-1.4-2.13-2.6-3-2.4_1.2-1.27-1-00.38-1.4-1.56-1.5-1-0.4-1.4-1.77-1.8-2-1-1.5-2.04-2.3-2-1.9-1.5-2.16-2.6-3-2.3-1.6-1.81-1.7-1-0.6殘差殘差平方和5.16.438.69.8254.77744.15.427.58.6186.32063.14.336.37.3126.44472.23.285.16.182.063381.22.163.94

16、.849.916570.10.972.63.532.27474-10.051.62.430.56313-2_1.300.846.99263-3-2-1063.989734.45.77.88.9204.33183.44.626.67.7140.76872.43.525.46.490.60161.42.384.15.153.578330.31.192.83.731.3411-10.281.82.625.79425-2-10.31.135.15702-2-1.7-00.347.692563.54.726.77.8146.39022.63.735.66.699.232061.62.614.45.359

17、.888460.51.4233.933.98166-00.5122.825.50261-2-0.80.51.330.22554-2-1.4-00.540.120972.63.715.66.799.505671.72.754.55.565.015170.71.623.34.238.08646-00.742.33.127.22392-1-0.50.81.627.51617-2-1.20.10.834.882991.62.634.45.463.227870.71.663.34.340.77641-00.852.43.329.89965-1-0.411.827.58219-2-10.31.133.01

18、0830.51.463.14.140.340210181223970.347-0.72-1.7-2.01-2-2-1.1-00.772.43.332.606650181227660.853-0.45-1.7-2.25-2.5-2-1.9-1-0.31.11.930.60740181229511.163-0.27-1.7-2.35-2.7-3-2.3-2-0.90.41.234.955110214323970.07-1.01-2-2.32-2.3-2-1.4-00.4222.935.079960214327660.407-0.86-2.1-2.58-2.8-3-2.1-1-0.50.91.837

19、.158840214329510.665-0.72-2.1-2.68-3-3-2.5-2-10.41.241.886250239727660.15-1.12-2.4-2.85-3-3-2.4-2-0.80.71.544.484380239729510.333-1.03-2.4-2.97-3.2-3-2.8-2-1.30.10.950.621290276629510.039-1.37-2.8-3.38-3.7-4-3.3-3-1.8-00.468.573552636419090.5210.057-0.10.0730.591.42.423.85.0278.1162.477726364112080.

20、8340.173-0.3-0.240.10.71.632.93.995.86.9109.2126364115021.1440.294-0.4-0.53-0.40.10.88234.75.770.3191126364118121.4690.423-0.5-0.82-0.8-10.111.11.993.64.542.2225226364121431.8140.56-0.7-1.13-1.3-1-0.70.10.922.43.225.7114626364123972.0780.666-0.8-1.37-1.7-2-1.3-10.111.52.222.1551826364127662.4680.822

21、-0.9-1.72-2.2-2-2.2-2-1.10.10.831.1400226364129512.6660.901-1-1.9-2.5-3-2.7-2-1.7-10.142.131926390912080.6940.079-0.3-0.230.150.81.7534.176.17.1117.713626390915021.0580.25-0.4-0.48-0.30.21.032.23.2155.977.5642326390918121.4840.464-0.5-0.73-0.7-00.271.22.183.84.746.4736526390921431.9490.702-0.5-0.98-

22、1.2-1-0.50.31.12.63.427.41592639092397最,2.306小0.888-0.6-1.16-1.5-1-1.1-00.281.62.422.1578326390927662.831.162-0.6-1.43-2-2-2-1-0.90.3128.3869526390929513.0941.3-0.6-1.56-2.2-2-2.4-2-1.5-00.337.68253263120815020.8490.083-0.5-0.55-0.30.21.062.23.295.16.180.7826263120818121.2640.303-0.5-0.76-0.7-00.381

23、.42.3744.951.38812263120821431.8070.613-0.5-0.95-1.1-1-0.40.51.322.83.730.1971263120823972.250.874-0.5-1.09-1.4-1-1-00.511.92.722.87075263120827662.9171.273-0.5-1.27-1.8-2-1.8-1-0.70.51.326.03487263120829513.2561.477-0.4-1.36-2-2-2.2-2-1.2-00.533.70231263150218121.0290.096-0.7-0.9-0.8-00.311.42.3344

24、.950.9454263150221431.5040.368-0.7-1.07-1.1-1-0.40.51.4233.831.95058263150223971.9630.651-0.7-1.17-1.4-1-0.9-00.672.12.923.69936263150227662.7141.128-0.5-1.29-1.8-2-1.7-1-0.50.81.524.23767263150229513.1121.385-0.5-1.34-1.9-2-2.1-2-10.10.830.39964263181221431.2280.112-0.9-1.28-1.3-1-0.50.41.292.83.73

25、0.94469263181223971.5930.33-0.9-1.38-1.6-1-1-00.652.12.924.26015263181227662.3110.794-0.8-1.47-1.9-2-1.7-1-0.40.91.723.65512263181229512.7271.071-0.7-1.49-2-2-2-2-0.90.3128.47493263214323971.3350.068-1.2-1.65-1.8-2-1.2-00.371.82.724.44539263214327661.8590.389-1.1-1.77-2.1-2-1.9-1-0.50.81.625.5635626

26、3214329512.2240.631-1-1.79-2.3-2-2.2-2-10.3129.80368263239727661.6160.146-1.4-2.01-2.4-2一2.1-1-0.70.61.429.1563263239729511.8940.321-1.3-2.06-2.5_3-2.4-2-1.10.10.833.8491263276629511.6440.038-1.6-2.41-2.9-3-2.8-2-1.6-o0.344.708264190912081.6720.7910.05-0.10.040.51.232.33.355.981.6720164190915022.219

