函數
>
數據分析
>
曲線擬合
>
範例：指數遞歸
範例：指數遞歸
使用
expfit
函數執行指數遞歸。指數函數經常用在從衰變或成長到穩定狀態的過程，例如輻射衰變、RC 電路的暫態回應，或過阻尼彈簧的位移。
1.
定義適當的數據集。
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<real xml:lang="zh_TW">17</real> <real xml:lang="zh_TW">18</real> <real xml:lang="zh_TW">19</real> <real xml:lang="zh_TW">20</real> <real xml:lang="zh_TW">21</real> <real xml:lang="zh_TW">22</real> <real xml:lang="zh_TW">23</real> <real xml:lang="zh_TW">24</real> <real xml:lang="zh_TW">25</real> <real xml:lang="zh_TW">26</real> <real xml:lang="zh_TW">27</real> <real xml:lang="zh_TW">28</real> <real xml:lang="zh_TW">29</real> <real xml:lang="zh_TW">30</real> <real xml:lang="zh_TW">31</real> <real xml:lang="zh_TW">32</real> <real xml:lang="zh_TW">33</real> <real xml:lang="zh_TW">34</real> <real xml:lang="zh_TW">35</real> <real xml:lang="zh_TW">36</real> <real xml:lang="zh_TW">37</real> <real xml:lang="zh_TW">38</real> <real xml:lang="zh_TW">39</real> <real xml:lang="zh_TW">40</real> <real xml:lang="zh_TW">41</real> <real xml:lang="zh_TW">42</real> <real xml:lang="zh_TW">43</real> <real xml:lang="zh_TW">44</real> <real xml:lang="zh_TW">45</real> <real xml:lang="zh_TW">46</real> <real xml:lang="zh_TW">47</real> <real xml:lang="zh_TW">48</real> <real xml:lang="zh_TW">49</real> <real xml:lang="zh_TW">50</real> <real xml:lang="zh_TW">51</real> <real xml:lang="zh_TW">52</real> <real xml:lang="zh_TW">53</real> <real xml:lang="zh_TW">54</real> <real xml:lang="zh_TW">55</real> <real xml:lang="zh_TW">56</real> <real xml:lang="zh_TW">57</real> <real xml:lang="zh_TW">58</real> <real xml:lang="zh_TW">59</real> <real xml:lang="zh_TW">60</real> <real xml:lang="zh_TW">61</real> <real xml:lang="zh_TW">62</real> <real xml:lang="zh_TW">63</real> <real xml:lang="zh_TW">64</real> <real xml:lang="zh_TW">65</real> <real xml:lang="zh_TW">66</real> <real xml:lang="zh_TW">67</real> <real xml:lang="zh_TW">68</real> <real xml:lang="zh_TW">69</real> <real xml:lang="zh_TW">70</real> <real xml:lang="zh_TW">71</real> <real xml:lang="zh_TW">72</real> <real xml:lang="zh_TW">73</real> <real xml:lang="zh_TW">74</real> <real xml:lang="zh_TW">75</real> <real xml:lang="zh_TW">76</real> <real xml:lang="zh_TW">77</real> <real xml:lang="zh_TW">78</real> <real xml:lang="zh_TW">79</real> <real xml:lang="zh_TW">80</real> <real xml:lang="zh_TW">81</real> <real xml:lang="zh_TW">82</real> <real xml:lang="zh_TW">83</real> <real xml:lang="zh_TW">84</real> <real xml:lang="zh_TW">85</real> <real xml:lang="zh_TW">86</real> <real xml:lang="zh_TW">87</real> <real xml:lang="zh_TW">88</real> <real xml:lang="zh_TW">89</real> <real xml:lang="zh_TW">90</real> <real xml:lang="zh_TW">91</real> <real xml:lang="zh_TW">92</real> <real xml:lang="zh_TW">93</real> <real xml:lang="zh_TW">94</real> <real xml:lang="zh_TW">95</real> <real xml:lang="zh_TW">96</real> <real xml:lang="zh_TW">97</real> <real xml:lang="zh_TW">98</real> <real xml:lang="zh_TW">99</real> <real xml:lang="zh_TW">100</real> <real xml:lang="zh_TW">101</real> <real xml:lang="zh_TW">102</real> <real xml:lang="zh_TW">103</real> <real xml:lang="zh_TW">104</real> <real xml:lang="zh_TW">105</real> <real xml:lang="zh_TW">106</real> <real xml:lang="zh_TW">107</real> <real xml:lang="zh_TW">108</real> <real xml:lang="zh_TW">109</real> <real xml:lang="zh_TW">110</real> <real xml:lang="zh_TW">111</real> <real xml:lang="zh_TW">112</real> <real xml:lang="zh_TW">113</real> <real xml:lang="zh_TW">114</real> <real xml:lang="zh_TW">115</real> <real