27、1.1370.14-0.19-0.20.10.641.62.4744.955.629464190918122.7821.4940.22-0.28-0.5-00.060.81.6233.937.9414764190921433.3771.8720.32-0.37-0.8-0.60.10.7322.828.9048264190923973.8312.1610.39-0.44-1-1-10.051.21.928.5624164190927664.5042.5890.49-0.54_1.3-2-1.7-1-0.90.10.738.4078164190929514.8462.8060.55-0.59-1

28、.5-2_2.1-2-1.5-10.148.10463641120815022.0341.0040.07-0.21-0.20.10.751.72.644.25.160.13357641120818122.7091.4570.23-0.24-0.4-00.1911.83.24.141.10882641120821433.4671.9740.43-0.25-0.6-0.40.20.892.22.931.33104641120823974.0462.370.59-0.25-0.8-0.9-o0.221.42.131.11637641120827664.8932.9510.82-0.24-1-1.5-

29、1-0.80.20.841.51772641120829515.3193.2440.93-0.24-1.1-2-1.8-2-1.2-o0.251.45373641150218122.4471.2460.09-0.34-0.5-00.2111.883.44.241.64453641150221433.2091.7770.31-0.31-0.7-0.30.31.052.43.131.391641150223973.8692.2450.53-0.26-0.8-0.7-o0.41.62.330.36679641150227664.8742.9640.87-0.16-0.9_1.3-1-0.60.41.

30、140.33692641150229515.3853.3321.05-0.1-1-1.6-1-1-o0.450.3721641181221432.9061.5140.1-0.48-0.8-0.40.31.032.43.229.25432641181223973.4881.9290.3-0.43-0.9-0.8-o0.471.72.427.58358641181227664.5262.6860.69-0.28-1_1.3-1-0.40.61.335.63958641181229515.0923.1040.91-0.19-1-1.6-1-0.90.10.744.93083641214323973.

31、2241.6770.06-0.66-1.1-0.9-o0.31.52.324.97741641214327664.0512.2750.36-0.56-1.2-1.4-1-0.50.61.330.91065641214329514.5822.670.57-0.46-1.2-2-1.6-1-0.90.10.838.50936641239727663.8052.0390.14-0.77-1.4-2-1.6-1-0.60.51.229.79839641239729514.2342.3530.3-0.7-1.4-2-1.8-1-100.736.12521641276629514.0262.1210.04

32、-0.99-1.7-2_2.1-2-1.3-o0.339.09798909120815022.9281.70.510.055-0.10.10.531.32.163.64.551.43719909120818123.7452.2890.790.149-0.2-00.070.71.412.73.541.92665909120821434.5932.9021.10.252-0.3-0.40.10.641.82.541.02116909120823975.2313.3641.330.33-0.4-0.8-o0.071.11.746.15263909120827666.1724.0441.660.445

33、-0.5_1.3-1-0.80.10.662.684909120829516.6474.3881.830.504-0.6-1.5-1-1.2-o0.175.08615909150218123.4962.0970.680.079-0.2-o0.130.81.552.93.742.10071909150221434.4722.8211.070.257-0.3-o-0.30.20.7922.741.14717909150223975.253.4031.390.412-0.3-0.6-o0.221.31.947.40133909150227666.3934.261.870.647-0.3-1.1-1-

34、0.60.20.866.91806909150229516.9664.6892.110.767-0.3_1.3-1-1-o0.281.184909181221434.1322.5380.860.098-0.4-0.30.20.852.12.837.50685909181223974.8893.111.190.269-0.4-0.6-o0.341.42.142.46651909181227666.1534.0771.770.59-0.3_1-1-1-0.40.4161.81572909181229516.8154.5862.070.766-0.3-1-1.2_1-0.8-o0.576.92224

35、909214323974.5882.8390.960.062-0.6-1-0.7-o0.241.3237.57287909214327665.6613.661.450.335-0.5-1-1.1-0.40.51.152.34147909214329516.324.1711.760.526-0.4-1-1.2-0.80.10.665.71702909239727665.393.411.220.132-0.7-1-1.2-0.50.4147.85084909239729515.9423.8361.490.286-0.6-1-1.4-0.800.658.54861909276629515.7393.

36、6181.250.035-0.9-1-1.7-2-1.1-o0.357.042391208150218124.7533.1161.380.5820.06-00.050.61.162.3350.901921208150221435.89341.930.9060.16-o-0.30.10.531.52.162.592041208150223976.7384.6552.331.1490.23-o-0.5-o0.060.91.577.374381208150227667.9725.6122.921.5040.34-o-0.8-0.60.10.5108.13231208150229518.5936.09

37、33.221.6820.4-o-1-0.9-o0.1127.69271208181221435.5723.7411.750.7830.09-o-0.20.10.641.72.357.421871208181223976.534.4962.241.0960.22-o-0.4-o0.191.11.773.389591208181227668.0075.66431.5940.45-o-0.7-0.50.20.7109.68461208181229518.7496.2533.381.8470.56-o-0.8-0.8-o0.2133.10261208214323976.1574.1741.980.8870.06-o-0.5-o0.181.11.764.696581208214327667.545.2762.711.3760.3-o-0.7-0.40.40.996.102361208214329518.3425.9193.151.6750.46-o-0.8-0.7-o0.4119.65231208239727667.2154.9882.481.1690.13-1-0.8-0.40.30.886.906831208239729517.9225.5552.861.4330.26-o-0.9-0.7-o0.41