xml:lang="zh_TW">116</real> <real xml:lang="zh_TW">117</real> <real xml:lang="zh_TW">118</real> <real xml:lang="zh_TW">119</real> <real xml:lang="zh_TW">120</real> <real xml:lang="zh_TW">121</real> <real xml:lang="zh_TW">122</real> <real xml:lang="zh_TW">123</real> <real xml:lang="zh_TW">124</real> <real xml:lang="zh_TW">125</real> <real xml:lang="zh_TW">126</real> <real xml:lang="zh_TW">127</real> <real xml:lang="zh_TW">128</real> <real xml:lang="zh_TW">129</real> <real xml:lang="zh_TW">130</real> <real xml:lang="zh_TW">131</real> <real xml:lang="zh_TW">132</real> <real xml:lang="zh_TW">133</real> <real xml:lang="zh_TW">134</real> <real xml:lang="zh_TW">135</real> <real xml:lang="zh_TW">136</real> <real xml:lang="zh_TW">137</real> <real xml:lang="zh_TW">138</real> <real xml:lang="zh_TW">139</real> <real xml:lang="zh_TW">140</real> <real xml:lang="zh_TW">141</real> <real xml:lang="zh_TW">142</real> <real xml:lang="zh_TW">143</real> <real xml:lang="zh_TW">144</real> <real xml:lang="zh_TW">145</real> <real xml:lang="zh_TW">146</real> <real xml:lang="zh_TW">147</real> <real xml:lang="zh_TW">148</real> <real xml:lang="zh_TW">149</real> <real xml:lang="zh_TW">150</real> <real xml:lang="zh_TW">3.248</real> <real xml:lang="zh_TW">5.381</real> <real xml:lang="zh_TW">6.762</real> <real xml:lang="zh_TW">6.585</real> <real xml:lang="zh_TW">5.11</real> <real xml:lang="zh_TW">3.565</real> <real xml:lang="zh_TW">4.23</real> <real xml:lang="zh_TW">5.004</real> <real xml:lang="zh_TW">4.583</real> <real xml:lang="zh_TW">6.015</real> <real xml:lang="zh_TW">5.864</real> <real xml:lang="zh_TW">6.768</real> <real xml:lang="zh_TW">5.819</real> <real xml:lang="zh_TW">8.104</real> <real xml:lang="zh_TW">7.086</real> <real xml:lang="zh_TW">4.317</real> <real xml:lang="zh_TW">8.449</real> <real xml:lang="zh_TW">5.747</real> <real xml:lang="zh_TW">4.434</real> <real xml:lang="zh_TW">6.343</real> <real xml:lang="zh_TW">4.631</real> <real xml:lang="zh_TW">8.856</real> <real xml:lang="zh_TW">6.668</real> <real xml:lang="zh_TW">7.81</real> <real xml:lang="zh_TW">9.225</real> <real xml:lang="zh_TW">7.921</real> <real xml:lang="zh_TW">9.012</real> <real xml:lang="zh_TW">8.101</real> <real xml:lang="zh_TW">8.098</real> <real xml:lang="zh_TW">8.722</real> <real xml:lang="zh_TW">5.518</real> <real xml:lang="zh_TW">7.285</real> <real xml:lang="zh_TW">9.544</real> <real xml:lang="zh_TW">10.15</real> <real xml:lang="zh_TW">9.55</real> <real xml:lang="zh_TW">10.51</real> <real xml:lang="zh_TW">8.841</real> <real xml:lang="zh_TW">10.19</real> <real xml:lang="zh_TW">7.402</real> <real xml:lang="zh_TW">10.53</real> <real xml:lang="zh_TW">10.92</real> <real xml:lang="zh_TW">9.955</real> <real xml:lang="zh_TW">10.03</real> <real xml:lang="zh_TW">8.343</real> <real xml:lang="zh_TW">11.17</real> <real xml:lang="zh_TW">10.95</real> <real xml:lang="zh_TW">7.935</real> <real xml:lang="zh_TW">11.96</real> <real xml:lang="zh_TW">9.372</real> <real xml:lang="zh_TW">10.36</real> <real xml:lang="zh_TW">9.114</real> <real xml:lang="zh_TW">13.29</real> <real xml:lang="zh_TW">11.73</real> <real xml:lang="zh_TW">9.53</real> <real xml:lang="zh_TW">10.21</real> <real xml:lang="zh_TW">13.21</real> <real xml:lang="zh_TW">11.05</real> <real xml:lang="zh_TW">10.54</real> <real xml:lang="zh_TW">10.73</real> <real xml:lang="zh_TW">14.62</real> <real xml:lang="zh_TW">10.07</real> <real xml:lang="zh_TW">10.92</real> <real xml:lang="zh_TW">10.41</real> <real xml:lang="zh_TW">13.8</real> <real xml:lang="zh_TW">12.86</real> <real xml:lang="zh_TW">11.46</real> <real xml:lang="zh_TW">14.73</real> <real xml:lang="zh_TW">14.74</real> <real xml:lang="zh_TW">14.37</real> <real xml:lang="zh_TW">12.19</real> <real xml:lang="zh_TW">13.47</real> <real xml:lang="zh_TW">16.73</real> <real xml:lang="zh_TW">13.1</real> <real xml:lang="zh_TW">16.46</real> <real xml:lang="zh_TW">17.5</real> <real xml:lang="zh_TW">14.91</real> <real xml:lang="zh_TW">16.94</real> <real xml:lang="zh_TW">18.14</real> <real xml:lang="zh_TW">18.64</real> <real xml:lang="zh_TW">19.13</real> <real xml:lang="zh_TW">17.06</real> <real xml:lang="zh_TW">16.93</real> <real xml:lang="zh_TW">17.6</real> <real xml:lang="zh_TW">15.78</real> <real xml:lang="zh_TW">19.96</real> <real xml:lang="zh_TW">16.87</real> <real xml:lang="zh_TW">20.17</real> <real xml:lang="zh_TW">17.86</real> <real xml:lang="zh_TW">17.68</real> <real xml:lang="zh_TW">17.88</real> <real xml:lang="zh_TW">21.39</real> <real xml:lang="zh_TW">23.09</real> <real xml:lang="zh_TW">21.05</real> <real xml:lang="zh_TW">20.48</real> <real xml:lang="zh_TW">22.93</real> <real xml:lang="zh_TW">20.17</real> <real xml:lang="zh_TW">21.16</real> <real xml:lang="zh_TW">23.63</real> <real xml:lang="zh_TW">23.62</real> <real xml:lang="zh_TW">23.23</real> <real xml:lang="zh_TW">23.88</real> <real xml:lang="zh_TW">25.65</real> <real xml:lang="zh_TW">27.99</real> <real xml:lang="zh_TW">24.29</real> <real xml:lang="zh_TW">27.57</real> <real xml:lang="zh_TW">28.38</real> <real xml:lang="zh_TW">28.1</real> <real xml:lang="zh_TW">28.94</real> <real xml:lang="zh_TW">28.85</real> <real xml:lang="zh_TW">29.28</real> <real xml:lang="zh_TW">28.31</real> <real xml:lang="zh_TW">30.09</real> <real xml:lang="zh_TW">31.53</real> <real xml:lang="zh_TW">31.87</real> <real xml:lang="zh_TW">33.32</real> <real xml:lang="zh_TW">34.87</real> <real xml:lang="zh_TW">33.67</real> <real xml:lang="zh_TW">34.1</real> <real xml:lang="zh_TW">34.3</real> <real xml:lang="zh_TW">34.66</real> <real xml:lang="zh_TW">37.77</real> <real xml:lang="zh_TW">35</real> <real xml:lang="zh_TW">38.07</real> <real xml:lang="zh_TW">38.24</real> <real xml:lang="zh_TW">39.03</real> <real xml:lang="zh_TW">39.33</real> <real xml:lang="zh_TW">39.52</real> <real xml:lang="zh_TW">43.02</real> <real xml:lang="zh_TW">41.81</real> <real xml:lang="zh_TW">40.46</real> <real xml:lang="zh_TW">42.05</real> <real xml:lang="zh_TW">44.48</real> <real xml:lang="zh_TW">44.18</real> <real xml:lang="zh_TW">46.28</real> <real xml:lang="zh_TW">46.24</real> <real xml:lang="zh_TW">48.65</real> <real xml:lang="zh_TW">45.56</real> <real xml:lang="zh_TW">48.34</real> <real xml:lang="zh_TW">51.54</real> <real xml:lang="zh_TW">52.81</real> <real xml:lang="zh_TW">52.59</real> <real xml:lang="zh_TW">54</real> <real xml:lang="zh_TW">56.13</real> <real xml:lang="zh_TW">54.2</real> <real xml:lang="zh_TW">57.97</real> <real xml:lang="zh_TW">56.8</real> <real xml:lang="zh_TW">59.97</real> <real xml:lang="zh_TW">56.94</real> <real xml:lang="zh_TW">62.57</real> <real xml:lang="zh_TW">59.89</real> <real xml:lang="zh_TW">60.84</real> </matrix> </define> </math> </region>
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<region id="ID0EWMDU" top="345.6" left="19.200000000000003" xml:lang="zh_TW" height="27.752000000000002" width="85.966666666666669" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="33" xml:lang="zh_TW"> <define xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve" xml:lang="zh_TW">vy</id> <apply xml:lang="zh_TW"> <matcol xml:lang="zh_TW" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">data</id> <real xml:lang="zh_TW">1</real> </apply> </define> <resultFormat xml:lang="zh_TW"> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="zh_TW" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="zh_TW" /> </resultFormat> </math> </region>
2.
呼叫
expfit
函數求解指數擬合的參數。
<region id="ID0EQODU" top="441.60000000000008" left="19.200000000000003" xml:lang="zh_TW" height="25.6" width="129.3" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="35" xml:lang="zh_TW"> <define xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve" xml:lang="zh_TW">β</id> <apply xml:lang="zh_TW"> <id labels="FUNCTION" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">expfit</id> <sequence xml:lang="zh_TW"> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">vx</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">vy</id> </sequence> </apply> </define> <resultFormat xml:lang="zh_TW"> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="zh_TW" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="zh_TW" /> </resultFormat> </math> </region>
<region id="ID0ECPDU" top="480.00000000000006" left="19.200000000000003" xml:lang="zh_TW" height="62.800000000000004" width="92.983333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="37" xml:lang="zh_TW"> <eval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">β</id> <unitOverride xml:lang="zh_TW"> <placeholder xml:lang="zh_TW" /> </unitOverride> </eval> <resultFormat xml:lang="zh_TW"> <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="zh_TW" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="zh_TW" /> </resultFormat> </math> </region>
參數會擬合下列指數方程式：
<region id="ID0EQQDU" top="614.40000000000009" left="19.200000000000003" xml:lang="zh_TW" height="47.352000000000004" width="139.25333333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="39" xml:lang="zh_TW"> <define xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50"> <function xml:lang="zh_TW"> <id labels="*" xml:space="preserve" xml:lang="zh_TW">f</id> <boundVars xml:lang="zh_TW"> <id labels="*" xml:space="preserve" xml:lang="zh_TW">x</id> </boundVars> </function> <apply xml:lang="zh_TW"> <plus xml:lang="zh_TW" /> <apply xml:lang="zh_TW"> <mult xml:lang="zh_TW" /> <apply xml:lang="zh_TW"> <indexer xml:lang="zh_TW" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">β</id> <real xml:lang="zh_TW">0</real> </apply> <apply xml:lang="zh_TW"> <pow xml:lang="zh_TW" /> <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">e</id> <apply xml:lang="zh_TW"> <mult xml:lang="zh_TW" /> <apply xml:lang="zh_TW"> <indexer xml:lang="zh_TW" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">β</id> <real xml:lang="zh_TW">1</real> </apply> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">x</id> </apply> </apply> </apply> <apply xml:lang="zh_TW"> <indexer xml:lang="zh_TW" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">β</id> <real xml:lang="zh_TW">2</real> </apply> </apply> </define> <resultFormat xml:lang="zh_TW"> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="zh_TW" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="zh_TW" /> </resultFormat> </math> </region>
3.
繪製數據與指數擬合。
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由於
expfit
函數不需要估值，因此在非線性擬合函數中並不常見，但是若不喜歡從內部估值常式取得的結果，則可提供該函數。內部常式會從擬合直線與
y
值開始。
複製運算式
